diff --git "a/data/sensemath_v2_d2.json" "b/data/sensemath_v2_d2.json" new file mode 100644--- /dev/null +++ "b/data/sensemath_v2_d2.json" @@ -0,0 +1,28602 @@ +[ + { + "family_id": "magnitude_estimation_000001", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 98", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 98?", + "options": [ + "9614", + "9594", + "9604", + "9624" + ], + "answer": "9604", + "correct_value": 9604, + "explanation": "Round to 100: 98≈100, 98≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99 * 105", + "ns_shortcut": "Round 99≈100, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 105?", + "options": [ + "10405", + "10385", + "10415", + "10395" + ], + "answer": "10395", + "correct_value": 10395, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "57 * 52", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 57 * 52?", + "options": [ + "2974", + "2984", + "2954", + "2964" + ], + "answer": "2964", + "correct_value": 2964, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000002", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 * 99", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 99?", + "options": [ + "9811", + "9791", + "9821", + "9801" + ], + "answer": "9801", + "correct_value": 9801, + "explanation": "Round to 100: 99≈100, 99≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102 * 103", + "ns_shortcut": "Round 102≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 103?", + "options": [ + "10506", + "10516", + "10526", + "10496" + ], + "answer": "10506", + "correct_value": 10506, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "64 * 66", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 64 * 66?", + "options": [ + "4244", + "4234", + "4214", + "4224" + ], + "answer": "4224", + "correct_value": 4224, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000003", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 101", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 101?", + "options": [ + "10221", + "10191", + "10201", + "10211" + ], + "answer": "10201", + "correct_value": 10201, + "explanation": "Round to 100: 101≈100, 101≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98 * 103", + "ns_shortcut": "Round 98≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 103?", + "options": [ + "10114", + "10104", + "10094", + "10084" + ], + "answer": "10094", + "correct_value": 10094, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "66 * 50", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 66 * 50?", + "options": [ + "3290", + "3310", + "3300", + "3320" + ], + "answer": "3300", + "correct_value": 3300, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000004", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 102", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 102?", + "options": [ + "9986", + "10006", + "10016", + "9996" + ], + "answer": "9996", + "correct_value": 9996, + "explanation": "Round to 100: 98≈100, 102≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101 * 105", + "ns_shortcut": "Round 101≈100, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 105?", + "options": [ + "10595", + "10605", + "10625", + "10615" + ], + "answer": "10605", + "correct_value": 10605, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "40 * 60", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 40 * 60?", + "options": [ + "2420", + "2400", + "2390", + "2410" + ], + "answer": "2400", + "correct_value": 2400, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000005", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 * 98", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 98?", + "options": [ + "9702", + "9722", + "9712", + "9692" + ], + "answer": "9702", + "correct_value": 9702, + "explanation": "Round to 100: 99≈100, 98≈100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101 * 103", + "ns_shortcut": "Round 101≈100, then adjust.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 103?", + "options": [ + "10393", + "10403", + "10413", + "10423" + ], + "answer": "10403", + "correct_value": 10403, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "60 * 47", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 60 * 47?", + "options": [ + "2830", + "2810", + "2840", + "2820" + ], + "answer": "2820", + "correct_value": 2820, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000006", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 * 99", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 99?", + "options": [ + "9801", + "9821", + "9811", + "9791" + ], + "answer": "9801", + "correct_value": 9801, + "explanation": "Round to 100: 99≈100, 99≈100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99 * 105", + "ns_shortcut": "Round 99≈100, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 105?", + "options": [ + "10405", + "10385", + "10395", + "10415" + ], + "answer": "10395", + "correct_value": 10395, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "65 * 40", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 65 * 40?", + "options": [ + "2600", + "2590", + "2610", + "2620" + ], + "answer": "2600", + "correct_value": 2600, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000007", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 99", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 99?", + "options": [ + "9692", + "9712", + "9702", + "9722" + ], + "answer": "9702", + "correct_value": 9702, + "explanation": "Round to 100: 98≈100, 99≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102 * 105", + "ns_shortcut": "Round 102≈100, then adjust.", + "shortcut_strength": 0.965, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 105?", + "options": [ + "10730", + "10720", + "10710", + "10700" + ], + "answer": "10710", + "correct_value": 10710, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "43 * 46", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 43 * 46?", + "options": [ + "1978", + "1988", + "1968", + "1998" + ], + "answer": "1978", + "correct_value": 1978, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000008", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 101", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 101?", + "options": [ + "10322", + "10292", + "10302", + "10312" + ], + "answer": "10302", + "correct_value": 10302, + "explanation": "Round to 100: 102≈100, 101≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98 * 103", + "ns_shortcut": "Round 98≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 103?", + "options": [ + "10104", + "10114", + "10084", + "10094" + ], + "answer": "10094", + "correct_value": 10094, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "64 * 60", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 64 * 60?", + "options": [ + "3850", + "3840", + "3860", + "3830" + ], + "answer": "3840", + "correct_value": 3840, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000009", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 102", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 102?", + "options": [ + "10404", + "10414", + "10424", + "10394" + ], + "answer": "10404", + "correct_value": 10404, + "explanation": "Round to 100: 102≈100, 102≈100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98 * 105", + "ns_shortcut": "Round 98≈100, then adjust.", + "shortcut_strength": 0.965, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 105?", + "options": [ + "10280", + "10300", + "10310", + "10290" + ], + "answer": "10290", + "correct_value": 10290, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "63 * 59", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 63 * 59?", + "options": [ + "3707", + "3727", + "3717", + "3737" + ], + "answer": "3717", + "correct_value": 3717, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000010", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 99", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 99?", + "options": [ + "10118", + "10108", + "10098", + "10088" + ], + "answer": "10098", + "correct_value": 10098, + "explanation": "Round to 100: 102≈100, 99≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99 * 105", + "ns_shortcut": "Round 99≈100, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 105?", + "options": [ + "10385", + "10405", + "10415", + "10395" + ], + "answer": "10395", + "correct_value": 10395, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "65 * 59", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 65 * 59?", + "options": [ + "3855", + "3825", + "3835", + "3845" + ], + "answer": "3835", + "correct_value": 3835, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000011", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 * 98", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 98?", + "options": [ + "9692", + "9702", + "9712", + "9722" + ], + "answer": "9702", + "correct_value": 9702, + "explanation": "Round to 100: 99≈100, 98≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98 * 104", + "ns_shortcut": "Round 98≈100, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 104?", + "options": [ + "10202", + "10182", + "10192", + "10212" + ], + "answer": "10192", + "correct_value": 10192, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "46 * 67", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 46 * 67?", + "options": [ + "3102", + "3092", + "3072", + "3082" + ], + "answer": "3082", + "correct_value": 3082, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000012", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 99", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 99?", + "options": [ + "9702", + "9692", + "9722", + "9712" + ], + "answer": "9702", + "correct_value": 9702, + "explanation": "Round to 100: 98≈100, 99≈100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99 * 104", + "ns_shortcut": "Round 99≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 104?", + "options": [ + "10306", + "10316", + "10296", + "10286" + ], + "answer": "10296", + "correct_value": 10296, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "61 * 48", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 61 * 48?", + "options": [ + "2928", + "2938", + "2948", + "2918" + ], + "answer": "2928", + "correct_value": 2928, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000013", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 102", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 102?", + "options": [ + "10414", + "10424", + "10394", + "10404" + ], + "answer": "10404", + "correct_value": 10404, + "explanation": "Round to 100: 102≈100, 102≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98 * 104", + "ns_shortcut": "Round 98≈100, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 104?", + "options": [ + "10202", + "10182", + "10212", + "10192" + ], + "answer": "10192", + "correct_value": 10192, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "57 * 46", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 57 * 46?", + "options": [ + "2632", + "2642", + "2612", + "2622" + ], + "answer": "2622", + "correct_value": 2622, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000014", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 * 98", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 98?", + "options": [ + "9722", + "9692", + "9712", + "9702" + ], + "answer": "9702", + "correct_value": 9702, + "explanation": "Round to 100: 99≈100, 98≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99 * 104", + "ns_shortcut": "Round 99≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 104?", + "options": [ + "10306", + "10316", + "10296", + "10286" + ], + "answer": "10296", + "correct_value": 10296, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "62 * 68", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 62 * 68?", + "options": [ + "4236", + "4226", + "4216", + "4206" + ], + "answer": "4216", + "correct_value": 4216, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000015", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 99", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 99?", + "options": [ + "10118", + "10108", + "10088", + "10098" + ], + "answer": "10098", + "correct_value": 10098, + "explanation": "Round to 100: 102≈100, 99≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101 * 104", + "ns_shortcut": "Round 101≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 104?", + "options": [ + "10524", + "10504", + "10514", + "10494" + ], + "answer": "10504", + "correct_value": 10504, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "60 * 59", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 60 * 59?", + "options": [ + "3560", + "3550", + "3530", + "3540" + ], + "answer": "3540", + "correct_value": 3540, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000016", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 98", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 98?", + "options": [ + "9888", + "9918", + "9898", + "9908" + ], + "answer": "9898", + "correct_value": 9898, + "explanation": "Round to 100: 101≈100, 98≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102 * 103", + "ns_shortcut": "Round 102≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 103?", + "options": [ + "10526", + "10516", + "10506", + "10496" + ], + "answer": "10506", + "correct_value": 10506, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "53 * 46", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 53 * 46?", + "options": [ + "2458", + "2428", + "2448", + "2438" + ], + "answer": "2438", + "correct_value": 2438, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000017", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 98", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 98?", + "options": [ + "9614", + "9624", + "9594", + "9604" + ], + "answer": "9604", + "correct_value": 9604, + "explanation": "Round to 100: 98≈100, 98≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102 * 104", + "ns_shortcut": "Round 102≈100, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 104?", + "options": [ + "10628", + "10618", + "10608", + "10598" + ], + "answer": "10608", + "correct_value": 10608, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "67 * 40", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 67 * 40?", + "options": [ + "2700", + "2680", + "2670", + "2690" + ], + "answer": "2680", + "correct_value": 2680, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000018", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 101", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 101?", + "options": [ + "10302", + "10322", + "10312", + "10292" + ], + "answer": "10302", + "correct_value": 10302, + "explanation": "Round to 100: 102≈100, 101≈100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102 * 103", + "ns_shortcut": "Round 102≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 103?", + "options": [ + "10526", + "10506", + "10496", + "10516" + ], + "answer": "10506", + "correct_value": 10506, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "69 * 40", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 69 * 40?", + "options": [ + "2750", + "2770", + "2780", + "2760" + ], + "answer": "2760", + "correct_value": 2760, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000019", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 98", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 98?", + "options": [ + "9614", + "9604", + "9624", + "9594" + ], + "answer": "9604", + "correct_value": 9604, + "explanation": "Round to 100: 98≈100, 98≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98 * 105", + "ns_shortcut": "Round 98≈100, then adjust.", + "shortcut_strength": 0.965, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 105?", + "options": [ + "10300", + "10310", + "10280", + "10290" + ], + "answer": "10290", + "correct_value": 10290, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "58 * 41", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 58 * 41?", + "options": [ + "2398", + "2378", + "2368", + "2388" + ], + "answer": "2378", + "correct_value": 2378, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000020", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 * 98", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 98?", + "options": [ + "9692", + "9722", + "9712", + "9702" + ], + "answer": "9702", + "correct_value": 9702, + "explanation": "Round to 100: 99≈100, 98≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101 * 103", + "ns_shortcut": "Round 101≈100, then adjust.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 103?", + "options": [ + "10393", + "10413", + "10403", + "10423" + ], + "answer": "10403", + "correct_value": 10403, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "51 * 43", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 51 * 43?", + "options": [ + "2193", + "2183", + "2203", + "2213" + ], + "answer": "2193", + "correct_value": 2193, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000021", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 98", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 98?", + "options": [ + "9614", + "9604", + "9624", + "9594" + ], + "answer": "9604", + "correct_value": 9604, + "explanation": "Round to 100: 98≈100, 98≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98 * 105", + "ns_shortcut": "Round 98≈100, then adjust.", + "shortcut_strength": 0.965, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 105?", + "options": [ + "10290", + "10280", + "10310", + "10300" + ], + "answer": "10290", + "correct_value": 10290, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "43 * 68", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 43 * 68?", + "options": [ + "2934", + "2914", + "2924", + "2944" + ], + "answer": "2924", + "correct_value": 2924, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000022", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 101", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 101?", + "options": [ + "10201", + "10221", + "10191", + "10211" + ], + "answer": "10201", + "correct_value": 10201, + "explanation": "Round to 100: 101≈100, 101≈100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98 * 105", + "ns_shortcut": "Round 98≈100, then adjust.", + "shortcut_strength": 0.965, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 105?", + "options": [ + "10300", + "10290", + "10280", + "10310" + ], + "answer": "10290", + "correct_value": 10290, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "69 * 67", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 69 * 67?", + "options": [ + "4643", + "4623", + "4613", + "4633" + ], + "answer": "4623", + "correct_value": 4623, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000023", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 99", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 99?", + "options": [ + "9692", + "9702", + "9712", + "9722" + ], + "answer": "9702", + "correct_value": 9702, + "explanation": "Round to 100: 98≈100, 99≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101 * 103", + "ns_shortcut": "Round 101≈100, then adjust.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 103?", + "options": [ + "10403", + "10413", + "10393", + "10423" + ], + "answer": "10403", + "correct_value": 10403, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "47 * 67", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 47 * 67?", + "options": [ + "3169", + "3159", + "3139", + "3149" + ], + "answer": "3149", + "correct_value": 3149, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000024", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 98", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 98?", + "options": [ + "9918", + "9888", + "9908", + "9898" + ], + "answer": "9898", + "correct_value": 9898, + "explanation": "Round to 100: 101≈100, 98≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101 * 103", + "ns_shortcut": "Round 101≈100, then adjust.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 103?", + "options": [ + "10403", + "10413", + "10423", + "10393" + ], + "answer": "10403", + "correct_value": 10403, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "56 * 39", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 56 * 39?", + "options": [ + "2194", + "2204", + "2174", + "2184" + ], + "answer": "2184", + "correct_value": 2184, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000025", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 101", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 101?", + "options": [ + "9918", + "9898", + "9888", + "9908" + ], + "answer": "9898", + "correct_value": 9898, + "explanation": "Round to 100: 98≈100, 101≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98 * 104", + "ns_shortcut": "Round 98≈100, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 104?", + "options": [ + "10182", + "10202", + "10192", + "10212" + ], + "answer": "10192", + "correct_value": 10192, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "45 * 60", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 45 * 60?", + "options": [ + "2690", + "2700", + "2720", + "2710" + ], + "answer": "2700", + "correct_value": 2700, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000026", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 102", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 102?", + "options": [ + "10292", + "10322", + "10302", + "10312" + ], + "answer": "10302", + "correct_value": 10302, + "explanation": "Round to 100: 101≈100, 102≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99 * 104", + "ns_shortcut": "Round 99≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 104?", + "options": [ + "10316", + "10306", + "10286", + "10296" + ], + "answer": "10296", + "correct_value": 10296, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "64 * 68", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 64 * 68?", + "options": [ + "4362", + "4352", + "4372", + "4342" + ], + "answer": "4352", + "correct_value": 4352, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000027", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 99", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 99?", + "options": [ + "10019", + "9989", + "10009", + "9999" + ], + "answer": "9999", + "correct_value": 9999, + "explanation": "Round to 100: 101≈100, 99≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102 * 103", + "ns_shortcut": "Round 102≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 103?", + "options": [ + "10506", + "10526", + "10496", + "10516" + ], + "answer": "10506", + "correct_value": 10506, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "48 * 65", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 48 * 65?", + "options": [ + "3110", + "3140", + "3120", + "3130" + ], + "answer": "3120", + "correct_value": 3120, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000028", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 * 102", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 102?", + "options": [ + "10088", + "10098", + "10108", + "10118" + ], + "answer": "10098", + "correct_value": 10098, + "explanation": "Round to 100: 99≈100, 102≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101 * 104", + "ns_shortcut": "Round 101≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 104?", + "options": [ + "10504", + "10514", + "10524", + "10494" + ], + "answer": "10504", + "correct_value": 10504, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "69 * 39", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 69 * 39?", + "options": [ + "2711", + "2681", + "2701", + "2691" + ], + "answer": "2691", + "correct_value": 2691, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000029", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 98", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 98?", + "options": [ + "9888", + "9908", + "9918", + "9898" + ], + "answer": "9898", + "correct_value": 9898, + "explanation": "Round to 100: 101≈100, 98≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101 * 105", + "ns_shortcut": "Round 101≈100, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 105?", + "options": [ + "10595", + "10605", + "10615", + "10625" + ], + "answer": "10605", + "correct_value": 10605, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "51 * 67", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 51 * 67?", + "options": [ + "3437", + "3417", + "3407", + "3427" + ], + "answer": "3417", + "correct_value": 3417, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000030", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 98", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 98?", + "options": [ + "9996", + "10016", + "9986", + "10006" + ], + "answer": "9996", + "correct_value": 9996, + "explanation": "Round to 100: 102≈100, 98≈100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98 * 105", + "ns_shortcut": "Round 98≈100, then adjust.", + "shortcut_strength": 0.965, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 105?", + "options": [ + "10290", + "10300", + "10280", + "10310" + ], + "answer": "10290", + "correct_value": 10290, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "60 * 66", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 60 * 66?", + "options": [ + "3950", + "3970", + "3980", + "3960" + ], + "answer": "3960", + "correct_value": 3960, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000031", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 102", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 102?", + "options": [ + "10302", + "10292", + "10312", + "10322" + ], + "answer": "10302", + "correct_value": 10302, + "explanation": "Round to 100: 101≈100, 102≈100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99 * 103", + "ns_shortcut": "Round 99≈100, then adjust.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 103?", + "options": [ + "10197", + "10207", + "10217", + "10187" + ], + "answer": "10197", + "correct_value": 10197, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "60 * 62", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 60 * 62?", + "options": [ + "3720", + "3710", + "3740", + "3730" + ], + "answer": "3720", + "correct_value": 3720, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000032", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 98", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 98?", + "options": [ + "9986", + "9996", + "10006", + "10016" + ], + "answer": "9996", + "correct_value": 9996, + "explanation": "Round to 100: 102≈100, 98≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102 * 105", + "ns_shortcut": "Round 102≈100, then adjust.", + "shortcut_strength": 0.965, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 105?", + "options": [ + "10710", + "10700", + "10720", + "10730" + ], + "answer": "10710", + "correct_value": 10710, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "53 * 60", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 53 * 60?", + "options": [ + "3180", + "3170", + "3200", + "3190" + ], + "answer": "3180", + "correct_value": 3180, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000033", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 * 98", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 98?", + "options": [ + "9702", + "9712", + "9692", + "9722" + ], + "answer": "9702", + "correct_value": 9702, + "explanation": "Round to 100: 99≈100, 98≈100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98 * 103", + "ns_shortcut": "Round 98≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 103?", + "options": [ + "10104", + "10084", + "10094", + "10114" + ], + "answer": "10094", + "correct_value": 10094, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "43 * 39", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 43 * 39?", + "options": [ + "1697", + "1687", + "1667", + "1677" + ], + "answer": "1677", + "correct_value": 1677, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000034", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 102", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 102?", + "options": [ + "10006", + "9996", + "10016", + "9986" + ], + "answer": "9996", + "correct_value": 9996, + "explanation": "Round to 100: 98≈100, 102≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102 * 104", + "ns_shortcut": "Round 102≈100, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 104?", + "options": [ + "10618", + "10628", + "10608", + "10598" + ], + "answer": "10608", + "correct_value": 10608, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "59 * 61", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 59 * 61?", + "options": [ + "3619", + "3609", + "3589", + "3599" + ], + "answer": "3599", + "correct_value": 3599, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000035", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 * 99", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 99?", + "options": [ + "9811", + "9821", + "9801", + "9791" + ], + "answer": "9801", + "correct_value": 9801, + "explanation": "Round to 100: 99≈100, 99≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98 * 104", + "ns_shortcut": "Round 98≈100, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 104?", + "options": [ + "10192", + "10212", + "10182", + "10202" + ], + "answer": "10192", + "correct_value": 10192, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "69 * 63", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 69 * 63?", + "options": [ + "4347", + "4357", + "4367", + "4337" + ], + "answer": "4347", + "correct_value": 4347, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000036", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 99", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 99?", + "options": [ + "10098", + "10118", + "10108", + "10088" + ], + "answer": "10098", + "correct_value": 10098, + "explanation": "Round to 100: 102≈100, 99≈100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99 * 105", + "ns_shortcut": "Round 99≈100, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 105?", + "options": [ + "10395", + "10405", + "10385", + "10415" + ], + "answer": "10395", + "correct_value": 10395, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "59 * 46", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 59 * 46?", + "options": [ + "2704", + "2714", + "2724", + "2734" + ], + "answer": "2714", + "correct_value": 2714, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000037", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 99", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 99?", + "options": [ + "9999", + "9989", + "10009", + "10019" + ], + "answer": "9999", + "correct_value": 9999, + "explanation": "Round to 100: 101≈100, 99≈100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98 * 103", + "ns_shortcut": "Round 98≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 103?", + "options": [ + "10114", + "10084", + "10094", + "10104" + ], + "answer": "10094", + "correct_value": 10094, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "61 * 43", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 61 * 43?", + "options": [ + "2643", + "2613", + "2623", + "2633" + ], + "answer": "2623", + "correct_value": 2623, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000038", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 101", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 101?", + "options": [ + "10322", + "10302", + "10312", + "10292" + ], + "answer": "10302", + "correct_value": 10302, + "explanation": "Round to 100: 102≈100, 101≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99 * 104", + "ns_shortcut": "Round 99≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 104?", + "options": [ + "10306", + "10296", + "10316", + "10286" + ], + "answer": "10296", + "correct_value": 10296, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "66 * 67", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 66 * 67?", + "options": [ + "4442", + "4422", + "4432", + "4412" + ], + "answer": "4422", + "correct_value": 4422, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000039", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 101", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 101?", + "options": [ + "10221", + "10201", + "10211", + "10191" + ], + "answer": "10201", + "correct_value": 10201, + "explanation": "Round to 100: 101≈100, 101≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102 * 103", + "ns_shortcut": "Round 102≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 103?", + "options": [ + "10506", + "10496", + "10516", + "10526" + ], + "answer": "10506", + "correct_value": 10506, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "39 * 51", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 39 * 51?", + "options": [ + "1989", + "1999", + "1979", + "2009" + ], + "answer": "1989", + "correct_value": 1989, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000040", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 * 102", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 102?", + "options": [ + "10088", + "10098", + "10118", + "10108" + ], + "answer": "10098", + "correct_value": 10098, + "explanation": "Round to 100: 99≈100, 102≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102 * 105", + "ns_shortcut": "Round 102≈100, then adjust.", + "shortcut_strength": 0.965, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 105?", + "options": [ + "10730", + "10720", + "10700", + "10710" + ], + "answer": "10710", + "correct_value": 10710, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "43 * 66", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 43 * 66?", + "options": [ + "2848", + "2858", + "2838", + "2828" + ], + "answer": "2838", + "correct_value": 2838, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000041", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 102", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 102?", + "options": [ + "10292", + "10312", + "10302", + "10322" + ], + "answer": "10302", + "correct_value": 10302, + "explanation": "Round to 100: 101≈100, 102≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101 * 104", + "ns_shortcut": "Round 101≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 104?", + "options": [ + "10504", + "10524", + "10514", + "10494" + ], + "answer": "10504", + "correct_value": 10504, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "47 * 65", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 47 * 65?", + "options": [ + "3075", + "3045", + "3065", + "3055" + ], + "answer": "3055", + "correct_value": 3055, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000042", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 101", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 101?", + "options": [ + "9888", + "9898", + "9918", + "9908" + ], + "answer": "9898", + "correct_value": 9898, + "explanation": "Round to 100: 98≈100, 101≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98 * 103", + "ns_shortcut": "Round 98≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 103?", + "options": [ + "10084", + "10114", + "10094", + "10104" + ], + "answer": "10094", + "correct_value": 10094, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "58 * 43", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 58 * 43?", + "options": [ + "2514", + "2484", + "2494", + "2504" + ], + "answer": "2494", + "correct_value": 2494, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000043", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 99", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 99?", + "options": [ + "9702", + "9712", + "9722", + "9692" + ], + "answer": "9702", + "correct_value": 9702, + "explanation": "Round to 100: 98≈100, 99≈100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101 * 103", + "ns_shortcut": "Round 101≈100, then adjust.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 103?", + "options": [ + "10423", + "10413", + "10393", + "10403" + ], + "answer": "10403", + "correct_value": 10403, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "42 * 57", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 42 * 57?", + "options": [ + "2384", + "2414", + "2404", + "2394" + ], + "answer": "2394", + "correct_value": 2394, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000044", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 99", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 99?", + "options": [ + "10108", + "10088", + "10118", + "10098" + ], + "answer": "10098", + "correct_value": 10098, + "explanation": "Round to 100: 102≈100, 99≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98 * 105", + "ns_shortcut": "Round 98≈100, then adjust.", + "shortcut_strength": 0.965, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 105?", + "options": [ + "10300", + "10290", + "10280", + "10310" + ], + "answer": "10290", + "correct_value": 10290, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "64 * 57", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 64 * 57?", + "options": [ + "3638", + "3668", + "3658", + "3648" + ], + "answer": "3648", + "correct_value": 3648, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000045", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 98", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 98?", + "options": [ + "9888", + "9908", + "9898", + "9918" + ], + "answer": "9898", + "correct_value": 9898, + "explanation": "Round to 100: 101≈100, 98≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102 * 103", + "ns_shortcut": "Round 102≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 103?", + "options": [ + "10506", + "10526", + "10516", + "10496" + ], + "answer": "10506", + "correct_value": 10506, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "61 * 43", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 61 * 43?", + "options": [ + "2643", + "2633", + "2623", + "2613" + ], + "answer": "2623", + "correct_value": 2623, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000046", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 102", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 102?", + "options": [ + "10404", + "10414", + "10424", + "10394" + ], + "answer": "10404", + "correct_value": 10404, + "explanation": "Round to 100: 102≈100, 102≈100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101 * 103", + "ns_shortcut": "Round 101≈100, then adjust.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 103?", + "options": [ + "10393", + "10423", + "10413", + "10403" + ], + "answer": "10403", + "correct_value": 10403, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "46 * 63", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 46 * 63?", + "options": [ + "2898", + "2908", + "2918", + "2888" + ], + "answer": "2898", + "correct_value": 2898, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000047", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 98", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 98?", + "options": [ + "9918", + "9898", + "9908", + "9888" + ], + "answer": "9898", + "correct_value": 9898, + "explanation": "Round to 100: 101≈100, 98≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102 * 103", + "ns_shortcut": "Round 102≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 103?", + "options": [ + "10506", + "10496", + "10516", + "10526" + ], + "answer": "10506", + "correct_value": 10506, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "47 * 67", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 47 * 67?", + "options": [ + "3159", + "3149", + "3139", + "3169" + ], + "answer": "3149", + "correct_value": 3149, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000048", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 * 98", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 98?", + "options": [ + "9702", + "9692", + "9722", + "9712" + ], + "answer": "9702", + "correct_value": 9702, + "explanation": "Round to 100: 99≈100, 98≈100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "102 * 105", + "ns_shortcut": "Round 102≈100, then adjust.", + "shortcut_strength": 0.965, + "pure_math": { + "question": "Which of the following is the best estimate for 102 * 105?", + "options": [ + "10710", + "10700", + "10730", + "10720" + ], + "answer": "10710", + "correct_value": 10710, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "59 * 61", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 59 * 61?", + "options": [ + "3619", + "3599", + "3609", + "3589" + ], + "answer": "3599", + "correct_value": 3599, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000049", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 101", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 98 * 101?", + "options": [ + "9918", + "9898", + "9888", + "9908" + ], + "answer": "9898", + "correct_value": 9898, + "explanation": "Round to 100: 98≈100, 101≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "101 * 105", + "ns_shortcut": "Round 101≈100, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 105?", + "options": [ + "10605", + "10595", + "10615", + "10625" + ], + "answer": "10605", + "correct_value": 10605, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "52 * 62", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 52 * 62?", + "options": [ + "3224", + "3214", + "3234", + "3244" + ], + "answer": "3224", + "correct_value": 3224, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000050", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 101", + "ns_shortcut": "Round both to 100 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 101 * 101?", + "options": [ + "10211", + "10191", + "10201", + "10221" + ], + "answer": "10201", + "correct_value": 10201, + "explanation": "Round to 100: 101≈100, 101≈100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99 * 104", + "ns_shortcut": "Round 99≈100, then adjust.", + "shortcut_strength": 0.975, + "pure_math": { + "question": "Which of the following is the best estimate for 99 * 104?", + "options": [ + "10306", + "10286", + "10316", + "10296" + ], + "answer": "10296", + "correct_value": 10296, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "45 * 45", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 45 * 45?", + "options": [ + "2025", + "2045", + "2035", + "2015" + ], + "answer": "2025", + "correct_value": 2025, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000051", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 34", + "ns_shortcut": "Use (100-2)*m = 100*m - 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 98 * 34?", + "options": [ + "3322", + "3342", + "3332", + "3352" + ], + "answer": "3332", + "correct_value": 3332, + "explanation": "Distributive: (100-2)*34 = 100*34 - 2*34." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "96 * 48", + "ns_shortcut": "Use (100-4)*m, correction is 4*48.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 96 * 48?", + "options": [ + "4618", + "4598", + "4628", + "4608" + ], + "answer": "4608", + "correct_value": 4608, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "47 * 40", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 47 * 40?", + "options": [ + "1870", + "1890", + "1900", + "1880" + ], + "answer": "1880", + "correct_value": 1880, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000052", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 * 91", + "ns_shortcut": "Use (100-1)*m = 100*m - 1*m.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "What is 99 * 91?", + "options": [ + "9019", + "9029", + "9009", + "8999" + ], + "answer": "9009", + "correct_value": 9009, + "explanation": "Distributive: (100-1)*91 = 100*91 - 1*91." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "104 * 46", + "ns_shortcut": "Use (100+4)*m, correction is 4*46.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 104 * 46?", + "options": [ + "4784", + "4804", + "4794", + "4774" + ], + "answer": "4784", + "correct_value": 4784, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "49 * 44", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 49 * 44?", + "options": [ + "2146", + "2176", + "2166", + "2156" + ], + "answer": "2156", + "correct_value": 2156, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000053", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 18", + "ns_shortcut": "Use (100-2)*m = 100*m - 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 98 * 18?", + "options": [ + "1784", + "1764", + "1774", + "1754" + ], + "answer": "1764", + "correct_value": 1764, + "explanation": "Distributive: (100-2)*18 = 100*18 - 2*18." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "104 * 90", + "ns_shortcut": "Use (100+4)*m, correction is 4*90.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 104 * 90?", + "options": [ + "9370", + "9380", + "9350", + "9360" + ], + "answer": "9360", + "correct_value": 9360, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "64 * 57", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 64 * 57?", + "options": [ + "3658", + "3638", + "3648", + "3668" + ], + "answer": "3648", + "correct_value": 3648, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000054", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 81", + "ns_shortcut": "Use (100-2)*m = 100*m - 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 98 * 81?", + "options": [ + "7948", + "7938", + "7958", + "7928" + ], + "answer": "7938", + "correct_value": 7938, + "explanation": "Distributive: (100-2)*81 = 100*81 - 2*81." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "104 * 58", + "ns_shortcut": "Use (100+4)*m, correction is 4*58.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 104 * 58?", + "options": [ + "6032", + "6052", + "6042", + "6022" + ], + "answer": "6032", + "correct_value": 6032, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "66 * 57", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 66 * 57?", + "options": [ + "3782", + "3762", + "3772", + "3752" + ], + "answer": "3762", + "correct_value": 3762, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000055", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 36", + "ns_shortcut": "Use (100-2)*m = 100*m - 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 98 * 36?", + "options": [ + "3548", + "3528", + "3518", + "3538" + ], + "answer": "3528", + "correct_value": 3528, + "explanation": "Distributive: (100-2)*36 = 100*36 - 2*36." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "96 * 40", + "ns_shortcut": "Use (100-4)*m, correction is 4*40.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 96 * 40?", + "options": [ + "3830", + "3840", + "3860", + "3850" + ], + "answer": "3840", + "correct_value": 3840, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "44 * 39", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 44 * 39?", + "options": [ + "1716", + "1736", + "1726", + "1706" + ], + "answer": "1716", + "correct_value": 1716, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000056", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 14", + "ns_shortcut": "Use (100+1)*m = 100*m + 1*m.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "What is 101 * 14?", + "options": [ + "1414", + "1434", + "1404", + "1424" + ], + "answer": "1414", + "correct_value": 1414, + "explanation": "Distributive: (100+1)*14 = 100*14 + 1*14." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "105 * 68", + "ns_shortcut": "Use (100+5)*m, correction is 5*68.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "What is 105 * 68?", + "options": [ + "7150", + "7130", + "7160", + "7140" + ], + "answer": "7140", + "correct_value": 7140, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "68 * 47", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 68 * 47?", + "options": [ + "3216", + "3196", + "3186", + "3206" + ], + "answer": "3196", + "correct_value": 3196, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000057", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 * 92", + "ns_shortcut": "Use (100-1)*m = 100*m - 1*m.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "What is 99 * 92?", + "options": [ + "9128", + "9108", + "9098", + "9118" + ], + "answer": "9108", + "correct_value": 9108, + "explanation": "Distributive: (100-1)*92 = 100*92 - 1*92." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "95 * 17", + "ns_shortcut": "Use (100-5)*m, correction is 5*17.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "What is 95 * 17?", + "options": [ + "1615", + "1635", + "1605", + "1625" + ], + "answer": "1615", + "correct_value": 1615, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "53 * 42", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 53 * 42?", + "options": [ + "2226", + "2236", + "2246", + "2216" + ], + "answer": "2226", + "correct_value": 2226, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000058", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 44", + "ns_shortcut": "Use (100+2)*m = 100*m + 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 102 * 44?", + "options": [ + "4508", + "4488", + "4498", + "4478" + ], + "answer": "4488", + "correct_value": 4488, + "explanation": "Distributive: (100+2)*44 = 100*44 + 2*44." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "104 * 15", + "ns_shortcut": "Use (100+4)*m, correction is 4*15.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 104 * 15?", + "options": [ + "1560", + "1570", + "1580", + "1550" + ], + "answer": "1560", + "correct_value": 1560, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "58 * 47", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 58 * 47?", + "options": [ + "2736", + "2746", + "2716", + "2726" + ], + "answer": "2726", + "correct_value": 2726, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000059", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 96", + "ns_shortcut": "Use (100-2)*m = 100*m - 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 98 * 96?", + "options": [ + "9418", + "9408", + "9398", + "9428" + ], + "answer": "9408", + "correct_value": 9408, + "explanation": "Distributive: (100-2)*96 = 100*96 - 2*96." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "96 * 70", + "ns_shortcut": "Use (100-4)*m, correction is 4*70.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 96 * 70?", + "options": [ + "6740", + "6720", + "6730", + "6710" + ], + "answer": "6720", + "correct_value": 6720, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "57 * 52", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 57 * 52?", + "options": [ + "2984", + "2974", + "2964", + "2954" + ], + "answer": "2964", + "correct_value": 2964, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000060", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 62", + "ns_shortcut": "Use (100+1)*m = 100*m + 1*m.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "What is 101 * 62?", + "options": [ + "6252", + "6262", + "6272", + "6282" + ], + "answer": "6262", + "correct_value": 6262, + "explanation": "Distributive: (100+1)*62 = 100*62 + 1*62." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "96 * 52", + "ns_shortcut": "Use (100-4)*m, correction is 4*52.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 96 * 52?", + "options": [ + "4992", + "5002", + "5012", + "4982" + ], + "answer": "4992", + "correct_value": 4992, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "48 * 60", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 48 * 60?", + "options": [ + "2890", + "2880", + "2900", + "2870" + ], + "answer": "2880", + "correct_value": 2880, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000061", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 21", + "ns_shortcut": "Use (100+2)*m = 100*m + 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 102 * 21?", + "options": [ + "2142", + "2132", + "2152", + "2162" + ], + "answer": "2142", + "correct_value": 2142, + "explanation": "Distributive: (100+2)*21 = 100*21 + 2*21." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "95 * 61", + "ns_shortcut": "Use (100-5)*m, correction is 5*61.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "What is 95 * 61?", + "options": [ + "5785", + "5805", + "5815", + "5795" + ], + "answer": "5795", + "correct_value": 5795, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "52 * 40", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 52 * 40?", + "options": [ + "2080", + "2070", + "2100", + "2090" + ], + "answer": "2080", + "correct_value": 2080, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000062", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 90", + "ns_shortcut": "Use (100+2)*m = 100*m + 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 102 * 90?", + "options": [ + "9190", + "9200", + "9180", + "9170" + ], + "answer": "9180", + "correct_value": 9180, + "explanation": "Distributive: (100+2)*90 = 100*90 + 2*90." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "97 * 80", + "ns_shortcut": "Use (100-3)*m, correction is 3*80.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "What is 97 * 80?", + "options": [ + "7760", + "7780", + "7750", + "7770" + ], + "answer": "7760", + "correct_value": 7760, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "67 * 61", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 67 * 61?", + "options": [ + "4097", + "4107", + "4087", + "4077" + ], + "answer": "4087", + "correct_value": 4087, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000063", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 * 30", + "ns_shortcut": "Use (100-1)*m = 100*m - 1*m.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "What is 99 * 30?", + "options": [ + "2980", + "2970", + "2960", + "2990" + ], + "answer": "2970", + "correct_value": 2970, + "explanation": "Distributive: (100-1)*30 = 100*30 - 1*30." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "104 * 62", + "ns_shortcut": "Use (100+4)*m, correction is 4*62.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 104 * 62?", + "options": [ + "6458", + "6468", + "6438", + "6448" + ], + "answer": "6448", + "correct_value": 6448, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "53 * 69", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 53 * 69?", + "options": [ + "3667", + "3647", + "3677", + "3657" + ], + "answer": "3657", + "correct_value": 3657, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000064", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 47", + "ns_shortcut": "Use (100+2)*m = 100*m + 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 102 * 47?", + "options": [ + "4794", + "4784", + "4804", + "4814" + ], + "answer": "4794", + "correct_value": 4794, + "explanation": "Distributive: (100+2)*47 = 100*47 + 2*47." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "103 * 41", + "ns_shortcut": "Use (100+3)*m, correction is 3*41.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "What is 103 * 41?", + "options": [ + "4233", + "4223", + "4243", + "4213" + ], + "answer": "4223", + "correct_value": 4223, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "51 * 45", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 51 * 45?", + "options": [ + "2315", + "2285", + "2295", + "2305" + ], + "answer": "2295", + "correct_value": 2295, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000065", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 53", + "ns_shortcut": "Use (100+2)*m = 100*m + 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 102 * 53?", + "options": [ + "5396", + "5416", + "5406", + "5426" + ], + "answer": "5406", + "correct_value": 5406, + "explanation": "Distributive: (100+2)*53 = 100*53 + 2*53." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "105 * 48", + "ns_shortcut": "Use (100+5)*m, correction is 5*48.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "What is 105 * 48?", + "options": [ + "5060", + "5050", + "5040", + "5030" + ], + "answer": "5040", + "correct_value": 5040, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "56 * 54", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 56 * 54?", + "options": [ + "3024", + "3014", + "3034", + "3044" + ], + "answer": "3024", + "correct_value": 3024, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000066", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 51", + "ns_shortcut": "Use (100+2)*m = 100*m + 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 102 * 51?", + "options": [ + "5192", + "5202", + "5222", + "5212" + ], + "answer": "5202", + "correct_value": 5202, + "explanation": "Distributive: (100+2)*51 = 100*51 + 2*51." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "104 * 59", + "ns_shortcut": "Use (100+4)*m, correction is 4*59.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 104 * 59?", + "options": [ + "6156", + "6136", + "6126", + "6146" + ], + "answer": "6136", + "correct_value": 6136, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "49 * 62", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 49 * 62?", + "options": [ + "3038", + "3048", + "3058", + "3028" + ], + "answer": "3038", + "correct_value": 3038, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000067", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 94", + "ns_shortcut": "Use (100+2)*m = 100*m + 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 102 * 94?", + "options": [ + "9608", + "9588", + "9578", + "9598" + ], + "answer": "9588", + "correct_value": 9588, + "explanation": "Distributive: (100+2)*94 = 100*94 + 2*94." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "96 * 74", + "ns_shortcut": "Use (100-4)*m, correction is 4*74.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 96 * 74?", + "options": [ + "7094", + "7114", + "7104", + "7124" + ], + "answer": "7104", + "correct_value": 7104, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "42 * 55", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 42 * 55?", + "options": [ + "2320", + "2330", + "2310", + "2300" + ], + "answer": "2310", + "correct_value": 2310, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000068", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 93", + "ns_shortcut": "Use (100+2)*m = 100*m + 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 102 * 93?", + "options": [ + "9506", + "9476", + "9486", + "9496" + ], + "answer": "9486", + "correct_value": 9486, + "explanation": "Distributive: (100+2)*93 = 100*93 + 2*93." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "97 * 53", + "ns_shortcut": "Use (100-3)*m, correction is 3*53.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "What is 97 * 53?", + "options": [ + "5151", + "5141", + "5161", + "5131" + ], + "answer": "5141", + "correct_value": 5141, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "66 * 49", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 66 * 49?", + "options": [ + "3234", + "3254", + "3244", + "3224" + ], + "answer": "3234", + "correct_value": 3234, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000069", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 89", + "ns_shortcut": "Use (100-2)*m = 100*m - 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 98 * 89?", + "options": [ + "8742", + "8732", + "8712", + "8722" + ], + "answer": "8722", + "correct_value": 8722, + "explanation": "Distributive: (100-2)*89 = 100*89 - 2*89." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "96 * 21", + "ns_shortcut": "Use (100-4)*m, correction is 4*21.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 96 * 21?", + "options": [ + "2026", + "2036", + "2016", + "2006" + ], + "answer": "2016", + "correct_value": 2016, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "53 * 44", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 53 * 44?", + "options": [ + "2342", + "2322", + "2332", + "2352" + ], + "answer": "2332", + "correct_value": 2332, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000070", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 17", + "ns_shortcut": "Use (100+1)*m = 100*m + 1*m.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "What is 101 * 17?", + "options": [ + "1717", + "1707", + "1727", + "1737" + ], + "answer": "1717", + "correct_value": 1717, + "explanation": "Distributive: (100+1)*17 = 100*17 + 1*17." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "103 * 28", + "ns_shortcut": "Use (100+3)*m, correction is 3*28.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "What is 103 * 28?", + "options": [ + "2884", + "2874", + "2904", + "2894" + ], + "answer": "2884", + "correct_value": 2884, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "57 * 60", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 57 * 60?", + "options": [ + "3410", + "3440", + "3420", + "3430" + ], + "answer": "3420", + "correct_value": 3420, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000071", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 88", + "ns_shortcut": "Use (100-2)*m = 100*m - 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 98 * 88?", + "options": [ + "8634", + "8624", + "8644", + "8614" + ], + "answer": "8624", + "correct_value": 8624, + "explanation": "Distributive: (100-2)*88 = 100*88 - 2*88." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "103 * 84", + "ns_shortcut": "Use (100+3)*m, correction is 3*84.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "What is 103 * 84?", + "options": [ + "8672", + "8642", + "8652", + "8662" + ], + "answer": "8652", + "correct_value": 8652, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "59 * 47", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 59 * 47?", + "options": [ + "2793", + "2773", + "2783", + "2763" + ], + "answer": "2773", + "correct_value": 2773, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000072", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 46", + "ns_shortcut": "Use (100+2)*m = 100*m + 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 102 * 46?", + "options": [ + "4712", + "4682", + "4692", + "4702" + ], + "answer": "4692", + "correct_value": 4692, + "explanation": "Distributive: (100+2)*46 = 100*46 + 2*46." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "97 * 85", + "ns_shortcut": "Use (100-3)*m, correction is 3*85.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "What is 97 * 85?", + "options": [ + "8245", + "8255", + "8235", + "8265" + ], + "answer": "8245", + "correct_value": 8245, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "61 * 47", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 61 * 47?", + "options": [ + "2887", + "2867", + "2877", + "2857" + ], + "answer": "2867", + "correct_value": 2867, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000073", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 71", + "ns_shortcut": "Use (100+2)*m = 100*m + 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 102 * 71?", + "options": [ + "7262", + "7252", + "7232", + "7242" + ], + "answer": "7242", + "correct_value": 7242, + "explanation": "Distributive: (100+2)*71 = 100*71 + 2*71." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "104 * 55", + "ns_shortcut": "Use (100+4)*m, correction is 4*55.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 104 * 55?", + "options": [ + "5720", + "5730", + "5710", + "5740" + ], + "answer": "5720", + "correct_value": 5720, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "39 * 65", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 39 * 65?", + "options": [ + "2555", + "2535", + "2545", + "2525" + ], + "answer": "2535", + "correct_value": 2535, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000074", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 87", + "ns_shortcut": "Use (100-2)*m = 100*m - 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 98 * 87?", + "options": [ + "8546", + "8526", + "8536", + "8516" + ], + "answer": "8526", + "correct_value": 8526, + "explanation": "Distributive: (100-2)*87 = 100*87 - 2*87." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "104 * 55", + "ns_shortcut": "Use (100+4)*m, correction is 4*55.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 104 * 55?", + "options": [ + "5710", + "5720", + "5740", + "5730" + ], + "answer": "5720", + "correct_value": 5720, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "58 * 47", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 58 * 47?", + "options": [ + "2716", + "2726", + "2746", + "2736" + ], + "answer": "2726", + "correct_value": 2726, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000075", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 49", + "ns_shortcut": "Use (100-2)*m = 100*m - 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 98 * 49?", + "options": [ + "4822", + "4812", + "4802", + "4792" + ], + "answer": "4802", + "correct_value": 4802, + "explanation": "Distributive: (100-2)*49 = 100*49 - 2*49." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "105 * 26", + "ns_shortcut": "Use (100+5)*m, correction is 5*26.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "What is 105 * 26?", + "options": [ + "2720", + "2750", + "2740", + "2730" + ], + "answer": "2730", + "correct_value": 2730, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "62 * 52", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 62 * 52?", + "options": [ + "3214", + "3244", + "3224", + "3234" + ], + "answer": "3224", + "correct_value": 3224, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000076", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 77", + "ns_shortcut": "Use (100+1)*m = 100*m + 1*m.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "What is 101 * 77?", + "options": [ + "7767", + "7787", + "7777", + "7797" + ], + "answer": "7777", + "correct_value": 7777, + "explanation": "Distributive: (100+1)*77 = 100*77 + 1*77." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "103 * 47", + "ns_shortcut": "Use (100+3)*m, correction is 3*47.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "What is 103 * 47?", + "options": [ + "4831", + "4851", + "4841", + "4861" + ], + "answer": "4841", + "correct_value": 4841, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "69 * 41", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 69 * 41?", + "options": [ + "2849", + "2819", + "2829", + "2839" + ], + "answer": "2829", + "correct_value": 2829, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000077", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 21", + "ns_shortcut": "Use (100+1)*m = 100*m + 1*m.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "What is 101 * 21?", + "options": [ + "2141", + "2131", + "2121", + "2111" + ], + "answer": "2121", + "correct_value": 2121, + "explanation": "Distributive: (100+1)*21 = 100*21 + 1*21." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "104 * 25", + "ns_shortcut": "Use (100+4)*m, correction is 4*25.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 104 * 25?", + "options": [ + "2600", + "2620", + "2610", + "2590" + ], + "answer": "2600", + "correct_value": 2600, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "57 * 51", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 57 * 51?", + "options": [ + "2917", + "2897", + "2907", + "2927" + ], + "answer": "2907", + "correct_value": 2907, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000078", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 13", + "ns_shortcut": "Use (100+2)*m = 100*m + 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 102 * 13?", + "options": [ + "1336", + "1326", + "1316", + "1346" + ], + "answer": "1326", + "correct_value": 1326, + "explanation": "Distributive: (100+2)*13 = 100*13 + 2*13." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "105 * 18", + "ns_shortcut": "Use (100+5)*m, correction is 5*18.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "What is 105 * 18?", + "options": [ + "1890", + "1900", + "1910", + "1880" + ], + "answer": "1890", + "correct_value": 1890, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "59 * 52", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 59 * 52?", + "options": [ + "3078", + "3088", + "3058", + "3068" + ], + "answer": "3068", + "correct_value": 3068, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000079", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 48", + "ns_shortcut": "Use (100-2)*m = 100*m - 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 98 * 48?", + "options": [ + "4714", + "4724", + "4704", + "4694" + ], + "answer": "4704", + "correct_value": 4704, + "explanation": "Distributive: (100-2)*48 = 100*48 - 2*48." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "96 * 78", + "ns_shortcut": "Use (100-4)*m, correction is 4*78.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 96 * 78?", + "options": [ + "7488", + "7508", + "7478", + "7498" + ], + "answer": "7488", + "correct_value": 7488, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "50 * 60", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 50 * 60?", + "options": [ + "3010", + "3000", + "3020", + "2990" + ], + "answer": "3000", + "correct_value": 3000, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000080", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 93", + "ns_shortcut": "Use (100+2)*m = 100*m + 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 102 * 93?", + "options": [ + "9476", + "9506", + "9496", + "9486" + ], + "answer": "9486", + "correct_value": 9486, + "explanation": "Distributive: (100+2)*93 = 100*93 + 2*93." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "97 * 14", + "ns_shortcut": "Use (100-3)*m, correction is 3*14.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "What is 97 * 14?", + "options": [ + "1348", + "1368", + "1358", + "1378" + ], + "answer": "1358", + "correct_value": 1358, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "53 * 69", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 53 * 69?", + "options": [ + "3677", + "3657", + "3647", + "3667" + ], + "answer": "3657", + "correct_value": 3657, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000081", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 79", + "ns_shortcut": "Use (100+1)*m = 100*m + 1*m.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "What is 101 * 79?", + "options": [ + "7969", + "7989", + "7979", + "7999" + ], + "answer": "7979", + "correct_value": 7979, + "explanation": "Distributive: (100+1)*79 = 100*79 + 1*79." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "97 * 34", + "ns_shortcut": "Use (100-3)*m, correction is 3*34.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "What is 97 * 34?", + "options": [ + "3318", + "3288", + "3308", + "3298" + ], + "answer": "3298", + "correct_value": 3298, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "60 * 45", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 60 * 45?", + "options": [ + "2720", + "2710", + "2690", + "2700" + ], + "answer": "2700", + "correct_value": 2700, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000082", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 * 26", + "ns_shortcut": "Use (100-1)*m = 100*m - 1*m.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "What is 99 * 26?", + "options": [ + "2564", + "2594", + "2574", + "2584" + ], + "answer": "2574", + "correct_value": 2574, + "explanation": "Distributive: (100-1)*26 = 100*26 - 1*26." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "104 * 37", + "ns_shortcut": "Use (100+4)*m, correction is 4*37.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 104 * 37?", + "options": [ + "3868", + "3848", + "3838", + "3858" + ], + "answer": "3848", + "correct_value": 3848, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "64 * 67", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 64 * 67?", + "options": [ + "4308", + "4278", + "4298", + "4288" + ], + "answer": "4288", + "correct_value": 4288, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000083", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 * 79", + "ns_shortcut": "Use (100-1)*m = 100*m - 1*m.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "What is 99 * 79?", + "options": [ + "7831", + "7811", + "7821", + "7841" + ], + "answer": "7821", + "correct_value": 7821, + "explanation": "Distributive: (100-1)*79 = 100*79 - 1*79." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "105 * 13", + "ns_shortcut": "Use (100+5)*m, correction is 5*13.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "What is 105 * 13?", + "options": [ + "1385", + "1355", + "1365", + "1375" + ], + "answer": "1365", + "correct_value": 1365, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "64 * 64", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 64 * 64?", + "options": [ + "4096", + "4086", + "4116", + "4106" + ], + "answer": "4096", + "correct_value": 4096, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000084", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 32", + "ns_shortcut": "Use (100-2)*m = 100*m - 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 98 * 32?", + "options": [ + "3146", + "3126", + "3136", + "3156" + ], + "answer": "3136", + "correct_value": 3136, + "explanation": "Distributive: (100-2)*32 = 100*32 - 2*32." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "105 * 63", + "ns_shortcut": "Use (100+5)*m, correction is 5*63.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "What is 105 * 63?", + "options": [ + "6625", + "6605", + "6635", + "6615" + ], + "answer": "6615", + "correct_value": 6615, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "54 * 53", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 54 * 53?", + "options": [ + "2872", + "2862", + "2852", + "2882" + ], + "answer": "2862", + "correct_value": 2862, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000085", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 74", + "ns_shortcut": "Use (100+2)*m = 100*m + 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 102 * 74?", + "options": [ + "7538", + "7548", + "7568", + "7558" + ], + "answer": "7548", + "correct_value": 7548, + "explanation": "Distributive: (100+2)*74 = 100*74 + 2*74." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "105 * 94", + "ns_shortcut": "Use (100+5)*m, correction is 5*94.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "What is 105 * 94?", + "options": [ + "9860", + "9870", + "9880", + "9890" + ], + "answer": "9870", + "correct_value": 9870, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "40 * 54", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 40 * 54?", + "options": [ + "2180", + "2160", + "2170", + "2150" + ], + "answer": "2160", + "correct_value": 2160, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000086", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 66", + "ns_shortcut": "Use (100+2)*m = 100*m + 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 102 * 66?", + "options": [ + "6752", + "6742", + "6722", + "6732" + ], + "answer": "6732", + "correct_value": 6732, + "explanation": "Distributive: (100+2)*66 = 100*66 + 2*66." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "104 * 28", + "ns_shortcut": "Use (100+4)*m, correction is 4*28.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 104 * 28?", + "options": [ + "2932", + "2922", + "2902", + "2912" + ], + "answer": "2912", + "correct_value": 2912, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "58 * 59", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 58 * 59?", + "options": [ + "3422", + "3412", + "3432", + "3442" + ], + "answer": "3422", + "correct_value": 3422, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000087", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 74", + "ns_shortcut": "Use (100+2)*m = 100*m + 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 102 * 74?", + "options": [ + "7558", + "7568", + "7548", + "7538" + ], + "answer": "7548", + "correct_value": 7548, + "explanation": "Distributive: (100+2)*74 = 100*74 + 2*74." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "105 * 93", + "ns_shortcut": "Use (100+5)*m, correction is 5*93.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "What is 105 * 93?", + "options": [ + "9765", + "9785", + "9755", + "9775" + ], + "answer": "9765", + "correct_value": 9765, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "67 * 46", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 67 * 46?", + "options": [ + "3082", + "3102", + "3092", + "3072" + ], + "answer": "3082", + "correct_value": 3082, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000088", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 63", + "ns_shortcut": "Use (100+2)*m = 100*m + 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 102 * 63?", + "options": [ + "6416", + "6446", + "6436", + "6426" + ], + "answer": "6426", + "correct_value": 6426, + "explanation": "Distributive: (100+2)*63 = 100*63 + 2*63." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "97 * 95", + "ns_shortcut": "Use (100-3)*m, correction is 3*95.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "What is 97 * 95?", + "options": [ + "9205", + "9215", + "9225", + "9235" + ], + "answer": "9215", + "correct_value": 9215, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "60 * 68", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 60 * 68?", + "options": [ + "4090", + "4100", + "4080", + "4070" + ], + "answer": "4080", + "correct_value": 4080, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000089", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 21", + "ns_shortcut": "Use (100-2)*m = 100*m - 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 98 * 21?", + "options": [ + "2078", + "2068", + "2058", + "2048" + ], + "answer": "2058", + "correct_value": 2058, + "explanation": "Distributive: (100-2)*21 = 100*21 - 2*21." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "96 * 81", + "ns_shortcut": "Use (100-4)*m, correction is 4*81.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 96 * 81?", + "options": [ + "7766", + "7786", + "7796", + "7776" + ], + "answer": "7776", + "correct_value": 7776, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "60 * 42", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 60 * 42?", + "options": [ + "2520", + "2540", + "2530", + "2510" + ], + "answer": "2520", + "correct_value": 2520, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000090", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 46", + "ns_shortcut": "Use (100-2)*m = 100*m - 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 98 * 46?", + "options": [ + "4498", + "4508", + "4518", + "4528" + ], + "answer": "4508", + "correct_value": 4508, + "explanation": "Distributive: (100-2)*46 = 100*46 - 2*46." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "104 * 74", + "ns_shortcut": "Use (100+4)*m, correction is 4*74.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 104 * 74?", + "options": [ + "7716", + "7686", + "7696", + "7706" + ], + "answer": "7696", + "correct_value": 7696, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "46 * 66", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 46 * 66?", + "options": [ + "3026", + "3056", + "3046", + "3036" + ], + "answer": "3036", + "correct_value": 3036, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000091", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 45", + "ns_shortcut": "Use (100-2)*m = 100*m - 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 98 * 45?", + "options": [ + "4430", + "4410", + "4400", + "4420" + ], + "answer": "4410", + "correct_value": 4410, + "explanation": "Distributive: (100-2)*45 = 100*45 - 2*45." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "104 * 94", + "ns_shortcut": "Use (100+4)*m, correction is 4*94.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 104 * 94?", + "options": [ + "9786", + "9766", + "9776", + "9796" + ], + "answer": "9776", + "correct_value": 9776, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "47 * 61", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 47 * 61?", + "options": [ + "2867", + "2857", + "2877", + "2887" + ], + "answer": "2867", + "correct_value": 2867, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000092", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 33", + "ns_shortcut": "Use (100-2)*m = 100*m - 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 98 * 33?", + "options": [ + "3234", + "3224", + "3254", + "3244" + ], + "answer": "3234", + "correct_value": 3234, + "explanation": "Distributive: (100-2)*33 = 100*33 - 2*33." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "95 * 28", + "ns_shortcut": "Use (100-5)*m, correction is 5*28.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "What is 95 * 28?", + "options": [ + "2670", + "2680", + "2650", + "2660" + ], + "answer": "2660", + "correct_value": 2660, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "51 * 52", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 51 * 52?", + "options": [ + "2672", + "2652", + "2662", + "2642" + ], + "answer": "2652", + "correct_value": 2652, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000093", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 49", + "ns_shortcut": "Use (100+1)*m = 100*m + 1*m.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "What is 101 * 49?", + "options": [ + "4959", + "4949", + "4939", + "4969" + ], + "answer": "4949", + "correct_value": 4949, + "explanation": "Distributive: (100+1)*49 = 100*49 + 1*49." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "95 * 29", + "ns_shortcut": "Use (100-5)*m, correction is 5*29.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "What is 95 * 29?", + "options": [ + "2745", + "2755", + "2775", + "2765" + ], + "answer": "2755", + "correct_value": 2755, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "60 * 41", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 60 * 41?", + "options": [ + "2460", + "2450", + "2470", + "2480" + ], + "answer": "2460", + "correct_value": 2460, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000094", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 87", + "ns_shortcut": "Use (100+2)*m = 100*m + 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 102 * 87?", + "options": [ + "8874", + "8894", + "8884", + "8864" + ], + "answer": "8874", + "correct_value": 8874, + "explanation": "Distributive: (100+2)*87 = 100*87 + 2*87." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "96 * 75", + "ns_shortcut": "Use (100-4)*m, correction is 4*75.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 96 * 75?", + "options": [ + "7190", + "7220", + "7210", + "7200" + ], + "answer": "7200", + "correct_value": 7200, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "42 * 48", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 42 * 48?", + "options": [ + "2016", + "2026", + "2036", + "2006" + ], + "answer": "2016", + "correct_value": 2016, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000095", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 38", + "ns_shortcut": "Use (100-2)*m = 100*m - 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 98 * 38?", + "options": [ + "3734", + "3724", + "3744", + "3714" + ], + "answer": "3724", + "correct_value": 3724, + "explanation": "Distributive: (100-2)*38 = 100*38 - 2*38." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "95 * 36", + "ns_shortcut": "Use (100-5)*m, correction is 5*36.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "What is 95 * 36?", + "options": [ + "3430", + "3410", + "3420", + "3440" + ], + "answer": "3420", + "correct_value": 3420, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "63 * 47", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 63 * 47?", + "options": [ + "2951", + "2961", + "2971", + "2981" + ], + "answer": "2961", + "correct_value": 2961, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000096", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 * 73", + "ns_shortcut": "Use (100-2)*m = 100*m - 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 98 * 73?", + "options": [ + "7164", + "7174", + "7154", + "7144" + ], + "answer": "7154", + "correct_value": 7154, + "explanation": "Distributive: (100-2)*73 = 100*73 - 2*73." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "103 * 78", + "ns_shortcut": "Use (100+3)*m, correction is 3*78.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "What is 103 * 78?", + "options": [ + "8034", + "8024", + "8054", + "8044" + ], + "answer": "8034", + "correct_value": 8034, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "41 * 51", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 41 * 51?", + "options": [ + "2111", + "2081", + "2101", + "2091" + ], + "answer": "2091", + "correct_value": 2091, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000097", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "102 * 19", + "ns_shortcut": "Use (100+2)*m = 100*m + 2*m.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "What is 102 * 19?", + "options": [ + "1938", + "1948", + "1928", + "1958" + ], + "answer": "1938", + "correct_value": 1938, + "explanation": "Distributive: (100+2)*19 = 100*19 + 2*19." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "104 * 61", + "ns_shortcut": "Use (100+4)*m, correction is 4*61.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 104 * 61?", + "options": [ + "6364", + "6344", + "6354", + "6334" + ], + "answer": "6344", + "correct_value": 6344, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "51 * 39", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 51 * 39?", + "options": [ + "1979", + "1999", + "1989", + "2009" + ], + "answer": "1989", + "correct_value": 1989, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000098", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "101 * 21", + "ns_shortcut": "Use (100+1)*m = 100*m + 1*m.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "What is 101 * 21?", + "options": [ + "2131", + "2141", + "2121", + "2111" + ], + "answer": "2121", + "correct_value": 2121, + "explanation": "Distributive: (100+1)*21 = 100*21 + 1*21." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "103 * 85", + "ns_shortcut": "Use (100+3)*m, correction is 3*85.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "What is 103 * 85?", + "options": [ + "8765", + "8755", + "8775", + "8745" + ], + "answer": "8755", + "correct_value": 8755, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "51 * 66", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 51 * 66?", + "options": [ + "3366", + "3376", + "3356", + "3386" + ], + "answer": "3366", + "correct_value": 3366, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000099", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 * 52", + "ns_shortcut": "Use (100-1)*m = 100*m - 1*m.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "What is 99 * 52?", + "options": [ + "5138", + "5168", + "5148", + "5158" + ], + "answer": "5148", + "correct_value": 5148, + "explanation": "Distributive: (100-1)*52 = 100*52 - 1*52." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "104 * 75", + "ns_shortcut": "Use (100+4)*m, correction is 4*75.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 104 * 75?", + "options": [ + "7800", + "7820", + "7790", + "7810" + ], + "answer": "7800", + "correct_value": 7800, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "62 * 69", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 62 * 69?", + "options": [ + "4268", + "4298", + "4278", + "4288" + ], + "answer": "4278", + "correct_value": 4278, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000100", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 * 42", + "ns_shortcut": "Use (100-1)*m = 100*m - 1*m.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "What is 99 * 42?", + "options": [ + "4148", + "4178", + "4158", + "4168" + ], + "answer": "4158", + "correct_value": 4158, + "explanation": "Distributive: (100-1)*42 = 100*42 - 1*42." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "105 * 19", + "ns_shortcut": "Use (100+5)*m, correction is 5*19.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "What is 105 * 19?", + "options": [ + "1995", + "1985", + "2015", + "2005" + ], + "answer": "1995", + "correct_value": 1995, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "53 * 63", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 53 * 63?", + "options": [ + "3339", + "3349", + "3359", + "3329" + ], + "answer": "3339", + "correct_value": 3339, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "relative_distance_000101", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 42/90 or 27/50?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.07333333333333336, + "pure_math": { + "question": "Which is larger: 42/90 or 27/50?", + "options": [ + "42/90", + "27/50", + "35/66", + "11/18" + ], + "answer": "27/50", + "correct_value": 0.54, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 26/55 or 41/74?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.08132678132678134, + "pure_math": { + "question": "Which is larger: 26/55 or 41/74?", + "options": [ + "8/19", + "26/55", + "45/69", + "41/74" + ], + "answer": "41/74", + "correct_value": 0.5540540540540541, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 26/46 or 12/19?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 26/46 or 12/19?", + "options": [ + "28/92", + "12/19", + "26/46", + "60/88" + ], + "answer": "12/19", + "correct_value": 0.631578947368421, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000102", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 32/67 or 43/83?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.040460348858119055, + "pure_math": { + "question": "Which is larger: 32/67 or 43/83?", + "options": [ + "31/51", + "49/70", + "43/83", + "32/67" + ], + "answer": "43/83", + "correct_value": 0.5180722891566265, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 11/23 or 29/53?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.0689089417555373, + "pure_math": { + "question": "Which is larger: 11/23 or 29/53?", + "options": [ + "13/20", + "11/23", + "34/92", + "29/53" + ], + "answer": "29/53", + "correct_value": 0.5471698113207547, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 37/66 or 45/77?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 37/66 or 45/77?", + "options": [ + "45/77", + "25/56", + "37/66", + "29/76" + ], + "answer": "45/77", + "correct_value": 0.5844155844155844, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000103", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 25/53 or 49/96?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.038718553459119454, + "pure_math": { + "question": "Which is larger: 25/53 or 49/96?", + "options": [ + "15/52", + "25/53", + "49/96", + "63/90" + ], + "answer": "49/96", + "correct_value": 0.5104166666666666, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 22/46 or 23/42?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.06935817805383027, + "pure_math": { + "question": "Which is larger: 22/46 or 23/42?", + "options": [ + "22/46", + "12/29", + "61/94", + "23/42" + ], + "answer": "23/42", + "correct_value": 0.5476190476190477, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 30/54 or 29/49?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 30/54 or 29/49?", + "options": [ + "48/94", + "29/49", + "67/93", + "30/54" + ], + "answer": "29/49", + "correct_value": 0.5918367346938775, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000104", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 22/49 or 44/81?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.09423028470647515, + "pure_math": { + "question": "Which is larger: 22/49 or 44/81?", + "options": [ + "16/26", + "22/49", + "39/52", + "44/81" + ], + "answer": "44/81", + "correct_value": 0.5432098765432098, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 36/75 or 31/56?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07357142857142862, + "pure_math": { + "question": "Which is larger: 36/75 or 31/56?", + "options": [ + "31/56", + "5/12", + "17/49", + "36/75" + ], + "answer": "31/56", + "correct_value": 0.5535714285714286, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 19/34 or 23/38?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 19/34 or 23/38?", + "options": [ + "19/34", + "8/19", + "8/27", + "23/38" + ], + "answer": "23/38", + "correct_value": 0.6052631578947368, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000105", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 46/94 or 28/53?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.038940184664793265, + "pure_math": { + "question": "Which is larger: 46/94 or 28/53?", + "options": [ + "9/29", + "28/53", + "45/86", + "46/94" + ], + "answer": "28/53", + "correct_value": 0.5283018867924528, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 32/66 or 8/15?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.048484848484848464, + "pure_math": { + "question": "Which is larger: 32/66 or 8/15?", + "options": [ + "8/15", + "38/62", + "70/96", + "32/66" + ], + "answer": "8/15", + "correct_value": 0.5333333333333333, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 37/67 or 23/39?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 37/67 or 23/39?", + "options": [ + "23/39", + "34/78", + "27/49", + "37/67" + ], + "answer": "23/39", + "correct_value": 0.5897435897435898, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000106", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 32/69 or 26/46?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.10144927536231879, + "pure_math": { + "question": "Which is larger: 32/69 or 26/46?", + "options": [ + "45/77", + "26/46", + "13/30", + "32/69" + ], + "answer": "26/46", + "correct_value": 0.5652173913043478, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 44/92 or 39/71?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07103490508266996, + "pure_math": { + "question": "Which is larger: 44/92 or 39/71?", + "options": [ + "31/57", + "16/23", + "44/92", + "39/71" + ], + "answer": "39/71", + "correct_value": 0.5492957746478874, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 17/30 or 26/44?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 17/30 or 26/44?", + "options": [ + "17/30", + "48/67", + "26/44", + "20/28" + ], + "answer": "26/44", + "correct_value": 0.5909090909090909, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000107", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 30/67 or 29/54?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.08927584300718633, + "pure_math": { + "question": "Which is larger: 30/67 or 29/54?", + "options": [ + "10/16", + "29/54", + "41/74", + "30/67" + ], + "answer": "29/54", + "correct_value": 0.5370370370370371, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 27/57 or 21/38?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07894736842105271, + "pure_math": { + "question": "Which is larger: 27/57 or 21/38?", + "options": [ + "27/57", + "16/22", + "21/38", + "5/13" + ], + "answer": "21/38", + "correct_value": 0.5526315789473685, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 8/14 or 27/46?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 8/14 or 27/46?", + "options": [ + "8/14", + "25/70", + "45/70", + "27/46" + ], + "answer": "27/46", + "correct_value": 0.5869565217391305, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000108", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 18/42 or 22/37?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.16602316602316608, + "pure_math": { + "question": "Which is larger: 18/42 or 22/37?", + "options": [ + "23/64", + "18/42", + "22/37", + "54/87" + ], + "answer": "22/37", + "correct_value": 0.5945945945945946, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 12/25 or 9/16?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.08250000000000002, + "pure_math": { + "question": "Which is larger: 12/25 or 9/16?", + "options": [ + "12/25", + "23/40", + "9/16", + "61/91" + ], + "answer": "9/16", + "correct_value": 0.5625, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 41/73 or 50/86?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 41/73 or 50/86?", + "options": [ + "54/90", + "50/86", + "20/78", + "41/73" + ], + "answer": "50/86", + "correct_value": 0.5813953488372093, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000109", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 20/47 or 35/64?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.12134308510638298, + "pure_math": { + "question": "Which is larger: 20/47 or 35/64?", + "options": [ + "46/83", + "35/64", + "20/55", + "20/47" + ], + "answer": "35/64", + "correct_value": 0.546875, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 37/77 or 31/56?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07305194805194809, + "pure_math": { + "question": "Which is larger: 37/77 or 31/56?", + "options": [ + "51/79", + "12/18", + "37/77", + "31/56" + ], + "answer": "31/56", + "correct_value": 0.5535714285714286, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 9/15 or 53/91?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 9/15 or 53/91?", + "options": [ + "10/42", + "53/91", + "9/15", + "25/59" + ], + "answer": "9/15", + "correct_value": 0.6, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000110", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 19/40 or 48/94?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03563829787234041, + "pure_math": { + "question": "Which is larger: 19/40 or 48/94?", + "options": [ + "48/94", + "19/40", + "6/27", + "42/84" + ], + "answer": "48/94", + "correct_value": 0.5106382978723404, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 44/92 or 18/32?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.0842391304347826, + "pure_math": { + "question": "Which is larger: 44/92 or 18/32?", + "options": [ + "54/99", + "53/97", + "18/32", + "44/92" + ], + "answer": "18/32", + "correct_value": 0.5625, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 38/68 or 9/14?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 38/68 or 9/14?", + "options": [ + "6/15", + "17/47", + "38/68", + "9/14" + ], + "answer": "9/14", + "correct_value": 0.6428571428571429, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000111", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 36/75 or 32/60?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.053333333333333344, + "pure_math": { + "question": "Which is larger: 36/75 or 32/60?", + "options": [ + "35/92", + "32/60", + "36/75", + "11/42" + ], + "answer": "32/60", + "correct_value": 0.5333333333333333, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 45/93 or 29/52?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07382133995037221, + "pure_math": { + "question": "Which is larger: 45/93 or 29/52?", + "options": [ + "29/52", + "45/93", + "14/44", + "27/57" + ], + "answer": "29/52", + "correct_value": 0.5576923076923077, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 33/59 or 31/53?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 33/59 or 31/53?", + "options": [ + "31/53", + "33/59", + "55/98", + "23/33" + ], + "answer": "31/53", + "correct_value": 0.5849056603773585, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000112", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 7/20 or 34/64?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.18125000000000002, + "pure_math": { + "question": "Which is larger: 7/20 or 34/64?", + "options": [ + "7/20", + "37/79", + "34/64", + "32/87" + ], + "answer": "34/64", + "correct_value": 0.53125, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 24/51 or 52/94?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.08260325406758451, + "pure_math": { + "question": "Which is larger: 24/51 or 52/94?", + "options": [ + "52/94", + "43/67", + "20/47", + "24/51" + ], + "answer": "52/94", + "correct_value": 0.5531914893617021, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 30/54 or 39/67?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 30/54 or 39/67?", + "options": [ + "30/54", + "39/67", + "49/88", + "8/15" + ], + "answer": "39/67", + "correct_value": 0.582089552238806, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000113", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 8/19 or 50/95?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.10526315789473684, + "pure_math": { + "question": "Which is larger: 8/19 or 50/95?", + "options": [ + "30/56", + "8/19", + "34/96", + "50/95" + ], + "answer": "50/95", + "correct_value": 0.5263157894736842, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 7/14 or 14/26?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.038461538461538436, + "pure_math": { + "question": "Which is larger: 7/14 or 14/26?", + "options": [ + "46/92", + "14/26", + "7/14", + "7/18" + ], + "answer": "14/26", + "correct_value": 0.5384615384615384, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 48/87 or 17/29?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 48/87 or 17/29?", + "options": [ + "48/87", + "52/96", + "17/29", + "25/57" + ], + "answer": "17/29", + "correct_value": 0.5862068965517241, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000114", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 36/77 or 30/56?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.06818181818181818, + "pure_math": { + "question": "Which is larger: 36/77 or 30/56?", + "options": [ + "36/77", + "13/41", + "40/93", + "30/56" + ], + "answer": "30/56", + "correct_value": 0.5357142857142857, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 35/73 or 42/76?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07317952415284795, + "pure_math": { + "question": "Which is larger: 35/73 or 42/76?", + "options": [ + "42/76", + "27/81", + "35/73", + "43/59" + ], + "answer": "42/76", + "correct_value": 0.5526315789473685, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 21/37 or 52/88?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 21/37 or 52/88?", + "options": [ + "21/37", + "33/46", + "30/72", + "52/88" + ], + "answer": "52/88", + "correct_value": 0.5909090909090909, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000115", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 32/67 or 18/32?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.08488805970149255, + "pure_math": { + "question": "Which is larger: 32/67 or 18/32?", + "options": [ + "32/67", + "18/32", + "33/50", + "15/21" + ], + "answer": "18/32", + "correct_value": 0.5625, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 38/80 or 43/79?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.06930379746835447, + "pure_math": { + "question": "Which is larger: 38/80 or 43/79?", + "options": [ + "38/80", + "43/79", + "12/30", + "20/47" + ], + "answer": "43/79", + "correct_value": 0.5443037974683544, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 14/25 or 21/35?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 14/25 or 21/35?", + "options": [ + "21/35", + "14/25", + "19/73", + "9/27" + ], + "answer": "21/35", + "correct_value": 0.6, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000116", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 31/69 or 43/80?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.0882246376811594, + "pure_math": { + "question": "Which is larger: 31/69 or 43/80?", + "options": [ + "43/80", + "31/69", + "6/21", + "24/49" + ], + "answer": "43/80", + "correct_value": 0.5375, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 30/62 or 46/83?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07034589972794403, + "pure_math": { + "question": "Which is larger: 30/62 or 46/83?", + "options": [ + "32/50", + "6/19", + "30/62", + "46/83" + ], + "answer": "46/83", + "correct_value": 0.5542168674698795, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 54/97 or 45/76?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 54/97 or 45/76?", + "options": [ + "17/54", + "60/80", + "45/76", + "54/97" + ], + "answer": "45/76", + "correct_value": 0.5921052631578947, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000117", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 34/74 or 12/17?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.24642289348171703, + "pure_math": { + "question": "Which is larger: 34/74 or 12/17?", + "options": [ + "34/74", + "12/17", + "12/31", + "11/29" + ], + "answer": "12/17", + "correct_value": 0.7058823529411765, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 26/54 or 42/76?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07115009746588702, + "pure_math": { + "question": "Which is larger: 26/54 or 42/76?", + "options": [ + "16/42", + "26/54", + "42/76", + "13/46" + ], + "answer": "42/76", + "correct_value": 0.5526315789473685, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 49/88 or 46/78?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 49/88 or 46/78?", + "options": [ + "13/21", + "33/92", + "46/78", + "49/88" + ], + "answer": "46/78", + "correct_value": 0.5897435897435898, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000118", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 38/82 or 9/15?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.1365853658536585, + "pure_math": { + "question": "Which is larger: 38/82 or 9/15?", + "options": [ + "38/82", + "9/15", + "7/15", + "36/93" + ], + "answer": "9/15", + "correct_value": 0.6, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 35/73 or 50/90?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07610350076103506, + "pure_math": { + "question": "Which is larger: 35/73 or 50/90?", + "options": [ + "37/79", + "35/73", + "50/90", + "42/92" + ], + "answer": "50/90", + "correct_value": 0.5555555555555556, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 27/48 or 40/68?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 27/48 or 40/68?", + "options": [ + "40/68", + "4/19", + "27/48", + "21/30" + ], + "answer": "40/68", + "correct_value": 0.5882352941176471, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000119", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 17/36 or 28/53?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.056079664570230614, + "pure_math": { + "question": "Which is larger: 17/36 or 28/53?", + "options": [ + "35/50", + "28/53", + "17/36", + "34/50" + ], + "answer": "28/53", + "correct_value": 0.5283018867924528, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 27/57 or 41/75?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07298245614035087, + "pure_math": { + "question": "Which is larger: 27/57 or 41/75?", + "options": [ + "26/81", + "27/57", + "41/75", + "19/50" + ], + "answer": "41/75", + "correct_value": 0.5466666666666666, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 37/67 or 24/41?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 37/67 or 24/41?", + "options": [ + "24/41", + "37/67", + "8/28", + "8/16" + ], + "answer": "24/41", + "correct_value": 0.5853658536585366, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000120", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 18/41 or 18/30?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.16097560975609754, + "pure_math": { + "question": "Which is larger: 18/41 or 18/30?", + "options": [ + "18/30", + "18/50", + "18/41", + "22/30" + ], + "answer": "18/30", + "correct_value": 0.6, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 35/73 or 7/12?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.10388127853881285, + "pure_math": { + "question": "Which is larger: 35/73 or 7/12?", + "options": [ + "11/21", + "47/74", + "7/12", + "35/73" + ], + "answer": "7/12", + "correct_value": 0.5833333333333334, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 9/16 or 10/16?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 9/16 or 10/16?", + "options": [ + "26/46", + "10/16", + "9/16", + "62/86" + ], + "answer": "10/16", + "correct_value": 0.625, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000121", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 8/23 or 35/65?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.1906354515050167, + "pure_math": { + "question": "Which is larger: 8/23 or 35/65?", + "options": [ + "38/56", + "26/61", + "8/23", + "35/65" + ], + "answer": "35/65", + "correct_value": 0.5384615384615384, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 39/82 or 45/81?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07994579945799463, + "pure_math": { + "question": "Which is larger: 39/82 or 45/81?", + "options": [ + "38/80", + "19/34", + "39/82", + "45/81" + ], + "answer": "45/81", + "correct_value": 0.5555555555555556, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 44/80 or 27/45?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 44/80 or 27/45?", + "options": [ + "65/89", + "29/88", + "27/45", + "44/80" + ], + "answer": "27/45", + "correct_value": 0.6, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000122", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 27/57 or 29/53?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.07348560079443894, + "pure_math": { + "question": "Which is larger: 27/57 or 29/53?", + "options": [ + "27/57", + "29/53", + "10/28", + "53/87" + ], + "answer": "29/53", + "correct_value": 0.5471698113207547, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 7/15 or 8/14?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.10476190476190472, + "pure_math": { + "question": "Which is larger: 7/15 or 8/14?", + "options": [ + "8/14", + "7/15", + "17/27", + "38/52" + ], + "answer": "8/14", + "correct_value": 0.5714285714285714, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 16/29 or 51/87?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 16/29 or 51/87?", + "options": [ + "31/51", + "16/29", + "51/87", + "26/75" + ], + "answer": "51/87", + "correct_value": 0.5862068965517241, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000123", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 43/88 or 26/48?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.05303030303030298, + "pure_math": { + "question": "Which is larger: 43/88 or 26/48?", + "options": [ + "50/78", + "26/48", + "43/88", + "50/74" + ], + "answer": "26/48", + "correct_value": 0.5416666666666666, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 27/57 or 29/52?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.08400809716599195, + "pure_math": { + "question": "Which is larger: 27/57 or 29/52?", + "options": [ + "34/63", + "27/57", + "31/96", + "29/52" + ], + "answer": "29/52", + "correct_value": 0.5576923076923077, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 24/43 or 54/93?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 24/43 or 54/93?", + "options": [ + "57/84", + "24/43", + "54/93", + "10/39" + ], + "answer": "54/93", + "correct_value": 0.5806451612903226, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000124", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 11/28 or 50/97?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.12260677466863029, + "pure_math": { + "question": "Which is larger: 11/28 or 50/97?", + "options": [ + "50/97", + "27/83", + "11/28", + "8/34" + ], + "answer": "50/97", + "correct_value": 0.5154639175257731, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 14/29 or 34/62?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.06562847608453831, + "pure_math": { + "question": "Which is larger: 14/29 or 34/62?", + "options": [ + "14/29", + "45/62", + "37/98", + "34/62" + ], + "answer": "34/62", + "correct_value": 0.5483870967741935, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 42/76 or 21/36?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 42/76 or 21/36?", + "options": [ + "40/81", + "21/36", + "42/76", + "53/94" + ], + "answer": "21/36", + "correct_value": 0.5833333333333334, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000125", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 32/68 or 42/78?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.0678733031674208, + "pure_math": { + "question": "Which is larger: 32/68 or 42/78?", + "options": [ + "42/78", + "32/68", + "65/97", + "34/55" + ], + "answer": "42/78", + "correct_value": 0.5384615384615384, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 16/34 or 23/41?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.09038737446197997, + "pure_math": { + "question": "Which is larger: 16/34 or 23/41?", + "options": [ + "19/48", + "23/41", + "16/34", + "28/45" + ], + "answer": "23/41", + "correct_value": 0.5609756097560976, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 54/98 or 42/72?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 54/98 or 42/72?", + "options": [ + "30/54", + "42/72", + "54/98", + "27/50" + ], + "answer": "42/72", + "correct_value": 0.5833333333333334, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000126", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 40/83 or 50/96?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03890562248995988, + "pure_math": { + "question": "Which is larger: 40/83 or 50/96?", + "options": [ + "41/60", + "50/96", + "37/54", + "40/83" + ], + "answer": "50/96", + "correct_value": 0.5208333333333334, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 22/46 or 11/20?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07173913043478264, + "pure_math": { + "question": "Which is larger: 22/46 or 11/20?", + "options": [ + "22/46", + "27/54", + "11/20", + "39/93" + ], + "answer": "11/20", + "correct_value": 0.55, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 43/78 or 26/44?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 43/78 or 26/44?", + "options": [ + "7/27", + "37/81", + "26/44", + "43/78" + ], + "answer": "26/44", + "correct_value": 0.5909090909090909, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000127", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 44/95 or 42/77?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.08229665071770331, + "pure_math": { + "question": "Which is larger: 44/95 or 42/77?", + "options": [ + "42/77", + "44/95", + "11/22", + "10/39" + ], + "answer": "42/77", + "correct_value": 0.5454545454545454, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 11/22 or 53/97?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.046391752577319534, + "pure_math": { + "question": "Which is larger: 11/22 or 53/97?", + "options": [ + "53/97", + "11/22", + "6/27", + "30/80" + ], + "answer": "53/97", + "correct_value": 0.5463917525773195, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 40/71 or 54/93?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 40/71 or 54/93?", + "options": [ + "40/71", + "17/35", + "21/51", + "54/93" + ], + "answer": "54/93", + "correct_value": 0.5806451612903226, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000128", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 36/78 or 9/14?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.18131868131868134, + "pure_math": { + "question": "Which is larger: 36/78 or 9/14?", + "options": [ + "9/32", + "36/78", + "9/14", + "40/60" + ], + "answer": "9/14", + "correct_value": 0.6428571428571429, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 7/14 or 6/11?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.045454545454545414, + "pure_math": { + "question": "Which is larger: 7/14 or 6/11?", + "options": [ + "6/11", + "27/87", + "7/14", + "30/48" + ], + "answer": "6/11", + "correct_value": 0.5454545454545454, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 51/92 or 46/79?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 51/92 or 46/79?", + "options": [ + "50/77", + "46/79", + "51/92", + "19/27" + ], + "answer": "46/79", + "correct_value": 0.5822784810126582, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000129", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 9/24 or 29/52?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.1826923076923077, + "pure_math": { + "question": "Which is larger: 9/24 or 29/52?", + "options": [ + "8/33", + "29/52", + "9/24", + "33/68" + ], + "answer": "29/52", + "correct_value": 0.5576923076923077, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 15/32 or 48/88?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07670454545454541, + "pure_math": { + "question": "Which is larger: 15/32 or 48/88?", + "options": [ + "48/64", + "15/32", + "48/88", + "47/91" + ], + "answer": "48/88", + "correct_value": 0.5454545454545454, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 53/95 or 13/22?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 53/95 or 13/22?", + "options": [ + "12/33", + "37/71", + "13/22", + "53/95" + ], + "answer": "13/22", + "correct_value": 0.5909090909090909, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000130", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 27/59 or 22/40?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.09237288135593225, + "pure_math": { + "question": "Which is larger: 27/59 or 22/40?", + "options": [ + "57/84", + "22/52", + "27/59", + "22/40" + ], + "answer": "22/40", + "correct_value": 0.55, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 47/98 or 22/40?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07040816326530619, + "pure_math": { + "question": "Which is larger: 47/98 or 22/40?", + "options": [ + "22/40", + "5/17", + "36/69", + "47/98" + ], + "answer": "22/40", + "correct_value": 0.55, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 54/98 or 15/25?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 54/98 or 15/25?", + "options": [ + "33/57", + "40/91", + "15/25", + "54/98" + ], + "answer": "15/25", + "correct_value": 0.6, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000131", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 33/71 or 18/29?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.15590092277804762, + "pure_math": { + "question": "Which is larger: 33/71 or 18/29?", + "options": [ + "45/63", + "44/87", + "18/29", + "33/71" + ], + "answer": "18/29", + "correct_value": 0.6206896551724138, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 37/77 or 37/67?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07171932545066878, + "pure_math": { + "question": "Which is larger: 37/77 or 37/67?", + "options": [ + "37/77", + "42/77", + "13/19", + "37/67" + ], + "answer": "37/67", + "correct_value": 0.5522388059701493, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 46/83 or 46/78?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 46/83 or 46/78?", + "options": [ + "46/83", + "9/29", + "46/78", + "28/51" + ], + "answer": "46/78", + "correct_value": 0.5897435897435898, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000132", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 39/84 or 39/72?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.07738095238095233, + "pure_math": { + "question": "Which is larger: 39/84 or 39/72?", + "options": [ + "19/67", + "39/72", + "39/84", + "25/68" + ], + "answer": "39/72", + "correct_value": 0.5416666666666666, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 39/81 or 38/69?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.06924315619967797, + "pure_math": { + "question": "Which is larger: 39/81 or 38/69?", + "options": [ + "39/81", + "6/12", + "38/69", + "43/96" + ], + "answer": "38/69", + "correct_value": 0.5507246376811594, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 47/84 or 12/19?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 47/84 or 12/19?", + "options": [ + "47/84", + "35/54", + "9/15", + "12/19" + ], + "answer": "12/19", + "correct_value": 0.631578947368421, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000133", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 6/18 or 37/70?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.19523809523809527, + "pure_math": { + "question": "Which is larger: 6/18 or 37/70?", + "options": [ + "37/70", + "26/62", + "6/18", + "18/27" + ], + "answer": "37/70", + "correct_value": 0.5285714285714286, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 32/67 or 32/58?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07411219763252702, + "pure_math": { + "question": "Which is larger: 32/67 or 32/58?", + "options": [ + "32/58", + "66/97", + "16/58", + "32/67" + ], + "answer": "32/58", + "correct_value": 0.5517241379310345, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 14/25 or 19/32?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 14/25 or 19/32?", + "options": [ + "27/77", + "44/78", + "19/32", + "14/25" + ], + "answer": "19/32", + "correct_value": 0.59375, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000134", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 23/48 or 38/69?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.07155797101449274, + "pure_math": { + "question": "Which is larger: 23/48 or 38/69?", + "options": [ + "46/79", + "43/78", + "38/69", + "23/48" + ], + "answer": "38/69", + "correct_value": 0.5507246376811594, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 26/54 or 16/29?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07024265644955302, + "pure_math": { + "question": "Which is larger: 26/54 or 16/29?", + "options": [ + "26/54", + "16/29", + "17/45", + "6/24" + ], + "answer": "16/29", + "correct_value": 0.5517241379310345, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 22/40 or 21/36?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 22/40 or 21/36?", + "options": [ + "22/40", + "21/36", + "5/18", + "23/86" + ], + "answer": "21/36", + "correct_value": 0.5833333333333334, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000135", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 18/41 or 10/15?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.2276422764227642, + "pure_math": { + "question": "Which is larger: 18/41 or 10/15?", + "options": [ + "10/15", + "10/37", + "26/39", + "18/41" + ], + "answer": "10/15", + "correct_value": 0.6666666666666666, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 19/39 or 46/83?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.06703738029039236, + "pure_math": { + "question": "Which is larger: 19/39 or 46/83?", + "options": [ + "64/86", + "19/39", + "46/83", + "30/50" + ], + "answer": "46/83", + "correct_value": 0.5542168674698795, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 16/28 or 55/94?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 16/28 or 55/94?", + "options": [ + "16/28", + "34/62", + "55/94", + "33/76" + ], + "answer": "55/94", + "correct_value": 0.5851063829787234, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000136", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 43/90 or 51/96?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.0534722222222222, + "pure_math": { + "question": "Which is larger: 43/90 or 51/96?", + "options": [ + "51/96", + "43/90", + "21/73", + "33/54" + ], + "answer": "51/96", + "correct_value": 0.53125, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 48/99 or 35/63?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07070707070707072, + "pure_math": { + "question": "Which is larger: 48/99 or 35/63?", + "options": [ + "35/63", + "24/48", + "24/62", + "48/99" + ], + "answer": "35/63", + "correct_value": 0.5555555555555556, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 22/40 or 23/38?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 22/40 or 23/38?", + "options": [ + "34/48", + "22/40", + "23/38", + "19/28" + ], + "answer": "23/38", + "correct_value": 0.6052631578947368, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000137", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 30/63 or 42/81?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.042328042328042326, + "pure_math": { + "question": "Which is larger: 30/63 or 42/81?", + "options": [ + "30/63", + "42/81", + "16/42", + "21/59" + ], + "answer": "42/81", + "correct_value": 0.5185185185185185, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 32/67 or 31/57?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.06624770882429959, + "pure_math": { + "question": "Which is larger: 32/67 or 31/57?", + "options": [ + "9/34", + "31/57", + "32/67", + "13/21" + ], + "answer": "31/57", + "correct_value": 0.543859649122807, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 11/19 or 20/34?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 11/19 or 20/34?", + "options": [ + "63/85", + "11/19", + "20/34", + "33/81" + ], + "answer": "20/34", + "correct_value": 0.5882352941176471, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000138", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 21/48 or 7/11?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.19886363636363635, + "pure_math": { + "question": "Which is larger: 21/48 or 7/11?", + "options": [ + "17/25", + "7/11", + "32/71", + "21/48" + ], + "answer": "7/11", + "correct_value": 0.6363636363636364, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 38/80 or 28/51?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07401960784313732, + "pure_math": { + "question": "Which is larger: 38/80 or 28/51?", + "options": [ + "28/51", + "51/69", + "38/80", + "23/46" + ], + "answer": "28/51", + "correct_value": 0.5490196078431373, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 31/56 or 44/75?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 31/56 or 44/75?", + "options": [ + "47/73", + "44/75", + "31/56", + "16/22" + ], + "answer": "44/75", + "correct_value": 0.5866666666666667, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000139", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 26/57 or 26/50?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.06385964912280706, + "pure_math": { + "question": "Which is larger: 26/57 or 26/50?", + "options": [ + "26/50", + "26/57", + "7/28", + "34/82" + ], + "answer": "26/50", + "correct_value": 0.52, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 23/47 or 16/29?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.062362435803374916, + "pure_math": { + "question": "Which is larger: 23/47 or 16/29?", + "options": [ + "16/29", + "18/35", + "21/39", + "23/47" + ], + "answer": "16/29", + "correct_value": 0.5517241379310345, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 24/42 or 42/72?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 24/42 or 42/72?", + "options": [ + "24/42", + "27/68", + "42/72", + "54/92" + ], + "answer": "42/72", + "correct_value": 0.5833333333333334, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000140", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 36/79 or 39/72?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.08597046413502107, + "pure_math": { + "question": "Which is larger: 36/79 or 39/72?", + "options": [ + "36/79", + "21/77", + "26/92", + "39/72" + ], + "answer": "39/72", + "correct_value": 0.5416666666666666, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 5/10 or 34/62?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.048387096774193505, + "pure_math": { + "question": "Which is larger: 5/10 or 34/62?", + "options": [ + "5/10", + "36/90", + "19/27", + "34/62" + ], + "answer": "34/62", + "correct_value": 0.5483870967741935, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 7/11 or 22/37?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 7/11 or 22/37?", + "options": [ + "7/11", + "22/37", + "47/74", + "15/57" + ], + "answer": "7/11", + "correct_value": 0.6363636363636364, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000141", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 5/12 or 8/10?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.38333333333333336, + "pure_math": { + "question": "Which is larger: 5/12 or 8/10?", + "options": [ + "8/10", + "5/12", + "7/12", + "15/62" + ], + "answer": "8/10", + "correct_value": 0.8, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 6/12 or 41/74?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.05405405405405406, + "pure_math": { + "question": "Which is larger: 6/12 or 41/74?", + "options": [ + "6/12", + "67/99", + "27/65", + "41/74" + ], + "answer": "41/74", + "correct_value": 0.5540540540540541, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 22/40 or 20/34?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 22/40 or 20/34?", + "options": [ + "21/42", + "20/34", + "22/40", + "55/89" + ], + "answer": "20/34", + "correct_value": 0.5882352941176471, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000142", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 26/57 or 34/61?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.10123669830313486, + "pure_math": { + "question": "Which is larger: 26/57 or 34/61?", + "options": [ + "26/57", + "33/48", + "17/41", + "34/61" + ], + "answer": "34/61", + "correct_value": 0.5573770491803278, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 10/21 or 34/61?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.08118657298985166, + "pure_math": { + "question": "Which is larger: 10/21 or 34/61?", + "options": [ + "10/21", + "50/80", + "26/58", + "34/61" + ], + "answer": "34/61", + "correct_value": 0.5573770491803278, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 55/99 or 18/31?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 55/99 or 18/31?", + "options": [ + "55/99", + "37/65", + "12/20", + "18/31" + ], + "answer": "18/31", + "correct_value": 0.5806451612903226, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000143", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 18/39 or 22/37?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.13305613305613306, + "pure_math": { + "question": "Which is larger: 18/39 or 22/37?", + "options": [ + "18/39", + "22/37", + "4/15", + "34/70" + ], + "answer": "22/37", + "correct_value": 0.5945945945945946, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 15/32 or 28/50?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.09125000000000005, + "pure_math": { + "question": "Which is larger: 15/32 or 28/50?", + "options": [ + "4/12", + "15/32", + "28/50", + "30/85" + ], + "answer": "28/50", + "correct_value": 0.56, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 17/30 or 48/82?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 17/30 or 48/82?", + "options": [ + "35/59", + "48/82", + "17/30", + "28/40" + ], + "answer": "48/82", + "correct_value": 0.5853658536585366, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000144", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 48/98 or 13/20?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.1602040816326531, + "pure_math": { + "question": "Which is larger: 48/98 or 13/20?", + "options": [ + "14/54", + "11/22", + "13/20", + "48/98" + ], + "answer": "13/20", + "correct_value": 0.65, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 42/87 or 48/88?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.06269592476489022, + "pure_math": { + "question": "Which is larger: 42/87 or 48/88?", + "options": [ + "48/88", + "12/51", + "42/87", + "46/86" + ], + "answer": "48/88", + "correct_value": 0.5454545454545454, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 22/39 or 32/55?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 22/39 or 32/55?", + "options": [ + "44/80", + "22/39", + "32/55", + "27/65" + ], + "answer": "32/55", + "correct_value": 0.5818181818181818, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000145", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 17/41 or 12/19?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.21694480102695762, + "pure_math": { + "question": "Which is larger: 17/41 or 12/19?", + "options": [ + "12/19", + "13/29", + "17/41", + "31/91" + ], + "answer": "12/19", + "correct_value": 0.631578947368421, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 11/23 or 12/21?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.09316770186335399, + "pure_math": { + "question": "Which is larger: 11/23 or 12/21?", + "options": [ + "11/23", + "39/66", + "12/21", + "4/10" + ], + "answer": "12/21", + "correct_value": 0.5714285714285714, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 12/21 or 20/33?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 12/21 or 20/33?", + "options": [ + "12/21", + "44/65", + "20/33", + "55/81" + ], + "answer": "20/33", + "correct_value": 0.6060606060606061, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000146", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 9/21 or 29/55?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.09870129870129868, + "pure_math": { + "question": "Which is larger: 9/21 or 29/55?", + "options": [ + "24/51", + "59/86", + "9/21", + "29/55" + ], + "answer": "29/55", + "correct_value": 0.5272727272727272, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 28/59 or 11/20?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.07542372881355935, + "pure_math": { + "question": "Which is larger: 28/59 or 11/20?", + "options": [ + "11/20", + "53/83", + "35/81", + "28/59" + ], + "answer": "11/20", + "correct_value": 0.55, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 18/31 or 16/27?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 18/31 or 16/27?", + "options": [ + "26/44", + "20/44", + "16/27", + "18/31" + ], + "answer": "16/27", + "correct_value": 0.5925925925925926, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000147", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 30/65 or 25/45?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.09401709401709402, + "pure_math": { + "question": "Which is larger: 30/65 or 25/45?", + "options": [ + "25/45", + "22/56", + "21/71", + "30/65" + ], + "answer": "25/45", + "correct_value": 0.5555555555555556, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 41/86 or 48/88?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.06871035940803377, + "pure_math": { + "question": "Which is larger: 41/86 or 48/88?", + "options": [ + "7/23", + "41/86", + "48/88", + "22/35" + ], + "answer": "48/88", + "correct_value": 0.5454545454545454, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 29/52 or 52/89?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 29/52 or 52/89?", + "options": [ + "27/52", + "52/89", + "29/52", + "39/58" + ], + "answer": "52/89", + "correct_value": 0.5842696629213483, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000148", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 23/48 or 29/51?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.08946078431372545, + "pure_math": { + "question": "Which is larger: 23/48 or 29/51?", + "options": [ + "29/51", + "23/35", + "23/48", + "17/50" + ], + "answer": "29/51", + "correct_value": 0.5686274509803921, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 47/98 or 50/91?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.06985871271585564, + "pure_math": { + "question": "Which is larger: 47/98 or 50/91?", + "options": [ + "50/91", + "54/72", + "47/98", + "20/41" + ], + "answer": "50/91", + "correct_value": 0.5494505494505495, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 31/56 or 14/24?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 31/56 or 14/24?", + "options": [ + "55/85", + "14/24", + "32/82", + "31/56" + ], + "answer": "14/24", + "correct_value": 0.5833333333333334, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000149", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 32/69 or 20/36?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.09178743961352659, + "pure_math": { + "question": "Which is larger: 32/69 or 20/36?", + "options": [ + "20/36", + "46/62", + "28/98", + "32/69" + ], + "answer": "20/36", + "correct_value": 0.5555555555555556, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 19/40 or 23/42?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.0726190476190477, + "pure_math": { + "question": "Which is larger: 19/40 or 23/42?", + "options": [ + "40/64", + "32/94", + "19/40", + "23/42" + ], + "answer": "23/42", + "correct_value": 0.5476190476190477, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 47/84 or 36/61?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 47/84 or 36/61?", + "options": [ + "51/78", + "50/98", + "47/84", + "36/61" + ], + "answer": "36/61", + "correct_value": 0.5901639344262295, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000150", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 10/23 or 24/41?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.1505832449628844, + "pure_math": { + "question": "Which is larger: 10/23 or 24/41?", + "options": [ + "62/92", + "10/23", + "18/55", + "24/41" + ], + "answer": "24/41", + "correct_value": 0.5853658536585366, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 25/52 or 36/65?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.0730769230769231, + "pure_math": { + "question": "Which is larger: 25/52 or 36/65?", + "options": [ + "9/22", + "36/65", + "5/23", + "25/52" + ], + "answer": "36/65", + "correct_value": 0.5538461538461539, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 37/67 or 34/57?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 37/67 or 34/57?", + "options": [ + "29/60", + "34/57", + "61/96", + "37/67" + ], + "answer": "34/57", + "correct_value": 0.5964912280701754, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "cancellation_identity_000151", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "71 + 28 - 27", + "ns_shortcut": "Near-cancellation: 28 - 27 = 1, so answer ≈ 71 + 1.", + "shortcut_strength": 0.9642857142857143, + "pure_math": { + "question": "Evaluate: 71 + 28 - 27", + "options": [ + "82", + "92", + "72", + "62" + ], + "answer": "72", + "correct_value": 72, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "71 + 28 - 38", + "ns_shortcut": "Near-cancellation: 28 - 38 = -10, offset is manageable.", + "shortcut_strength": 0.6428571428571428, + "pure_math": { + "question": "Evaluate: 71 + 28 - 38", + "options": [ + "61", + "51", + "71", + "81" + ], + "answer": "61", + "correct_value": 61, + "explanation": "B-C = -10; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "71 + 28 - 118", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 71 + 28 - 118", + "options": [ + "-19", + "-1", + "-11", + "-31" + ], + "answer": "-19", + "correct_value": -19, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000152", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "22 + 71 - 69", + "ns_shortcut": "Near-cancellation: 71 - 69 = 2, so answer ≈ 22 + 2.", + "shortcut_strength": 0.971830985915493, + "pure_math": { + "question": "Evaluate: 22 + 71 - 69", + "options": [ + "14", + "34", + "24", + "44" + ], + "answer": "24", + "correct_value": 24, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "22 + 71 - 77", + "ns_shortcut": "Near-cancellation: 71 - 77 = -6, offset is manageable.", + "shortcut_strength": 0.9154929577464789, + "pure_math": { + "question": "Evaluate: 22 + 71 - 77", + "options": [ + "36", + "6", + "26", + "16" + ], + "answer": "16", + "correct_value": 16, + "explanation": "B-C = -6; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "22 + 71 - 154", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 22 + 71 - 154", + "options": [ + "-79", + "-61", + "-49", + "-59" + ], + "answer": "-61", + "correct_value": -61, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000153", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "80 + 35 - 34", + "ns_shortcut": "Near-cancellation: 35 - 34 = 1, so answer ≈ 80 + 1.", + "shortcut_strength": 0.9714285714285714, + "pure_math": { + "question": "Evaluate: 80 + 35 - 34", + "options": [ + "101", + "81", + "91", + "71" + ], + "answer": "81", + "correct_value": 81, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "80 + 35 - 20", + "ns_shortcut": "Near-cancellation: 35 - 20 = 15, offset is manageable.", + "shortcut_strength": 0.5714285714285714, + "pure_math": { + "question": "Evaluate: 80 + 35 - 20", + "options": [ + "115", + "85", + "95", + "105" + ], + "answer": "95", + "correct_value": 95, + "explanation": "B-C = 15; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "80 + 35 - 100", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 80 + 35 - 100", + "options": [ + "15", + "35", + "5", + "25" + ], + "answer": "15", + "correct_value": 15, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000154", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "59 + 15 - 16", + "ns_shortcut": "Near-cancellation: 15 - 16 = -1, so answer ≈ 59 + -1.", + "shortcut_strength": 0.9333333333333333, + "pure_math": { + "question": "Evaluate: 59 + 15 - 16", + "options": [ + "78", + "58", + "48", + "68" + ], + "answer": "58", + "correct_value": 58, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "59 + 15 - 28", + "ns_shortcut": "Near-cancellation: 15 - 28 = -13, offset is manageable.", + "shortcut_strength": 0.1333333333333333, + "pure_math": { + "question": "Evaluate: 59 + 15 - 28", + "options": [ + "46", + "66", + "36", + "56" + ], + "answer": "46", + "correct_value": 46, + "explanation": "B-C = -13; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "59 + 15 - 67", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 59 + 15 - 67", + "options": [ + "-3", + "27", + "17", + "7" + ], + "answer": "7", + "correct_value": 7, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000155", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "77 + 55 - 53", + "ns_shortcut": "Near-cancellation: 55 - 53 = 2, so answer ≈ 77 + 2.", + "shortcut_strength": 0.9636363636363636, + "pure_math": { + "question": "Evaluate: 77 + 55 - 53", + "options": [ + "89", + "79", + "99", + "69" + ], + "answer": "79", + "correct_value": 79, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "77 + 55 - 45", + "ns_shortcut": "Near-cancellation: 55 - 45 = 10, offset is manageable.", + "shortcut_strength": 0.8181818181818181, + "pure_math": { + "question": "Evaluate: 77 + 55 - 45", + "options": [ + "97", + "87", + "107", + "77" + ], + "answer": "87", + "correct_value": 87, + "explanation": "B-C = 10; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "77 + 55 - 138", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 77 + 55 - 138", + "options": [ + "-4", + "-24", + "6", + "-6" + ], + "answer": "-6", + "correct_value": -6, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000156", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50 + 30 - 31", + "ns_shortcut": "Near-cancellation: 30 - 31 = -1, so answer ≈ 50 + -1.", + "shortcut_strength": 0.9666666666666667, + "pure_math": { + "question": "Evaluate: 50 + 30 - 31", + "options": [ + "59", + "39", + "49", + "69" + ], + "answer": "49", + "correct_value": 49, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "50 + 30 - 23", + "ns_shortcut": "Near-cancellation: 30 - 23 = 7, offset is manageable.", + "shortcut_strength": 0.7666666666666666, + "pure_math": { + "question": "Evaluate: 50 + 30 - 23", + "options": [ + "77", + "57", + "67", + "47" + ], + "answer": "57", + "correct_value": 57, + "explanation": "B-C = 7; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "50 + 30 - 106", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 50 + 30 - 106", + "options": [ + "-14", + "-24", + "-26", + "-44" + ], + "answer": "-26", + "correct_value": -26, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000157", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "91 + 83 - 84", + "ns_shortcut": "Near-cancellation: 83 - 84 = -1, so answer ≈ 91 + -1.", + "shortcut_strength": 0.9879518072289156, + "pure_math": { + "question": "Evaluate: 91 + 83 - 84", + "options": [ + "90", + "110", + "80", + "100" + ], + "answer": "90", + "correct_value": 90, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "91 + 83 - 90", + "ns_shortcut": "Near-cancellation: 83 - 90 = -7, offset is manageable.", + "shortcut_strength": 0.9156626506024097, + "pure_math": { + "question": "Evaluate: 91 + 83 - 90", + "options": [ + "74", + "84", + "94", + "104" + ], + "answer": "84", + "correct_value": 84, + "explanation": "B-C = -7; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "91 + 83 - 21", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 91 + 83 - 21", + "options": [ + "163", + "173", + "153", + "143" + ], + "answer": "153", + "correct_value": 153, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000158", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "59 + 86 - 84", + "ns_shortcut": "Near-cancellation: 86 - 84 = 2, so answer ≈ 59 + 2.", + "shortcut_strength": 0.9767441860465116, + "pure_math": { + "question": "Evaluate: 59 + 86 - 84", + "options": [ + "61", + "71", + "81", + "51" + ], + "answer": "61", + "correct_value": 61, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "59 + 86 - 74", + "ns_shortcut": "Near-cancellation: 86 - 74 = 12, offset is manageable.", + "shortcut_strength": 0.8604651162790697, + "pure_math": { + "question": "Evaluate: 59 + 86 - 74", + "options": [ + "81", + "91", + "71", + "61" + ], + "answer": "71", + "correct_value": 71, + "explanation": "B-C = 12; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "59 + 86 - 10", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 59 + 86 - 10", + "options": [ + "155", + "135", + "125", + "145" + ], + "answer": "135", + "correct_value": 135, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000159", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "45 + 43 - 42", + "ns_shortcut": "Near-cancellation: 43 - 42 = 1, so answer ≈ 45 + 1.", + "shortcut_strength": 0.9767441860465116, + "pure_math": { + "question": "Evaluate: 45 + 43 - 42", + "options": [ + "46", + "66", + "36", + "56" + ], + "answer": "46", + "correct_value": 46, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "45 + 43 - 57", + "ns_shortcut": "Near-cancellation: 43 - 57 = -14, offset is manageable.", + "shortcut_strength": 0.6744186046511628, + "pure_math": { + "question": "Evaluate: 45 + 43 - 57", + "options": [ + "21", + "51", + "41", + "31" + ], + "answer": "31", + "correct_value": 31, + "explanation": "B-C = -14; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "45 + 43 - 117", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 45 + 43 - 117", + "options": [ + "-21", + "-29", + "-11", + "-41" + ], + "answer": "-29", + "correct_value": -29, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000160", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "64 + 28 - 29", + "ns_shortcut": "Near-cancellation: 28 - 29 = -1, so answer ≈ 64 + -1.", + "shortcut_strength": 0.9642857142857143, + "pure_math": { + "question": "Evaluate: 64 + 28 - 29", + "options": [ + "53", + "73", + "63", + "83" + ], + "answer": "63", + "correct_value": 63, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "64 + 28 - 18", + "ns_shortcut": "Near-cancellation: 28 - 18 = 10, offset is manageable.", + "shortcut_strength": 0.6428571428571428, + "pure_math": { + "question": "Evaluate: 64 + 28 - 18", + "options": [ + "94", + "64", + "84", + "74" + ], + "answer": "74", + "correct_value": 74, + "explanation": "B-C = 10; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "64 + 28 - 117", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 64 + 28 - 117", + "options": [ + "-25", + "-35", + "-5", + "-15" + ], + "answer": "-25", + "correct_value": -25, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000161", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "88 + 69 - 70", + "ns_shortcut": "Near-cancellation: 69 - 70 = -1, so answer ≈ 88 + -1.", + "shortcut_strength": 0.9855072463768116, + "pure_math": { + "question": "Evaluate: 88 + 69 - 70", + "options": [ + "77", + "107", + "87", + "97" + ], + "answer": "87", + "correct_value": 87, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "88 + 69 - 83", + "ns_shortcut": "Near-cancellation: 69 - 83 = -14, offset is manageable.", + "shortcut_strength": 0.7971014492753623, + "pure_math": { + "question": "Evaluate: 88 + 69 - 83", + "options": [ + "74", + "84", + "64", + "94" + ], + "answer": "74", + "correct_value": 74, + "explanation": "B-C = -14; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "88 + 69 - 147", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 88 + 69 - 147", + "options": [ + "10", + "20", + "0", + "30" + ], + "answer": "10", + "correct_value": 10, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000162", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "47 + 88 - 87", + "ns_shortcut": "Near-cancellation: 88 - 87 = 1, so answer ≈ 47 + 1.", + "shortcut_strength": 0.9886363636363636, + "pure_math": { + "question": "Evaluate: 47 + 88 - 87", + "options": [ + "68", + "58", + "38", + "48" + ], + "answer": "48", + "correct_value": 48, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "47 + 88 - 97", + "ns_shortcut": "Near-cancellation: 88 - 97 = -9, offset is manageable.", + "shortcut_strength": 0.8977272727272727, + "pure_math": { + "question": "Evaluate: 47 + 88 - 97", + "options": [ + "28", + "38", + "58", + "48" + ], + "answer": "38", + "correct_value": 38, + "explanation": "B-C = -9; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "47 + 88 - 33", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 47 + 88 - 33", + "options": [ + "92", + "112", + "102", + "122" + ], + "answer": "102", + "correct_value": 102, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000163", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "72 + 33 - 32", + "ns_shortcut": "Near-cancellation: 33 - 32 = 1, so answer ≈ 72 + 1.", + "shortcut_strength": 0.9696969696969697, + "pure_math": { + "question": "Evaluate: 72 + 33 - 32", + "options": [ + "73", + "93", + "83", + "63" + ], + "answer": "73", + "correct_value": 73, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "72 + 33 - 48", + "ns_shortcut": "Near-cancellation: 33 - 48 = -15, offset is manageable.", + "shortcut_strength": 0.5454545454545454, + "pure_math": { + "question": "Evaluate: 72 + 33 - 48", + "options": [ + "47", + "77", + "67", + "57" + ], + "answer": "57", + "correct_value": 57, + "explanation": "B-C = -15; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "72 + 33 - 117", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 72 + 33 - 117", + "options": [ + "-12", + "2", + "-8", + "-28" + ], + "answer": "-12", + "correct_value": -12, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000164", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "84 + 15 - 17", + "ns_shortcut": "Near-cancellation: 15 - 17 = -2, so answer ≈ 84 + -2.", + "shortcut_strength": 0.8666666666666667, + "pure_math": { + "question": "Evaluate: 84 + 15 - 17", + "options": [ + "82", + "92", + "102", + "72" + ], + "answer": "82", + "correct_value": 82, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "84 + 15 - 28", + "ns_shortcut": "Near-cancellation: 15 - 28 = -13, offset is manageable.", + "shortcut_strength": 0.1333333333333333, + "pure_math": { + "question": "Evaluate: 84 + 15 - 28", + "options": [ + "91", + "81", + "71", + "61" + ], + "answer": "71", + "correct_value": 71, + "explanation": "B-C = -13; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "84 + 15 - 68", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 84 + 15 - 68", + "options": [ + "31", + "21", + "41", + "51" + ], + "answer": "31", + "correct_value": 31, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000165", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "53 + 85 - 86", + "ns_shortcut": "Near-cancellation: 85 - 86 = -1, so answer ≈ 53 + -1.", + "shortcut_strength": 0.9882352941176471, + "pure_math": { + "question": "Evaluate: 53 + 85 - 86", + "options": [ + "72", + "52", + "62", + "42" + ], + "answer": "52", + "correct_value": 52, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "53 + 85 - 91", + "ns_shortcut": "Near-cancellation: 85 - 91 = -6, offset is manageable.", + "shortcut_strength": 0.9294117647058824, + "pure_math": { + "question": "Evaluate: 53 + 85 - 91", + "options": [ + "57", + "67", + "47", + "37" + ], + "answer": "47", + "correct_value": 47, + "explanation": "B-C = -6; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "53 + 85 - 135", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 53 + 85 - 135", + "options": [ + "23", + "13", + "3", + "-7" + ], + "answer": "3", + "correct_value": 3, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000166", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "35 + 31 - 30", + "ns_shortcut": "Near-cancellation: 31 - 30 = 1, so answer ≈ 35 + 1.", + "shortcut_strength": 0.967741935483871, + "pure_math": { + "question": "Evaluate: 35 + 31 - 30", + "options": [ + "56", + "36", + "26", + "46" + ], + "answer": "36", + "correct_value": 36, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "35 + 31 - 22", + "ns_shortcut": "Near-cancellation: 31 - 22 = 9, offset is manageable.", + "shortcut_strength": 0.7096774193548387, + "pure_math": { + "question": "Evaluate: 35 + 31 - 22", + "options": [ + "64", + "34", + "54", + "44" + ], + "answer": "44", + "correct_value": 44, + "explanation": "B-C = 9; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "35 + 31 - 74", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 35 + 31 - 74", + "options": [ + "-2", + "-8", + "-22", + "8" + ], + "answer": "-8", + "correct_value": -8, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000167", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "64 + 26 - 28", + "ns_shortcut": "Near-cancellation: 26 - 28 = -2, so answer ≈ 64 + -2.", + "shortcut_strength": 0.9230769230769231, + "pure_math": { + "question": "Evaluate: 64 + 26 - 28", + "options": [ + "82", + "72", + "52", + "62" + ], + "answer": "62", + "correct_value": 62, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "64 + 26 - 33", + "ns_shortcut": "Near-cancellation: 26 - 33 = -7, offset is manageable.", + "shortcut_strength": 0.7307692307692308, + "pure_math": { + "question": "Evaluate: 64 + 26 - 33", + "options": [ + "57", + "47", + "77", + "67" + ], + "answer": "57", + "correct_value": 57, + "explanation": "B-C = -7; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "64 + 26 - 66", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 64 + 26 - 66", + "options": [ + "14", + "34", + "24", + "44" + ], + "answer": "24", + "correct_value": 24, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000168", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 + 31 - 29", + "ns_shortcut": "Near-cancellation: 31 - 29 = 2, so answer ≈ 98 + 2.", + "shortcut_strength": 0.935483870967742, + "pure_math": { + "question": "Evaluate: 98 + 31 - 29", + "options": [ + "120", + "110", + "90", + "100" + ], + "answer": "100", + "correct_value": 100, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98 + 31 - 37", + "ns_shortcut": "Near-cancellation: 31 - 37 = -6, offset is manageable.", + "shortcut_strength": 0.8064516129032258, + "pure_math": { + "question": "Evaluate: 98 + 31 - 37", + "options": [ + "82", + "112", + "102", + "92" + ], + "answer": "92", + "correct_value": 92, + "explanation": "B-C = -6; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "98 + 31 - 90", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 98 + 31 - 90", + "options": [ + "49", + "59", + "39", + "29" + ], + "answer": "39", + "correct_value": 39, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000169", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "18 + 82 - 84", + "ns_shortcut": "Near-cancellation: 82 - 84 = -2, so answer ≈ 18 + -2.", + "shortcut_strength": 0.975609756097561, + "pure_math": { + "question": "Evaluate: 18 + 82 - 84", + "options": [ + "16", + "36", + "6", + "26" + ], + "answer": "16", + "correct_value": 16, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "18 + 82 - 97", + "ns_shortcut": "Near-cancellation: 82 - 97 = -15, offset is manageable.", + "shortcut_strength": 0.8170731707317074, + "pure_math": { + "question": "Evaluate: 18 + 82 - 97", + "options": [ + "3", + "23", + "-7", + "13" + ], + "answer": "3", + "correct_value": 3, + "explanation": "B-C = -15; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "18 + 82 - 29", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 18 + 82 - 29", + "options": [ + "81", + "91", + "61", + "71" + ], + "answer": "71", + "correct_value": 71, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000170", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "85 + 64 - 63", + "ns_shortcut": "Near-cancellation: 64 - 63 = 1, so answer ≈ 85 + 1.", + "shortcut_strength": 0.984375, + "pure_math": { + "question": "Evaluate: 85 + 64 - 63", + "options": [ + "106", + "86", + "96", + "76" + ], + "answer": "86", + "correct_value": 86, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "85 + 64 - 73", + "ns_shortcut": "Near-cancellation: 64 - 73 = -9, offset is manageable.", + "shortcut_strength": 0.859375, + "pure_math": { + "question": "Evaluate: 85 + 64 - 73", + "options": [ + "96", + "86", + "66", + "76" + ], + "answer": "76", + "correct_value": 76, + "explanation": "B-C = -9; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "85 + 64 - 153", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 85 + 64 - 153", + "options": [ + "4", + "-16", + "-4", + "14" + ], + "answer": "-4", + "correct_value": -4, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000171", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "60 + 52 - 54", + "ns_shortcut": "Near-cancellation: 52 - 54 = -2, so answer ≈ 60 + -2.", + "shortcut_strength": 0.9615384615384616, + "pure_math": { + "question": "Evaluate: 60 + 52 - 54", + "options": [ + "58", + "78", + "48", + "68" + ], + "answer": "58", + "correct_value": 58, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "60 + 52 - 60", + "ns_shortcut": "Near-cancellation: 52 - 60 = -8, offset is manageable.", + "shortcut_strength": 0.8461538461538461, + "pure_math": { + "question": "Evaluate: 60 + 52 - 60", + "options": [ + "62", + "52", + "72", + "42" + ], + "answer": "52", + "correct_value": 52, + "explanation": "B-C = -8; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "60 + 52 - 125", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 60 + 52 - 125", + "options": [ + "-27", + "-13", + "-7", + "3" + ], + "answer": "-13", + "correct_value": -13, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000172", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "59 + 87 - 85", + "ns_shortcut": "Near-cancellation: 87 - 85 = 2, so answer ≈ 59 + 2.", + "shortcut_strength": 0.9770114942528736, + "pure_math": { + "question": "Evaluate: 59 + 87 - 85", + "options": [ + "71", + "61", + "51", + "81" + ], + "answer": "61", + "correct_value": 61, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "59 + 87 - 81", + "ns_shortcut": "Near-cancellation: 87 - 81 = 6, offset is manageable.", + "shortcut_strength": 0.9310344827586207, + "pure_math": { + "question": "Evaluate: 59 + 87 - 81", + "options": [ + "75", + "55", + "85", + "65" + ], + "answer": "65", + "correct_value": 65, + "explanation": "B-C = 6; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "59 + 87 - 127", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 59 + 87 - 127", + "options": [ + "9", + "39", + "19", + "29" + ], + "answer": "19", + "correct_value": 19, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000173", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "60 + 79 - 78", + "ns_shortcut": "Near-cancellation: 79 - 78 = 1, so answer ≈ 60 + 1.", + "shortcut_strength": 0.9873417721518988, + "pure_math": { + "question": "Evaluate: 60 + 79 - 78", + "options": [ + "61", + "81", + "71", + "51" + ], + "answer": "61", + "correct_value": 61, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "60 + 79 - 94", + "ns_shortcut": "Near-cancellation: 79 - 94 = -15, offset is manageable.", + "shortcut_strength": 0.810126582278481, + "pure_math": { + "question": "Evaluate: 60 + 79 - 94", + "options": [ + "65", + "45", + "35", + "55" + ], + "answer": "45", + "correct_value": 45, + "explanation": "B-C = -15; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "60 + 79 - 20", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 60 + 79 - 20", + "options": [ + "139", + "109", + "119", + "129" + ], + "answer": "119", + "correct_value": 119, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000174", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "60 + 90 - 92", + "ns_shortcut": "Near-cancellation: 90 - 92 = -2, so answer ≈ 60 + -2.", + "shortcut_strength": 0.9777777777777777, + "pure_math": { + "question": "Evaluate: 60 + 90 - 92", + "options": [ + "58", + "68", + "48", + "78" + ], + "answer": "58", + "correct_value": 58, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "60 + 90 - 95", + "ns_shortcut": "Near-cancellation: 90 - 95 = -5, offset is manageable.", + "shortcut_strength": 0.9444444444444444, + "pure_math": { + "question": "Evaluate: 60 + 90 - 95", + "options": [ + "75", + "45", + "55", + "65" + ], + "answer": "55", + "correct_value": 55, + "explanation": "B-C = -5; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "60 + 90 - 137", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 60 + 90 - 137", + "options": [ + "33", + "3", + "23", + "13" + ], + "answer": "13", + "correct_value": 13, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000175", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 + 16 - 17", + "ns_shortcut": "Near-cancellation: 16 - 17 = -1, so answer ≈ 25 + -1.", + "shortcut_strength": 0.9375, + "pure_math": { + "question": "Evaluate: 25 + 16 - 17", + "options": [ + "44", + "34", + "14", + "24" + ], + "answer": "24", + "correct_value": 24, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 + 16 - 10", + "ns_shortcut": "Near-cancellation: 16 - 10 = 6, offset is manageable.", + "shortcut_strength": 0.1875, + "pure_math": { + "question": "Evaluate: 25 + 16 - 10", + "options": [ + "21", + "41", + "31", + "51" + ], + "answer": "31", + "correct_value": 31, + "explanation": "B-C = 6; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "25 + 16 - 64", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 25 + 16 - 64", + "options": [ + "-7", + "-17", + "-37", + "-23" + ], + "answer": "-23", + "correct_value": -23, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000176", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "64 + 49 - 51", + "ns_shortcut": "Near-cancellation: 49 - 51 = -2, so answer ≈ 64 + -2.", + "shortcut_strength": 0.9591836734693877, + "pure_math": { + "question": "Evaluate: 64 + 49 - 51", + "options": [ + "72", + "62", + "52", + "82" + ], + "answer": "62", + "correct_value": 62, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "64 + 49 - 37", + "ns_shortcut": "Near-cancellation: 49 - 37 = 12, offset is manageable.", + "shortcut_strength": 0.7551020408163265, + "pure_math": { + "question": "Evaluate: 64 + 49 - 37", + "options": [ + "66", + "86", + "76", + "96" + ], + "answer": "76", + "correct_value": 76, + "explanation": "B-C = 12; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "64 + 49 - 125", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 64 + 49 - 125", + "options": [ + "-28", + "-8", + "-12", + "2" + ], + "answer": "-12", + "correct_value": -12, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000177", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "23 + 88 - 90", + "ns_shortcut": "Near-cancellation: 88 - 90 = -2, so answer ≈ 23 + -2.", + "shortcut_strength": 0.9772727272727273, + "pure_math": { + "question": "Evaluate: 23 + 88 - 90", + "options": [ + "11", + "21", + "41", + "31" + ], + "answer": "21", + "correct_value": 21, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "23 + 88 - 93", + "ns_shortcut": "Near-cancellation: 88 - 93 = -5, offset is manageable.", + "shortcut_strength": 0.9431818181818182, + "pure_math": { + "question": "Evaluate: 23 + 88 - 93", + "options": [ + "38", + "8", + "18", + "28" + ], + "answer": "18", + "correct_value": 18, + "explanation": "B-C = -5; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "23 + 88 - 151", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 23 + 88 - 151", + "options": [ + "-50", + "-20", + "-40", + "-30" + ], + "answer": "-40", + "correct_value": -40, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000178", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "80 + 68 - 69", + "ns_shortcut": "Near-cancellation: 68 - 69 = -1, so answer ≈ 80 + -1.", + "shortcut_strength": 0.9852941176470589, + "pure_math": { + "question": "Evaluate: 80 + 68 - 69", + "options": [ + "69", + "89", + "79", + "99" + ], + "answer": "79", + "correct_value": 79, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "80 + 68 - 59", + "ns_shortcut": "Near-cancellation: 68 - 59 = 9, offset is manageable.", + "shortcut_strength": 0.8676470588235294, + "pure_math": { + "question": "Evaluate: 80 + 68 - 59", + "options": [ + "79", + "99", + "109", + "89" + ], + "answer": "89", + "correct_value": 89, + "explanation": "B-C = 9; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "80 + 68 - 21", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 80 + 68 - 21", + "options": [ + "117", + "147", + "127", + "137" + ], + "answer": "127", + "correct_value": 127, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000179", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "61 + 23 - 21", + "ns_shortcut": "Near-cancellation: 23 - 21 = 2, so answer ≈ 61 + 2.", + "shortcut_strength": 0.9130434782608696, + "pure_math": { + "question": "Evaluate: 61 + 23 - 21", + "options": [ + "63", + "53", + "73", + "83" + ], + "answer": "63", + "correct_value": 63, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "61 + 23 - 36", + "ns_shortcut": "Near-cancellation: 23 - 36 = -13, offset is manageable.", + "shortcut_strength": 0.4347826086956522, + "pure_math": { + "question": "Evaluate: 61 + 23 - 36", + "options": [ + "68", + "58", + "48", + "38" + ], + "answer": "48", + "correct_value": 48, + "explanation": "B-C = -13; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "61 + 23 - 75", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 61 + 23 - 75", + "options": [ + "19", + "9", + "-1", + "29" + ], + "answer": "9", + "correct_value": 9, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000180", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "69 + 62 - 60", + "ns_shortcut": "Near-cancellation: 62 - 60 = 2, so answer ≈ 69 + 2.", + "shortcut_strength": 0.967741935483871, + "pure_math": { + "question": "Evaluate: 69 + 62 - 60", + "options": [ + "61", + "71", + "91", + "81" + ], + "answer": "71", + "correct_value": 71, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "69 + 62 - 67", + "ns_shortcut": "Near-cancellation: 62 - 67 = -5, offset is manageable.", + "shortcut_strength": 0.9193548387096774, + "pure_math": { + "question": "Evaluate: 69 + 62 - 67", + "options": [ + "74", + "84", + "64", + "54" + ], + "answer": "64", + "correct_value": 64, + "explanation": "B-C = -5; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "69 + 62 - 136", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 69 + 62 - 136", + "options": [ + "15", + "-15", + "5", + "-5" + ], + "answer": "-5", + "correct_value": -5, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000181", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "94 + 63 - 62", + "ns_shortcut": "Near-cancellation: 63 - 62 = 1, so answer ≈ 94 + 1.", + "shortcut_strength": 0.9841269841269842, + "pure_math": { + "question": "Evaluate: 94 + 63 - 62", + "options": [ + "105", + "115", + "85", + "95" + ], + "answer": "95", + "correct_value": 95, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "94 + 63 - 51", + "ns_shortcut": "Near-cancellation: 63 - 51 = 12, offset is manageable.", + "shortcut_strength": 0.8095238095238095, + "pure_math": { + "question": "Evaluate: 94 + 63 - 51", + "options": [ + "126", + "116", + "96", + "106" + ], + "answer": "106", + "correct_value": 106, + "explanation": "B-C = 12; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "94 + 63 - 18", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 94 + 63 - 18", + "options": [ + "139", + "129", + "159", + "149" + ], + "answer": "139", + "correct_value": 139, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000182", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 + 62 - 60", + "ns_shortcut": "Near-cancellation: 62 - 60 = 2, so answer ≈ 99 + 2.", + "shortcut_strength": 0.967741935483871, + "pure_math": { + "question": "Evaluate: 99 + 62 - 60", + "options": [ + "111", + "91", + "121", + "101" + ], + "answer": "101", + "correct_value": 101, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99 + 62 - 77", + "ns_shortcut": "Near-cancellation: 62 - 77 = -15, offset is manageable.", + "shortcut_strength": 0.7580645161290323, + "pure_math": { + "question": "Evaluate: 99 + 62 - 77", + "options": [ + "94", + "104", + "84", + "74" + ], + "answer": "84", + "correct_value": 84, + "explanation": "B-C = -15; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "99 + 62 - 129", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 99 + 62 - 129", + "options": [ + "32", + "52", + "22", + "42" + ], + "answer": "32", + "correct_value": 32, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000183", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50 + 65 - 64", + "ns_shortcut": "Near-cancellation: 65 - 64 = 1, so answer ≈ 50 + 1.", + "shortcut_strength": 0.9846153846153847, + "pure_math": { + "question": "Evaluate: 50 + 65 - 64", + "options": [ + "51", + "41", + "61", + "71" + ], + "answer": "51", + "correct_value": 51, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "50 + 65 - 71", + "ns_shortcut": "Near-cancellation: 65 - 71 = -6, offset is manageable.", + "shortcut_strength": 0.9076923076923077, + "pure_math": { + "question": "Evaluate: 50 + 65 - 71", + "options": [ + "34", + "64", + "44", + "54" + ], + "answer": "44", + "correct_value": 44, + "explanation": "B-C = -6; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "50 + 65 - 16", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 50 + 65 - 16", + "options": [ + "99", + "119", + "89", + "109" + ], + "answer": "99", + "correct_value": 99, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000184", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50 + 67 - 69", + "ns_shortcut": "Near-cancellation: 67 - 69 = -2, so answer ≈ 50 + -2.", + "shortcut_strength": 0.9701492537313433, + "pure_math": { + "question": "Evaluate: 50 + 67 - 69", + "options": [ + "38", + "58", + "48", + "68" + ], + "answer": "48", + "correct_value": 48, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "50 + 67 - 55", + "ns_shortcut": "Near-cancellation: 67 - 55 = 12, offset is manageable.", + "shortcut_strength": 0.8208955223880597, + "pure_math": { + "question": "Evaluate: 50 + 67 - 55", + "options": [ + "72", + "52", + "62", + "82" + ], + "answer": "62", + "correct_value": 62, + "explanation": "B-C = 12; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "50 + 67 - 117", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 50 + 67 - 117", + "options": [ + "20", + "10", + "-10", + "0" + ], + "answer": "0", + "correct_value": 0, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000185", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "28 + 26 - 27", + "ns_shortcut": "Near-cancellation: 26 - 27 = -1, so answer ≈ 28 + -1.", + "shortcut_strength": 0.9615384615384616, + "pure_math": { + "question": "Evaluate: 28 + 26 - 27", + "options": [ + "17", + "27", + "37", + "47" + ], + "answer": "27", + "correct_value": 27, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "28 + 26 - 17", + "ns_shortcut": "Near-cancellation: 26 - 17 = 9, offset is manageable.", + "shortcut_strength": 0.6538461538461539, + "pure_math": { + "question": "Evaluate: 28 + 26 - 17", + "options": [ + "57", + "27", + "37", + "47" + ], + "answer": "37", + "correct_value": 37, + "explanation": "B-C = 9; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "28 + 26 - 94", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 28 + 26 - 94", + "options": [ + "-20", + "-50", + "-30", + "-40" + ], + "answer": "-40", + "correct_value": -40, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000186", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "27 + 30 - 32", + "ns_shortcut": "Near-cancellation: 30 - 32 = -2, so answer ≈ 27 + -2.", + "shortcut_strength": 0.9333333333333333, + "pure_math": { + "question": "Evaluate: 27 + 30 - 32", + "options": [ + "35", + "45", + "15", + "25" + ], + "answer": "25", + "correct_value": 25, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "27 + 30 - 41", + "ns_shortcut": "Near-cancellation: 30 - 41 = -11, offset is manageable.", + "shortcut_strength": 0.6333333333333333, + "pure_math": { + "question": "Evaluate: 27 + 30 - 41", + "options": [ + "26", + "16", + "6", + "36" + ], + "answer": "16", + "correct_value": 16, + "explanation": "B-C = -11; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "27 + 30 - 107", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 27 + 30 - 107", + "options": [ + "-40", + "-30", + "-60", + "-50" + ], + "answer": "-50", + "correct_value": -50, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000187", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "35 + 71 - 72", + "ns_shortcut": "Near-cancellation: 71 - 72 = -1, so answer ≈ 35 + -1.", + "shortcut_strength": 0.9859154929577465, + "pure_math": { + "question": "Evaluate: 35 + 71 - 72", + "options": [ + "54", + "24", + "44", + "34" + ], + "answer": "34", + "correct_value": 34, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "35 + 71 - 83", + "ns_shortcut": "Near-cancellation: 71 - 83 = -12, offset is manageable.", + "shortcut_strength": 0.8309859154929577, + "pure_math": { + "question": "Evaluate: 35 + 71 - 83", + "options": [ + "43", + "33", + "13", + "23" + ], + "answer": "23", + "correct_value": 23, + "explanation": "B-C = -12; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "35 + 71 - 157", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 35 + 71 - 157", + "options": [ + "-39", + "-49", + "-51", + "-69" + ], + "answer": "-51", + "correct_value": -51, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000188", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "97 + 69 - 70", + "ns_shortcut": "Near-cancellation: 69 - 70 = -1, so answer ≈ 97 + -1.", + "shortcut_strength": 0.9855072463768116, + "pure_math": { + "question": "Evaluate: 97 + 69 - 70", + "options": [ + "106", + "86", + "96", + "116" + ], + "answer": "96", + "correct_value": 96, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "97 + 69 - 60", + "ns_shortcut": "Near-cancellation: 69 - 60 = 9, offset is manageable.", + "shortcut_strength": 0.8695652173913043, + "pure_math": { + "question": "Evaluate: 97 + 69 - 60", + "options": [ + "116", + "126", + "106", + "96" + ], + "answer": "106", + "correct_value": 106, + "explanation": "B-C = 9; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "97 + 69 - 148", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 97 + 69 - 148", + "options": [ + "18", + "8", + "28", + "38" + ], + "answer": "18", + "correct_value": 18, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000189", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "80 + 76 - 78", + "ns_shortcut": "Near-cancellation: 76 - 78 = -2, so answer ≈ 80 + -2.", + "shortcut_strength": 0.9736842105263158, + "pure_math": { + "question": "Evaluate: 80 + 76 - 78", + "options": [ + "68", + "98", + "88", + "78" + ], + "answer": "78", + "correct_value": 78, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "80 + 76 - 87", + "ns_shortcut": "Near-cancellation: 76 - 87 = -11, offset is manageable.", + "shortcut_strength": 0.8552631578947368, + "pure_math": { + "question": "Evaluate: 80 + 76 - 87", + "options": [ + "59", + "69", + "89", + "79" + ], + "answer": "69", + "correct_value": 69, + "explanation": "B-C = -11; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "80 + 76 - 156", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 80 + 76 - 156", + "options": [ + "0", + "10", + "20", + "-10" + ], + "answer": "0", + "correct_value": 0, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000190", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "71 + 84 - 82", + "ns_shortcut": "Near-cancellation: 84 - 82 = 2, so answer ≈ 71 + 2.", + "shortcut_strength": 0.9761904761904762, + "pure_math": { + "question": "Evaluate: 71 + 84 - 82", + "options": [ + "73", + "83", + "63", + "93" + ], + "answer": "73", + "correct_value": 73, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "71 + 84 - 75", + "ns_shortcut": "Near-cancellation: 84 - 75 = 9, offset is manageable.", + "shortcut_strength": 0.8928571428571429, + "pure_math": { + "question": "Evaluate: 71 + 84 - 75", + "options": [ + "100", + "70", + "80", + "90" + ], + "answer": "80", + "correct_value": 80, + "explanation": "B-C = 9; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "71 + 84 - 162", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 71 + 84 - 162", + "options": [ + "7", + "-23", + "-3", + "-7" + ], + "answer": "-7", + "correct_value": -7, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000191", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "70 + 82 - 81", + "ns_shortcut": "Near-cancellation: 82 - 81 = 1, so answer ≈ 70 + 1.", + "shortcut_strength": 0.9878048780487805, + "pure_math": { + "question": "Evaluate: 70 + 82 - 81", + "options": [ + "61", + "91", + "71", + "81" + ], + "answer": "71", + "correct_value": 71, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "70 + 82 - 97", + "ns_shortcut": "Near-cancellation: 82 - 97 = -15, offset is manageable.", + "shortcut_strength": 0.8170731707317074, + "pure_math": { + "question": "Evaluate: 70 + 82 - 97", + "options": [ + "75", + "55", + "45", + "65" + ], + "answer": "55", + "correct_value": 55, + "explanation": "B-C = -15; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "70 + 82 - 170", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 70 + 82 - 170", + "options": [ + "-2", + "-18", + "-12", + "-32" + ], + "answer": "-18", + "correct_value": -18, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000192", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "32 + 45 - 44", + "ns_shortcut": "Near-cancellation: 45 - 44 = 1, so answer ≈ 32 + 1.", + "shortcut_strength": 0.9777777777777777, + "pure_math": { + "question": "Evaluate: 32 + 45 - 44", + "options": [ + "43", + "33", + "23", + "53" + ], + "answer": "33", + "correct_value": 33, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "32 + 45 - 31", + "ns_shortcut": "Near-cancellation: 45 - 31 = 14, offset is manageable.", + "shortcut_strength": 0.6888888888888889, + "pure_math": { + "question": "Evaluate: 32 + 45 - 31", + "options": [ + "36", + "46", + "66", + "56" + ], + "answer": "46", + "correct_value": 46, + "explanation": "B-C = 14; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "32 + 45 - 116", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 32 + 45 - 116", + "options": [ + "-21", + "-39", + "-51", + "-31" + ], + "answer": "-39", + "correct_value": -39, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000193", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "21 + 22 - 23", + "ns_shortcut": "Near-cancellation: 22 - 23 = -1, so answer ≈ 21 + -1.", + "shortcut_strength": 0.9545454545454546, + "pure_math": { + "question": "Evaluate: 21 + 22 - 23", + "options": [ + "40", + "10", + "20", + "30" + ], + "answer": "20", + "correct_value": 20, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "21 + 22 - 32", + "ns_shortcut": "Near-cancellation: 22 - 32 = -10, offset is manageable.", + "shortcut_strength": 0.5454545454545454, + "pure_math": { + "question": "Evaluate: 21 + 22 - 32", + "options": [ + "31", + "1", + "11", + "21" + ], + "answer": "11", + "correct_value": 11, + "explanation": "B-C = -10; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "21 + 22 - 107", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 21 + 22 - 107", + "options": [ + "-56", + "-76", + "-64", + "-46" + ], + "answer": "-64", + "correct_value": -64, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000194", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "28 + 90 - 91", + "ns_shortcut": "Near-cancellation: 90 - 91 = -1, so answer ≈ 28 + -1.", + "shortcut_strength": 0.9888888888888889, + "pure_math": { + "question": "Evaluate: 28 + 90 - 91", + "options": [ + "17", + "37", + "47", + "27" + ], + "answer": "27", + "correct_value": 27, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "28 + 90 - 99", + "ns_shortcut": "Near-cancellation: 90 - 99 = -9, offset is manageable.", + "shortcut_strength": 0.8888888888888888, + "pure_math": { + "question": "Evaluate: 28 + 90 - 99", + "options": [ + "29", + "19", + "39", + "9" + ], + "answer": "19", + "correct_value": 19, + "explanation": "B-C = -9; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "28 + 90 - 151", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 28 + 90 - 151", + "options": [ + "-27", + "-47", + "-33", + "-17" + ], + "answer": "-33", + "correct_value": -33, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000195", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "65 + 77 - 76", + "ns_shortcut": "Near-cancellation: 77 - 76 = 1, so answer ≈ 65 + 1.", + "shortcut_strength": 0.987012987012987, + "pure_math": { + "question": "Evaluate: 65 + 77 - 76", + "options": [ + "66", + "56", + "86", + "76" + ], + "answer": "66", + "correct_value": 66, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "65 + 77 - 92", + "ns_shortcut": "Near-cancellation: 77 - 92 = -15, offset is manageable.", + "shortcut_strength": 0.8051948051948052, + "pure_math": { + "question": "Evaluate: 65 + 77 - 92", + "options": [ + "50", + "70", + "40", + "60" + ], + "answer": "50", + "correct_value": 50, + "explanation": "B-C = -15; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "65 + 77 - 163", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 65 + 77 - 163", + "options": [ + "-39", + "-9", + "-21", + "-19" + ], + "answer": "-21", + "correct_value": -21, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000196", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "99 + 88 - 86", + "ns_shortcut": "Near-cancellation: 88 - 86 = 2, so answer ≈ 99 + 2.", + "shortcut_strength": 0.9772727272727273, + "pure_math": { + "question": "Evaluate: 99 + 88 - 86", + "options": [ + "121", + "111", + "91", + "101" + ], + "answer": "101", + "correct_value": 101, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99 + 88 - 99", + "ns_shortcut": "Near-cancellation: 88 - 99 = -11, offset is manageable.", + "shortcut_strength": 0.8295454545454546, + "pure_math": { + "question": "Evaluate: 99 + 88 - 99", + "options": [ + "108", + "78", + "98", + "88" + ], + "answer": "88", + "correct_value": 88, + "explanation": "B-C = -11; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "99 + 88 - 27", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 99 + 88 - 27", + "options": [ + "160", + "180", + "150", + "170" + ], + "answer": "160", + "correct_value": 160, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000197", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "31 + 26 - 25", + "ns_shortcut": "Near-cancellation: 26 - 25 = 1, so answer ≈ 31 + 1.", + "shortcut_strength": 0.9615384615384616, + "pure_math": { + "question": "Evaluate: 31 + 26 - 25", + "options": [ + "52", + "42", + "32", + "22" + ], + "answer": "32", + "correct_value": 32, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "31 + 26 - 34", + "ns_shortcut": "Near-cancellation: 26 - 34 = -8, offset is manageable.", + "shortcut_strength": 0.6923076923076923, + "pure_math": { + "question": "Evaluate: 31 + 26 - 34", + "options": [ + "13", + "43", + "33", + "23" + ], + "answer": "23", + "correct_value": 23, + "explanation": "B-C = -8; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "31 + 26 - 116", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 31 + 26 - 116", + "options": [ + "-41", + "-51", + "-71", + "-59" + ], + "answer": "-59", + "correct_value": -59, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000198", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "66 + 68 - 69", + "ns_shortcut": "Near-cancellation: 68 - 69 = -1, so answer ≈ 66 + -1.", + "shortcut_strength": 0.9852941176470589, + "pure_math": { + "question": "Evaluate: 66 + 68 - 69", + "options": [ + "65", + "85", + "75", + "55" + ], + "answer": "65", + "correct_value": 65, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "66 + 68 - 55", + "ns_shortcut": "Near-cancellation: 68 - 55 = 13, offset is manageable.", + "shortcut_strength": 0.8088235294117647, + "pure_math": { + "question": "Evaluate: 66 + 68 - 55", + "options": [ + "99", + "89", + "79", + "69" + ], + "answer": "79", + "correct_value": 79, + "explanation": "B-C = 13; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "66 + 68 - 158", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 66 + 68 - 158", + "options": [ + "-6", + "-24", + "-36", + "-16" + ], + "answer": "-24", + "correct_value": -24, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000199", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "14 + 52 - 54", + "ns_shortcut": "Near-cancellation: 52 - 54 = -2, so answer ≈ 14 + -2.", + "shortcut_strength": 0.9615384615384616, + "pure_math": { + "question": "Evaluate: 14 + 52 - 54", + "options": [ + "2", + "12", + "22", + "32" + ], + "answer": "12", + "correct_value": 12, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "14 + 52 - 62", + "ns_shortcut": "Near-cancellation: 52 - 62 = -10, offset is manageable.", + "shortcut_strength": 0.8076923076923077, + "pure_math": { + "question": "Evaluate: 14 + 52 - 62", + "options": [ + "24", + "4", + "-6", + "14" + ], + "answer": "4", + "correct_value": 4, + "explanation": "B-C = -10; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "14 + 52 - 138", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 14 + 52 - 138", + "options": [ + "-88", + "-58", + "-68", + "-72" + ], + "answer": "-72", + "correct_value": -72, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000200", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "13 + 70 - 71", + "ns_shortcut": "Near-cancellation: 70 - 71 = -1, so answer ≈ 13 + -1.", + "shortcut_strength": 0.9857142857142858, + "pure_math": { + "question": "Evaluate: 13 + 70 - 71", + "options": [ + "22", + "32", + "12", + "2" + ], + "answer": "12", + "correct_value": 12, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "13 + 70 - 80", + "ns_shortcut": "Near-cancellation: 70 - 80 = -10, offset is manageable.", + "shortcut_strength": 0.8571428571428572, + "pure_math": { + "question": "Evaluate: 13 + 70 - 80", + "options": [ + "23", + "-7", + "3", + "13" + ], + "answer": "3", + "correct_value": 3, + "explanation": "B-C = -10; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "13 + 70 - 154", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 13 + 70 - 154", + "options": [ + "-89", + "-71", + "-69", + "-59" + ], + "answer": "-71", + "correct_value": -71, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000201", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24 * 4", + "ns_shortcut": "Round to 25×4 for easy mental computation.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Estimate 24 × 4.", + "options": [ + "116", + "86", + "106", + "96" + ], + "answer": "96", + "correct_value": 96, + "explanation": "Round to compatible pair 25×4 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 * 6", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.75, + "pure_math": { + "question": "Estimate 25 × 6.", + "options": [ + "140", + "170", + "150", + "160" + ], + "answer": "150", + "correct_value": 150, + "explanation": "One number rounds to 25, other is farther from 4." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "29 * 6", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 29 × 6.", + "options": [ + "164", + "184", + "174", + "194" + ], + "answer": "174", + "correct_value": 174, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000202", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26 * 5", + "ns_shortcut": "Round to 25×4 for easy mental computation.", + "shortcut_strength": 0.855, + "pure_math": { + "question": "Estimate 26 × 5.", + "options": [ + "120", + "140", + "130", + "150" + ], + "answer": "130", + "correct_value": 130, + "explanation": "Round to compatible pair 25×4 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "24 * 6", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.73, + "pure_math": { + "question": "Estimate 24 × 6.", + "options": [ + "164", + "144", + "134", + "154" + ], + "answer": "144", + "correct_value": 144, + "explanation": "One number rounds to 25, other is farther from 4." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "30 * 7", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 30 × 7.", + "options": [ + "230", + "210", + "220", + "200" + ], + "answer": "210", + "correct_value": 210, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000203", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50 * 2", + "ns_shortcut": "Round to 50×2 for easy mental computation.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Estimate 50 × 2.", + "options": [ + "100", + "110", + "90", + "120" + ], + "answer": "100", + "correct_value": 100, + "explanation": "Round to compatible pair 50×2 = 100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "50 * 4", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "Estimate 50 × 4.", + "options": [ + "210", + "200", + "220", + "190" + ], + "answer": "200", + "correct_value": 200, + "explanation": "One number rounds to 50, other is farther from 2." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "62 * 5", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 62 × 5.", + "options": [ + "330", + "300", + "320", + "310" + ], + "answer": "310", + "correct_value": 310, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000204", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 * 8", + "ns_shortcut": "Round to 25×8 for easy mental computation.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Estimate 25 × 8.", + "options": [ + "190", + "220", + "200", + "210" + ], + "answer": "200", + "correct_value": 200, + "explanation": "Round to compatible pair 25×8 = 200." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 * 9", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.9375, + "pure_math": { + "question": "Estimate 25 × 9.", + "options": [ + "235", + "245", + "225", + "215" + ], + "answer": "225", + "correct_value": 225, + "explanation": "One number rounds to 25, other is farther from 8." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "30 * 11", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 30 × 11.", + "options": [ + "350", + "340", + "320", + "330" + ], + "answer": "330", + "correct_value": 330, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000205", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26 * 9", + "ns_shortcut": "Round to 25×8 for easy mental computation.", + "shortcut_strength": 0.9175, + "pure_math": { + "question": "Estimate 26 × 9.", + "options": [ + "224", + "244", + "254", + "234" + ], + "answer": "234", + "correct_value": 234, + "explanation": "Round to compatible pair 25×8 = 200." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "24 * 7", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.9175, + "pure_math": { + "question": "Estimate 24 × 7.", + "options": [ + "178", + "188", + "158", + "168" + ], + "answer": "168", + "correct_value": 168, + "explanation": "One number rounds to 25, other is farther from 8." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "30 * 12", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 30 × 12.", + "options": [ + "360", + "350", + "370", + "380" + ], + "answer": "360", + "correct_value": 360, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000206", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51 * 2", + "ns_shortcut": "Round to 50×2 for easy mental computation.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Estimate 51 × 2.", + "options": [ + "92", + "102", + "112", + "122" + ], + "answer": "102", + "correct_value": 102, + "explanation": "Round to compatible pair 50×2 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "50 * 4", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "Estimate 50 × 4.", + "options": [ + "220", + "190", + "200", + "210" + ], + "answer": "200", + "correct_value": 200, + "explanation": "One number rounds to 50, other is farther from 2." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "59 * 5", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 59 × 5.", + "options": [ + "285", + "315", + "305", + "295" + ], + "answer": "295", + "correct_value": 295, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000207", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51 * 2", + "ns_shortcut": "Round to 50×2 for easy mental computation.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Estimate 51 × 2.", + "options": [ + "102", + "122", + "92", + "112" + ], + "answer": "102", + "correct_value": 102, + "explanation": "Round to compatible pair 50×2 = 100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "51 * 3", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.74, + "pure_math": { + "question": "Estimate 51 × 3.", + "options": [ + "173", + "153", + "143", + "163" + ], + "answer": "153", + "correct_value": 153, + "explanation": "One number rounds to 50, other is farther from 2." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "61 * 5", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 61 × 5.", + "options": [ + "315", + "295", + "305", + "325" + ], + "answer": "305", + "correct_value": 305, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000208", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26 * 4", + "ns_shortcut": "Round to 25×4 for easy mental computation.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Estimate 26 × 4.", + "options": [ + "94", + "124", + "104", + "114" + ], + "answer": "104", + "correct_value": 104, + "explanation": "Round to compatible pair 25×4 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 * 2", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.75, + "pure_math": { + "question": "Estimate 25 × 2.", + "options": [ + "70", + "40", + "60", + "50" + ], + "answer": "50", + "correct_value": 50, + "explanation": "One number rounds to 25, other is farther from 4." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "30 * 7", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 30 × 7.", + "options": [ + "210", + "220", + "230", + "200" + ], + "answer": "210", + "correct_value": 210, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000209", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 * 7", + "ns_shortcut": "Round to 25×8 for easy mental computation.", + "shortcut_strength": 0.9375, + "pure_math": { + "question": "Estimate 25 × 7.", + "options": [ + "175", + "185", + "195", + "165" + ], + "answer": "175", + "correct_value": 175, + "explanation": "Round to compatible pair 25×8 = 200." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "24 * 10", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.855, + "pure_math": { + "question": "Estimate 24 × 10.", + "options": [ + "260", + "240", + "230", + "250" + ], + "answer": "240", + "correct_value": 240, + "explanation": "One number rounds to 25, other is farther from 8." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "30 * 10", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 30 × 10.", + "options": [ + "310", + "320", + "300", + "290" + ], + "answer": "300", + "correct_value": 300, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000210", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 * 8", + "ns_shortcut": "Round to 25×8 for easy mental computation.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Estimate 25 × 8.", + "options": [ + "200", + "210", + "190", + "220" + ], + "answer": "200", + "correct_value": 200, + "explanation": "Round to compatible pair 25×8 = 200." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 * 9", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.9375, + "pure_math": { + "question": "Estimate 25 × 9.", + "options": [ + "215", + "225", + "245", + "235" + ], + "answer": "225", + "correct_value": 225, + "explanation": "One number rounds to 25, other is farther from 8." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "29 * 11", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 29 × 11.", + "options": [ + "319", + "329", + "309", + "339" + ], + "answer": "319", + "correct_value": 319, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000211", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51 * 3", + "ns_shortcut": "Round to 50×2 for easy mental computation.", + "shortcut_strength": 0.74, + "pure_math": { + "question": "Estimate 51 × 3.", + "options": [ + "153", + "173", + "143", + "163" + ], + "answer": "153", + "correct_value": 153, + "explanation": "Round to compatible pair 50×2 = 100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "50 * 0", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "Estimate 50 × 0.", + "options": [ + "-10", + "0", + "10", + "20" + ], + "answer": "0", + "correct_value": 0, + "explanation": "One number rounds to 50, other is farther from 2." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "62 * 4", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 62 × 4.", + "options": [ + "248", + "238", + "258", + "268" + ], + "answer": "248", + "correct_value": 248, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000212", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51 * 2", + "ns_shortcut": "Round to 50×2 for easy mental computation.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Estimate 51 × 2.", + "options": [ + "122", + "92", + "102", + "112" + ], + "answer": "102", + "correct_value": 102, + "explanation": "Round to compatible pair 50×2 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "50 * 0", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "Estimate 50 × 0.", + "options": [ + "0", + "-10", + "10", + "20" + ], + "answer": "0", + "correct_value": 0, + "explanation": "One number rounds to 50, other is farther from 2." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "61 * 5", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 61 × 5.", + "options": [ + "315", + "295", + "325", + "305" + ], + "answer": "305", + "correct_value": 305, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000213", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49 * 3", + "ns_shortcut": "Round to 50×2 for easy mental computation.", + "shortcut_strength": 0.74, + "pure_math": { + "question": "Estimate 49 × 3.", + "options": [ + "137", + "147", + "167", + "157" + ], + "answer": "147", + "correct_value": 147, + "explanation": "Round to compatible pair 50×2 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "50 * 3", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.75, + "pure_math": { + "question": "Estimate 50 × 3.", + "options": [ + "160", + "150", + "140", + "170" + ], + "answer": "150", + "correct_value": 150, + "explanation": "One number rounds to 50, other is farther from 2." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "60 * 4", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 60 × 4.", + "options": [ + "260", + "240", + "230", + "250" + ], + "answer": "240", + "correct_value": 240, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000214", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26 * 9", + "ns_shortcut": "Round to 25×8 for easy mental computation.", + "shortcut_strength": 0.9175, + "pure_math": { + "question": "Estimate 26 × 9.", + "options": [ + "234", + "244", + "224", + "254" + ], + "answer": "234", + "correct_value": 234, + "explanation": "Round to compatible pair 25×8 = 200." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 * 9", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.9375, + "pure_math": { + "question": "Estimate 25 × 9.", + "options": [ + "245", + "215", + "235", + "225" + ], + "answer": "225", + "correct_value": 225, + "explanation": "One number rounds to 25, other is farther from 8." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "28 * 12", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 28 × 12.", + "options": [ + "336", + "346", + "356", + "326" + ], + "answer": "336", + "correct_value": 336, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000215", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50 * 2", + "ns_shortcut": "Round to 50×2 for easy mental computation.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Estimate 50 × 2.", + "options": [ + "100", + "90", + "110", + "120" + ], + "answer": "100", + "correct_value": 100, + "explanation": "Round to compatible pair 50×2 = 100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "49 * 1", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.74, + "pure_math": { + "question": "Estimate 49 × 1.", + "options": [ + "69", + "49", + "59", + "39" + ], + "answer": "49", + "correct_value": 49, + "explanation": "One number rounds to 50, other is farther from 2." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "60 * 4", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 60 × 4.", + "options": [ + "230", + "260", + "250", + "240" + ], + "answer": "240", + "correct_value": 240, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000216", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 * 4", + "ns_shortcut": "Round to 25×4 for easy mental computation.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Estimate 25 × 4.", + "options": [ + "100", + "90", + "120", + "110" + ], + "answer": "100", + "correct_value": 100, + "explanation": "Round to compatible pair 25×4 = 100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "26 * 6", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.73, + "pure_math": { + "question": "Estimate 26 × 6.", + "options": [ + "166", + "176", + "156", + "146" + ], + "answer": "156", + "correct_value": 156, + "explanation": "One number rounds to 25, other is farther from 4." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "29 * 7", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 29 × 7.", + "options": [ + "223", + "193", + "213", + "203" + ], + "answer": "203", + "correct_value": 203, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000217", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 * 3", + "ns_shortcut": "Round to 25×4 for easy mental computation.", + "shortcut_strength": 0.875, + "pure_math": { + "question": "Estimate 25 × 3.", + "options": [ + "85", + "65", + "75", + "95" + ], + "answer": "75", + "correct_value": 75, + "explanation": "Round to compatible pair 25×4 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "26 * 3", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.855, + "pure_math": { + "question": "Estimate 26 × 3.", + "options": [ + "68", + "98", + "88", + "78" + ], + "answer": "78", + "correct_value": 78, + "explanation": "One number rounds to 25, other is farther from 4." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "31 * 7", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 31 × 7.", + "options": [ + "227", + "217", + "237", + "207" + ], + "answer": "217", + "correct_value": 217, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000218", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51 * 2", + "ns_shortcut": "Round to 50×2 for easy mental computation.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Estimate 51 × 2.", + "options": [ + "112", + "122", + "102", + "92" + ], + "answer": "102", + "correct_value": 102, + "explanation": "Round to compatible pair 50×2 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "50 * 1", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.75, + "pure_math": { + "question": "Estimate 50 × 1.", + "options": [ + "70", + "40", + "50", + "60" + ], + "answer": "50", + "correct_value": 50, + "explanation": "One number rounds to 50, other is farther from 2." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "55 * 5", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 55 × 5.", + "options": [ + "285", + "275", + "265", + "295" + ], + "answer": "275", + "correct_value": 275, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000219", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 * 8", + "ns_shortcut": "Round to 25×8 for easy mental computation.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Estimate 25 × 8.", + "options": [ + "190", + "220", + "210", + "200" + ], + "answer": "200", + "correct_value": 200, + "explanation": "Round to compatible pair 25×8 = 200." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 * 10", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.875, + "pure_math": { + "question": "Estimate 25 × 10.", + "options": [ + "240", + "270", + "250", + "260" + ], + "answer": "250", + "correct_value": 250, + "explanation": "One number rounds to 25, other is farther from 8." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "30 * 11", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 30 × 11.", + "options": [ + "320", + "340", + "350", + "330" + ], + "answer": "330", + "correct_value": 330, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000220", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24 * 8", + "ns_shortcut": "Round to 25×8 for easy mental computation.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Estimate 24 × 8.", + "options": [ + "202", + "182", + "192", + "212" + ], + "answer": "192", + "correct_value": 192, + "explanation": "Round to compatible pair 25×8 = 200." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 * 6", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.875, + "pure_math": { + "question": "Estimate 25 × 6.", + "options": [ + "150", + "170", + "140", + "160" + ], + "answer": "150", + "correct_value": 150, + "explanation": "One number rounds to 25, other is farther from 8." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "30 * 10", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 30 × 10.", + "options": [ + "290", + "310", + "320", + "300" + ], + "answer": "300", + "correct_value": 300, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000221", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26 * 4", + "ns_shortcut": "Round to 25×4 for easy mental computation.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Estimate 26 × 4.", + "options": [ + "114", + "94", + "124", + "104" + ], + "answer": "104", + "correct_value": 104, + "explanation": "Round to compatible pair 25×4 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "24 * 3", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.855, + "pure_math": { + "question": "Estimate 24 × 3.", + "options": [ + "82", + "62", + "92", + "72" + ], + "answer": "72", + "correct_value": 72, + "explanation": "One number rounds to 25, other is farther from 4." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "30 * 6", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 30 × 6.", + "options": [ + "180", + "200", + "190", + "170" + ], + "answer": "180", + "correct_value": 180, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000222", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50 * 2", + "ns_shortcut": "Round to 50×2 for easy mental computation.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Estimate 50 × 2.", + "options": [ + "100", + "90", + "110", + "120" + ], + "answer": "100", + "correct_value": 100, + "explanation": "Round to compatible pair 50×2 = 100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "50 * 1", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.75, + "pure_math": { + "question": "Estimate 50 × 1.", + "options": [ + "60", + "70", + "40", + "50" + ], + "answer": "50", + "correct_value": 50, + "explanation": "One number rounds to 50, other is farther from 2." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "60 * 5", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 60 × 5.", + "options": [ + "300", + "310", + "320", + "290" + ], + "answer": "300", + "correct_value": 300, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000223", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24 * 8", + "ns_shortcut": "Round to 25×8 for easy mental computation.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Estimate 24 × 8.", + "options": [ + "182", + "212", + "202", + "192" + ], + "answer": "192", + "correct_value": 192, + "explanation": "Round to compatible pair 25×8 = 200." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "26 * 10", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.855, + "pure_math": { + "question": "Estimate 26 × 10.", + "options": [ + "270", + "260", + "280", + "250" + ], + "answer": "260", + "correct_value": 260, + "explanation": "One number rounds to 25, other is farther from 8." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "28 * 10", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 28 × 10.", + "options": [ + "290", + "300", + "270", + "280" + ], + "answer": "280", + "correct_value": 280, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000224", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51 * 1", + "ns_shortcut": "Round to 50×2 for easy mental computation.", + "shortcut_strength": 0.74, + "pure_math": { + "question": "Estimate 51 × 1.", + "options": [ + "61", + "51", + "41", + "71" + ], + "answer": "51", + "correct_value": 51, + "explanation": "Round to compatible pair 50×2 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "49 * 3", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.74, + "pure_math": { + "question": "Estimate 49 × 3.", + "options": [ + "167", + "147", + "157", + "137" + ], + "answer": "147", + "correct_value": 147, + "explanation": "One number rounds to 50, other is farther from 2." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "56 * 5", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 56 × 5.", + "options": [ + "290", + "270", + "280", + "300" + ], + "answer": "280", + "correct_value": 280, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000225", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 * 8", + "ns_shortcut": "Round to 25×8 for easy mental computation.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Estimate 25 × 8.", + "options": [ + "200", + "220", + "210", + "190" + ], + "answer": "200", + "correct_value": 200, + "explanation": "Round to compatible pair 25×8 = 200." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 * 7", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.9375, + "pure_math": { + "question": "Estimate 25 × 7.", + "options": [ + "195", + "175", + "185", + "165" + ], + "answer": "175", + "correct_value": 175, + "explanation": "One number rounds to 25, other is farther from 8." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "28 * 11", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 28 × 11.", + "options": [ + "298", + "318", + "308", + "328" + ], + "answer": "308", + "correct_value": 308, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000226", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50 * 3", + "ns_shortcut": "Round to 50×2 for easy mental computation.", + "shortcut_strength": 0.75, + "pure_math": { + "question": "Estimate 50 × 3.", + "options": [ + "170", + "140", + "150", + "160" + ], + "answer": "150", + "correct_value": 150, + "explanation": "Round to compatible pair 50×2 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "51 * 0", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.49, + "pure_math": { + "question": "Estimate 51 × 0.", + "options": [ + "20", + "0", + "10", + "-10" + ], + "answer": "0", + "correct_value": 0, + "explanation": "One number rounds to 50, other is farther from 2." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "62 * 4", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 62 × 4.", + "options": [ + "238", + "268", + "258", + "248" + ], + "answer": "248", + "correct_value": 248, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000227", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 * 4", + "ns_shortcut": "Round to 25×4 for easy mental computation.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Estimate 25 × 4.", + "options": [ + "100", + "120", + "110", + "90" + ], + "answer": "100", + "correct_value": 100, + "explanation": "Round to compatible pair 25×4 = 100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 * 6", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.75, + "pure_math": { + "question": "Estimate 25 × 6.", + "options": [ + "150", + "160", + "170", + "140" + ], + "answer": "150", + "correct_value": 150, + "explanation": "One number rounds to 25, other is farther from 4." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "31 * 6", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 31 × 6.", + "options": [ + "196", + "176", + "206", + "186" + ], + "answer": "186", + "correct_value": 186, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000228", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26 * 5", + "ns_shortcut": "Round to 25×4 for easy mental computation.", + "shortcut_strength": 0.855, + "pure_math": { + "question": "Estimate 26 × 5.", + "options": [ + "140", + "150", + "120", + "130" + ], + "answer": "130", + "correct_value": 130, + "explanation": "Round to compatible pair 25×4 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 * 5", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.875, + "pure_math": { + "question": "Estimate 25 × 5.", + "options": [ + "135", + "125", + "115", + "145" + ], + "answer": "125", + "correct_value": 125, + "explanation": "One number rounds to 25, other is farther from 4." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "29 * 6", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 29 × 6.", + "options": [ + "194", + "184", + "174", + "164" + ], + "answer": "174", + "correct_value": 174, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000229", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 * 4", + "ns_shortcut": "Round to 25×4 for easy mental computation.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Estimate 25 × 4.", + "options": [ + "90", + "120", + "110", + "100" + ], + "answer": "100", + "correct_value": 100, + "explanation": "Round to compatible pair 25×4 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "24 * 3", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.855, + "pure_math": { + "question": "Estimate 24 × 3.", + "options": [ + "72", + "82", + "92", + "62" + ], + "answer": "72", + "correct_value": 72, + "explanation": "One number rounds to 25, other is farther from 4." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "29 * 6", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 29 × 6.", + "options": [ + "194", + "164", + "174", + "184" + ], + "answer": "174", + "correct_value": 174, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000230", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26 * 8", + "ns_shortcut": "Round to 25×8 for easy mental computation.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Estimate 26 × 8.", + "options": [ + "208", + "198", + "218", + "228" + ], + "answer": "208", + "correct_value": 208, + "explanation": "Round to compatible pair 25×8 = 200." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "26 * 7", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.9175, + "pure_math": { + "question": "Estimate 26 × 7.", + "options": [ + "182", + "192", + "202", + "172" + ], + "answer": "182", + "correct_value": 182, + "explanation": "One number rounds to 25, other is farther from 8." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "30 * 12", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 30 × 12.", + "options": [ + "350", + "380", + "360", + "370" + ], + "answer": "360", + "correct_value": 360, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000231", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26 * 8", + "ns_shortcut": "Round to 25×8 for easy mental computation.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Estimate 26 × 8.", + "options": [ + "218", + "208", + "198", + "228" + ], + "answer": "208", + "correct_value": 208, + "explanation": "Round to compatible pair 25×8 = 200." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "26 * 10", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.855, + "pure_math": { + "question": "Estimate 26 × 10.", + "options": [ + "270", + "280", + "260", + "250" + ], + "answer": "260", + "correct_value": 260, + "explanation": "One number rounds to 25, other is farther from 8." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "28 * 11", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 28 × 11.", + "options": [ + "298", + "328", + "308", + "318" + ], + "answer": "308", + "correct_value": 308, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000232", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 * 5", + "ns_shortcut": "Round to 25×4 for easy mental computation.", + "shortcut_strength": 0.875, + "pure_math": { + "question": "Estimate 25 × 5.", + "options": [ + "145", + "115", + "125", + "135" + ], + "answer": "125", + "correct_value": 125, + "explanation": "Round to compatible pair 25×4 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 * 6", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.75, + "pure_math": { + "question": "Estimate 25 × 6.", + "options": [ + "160", + "140", + "170", + "150" + ], + "answer": "150", + "correct_value": 150, + "explanation": "One number rounds to 25, other is farther from 4." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "31 * 6", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 31 × 6.", + "options": [ + "196", + "186", + "176", + "206" + ], + "answer": "186", + "correct_value": 186, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000233", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51 * 2", + "ns_shortcut": "Round to 50×2 for easy mental computation.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Estimate 51 × 2.", + "options": [ + "92", + "112", + "122", + "102" + ], + "answer": "102", + "correct_value": 102, + "explanation": "Round to compatible pair 50×2 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "49 * 0", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.49, + "pure_math": { + "question": "Estimate 49 × 0.", + "options": [ + "-10", + "0", + "20", + "10" + ], + "answer": "0", + "correct_value": 0, + "explanation": "One number rounds to 50, other is farther from 2." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "59 * 5", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 59 × 5.", + "options": [ + "285", + "295", + "315", + "305" + ], + "answer": "295", + "correct_value": 295, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000234", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 * 4", + "ns_shortcut": "Round to 25×4 for easy mental computation.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Estimate 25 × 4.", + "options": [ + "100", + "120", + "110", + "90" + ], + "answer": "100", + "correct_value": 100, + "explanation": "Round to compatible pair 25×4 = 100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 * 5", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.875, + "pure_math": { + "question": "Estimate 25 × 5.", + "options": [ + "115", + "125", + "135", + "145" + ], + "answer": "125", + "correct_value": 125, + "explanation": "One number rounds to 25, other is farther from 4." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "28 * 6", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 28 × 6.", + "options": [ + "178", + "168", + "158", + "188" + ], + "answer": "168", + "correct_value": 168, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000235", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26 * 8", + "ns_shortcut": "Round to 25×8 for easy mental computation.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Estimate 26 × 8.", + "options": [ + "218", + "208", + "198", + "228" + ], + "answer": "208", + "correct_value": 208, + "explanation": "Round to compatible pair 25×8 = 200." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 * 6", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.875, + "pure_math": { + "question": "Estimate 25 × 6.", + "options": [ + "170", + "160", + "150", + "140" + ], + "answer": "150", + "correct_value": 150, + "explanation": "One number rounds to 25, other is farther from 8." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "29 * 10", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 29 × 10.", + "options": [ + "290", + "280", + "300", + "310" + ], + "answer": "290", + "correct_value": 290, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000236", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49 * 2", + "ns_shortcut": "Round to 50×2 for easy mental computation.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Estimate 49 × 2.", + "options": [ + "88", + "98", + "108", + "118" + ], + "answer": "98", + "correct_value": 98, + "explanation": "Round to compatible pair 50×2 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "50 * 4", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "Estimate 50 × 4.", + "options": [ + "190", + "220", + "210", + "200" + ], + "answer": "200", + "correct_value": 200, + "explanation": "One number rounds to 50, other is farther from 2." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "59 * 4", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 59 × 4.", + "options": [ + "236", + "246", + "226", + "256" + ], + "answer": "236", + "correct_value": 236, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000237", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49 * 1", + "ns_shortcut": "Round to 50×2 for easy mental computation.", + "shortcut_strength": 0.74, + "pure_math": { + "question": "Estimate 49 × 1.", + "options": [ + "59", + "69", + "39", + "49" + ], + "answer": "49", + "correct_value": 49, + "explanation": "Round to compatible pair 50×2 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "50 * 1", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.75, + "pure_math": { + "question": "Estimate 50 × 1.", + "options": [ + "60", + "70", + "50", + "40" + ], + "answer": "50", + "correct_value": 50, + "explanation": "One number rounds to 50, other is farther from 2." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "61 * 4", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 61 × 4.", + "options": [ + "264", + "234", + "244", + "254" + ], + "answer": "244", + "correct_value": 244, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000238", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 * 8", + "ns_shortcut": "Round to 25×8 for easy mental computation.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Estimate 25 × 8.", + "options": [ + "220", + "210", + "190", + "200" + ], + "answer": "200", + "correct_value": 200, + "explanation": "Round to compatible pair 25×8 = 200." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 * 10", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.875, + "pure_math": { + "question": "Estimate 25 × 10.", + "options": [ + "250", + "270", + "260", + "240" + ], + "answer": "250", + "correct_value": 250, + "explanation": "One number rounds to 25, other is farther from 8." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "31 * 11", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 31 × 11.", + "options": [ + "361", + "331", + "351", + "341" + ], + "answer": "341", + "correct_value": 341, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000239", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 * 8", + "ns_shortcut": "Round to 25×8 for easy mental computation.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Estimate 25 × 8.", + "options": [ + "210", + "200", + "190", + "220" + ], + "answer": "200", + "correct_value": 200, + "explanation": "Round to compatible pair 25×8 = 200." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "26 * 6", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.855, + "pure_math": { + "question": "Estimate 26 × 6.", + "options": [ + "156", + "176", + "166", + "146" + ], + "answer": "156", + "correct_value": 156, + "explanation": "One number rounds to 25, other is farther from 8." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "28 * 11", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 28 × 11.", + "options": [ + "298", + "308", + "328", + "318" + ], + "answer": "308", + "correct_value": 308, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000240", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26 * 3", + "ns_shortcut": "Round to 25×4 for easy mental computation.", + "shortcut_strength": 0.855, + "pure_math": { + "question": "Estimate 26 × 3.", + "options": [ + "88", + "68", + "78", + "98" + ], + "answer": "78", + "correct_value": 78, + "explanation": "Round to compatible pair 25×4 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 * 2", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.75, + "pure_math": { + "question": "Estimate 25 × 2.", + "options": [ + "70", + "40", + "60", + "50" + ], + "answer": "50", + "correct_value": 50, + "explanation": "One number rounds to 25, other is farther from 4." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "30 * 6", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 30 × 6.", + "options": [ + "190", + "180", + "200", + "170" + ], + "answer": "180", + "correct_value": 180, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000241", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 * 4", + "ns_shortcut": "Round to 25×4 for easy mental computation.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Estimate 25 × 4.", + "options": [ + "110", + "120", + "90", + "100" + ], + "answer": "100", + "correct_value": 100, + "explanation": "Round to compatible pair 25×4 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 * 5", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.875, + "pure_math": { + "question": "Estimate 25 × 5.", + "options": [ + "135", + "115", + "125", + "145" + ], + "answer": "125", + "correct_value": 125, + "explanation": "One number rounds to 25, other is farther from 4." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "31 * 7", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 31 × 7.", + "options": [ + "227", + "207", + "217", + "237" + ], + "answer": "217", + "correct_value": 217, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000242", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 * 3", + "ns_shortcut": "Round to 25×4 for easy mental computation.", + "shortcut_strength": 0.875, + "pure_math": { + "question": "Estimate 25 × 3.", + "options": [ + "75", + "95", + "85", + "65" + ], + "answer": "75", + "correct_value": 75, + "explanation": "Round to compatible pair 25×4 = 100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 * 6", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.75, + "pure_math": { + "question": "Estimate 25 × 6.", + "options": [ + "160", + "170", + "150", + "140" + ], + "answer": "150", + "correct_value": 150, + "explanation": "One number rounds to 25, other is farther from 4." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "30 * 6", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 30 × 6.", + "options": [ + "180", + "170", + "200", + "190" + ], + "answer": "180", + "correct_value": 180, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000243", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50 * 3", + "ns_shortcut": "Round to 50×2 for easy mental computation.", + "shortcut_strength": 0.75, + "pure_math": { + "question": "Estimate 50 × 3.", + "options": [ + "150", + "170", + "160", + "140" + ], + "answer": "150", + "correct_value": 150, + "explanation": "Round to compatible pair 50×2 = 100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "51 * 1", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.74, + "pure_math": { + "question": "Estimate 51 × 1.", + "options": [ + "71", + "41", + "61", + "51" + ], + "answer": "51", + "correct_value": 51, + "explanation": "One number rounds to 50, other is farther from 2." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "58 * 5", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 58 × 5.", + "options": [ + "300", + "280", + "310", + "290" + ], + "answer": "290", + "correct_value": 290, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000244", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24 * 4", + "ns_shortcut": "Round to 25×4 for easy mental computation.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Estimate 24 × 4.", + "options": [ + "96", + "106", + "86", + "116" + ], + "answer": "96", + "correct_value": 96, + "explanation": "Round to compatible pair 25×4 = 100." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "24 * 5", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.855, + "pure_math": { + "question": "Estimate 24 × 5.", + "options": [ + "110", + "130", + "120", + "140" + ], + "answer": "120", + "correct_value": 120, + "explanation": "One number rounds to 25, other is farther from 4." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "28 * 7", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 28 × 7.", + "options": [ + "206", + "196", + "186", + "216" + ], + "answer": "196", + "correct_value": 196, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000245", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 * 7", + "ns_shortcut": "Round to 25×8 for easy mental computation.", + "shortcut_strength": 0.9375, + "pure_math": { + "question": "Estimate 25 × 7.", + "options": [ + "165", + "195", + "175", + "185" + ], + "answer": "175", + "correct_value": 175, + "explanation": "Round to compatible pair 25×8 = 200." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "26 * 7", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.9175, + "pure_math": { + "question": "Estimate 26 × 7.", + "options": [ + "192", + "202", + "182", + "172" + ], + "answer": "182", + "correct_value": 182, + "explanation": "One number rounds to 25, other is farther from 8." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "30 * 11", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 30 × 11.", + "options": [ + "320", + "350", + "330", + "340" + ], + "answer": "330", + "correct_value": 330, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000246", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26 * 9", + "ns_shortcut": "Round to 25×8 for easy mental computation.", + "shortcut_strength": 0.9175, + "pure_math": { + "question": "Estimate 26 × 9.", + "options": [ + "254", + "244", + "224", + "234" + ], + "answer": "234", + "correct_value": 234, + "explanation": "Round to compatible pair 25×8 = 200." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "24 * 9", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.9175, + "pure_math": { + "question": "Estimate 24 × 9.", + "options": [ + "216", + "226", + "236", + "206" + ], + "answer": "216", + "correct_value": 216, + "explanation": "One number rounds to 25, other is farther from 8." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "30 * 12", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 30 × 12.", + "options": [ + "350", + "370", + "380", + "360" + ], + "answer": "360", + "correct_value": 360, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000247", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49 * 3", + "ns_shortcut": "Round to 50×2 for easy mental computation.", + "shortcut_strength": 0.74, + "pure_math": { + "question": "Estimate 49 × 3.", + "options": [ + "167", + "147", + "157", + "137" + ], + "answer": "147", + "correct_value": 147, + "explanation": "Round to compatible pair 50×2 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "49 * 1", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.74, + "pure_math": { + "question": "Estimate 49 × 1.", + "options": [ + "39", + "59", + "69", + "49" + ], + "answer": "49", + "correct_value": 49, + "explanation": "One number rounds to 50, other is farther from 2." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "58 * 4", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 58 × 4.", + "options": [ + "252", + "222", + "232", + "242" + ], + "answer": "232", + "correct_value": 232, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000248", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 * 9", + "ns_shortcut": "Round to 25×8 for easy mental computation.", + "shortcut_strength": 0.9375, + "pure_math": { + "question": "Estimate 25 × 9.", + "options": [ + "215", + "245", + "235", + "225" + ], + "answer": "225", + "correct_value": 225, + "explanation": "Round to compatible pair 25×8 = 200." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 * 6", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.875, + "pure_math": { + "question": "Estimate 25 × 6.", + "options": [ + "160", + "170", + "140", + "150" + ], + "answer": "150", + "correct_value": 150, + "explanation": "One number rounds to 25, other is farther from 8." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "29 * 12", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 29 × 12.", + "options": [ + "338", + "368", + "348", + "358" + ], + "answer": "348", + "correct_value": 348, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000249", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 * 7", + "ns_shortcut": "Round to 25×8 for easy mental computation.", + "shortcut_strength": 0.9375, + "pure_math": { + "question": "Estimate 25 × 7.", + "options": [ + "175", + "165", + "195", + "185" + ], + "answer": "175", + "correct_value": 175, + "explanation": "Round to compatible pair 25×8 = 200." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "24 * 7", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.9175, + "pure_math": { + "question": "Estimate 24 × 7.", + "options": [ + "168", + "178", + "188", + "158" + ], + "answer": "168", + "correct_value": 168, + "explanation": "One number rounds to 25, other is farther from 8." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "31 * 10", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 31 × 10.", + "options": [ + "330", + "310", + "300", + "320" + ], + "answer": "310", + "correct_value": 310, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000250", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24 * 7", + "ns_shortcut": "Round to 25×8 for easy mental computation.", + "shortcut_strength": 0.9175, + "pure_math": { + "question": "Estimate 24 × 7.", + "options": [ + "178", + "168", + "188", + "158" + ], + "answer": "168", + "correct_value": 168, + "explanation": "Round to compatible pair 25×8 = 200." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "24 * 9", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.9175, + "pure_math": { + "question": "Estimate 24 × 9.", + "options": [ + "226", + "236", + "206", + "216" + ], + "answer": "216", + "correct_value": 216, + "explanation": "One number rounds to 25, other is farther from 8." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "31 * 10", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 31 × 10.", + "options": [ + "330", + "320", + "300", + "310" + ], + "answer": "310", + "correct_value": 310, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "landmark_comparison_000251", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 880 vs 195", + "ns_shortcut": "Use 10% as landmark: 10% of 880 = 88.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 880 greater than 195?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 88.0, + "explanation": "10% ≈ 10%. 10% of 880 = 88. Compare to 195." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "14% of 880 vs 195", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 14% of 880 greater than 195?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 123.2, + "explanation": "14% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 880 vs 195", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 880 greater than 195?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 730.4, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000252", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "52% of 790 vs 369", + "ns_shortcut": "Use 50% as landmark: 50% of 790 = 395.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 52% of 790 greater than 369?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 410.8, + "explanation": "52% ≈ 50%. 50% of 790 = 395. Compare to 369." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "55% of 790 vs 369", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 55% of 790 greater than 369?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 434.5, + "explanation": "55% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 790 vs 369", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 790 greater than 369?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 292.3, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000253", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24% of 990 vs 469", + "ns_shortcut": "Use 25% as landmark: 25% of 990 = 247.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 24% of 990 greater than 469?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 237.6, + "explanation": "24% ≈ 25%. 25% of 990 = 247. Compare to 469." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "30% of 990 vs 469", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 30% of 990 greater than 469?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 297.0, + "explanation": "30% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 990 vs 469", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 990 greater than 469?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 821.7, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000254", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50% of 990 vs 492", + "ns_shortcut": "Use 50% as landmark: 50% of 990 = 495.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 50% of 990 greater than 492?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 495.0, + "explanation": "50% ≈ 50%. 50% of 990 = 495. Compare to 492." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "56% of 990 vs 492", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 56% of 990 greater than 492?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 554.4, + "explanation": "56% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 990 vs 492", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 990 greater than 492?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 425.7, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000255", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "11% of 200 vs 58", + "ns_shortcut": "Use 10% as landmark: 10% of 200 = 20.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 11% of 200 greater than 58?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 22.0, + "explanation": "11% ≈ 10%. 10% of 200 = 20. Compare to 58." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7% of 200 vs 58", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 7% of 200 greater than 58?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 14.0, + "explanation": "7% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 200 vs 58", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 200 greater than 58?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 74.0, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000256", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49% of 830 vs 299", + "ns_shortcut": "Use 50% as landmark: 50% of 830 = 415.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 49% of 830 greater than 299?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 406.7, + "explanation": "49% ≈ 50%. 50% of 830 = 415. Compare to 299." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "45% of 830 vs 299", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 45% of 830 greater than 299?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 373.5, + "explanation": "45% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 830 vs 299", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 830 greater than 299?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 605.9, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000257", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "52% of 560 vs 139", + "ns_shortcut": "Use 50% as landmark: 50% of 560 = 280.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 52% of 560 greater than 139?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 291.2, + "explanation": "52% ≈ 50%. 50% of 560 = 280. Compare to 139." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "55% of 560 vs 139", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 55% of 560 greater than 139?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 308.0, + "explanation": "55% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 560 vs 139", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 560 greater than 139?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 464.8, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000258", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "27% of 640 vs 272", + "ns_shortcut": "Use 25% as landmark: 25% of 640 = 160.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 27% of 640 greater than 272?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 172.8, + "explanation": "27% ≈ 25%. 25% of 640 = 160. Compare to 272." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "29% of 640 vs 272", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 29% of 640 greater than 272?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 185.6, + "explanation": "29% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 640 vs 272", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 640 greater than 272?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 467.2, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000259", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 800 vs 360", + "ns_shortcut": "Use 10% as landmark: 10% of 800 = 80.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 800 greater than 360?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 80.0, + "explanation": "10% ≈ 10%. 10% of 800 = 80. Compare to 360." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "17% of 800 vs 360", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 17% of 800 greater than 360?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 136.0, + "explanation": "17% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 800 vs 360", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 800 greater than 360?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 664.0, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000260", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "27% of 710 vs 206", + "ns_shortcut": "Use 25% as landmark: 25% of 710 = 177.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 27% of 710 greater than 206?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 191.7, + "explanation": "27% ≈ 25%. 25% of 710 = 177. Compare to 206." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "29% of 710 vs 206", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 29% of 710 greater than 206?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 205.9, + "explanation": "29% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 710 vs 206", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 710 greater than 206?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 262.7, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000261", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 390 vs 135", + "ns_shortcut": "Use 10% as landmark: 10% of 390 = 39.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 390 greater than 135?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 39.0, + "explanation": "10% ≈ 10%. 10% of 390 = 39. Compare to 135." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3% of 390 vs 135", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 3% of 390 greater than 135?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 11.7, + "explanation": "3% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 390 vs 135", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 390 greater than 135?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 284.7, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000262", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 530 vs 246", + "ns_shortcut": "Use 10% as landmark: 10% of 530 = 53.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 530 greater than 246?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 53.0, + "explanation": "10% ≈ 10%. 10% of 530 = 53. Compare to 246." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "13% of 530 vs 246", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 13% of 530 greater than 246?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 68.9, + "explanation": "13% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 530 vs 246", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 530 greater than 246?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 196.1, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000263", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 280 vs 116", + "ns_shortcut": "Use 10% as landmark: 10% of 280 = 28.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 280 greater than 116?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 28.0, + "explanation": "10% ≈ 10%. 10% of 280 = 28. Compare to 116." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5% of 280 vs 116", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 5% of 280 greater than 116?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 14.0, + "explanation": "5% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 280 vs 116", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 280 greater than 116?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 120.4, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000264", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8% of 570 vs 134", + "ns_shortcut": "Use 10% as landmark: 10% of 570 = 57.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 8% of 570 greater than 134?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 45.6, + "explanation": "8% ≈ 10%. 10% of 570 = 57. Compare to 134." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "16% of 570 vs 134", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 16% of 570 greater than 134?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 91.2, + "explanation": "16% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 570 vs 134", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 570 greater than 134?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 473.1, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000265", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 930 vs 452", + "ns_shortcut": "Use 25% as landmark: 25% of 930 = 232.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 930 greater than 452?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 232.5, + "explanation": "25% ≈ 25%. 25% of 930 = 232. Compare to 452." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "31% of 930 vs 452", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 31% of 930 greater than 452?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 288.3, + "explanation": "31% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 930 vs 452", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 930 greater than 452?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 344.1, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000266", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 350 vs 154", + "ns_shortcut": "Use 10% as landmark: 10% of 350 = 35.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 350 greater than 154?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 35.0, + "explanation": "10% ≈ 10%. 10% of 350 = 35. Compare to 154." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3% of 350 vs 154", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 3% of 350 greater than 154?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 10.5, + "explanation": "3% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 350 vs 154", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 350 greater than 154?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 290.5, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000267", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "48% of 760 vs 330", + "ns_shortcut": "Use 50% as landmark: 50% of 760 = 380.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 48% of 760 greater than 330?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 364.8, + "explanation": "48% ≈ 50%. 50% of 760 = 380. Compare to 330." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "45% of 760 vs 330", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 45% of 760 greater than 330?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 342.0, + "explanation": "45% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 760 vs 330", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 760 greater than 330?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 326.8, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000268", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49% of 930 vs 186", + "ns_shortcut": "Use 50% as landmark: 50% of 930 = 465.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 49% of 930 greater than 186?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 455.7, + "explanation": "49% ≈ 50%. 50% of 930 = 465. Compare to 186." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "55% of 930 vs 186", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 55% of 930 greater than 186?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 511.5, + "explanation": "55% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 930 vs 186", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 930 greater than 186?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 678.9, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000269", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "48% of 610 vs 272", + "ns_shortcut": "Use 50% as landmark: 50% of 610 = 305.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 48% of 610 greater than 272?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 292.8, + "explanation": "48% ≈ 50%. 50% of 610 = 305. Compare to 272." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "47% of 610 vs 272", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 47% of 610 greater than 272?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 286.7, + "explanation": "47% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 610 vs 272", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 610 greater than 272?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 506.3, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000270", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 460 vs 181", + "ns_shortcut": "Use 25% as landmark: 25% of 460 = 115.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 460 greater than 181?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 115.0, + "explanation": "25% ≈ 25%. 25% of 460 = 115. Compare to 181." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "19% of 460 vs 181", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 19% of 460 greater than 181?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 87.4, + "explanation": "19% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 460 vs 181", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 460 greater than 181?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 335.8, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000271", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24% of 830 vs 167", + "ns_shortcut": "Use 25% as landmark: 25% of 830 = 207.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 24% of 830 greater than 167?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 199.2, + "explanation": "24% ≈ 25%. 25% of 830 = 207. Compare to 167." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "18% of 830 vs 167", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 18% of 830 greater than 167?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 149.4, + "explanation": "18% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 830 vs 167", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 830 greater than 167?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 307.1, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000272", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "23% of 740 vs 223", + "ns_shortcut": "Use 25% as landmark: 25% of 740 = 185.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 23% of 740 greater than 223?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 170.2, + "explanation": "23% ≈ 25%. 25% of 740 = 185. Compare to 223." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "32% of 740 vs 223", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 32% of 740 greater than 223?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 236.8, + "explanation": "32% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 740 vs 223", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 740 greater than 223?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 614.2, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000273", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "27% of 420 vs 172", + "ns_shortcut": "Use 25% as landmark: 25% of 420 = 105.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 27% of 420 greater than 172?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 113.4, + "explanation": "27% ≈ 25%. 25% of 420 = 105. Compare to 172." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "29% of 420 vs 172", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 29% of 420 greater than 172?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 121.8, + "explanation": "29% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 420 vs 172", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 420 greater than 172?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 281.4, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000274", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24% of 680 vs 310", + "ns_shortcut": "Use 25% as landmark: 25% of 680 = 170.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 24% of 680 greater than 310?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 163.2, + "explanation": "24% ≈ 25%. 25% of 680 = 170. Compare to 310." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "20% of 680 vs 310", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 20% of 680 greater than 310?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 136.0, + "explanation": "20% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 680 vs 310", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 680 greater than 310?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 251.6, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000275", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49% of 580 vs 278", + "ns_shortcut": "Use 50% as landmark: 50% of 580 = 290.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 49% of 580 greater than 278?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 284.2, + "explanation": "49% ≈ 50%. 50% of 580 = 290. Compare to 278." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "55% of 580 vs 278", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 55% of 580 greater than 278?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 319.0, + "explanation": "55% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 580 vs 278", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 580 greater than 278?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 481.4, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000276", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 780 vs 158", + "ns_shortcut": "Use 10% as landmark: 10% of 780 = 78.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 780 greater than 158?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 78.0, + "explanation": "10% ≈ 10%. 10% of 780 = 78. Compare to 158." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "13% of 780 vs 158", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 13% of 780 greater than 158?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 101.4, + "explanation": "13% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 780 vs 158", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 780 greater than 158?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 569.4, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000277", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49% of 960 vs 367", + "ns_shortcut": "Use 50% as landmark: 50% of 960 = 480.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 49% of 960 greater than 367?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 470.4, + "explanation": "49% ≈ 50%. 50% of 960 = 480. Compare to 367." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "55% of 960 vs 367", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 55% of 960 greater than 367?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 528.0, + "explanation": "55% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 960 vs 367", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 960 greater than 367?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 412.8, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000278", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49% of 830 vs 234", + "ns_shortcut": "Use 50% as landmark: 50% of 830 = 415.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 49% of 830 greater than 234?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 406.7, + "explanation": "49% ≈ 50%. 50% of 830 = 415. Compare to 234." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "46% of 830 vs 234", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 46% of 830 greater than 234?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 381.8, + "explanation": "46% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 830 vs 234", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 830 greater than 234?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 356.9, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000279", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50% of 430 vs 138", + "ns_shortcut": "Use 50% as landmark: 50% of 430 = 215.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 50% of 430 greater than 138?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 215.0, + "explanation": "50% ≈ 50%. 50% of 430 = 215. Compare to 138." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "54% of 430 vs 138", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 54% of 430 greater than 138?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 232.2, + "explanation": "54% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 430 vs 138", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 430 greater than 138?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 159.1, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000280", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12% of 250 vs 60", + "ns_shortcut": "Use 10% as landmark: 10% of 250 = 25.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 12% of 250 greater than 60?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 30.0, + "explanation": "12% ≈ 10%. 10% of 250 = 25. Compare to 60." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3% of 250 vs 60", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 3% of 250 greater than 60?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 7.5, + "explanation": "3% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 250 vs 60", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 250 greater than 60?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 92.5, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000281", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "11% of 710 vs 237", + "ns_shortcut": "Use 10% as landmark: 10% of 710 = 71.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 11% of 710 greater than 237?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 78.1, + "explanation": "11% ≈ 10%. 10% of 710 = 71. Compare to 237." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "16% of 710 vs 237", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 16% of 710 greater than 237?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 113.6, + "explanation": "16% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 710 vs 237", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 710 greater than 237?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 305.3, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000282", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "23% of 750 vs 221", + "ns_shortcut": "Use 25% as landmark: 25% of 750 = 187.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 23% of 750 greater than 221?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 172.5, + "explanation": "23% ≈ 25%. 25% of 750 = 187. Compare to 221." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "22% of 750 vs 221", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 22% of 750 greater than 221?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 165.0, + "explanation": "22% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 750 vs 221", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 750 greater than 221?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 622.5, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000283", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8% of 640 vs 250", + "ns_shortcut": "Use 10% as landmark: 10% of 640 = 64.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 8% of 640 greater than 250?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 51.2, + "explanation": "8% ≈ 10%. 10% of 640 = 64. Compare to 250." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "16% of 640 vs 250", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 16% of 640 greater than 250?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 102.4, + "explanation": "16% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 640 vs 250", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 640 greater than 250?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 275.2, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000284", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "11% of 680 vs 166", + "ns_shortcut": "Use 10% as landmark: 10% of 680 = 68.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 11% of 680 greater than 166?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 74.8, + "explanation": "11% ≈ 10%. 10% of 680 = 68. Compare to 166." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7% of 680 vs 166", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 7% of 680 greater than 166?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 47.6, + "explanation": "7% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 680 vs 166", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 680 greater than 166?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 292.4, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000285", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51% of 580 vs 264", + "ns_shortcut": "Use 50% as landmark: 50% of 580 = 290.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 51% of 580 greater than 264?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 295.8, + "explanation": "51% ≈ 50%. 50% of 580 = 290. Compare to 264." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "54% of 580 vs 264", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 54% of 580 greater than 264?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 313.2, + "explanation": "54% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 580 vs 264", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 580 greater than 264?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 481.4, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000286", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "27% of 430 vs 146", + "ns_shortcut": "Use 25% as landmark: 25% of 430 = 107.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 27% of 430 greater than 146?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 116.1, + "explanation": "27% ≈ 25%. 25% of 430 = 107. Compare to 146." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "30% of 430 vs 146", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 30% of 430 greater than 146?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 129.0, + "explanation": "30% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 430 vs 146", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 430 greater than 146?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 184.9, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000287", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "52% of 730 vs 254", + "ns_shortcut": "Use 50% as landmark: 50% of 730 = 365.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 52% of 730 greater than 254?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 379.6, + "explanation": "52% ≈ 50%. 50% of 730 = 365. Compare to 254." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "55% of 730 vs 254", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 55% of 730 greater than 254?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 401.5, + "explanation": "55% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 730 vs 254", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 730 greater than 254?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 532.9, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000288", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "48% of 450 vs 205", + "ns_shortcut": "Use 50% as landmark: 50% of 450 = 225.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 48% of 450 greater than 205?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 216.0, + "explanation": "48% ≈ 50%. 50% of 450 = 225. Compare to 205." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "47% of 450 vs 205", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 47% of 450 greater than 205?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 211.5, + "explanation": "47% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 450 vs 205", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 450 greater than 205?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 301.5, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000289", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 400 vs 151", + "ns_shortcut": "Use 25% as landmark: 25% of 400 = 100.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 400 greater than 151?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 100.0, + "explanation": "25% ≈ 25%. 25% of 400 = 100. Compare to 151." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "21% of 400 vs 151", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 21% of 400 greater than 151?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 84.0, + "explanation": "21% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 400 vs 151", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 400 greater than 151?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 292.0, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000290", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 110 vs 33", + "ns_shortcut": "Use 25% as landmark: 25% of 110 = 27.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 110 greater than 33?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 27.5, + "explanation": "25% ≈ 25%. 25% of 110 = 27. Compare to 33." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "31% of 110 vs 33", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 31% of 110 greater than 33?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 34.1, + "explanation": "31% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 110 vs 33", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 110 greater than 33?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 73.7, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000291", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9% of 420 vs 155", + "ns_shortcut": "Use 10% as landmark: 10% of 420 = 42.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 9% of 420 greater than 155?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 37.8, + "explanation": "9% ≈ 10%. 10% of 420 = 42. Compare to 155." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6% of 420 vs 155", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 6% of 420 greater than 155?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 25.2, + "explanation": "6% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 420 vs 155", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 420 greater than 155?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 155.4, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000292", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "27% of 370 vs 178", + "ns_shortcut": "Use 25% as landmark: 25% of 370 = 92.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 27% of 370 greater than 178?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 99.9, + "explanation": "27% ≈ 25%. 25% of 370 = 92. Compare to 178." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "19% of 370 vs 178", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 19% of 370 greater than 178?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 70.3, + "explanation": "19% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 370 vs 178", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 370 greater than 178?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 247.9, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000293", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26% of 430 vs 178", + "ns_shortcut": "Use 25% as landmark: 25% of 430 = 107.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 26% of 430 greater than 178?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 111.8, + "explanation": "26% ≈ 25%. 25% of 430 = 107. Compare to 178." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "19% of 430 vs 178", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 19% of 430 greater than 178?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 81.7, + "explanation": "19% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 430 vs 178", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 430 greater than 178?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 159.1, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000294", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49% of 540 vs 263", + "ns_shortcut": "Use 50% as landmark: 50% of 540 = 270.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 49% of 540 greater than 263?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 264.6, + "explanation": "49% ≈ 50%. 50% of 540 = 270. Compare to 263." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "55% of 540 vs 263", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 55% of 540 greater than 263?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 297.0, + "explanation": "55% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 540 vs 263", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 540 greater than 263?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 448.2, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000295", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 720 vs 342", + "ns_shortcut": "Use 10% as landmark: 10% of 720 = 72.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 720 greater than 342?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 72.0, + "explanation": "10% ≈ 10%. 10% of 720 = 72. Compare to 342." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "17% of 720 vs 342", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 17% of 720 greater than 342?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 122.4, + "explanation": "17% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 720 vs 342", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 720 greater than 342?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 482.4, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000296", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "48% of 690 vs 299", + "ns_shortcut": "Use 50% as landmark: 50% of 690 = 345.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 48% of 690 greater than 299?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 331.2, + "explanation": "48% ≈ 50%. 50% of 690 = 345. Compare to 299." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "56% of 690 vs 299", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 56% of 690 greater than 299?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 386.4, + "explanation": "56% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 690 vs 299", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 690 greater than 299?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 572.7, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000297", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50% of 830 vs 269", + "ns_shortcut": "Use 50% as landmark: 50% of 830 = 415.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 50% of 830 greater than 269?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 415.0, + "explanation": "50% ≈ 50%. 50% of 830 = 415. Compare to 269." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "54% of 830 vs 269", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 54% of 830 greater than 269?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 448.2, + "explanation": "54% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 830 vs 269", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 830 greater than 269?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 556.1, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000298", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12% of 500 vs 174", + "ns_shortcut": "Use 10% as landmark: 10% of 500 = 50.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 12% of 500 greater than 174?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 60.0, + "explanation": "12% ≈ 10%. 10% of 500 = 50. Compare to 174." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "15% of 500 vs 174", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 15% of 500 greater than 174?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 75.0, + "explanation": "15% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 500 vs 174", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 500 greater than 174?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 365.0, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000299", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "11% of 560 vs 262", + "ns_shortcut": "Use 10% as landmark: 10% of 560 = 56.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 11% of 560 greater than 262?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 61.6, + "explanation": "11% ≈ 10%. 10% of 560 = 56. Compare to 262." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6% of 560 vs 262", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 6% of 560 greater than 262?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 33.6, + "explanation": "6% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 560 vs 262", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 560 greater than 262?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 207.2, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000300", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12% of 260 vs 82", + "ns_shortcut": "Use 10% as landmark: 10% of 260 = 26.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 12% of 260 greater than 82?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 31.2, + "explanation": "12% ≈ 10%. 10% of 260 = 26. Compare to 82." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6% of 260 vs 82", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 6% of 260 greater than 82?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 15.6, + "explanation": "6% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 260 vs 82", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 260 greater than 82?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 111.8, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "equation_reasoning_000301", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "23 + 22 = ___ + 18", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 23 + 22 = ___ + 18", + "options": [ + "37", + "27", + "47", + "17" + ], + "answer": "27", + "correct_value": 27, + "explanation": "Blank = A + B − C = 23 + 22 − 18 = 27." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "13 + 60 + ___ = 60 + 77 + 20", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 13 + 60 + ___ = 60 + 77 + 20", + "options": [ + "104", + "74", + "94", + "84" + ], + "answer": "84", + "correct_value": 84, + "explanation": "B cancels; blank = C + D − A = 77 + 20 − 13 = 84." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "20 + 93 + ___ = 59 + 49", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 20 + 93 + ___ = 59 + 49", + "options": [ + "15", + "-15", + "-5", + "5" + ], + "answer": "-5", + "correct_value": -5, + "explanation": "No matching terms; blank = 59 + 49 − 20 − 93 = -5." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000302", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "90 + 26 = ___ + 26", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 90 + 26 = ___ + 26", + "options": [ + "80", + "110", + "90", + "100" + ], + "answer": "90", + "correct_value": 90, + "explanation": "Blank = A + B − C = 90 + 26 − 26 = 90." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "94 + 55 + ___ = 55 + 51 + 70", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 94 + 55 + ___ = 55 + 51 + 70", + "options": [ + "27", + "37", + "47", + "17" + ], + "answer": "27", + "correct_value": 27, + "explanation": "B cancels; blank = C + D − A = 51 + 70 − 94 = 27." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "80 + 66 + ___ = 27 + 99", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 80 + 66 + ___ = 27 + 99", + "options": [ + "-20", + "-30", + "0", + "-10" + ], + "answer": "-20", + "correct_value": -20, + "explanation": "No matching terms; blank = 27 + 99 − 80 − 66 = -20." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000303", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "77 + 91 = ___ + 91", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 77 + 91 = ___ + 91", + "options": [ + "77", + "67", + "87", + "97" + ], + "answer": "77", + "correct_value": 77, + "explanation": "Blank = A + B − C = 77 + 91 − 91 = 77." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "79 + 39 + ___ = 39 + 21 + 66", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 79 + 39 + ___ = 39 + 21 + 66", + "options": [ + "8", + "-2", + "18", + "28" + ], + "answer": "8", + "correct_value": 8, + "explanation": "B cancels; blank = C + D − A = 21 + 66 − 79 = 8." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "43 + 29 + ___ = 48 + 11", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 43 + 29 + ___ = 48 + 11", + "options": [ + "-7", + "-27", + "3", + "-13" + ], + "answer": "-13", + "correct_value": -13, + "explanation": "No matching terms; blank = 48 + 11 − 43 − 29 = -13." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000304", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "46 × (70 + 78) = 46 × ___ + 46 × 78", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 46 × (70 + 78) = 46 × ___ + 46 × 78", + "options": [ + "60", + "80", + "90", + "70" + ], + "answer": "70", + "correct_value": 70, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 70." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "71 + 48 + ___ = 48 + 11 + 65", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 71 + 48 + ___ = 48 + 11 + 65", + "options": [ + "15", + "5", + "-5", + "25" + ], + "answer": "5", + "correct_value": 5, + "explanation": "B cancels; blank = C + D − A = 11 + 65 − 71 = 5." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "10 + 80 + ___ = 57 + 34", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 10 + 80 + ___ = 57 + 34", + "options": [ + "1", + "11", + "21", + "-9" + ], + "answer": "1", + "correct_value": 1, + "explanation": "No matching terms; blank = 57 + 34 − 10 − 80 = 1." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000305", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "41 + 30 = ___ + 92", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 41 + 30 = ___ + 92", + "options": [ + "-9", + "-21", + "-19", + "-39" + ], + "answer": "-21", + "correct_value": -21, + "explanation": "Blank = A + B − C = 41 + 30 − 92 = -21." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "43 + 20 + ___ = 20 + 64 + 90", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 43 + 20 + ___ = 20 + 64 + 90", + "options": [ + "111", + "101", + "131", + "121" + ], + "answer": "111", + "correct_value": 111, + "explanation": "B cancels; blank = C + D − A = 64 + 90 − 43 = 111." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "46 + 93 + ___ = 78 + 44", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 46 + 93 + ___ = 78 + 44", + "options": [ + "-13", + "-3", + "-17", + "-33" + ], + "answer": "-17", + "correct_value": -17, + "explanation": "No matching terms; blank = 78 + 44 − 46 − 93 = -17." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000306", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "27 × 47 × 24 = ___ × 24 × 27", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 27 × 47 × 24 = ___ × 24 × 27", + "options": [ + "47", + "57", + "67", + "37" + ], + "answer": "47", + "correct_value": 47, + "explanation": "A and C appear on both sides; blank = B = 47." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "44 + 99 + ___ = 99 + 89 + 72", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 44 + 99 + ___ = 99 + 89 + 72", + "options": [ + "137", + "117", + "107", + "127" + ], + "answer": "117", + "correct_value": 117, + "explanation": "B cancels; blank = C + D − A = 89 + 72 − 44 = 117." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "29 + 16 + ___ = 78 + 12", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 29 + 16 + ___ = 78 + 12", + "options": [ + "65", + "35", + "45", + "55" + ], + "answer": "45", + "correct_value": 45, + "explanation": "No matching terms; blank = 78 + 12 − 29 − 16 = 45." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000307", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26 × (66 + 11) = 26 × ___ + 26 × 11", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 26 × (66 + 11) = 26 × ___ + 26 × 11", + "options": [ + "56", + "86", + "66", + "76" + ], + "answer": "66", + "correct_value": 66, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 66." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "11 + 73 + ___ = 73 + 37 + 53", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 11 + 73 + ___ = 73 + 37 + 53", + "options": [ + "79", + "99", + "89", + "69" + ], + "answer": "79", + "correct_value": 79, + "explanation": "B cancels; blank = C + D − A = 37 + 53 − 11 = 79." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "57 + 69 + ___ = 99 + 14", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 57 + 69 + ___ = 99 + 14", + "options": [ + "-7", + "-27", + "-13", + "3" + ], + "answer": "-13", + "correct_value": -13, + "explanation": "No matching terms; blank = 99 + 14 − 57 − 69 = -13." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000308", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "44 + 60 + ___ = 60 + 19 + 44", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 44 + 60 + ___ = 60 + 19 + 44", + "options": [ + "29", + "39", + "19", + "9" + ], + "answer": "19", + "correct_value": 19, + "explanation": "A and B appear on both sides and cancel; blank = C = 19." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "42 + 58 + ___ = 58 + 11 + 25", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 42 + 58 + ___ = 58 + 11 + 25", + "options": [ + "-6", + "6", + "-24", + "-4" + ], + "answer": "-6", + "correct_value": -6, + "explanation": "B cancels; blank = C + D − A = 11 + 25 − 42 = -6." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "41 + 14 + ___ = 35 + 68", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 41 + 14 + ___ = 35 + 68", + "options": [ + "58", + "38", + "68", + "48" + ], + "answer": "48", + "correct_value": 48, + "explanation": "No matching terms; blank = 35 + 68 − 41 − 14 = 48." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000309", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "59 + 97 = ___ + 49", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 59 + 97 = ___ + 49", + "options": [ + "117", + "97", + "107", + "127" + ], + "answer": "107", + "correct_value": 107, + "explanation": "Blank = A + B − C = 59 + 97 − 49 = 107." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "80 + 76 + ___ = 76 + 25 + 42", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 80 + 76 + ___ = 76 + 25 + 42", + "options": [ + "-13", + "-7", + "3", + "-27" + ], + "answer": "-13", + "correct_value": -13, + "explanation": "B cancels; blank = C + D − A = 25 + 42 − 80 = -13." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "35 + 56 + ___ = 16 + 95", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 35 + 56 + ___ = 16 + 95", + "options": [ + "10", + "40", + "20", + "30" + ], + "answer": "20", + "correct_value": 20, + "explanation": "No matching terms; blank = 16 + 95 − 35 − 56 = 20." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000310", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51 + 11 = ___ + 31", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 51 + 11 = ___ + 31", + "options": [ + "41", + "31", + "21", + "51" + ], + "answer": "31", + "correct_value": 31, + "explanation": "Blank = A + B − C = 51 + 11 − 31 = 31." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "60 + 41 + ___ = 41 + 11 + 15", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 60 + 41 + ___ = 41 + 11 + 15", + "options": [ + "-26", + "-34", + "-16", + "-46" + ], + "answer": "-34", + "correct_value": -34, + "explanation": "B cancels; blank = C + D − A = 11 + 15 − 60 = -34." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "57 + 21 + ___ = 41 + 30", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 57 + 21 + ___ = 41 + 30", + "options": [ + "7", + "-3", + "-7", + "-23" + ], + "answer": "-7", + "correct_value": -7, + "explanation": "No matching terms; blank = 41 + 30 − 57 − 21 = -7." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000311", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "17 + 55 + ___ = 55 + 92 + 17", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 17 + 55 + ___ = 55 + 92 + 17", + "options": [ + "82", + "92", + "112", + "102" + ], + "answer": "92", + "correct_value": 92, + "explanation": "A and B appear on both sides and cancel; blank = C = 92." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "26 + 72 + ___ = 72 + 39 + 50", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 26 + 72 + ___ = 72 + 39 + 50", + "options": [ + "53", + "63", + "73", + "83" + ], + "answer": "63", + "correct_value": 63, + "explanation": "B cancels; blank = C + D − A = 39 + 50 − 26 = 63." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "12 + 98 + ___ = 58 + 55", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 12 + 98 + ___ = 58 + 55", + "options": [ + "23", + "3", + "-7", + "13" + ], + "answer": "3", + "correct_value": 3, + "explanation": "No matching terms; blank = 58 + 55 − 12 − 98 = 3." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000312", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "66 + 81 = ___ + 47", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 66 + 81 = ___ + 47", + "options": [ + "110", + "120", + "100", + "90" + ], + "answer": "100", + "correct_value": 100, + "explanation": "Blank = A + B − C = 66 + 81 − 47 = 100." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "79 + 54 + ___ = 54 + 51 + 69", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 79 + 54 + ___ = 54 + 51 + 69", + "options": [ + "51", + "41", + "31", + "61" + ], + "answer": "41", + "correct_value": 41, + "explanation": "B cancels; blank = C + D − A = 51 + 69 − 79 = 41." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "18 + 77 + ___ = 29 + 67", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 18 + 77 + ___ = 29 + 67", + "options": [ + "1", + "-9", + "11", + "21" + ], + "answer": "1", + "correct_value": 1, + "explanation": "No matching terms; blank = 29 + 67 − 18 − 77 = 1." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000313", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10 + 91 = ___ + 74", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 10 + 91 = ___ + 74", + "options": [ + "37", + "27", + "47", + "17" + ], + "answer": "27", + "correct_value": 27, + "explanation": "Blank = A + B − C = 10 + 91 − 74 = 27." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "12 + 89 + ___ = 89 + 56 + 89", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 12 + 89 + ___ = 89 + 56 + 89", + "options": [ + "153", + "133", + "123", + "143" + ], + "answer": "133", + "correct_value": 133, + "explanation": "B cancels; blank = C + D − A = 56 + 89 − 12 = 133." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "94 + 78 + ___ = 59 + 34", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 94 + 78 + ___ = 59 + 34", + "options": [ + "-71", + "-79", + "-61", + "-91" + ], + "answer": "-79", + "correct_value": -79, + "explanation": "No matching terms; blank = 59 + 34 − 94 − 78 = -79." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000314", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51 × (62 + 92) = 51 × ___ + 51 × 92", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 51 × (62 + 92) = 51 × ___ + 51 × 92", + "options": [ + "72", + "52", + "62", + "82" + ], + "answer": "62", + "correct_value": 62, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 62." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "76 + 94 + ___ = 94 + 18 + 15", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 76 + 94 + ___ = 94 + 18 + 15", + "options": [ + "-57", + "-27", + "-43", + "-37" + ], + "answer": "-43", + "correct_value": -43, + "explanation": "B cancels; blank = C + D − A = 18 + 15 − 76 = -43." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "35 + 23 + ___ = 72 + 34", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 35 + 23 + ___ = 72 + 34", + "options": [ + "68", + "58", + "38", + "48" + ], + "answer": "48", + "correct_value": 48, + "explanation": "No matching terms; blank = 72 + 34 − 35 − 23 = 48." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000315", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "77 + 89 = ___ + 99", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 77 + 89 = ___ + 99", + "options": [ + "57", + "77", + "67", + "87" + ], + "answer": "67", + "correct_value": 67, + "explanation": "Blank = A + B − C = 77 + 89 − 99 = 67." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "38 + 87 + ___ = 87 + 19 + 11", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 38 + 87 + ___ = 87 + 19 + 11", + "options": [ + "-8", + "-22", + "8", + "-2" + ], + "answer": "-8", + "correct_value": -8, + "explanation": "B cancels; blank = C + D − A = 19 + 11 − 38 = -8." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "20 + 69 + ___ = 91 + 17", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 20 + 69 + ___ = 91 + 17", + "options": [ + "19", + "29", + "9", + "39" + ], + "answer": "19", + "correct_value": 19, + "explanation": "No matching terms; blank = 91 + 17 − 20 − 69 = 19." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000316", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "17 × (88 + 44) = 17 × ___ + 17 × 44", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 17 × (88 + 44) = 17 × ___ + 17 × 44", + "options": [ + "98", + "78", + "108", + "88" + ], + "answer": "88", + "correct_value": 88, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 88." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "48 + 19 + ___ = 19 + 37 + 76", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 48 + 19 + ___ = 19 + 37 + 76", + "options": [ + "75", + "85", + "55", + "65" + ], + "answer": "65", + "correct_value": 65, + "explanation": "B cancels; blank = C + D − A = 37 + 76 − 48 = 65." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "78 + 59 + ___ = 34 + 34", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 78 + 59 + ___ = 34 + 34", + "options": [ + "-81", + "-69", + "-51", + "-61" + ], + "answer": "-69", + "correct_value": -69, + "explanation": "No matching terms; blank = 34 + 34 − 78 − 59 = -69." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000317", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "23 + 91 = ___ + 83", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 23 + 91 = ___ + 83", + "options": [ + "31", + "41", + "21", + "51" + ], + "answer": "31", + "correct_value": 31, + "explanation": "Blank = A + B − C = 23 + 91 − 83 = 31." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "99 + 95 + ___ = 95 + 64 + 23", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 99 + 95 + ___ = 95 + 64 + 23", + "options": [ + "2", + "-28", + "-12", + "-8" + ], + "answer": "-12", + "correct_value": -12, + "explanation": "B cancels; blank = C + D − A = 64 + 23 − 99 = -12." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "74 + 53 + ___ = 94 + 30", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 74 + 53 + ___ = 94 + 30", + "options": [ + "-3", + "13", + "3", + "-17" + ], + "answer": "-3", + "correct_value": -3, + "explanation": "No matching terms; blank = 94 + 30 − 74 − 53 = -3." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000318", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "57 × (62 + 91) = 57 × ___ + 57 × 91", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 57 × (62 + 91) = 57 × ___ + 57 × 91", + "options": [ + "62", + "72", + "82", + "52" + ], + "answer": "62", + "correct_value": 62, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 62." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "87 + 40 + ___ = 40 + 82 + 79", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 87 + 40 + ___ = 40 + 82 + 79", + "options": [ + "74", + "94", + "64", + "84" + ], + "answer": "74", + "correct_value": 74, + "explanation": "B cancels; blank = C + D − A = 82 + 79 − 87 = 74." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "56 + 67 + ___ = 67 + 60", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 56 + 67 + ___ = 67 + 60", + "options": [ + "14", + "-6", + "24", + "4" + ], + "answer": "4", + "correct_value": 4, + "explanation": "No matching terms; blank = 67 + 60 − 56 − 67 = 4." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000319", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "86 × (12 + 76) = 86 × ___ + 86 × 76", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 86 × (12 + 76) = 86 × ___ + 86 × 76", + "options": [ + "32", + "12", + "2", + "22" + ], + "answer": "12", + "correct_value": 12, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 12." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "48 + 38 + ___ = 38 + 37 + 50", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 48 + 38 + ___ = 38 + 37 + 50", + "options": [ + "29", + "49", + "59", + "39" + ], + "answer": "39", + "correct_value": 39, + "explanation": "B cancels; blank = C + D − A = 37 + 50 − 48 = 39." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "80 + 75 + ___ = 81 + 83", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 80 + 75 + ___ = 81 + 83", + "options": [ + "-1", + "29", + "9", + "19" + ], + "answer": "9", + "correct_value": 9, + "explanation": "No matching terms; blank = 81 + 83 − 80 − 75 = 9." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000320", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "55 + 29 = ___ + 27", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 55 + 29 = ___ + 27", + "options": [ + "47", + "57", + "67", + "77" + ], + "answer": "57", + "correct_value": 57, + "explanation": "Blank = A + B − C = 55 + 29 − 27 = 57." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "39 + 42 + ___ = 42 + 48 + 21", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 39 + 42 + ___ = 42 + 48 + 21", + "options": [ + "40", + "30", + "50", + "20" + ], + "answer": "30", + "correct_value": 30, + "explanation": "B cancels; blank = C + D − A = 48 + 21 − 39 = 30." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "19 + 51 + ___ = 71 + 32", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 19 + 51 + ___ = 71 + 32", + "options": [ + "43", + "33", + "53", + "23" + ], + "answer": "33", + "correct_value": 33, + "explanation": "No matching terms; blank = 71 + 32 − 19 − 51 = 33." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000321", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "14 × (83 + 58) = 14 × ___ + 14 × 58", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 14 × (83 + 58) = 14 × ___ + 14 × 58", + "options": [ + "103", + "93", + "73", + "83" + ], + "answer": "83", + "correct_value": 83, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 83." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "54 + 65 + ___ = 65 + 63 + 25", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 54 + 65 + ___ = 65 + 63 + 25", + "options": [ + "24", + "34", + "54", + "44" + ], + "answer": "34", + "correct_value": 34, + "explanation": "B cancels; blank = C + D − A = 63 + 25 − 54 = 34." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "91 + 90 + ___ = 58 + 16", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 91 + 90 + ___ = 58 + 16", + "options": [ + "-103", + "-107", + "-123", + "-93" + ], + "answer": "-107", + "correct_value": -107, + "explanation": "No matching terms; blank = 58 + 16 − 91 − 90 = -107." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000322", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "79 × (82 + 29) = 79 × ___ + 79 × 29", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 79 × (82 + 29) = 79 × ___ + 79 × 29", + "options": [ + "92", + "82", + "72", + "102" + ], + "answer": "82", + "correct_value": 82, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 82." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "29 + 16 + ___ = 16 + 96 + 60", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 29 + 16 + ___ = 16 + 96 + 60", + "options": [ + "137", + "127", + "147", + "117" + ], + "answer": "127", + "correct_value": 127, + "explanation": "B cancels; blank = C + D − A = 96 + 60 − 29 = 127." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "38 + 65 + ___ = 47 + 60", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 38 + 65 + ___ = 47 + 60", + "options": [ + "24", + "-6", + "4", + "14" + ], + "answer": "4", + "correct_value": 4, + "explanation": "No matching terms; blank = 47 + 60 − 38 − 65 = 4." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000323", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "70 + 83 + ___ = 83 + 16 + 70", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 70 + 83 + ___ = 83 + 16 + 70", + "options": [ + "36", + "16", + "26", + "6" + ], + "answer": "16", + "correct_value": 16, + "explanation": "A and B appear on both sides and cancel; blank = C = 16." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "61 + 74 + ___ = 74 + 65 + 99", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 61 + 74 + ___ = 74 + 65 + 99", + "options": [ + "123", + "93", + "103", + "113" + ], + "answer": "103", + "correct_value": 103, + "explanation": "B cancels; blank = C + D − A = 65 + 99 − 61 = 103." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "81 + 35 + ___ = 72 + 60", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 81 + 35 + ___ = 72 + 60", + "options": [ + "6", + "16", + "36", + "26" + ], + "answer": "16", + "correct_value": 16, + "explanation": "No matching terms; blank = 72 + 60 − 81 − 35 = 16." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000324", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "19 × 44 × 65 = ___ × 65 × 19", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 19 × 44 × 65 = ___ × 65 × 19", + "options": [ + "54", + "44", + "34", + "64" + ], + "answer": "44", + "correct_value": 44, + "explanation": "A and C appear on both sides; blank = B = 44." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "24 + 95 + ___ = 95 + 62 + 64", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 24 + 95 + ___ = 95 + 62 + 64", + "options": [ + "92", + "102", + "112", + "122" + ], + "answer": "102", + "correct_value": 102, + "explanation": "B cancels; blank = C + D − A = 62 + 64 − 24 = 102." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "51 + 54 + ___ = 86 + 93", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 51 + 54 + ___ = 86 + 93", + "options": [ + "94", + "64", + "84", + "74" + ], + "answer": "74", + "correct_value": 74, + "explanation": "No matching terms; blank = 86 + 93 − 51 − 54 = 74." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000325", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "88 × (66 + 89) = 88 × ___ + 88 × 89", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 88 × (66 + 89) = 88 × ___ + 88 × 89", + "options": [ + "86", + "66", + "56", + "76" + ], + "answer": "66", + "correct_value": 66, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 66." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "39 + 35 + ___ = 35 + 97 + 57", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 39 + 35 + ___ = 35 + 97 + 57", + "options": [ + "105", + "115", + "135", + "125" + ], + "answer": "115", + "correct_value": 115, + "explanation": "B cancels; blank = C + D − A = 97 + 57 − 39 = 115." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "24 + 16 + ___ = 41 + 89", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 24 + 16 + ___ = 41 + 89", + "options": [ + "110", + "90", + "80", + "100" + ], + "answer": "90", + "correct_value": 90, + "explanation": "No matching terms; blank = 41 + 89 − 24 − 16 = 90." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000326", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 × (24 + 49) = 98 × ___ + 98 × 49", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 98 × (24 + 49) = 98 × ___ + 98 × 49", + "options": [ + "44", + "24", + "34", + "14" + ], + "answer": "24", + "correct_value": 24, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 24." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "52 + 36 + ___ = 36 + 78 + 37", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 52 + 36 + ___ = 36 + 78 + 37", + "options": [ + "63", + "53", + "83", + "73" + ], + "answer": "63", + "correct_value": 63, + "explanation": "B cancels; blank = C + D − A = 78 + 37 − 52 = 63." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "80 + 71 + ___ = 89 + 28", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 80 + 71 + ___ = 89 + 28", + "options": [ + "-16", + "-46", + "-26", + "-34" + ], + "answer": "-34", + "correct_value": -34, + "explanation": "No matching terms; blank = 89 + 28 − 80 − 71 = -34." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000327", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "31 × (17 + 46) = 31 × ___ + 31 × 46", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 31 × (17 + 46) = 31 × ___ + 31 × 46", + "options": [ + "17", + "7", + "37", + "27" + ], + "answer": "17", + "correct_value": 17, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 17." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "16 + 69 + ___ = 69 + 93 + 11", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 16 + 69 + ___ = 69 + 93 + 11", + "options": [ + "78", + "108", + "98", + "88" + ], + "answer": "88", + "correct_value": 88, + "explanation": "B cancels; blank = C + D − A = 93 + 11 − 16 = 88." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "94 + 94 + ___ = 56 + 23", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 94 + 94 + ___ = 56 + 23", + "options": [ + "-121", + "-91", + "-101", + "-109" + ], + "answer": "-109", + "correct_value": -109, + "explanation": "No matching terms; blank = 56 + 23 − 94 − 94 = -109." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000328", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "58 × 75 × 22 = ___ × 22 × 58", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 58 × 75 × 22 = ___ × 22 × 58", + "options": [ + "65", + "75", + "85", + "95" + ], + "answer": "75", + "correct_value": 75, + "explanation": "A and C appear on both sides; blank = B = 75." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "33 + 35 + ___ = 35 + 74 + 68", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 33 + 35 + ___ = 35 + 74 + 68", + "options": [ + "119", + "129", + "99", + "109" + ], + "answer": "109", + "correct_value": 109, + "explanation": "B cancels; blank = C + D − A = 74 + 68 − 33 = 109." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "77 + 49 + ___ = 52 + 72", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 77 + 49 + ___ = 52 + 72", + "options": [ + "2", + "-2", + "12", + "-18" + ], + "answer": "-2", + "correct_value": -2, + "explanation": "No matching terms; blank = 52 + 72 − 77 − 49 = -2." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000329", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "54 + 97 + ___ = 97 + 55 + 54", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 54 + 97 + ___ = 97 + 55 + 54", + "options": [ + "55", + "65", + "45", + "75" + ], + "answer": "55", + "correct_value": 55, + "explanation": "A and B appear on both sides and cancel; blank = C = 55." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "40 + 74 + ___ = 74 + 45 + 17", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 40 + 74 + ___ = 74 + 45 + 17", + "options": [ + "12", + "22", + "32", + "42" + ], + "answer": "22", + "correct_value": 22, + "explanation": "B cancels; blank = C + D − A = 45 + 17 − 40 = 22." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "81 + 45 + ___ = 38 + 29", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 81 + 45 + ___ = 38 + 29", + "options": [ + "-71", + "-59", + "-51", + "-41" + ], + "answer": "-59", + "correct_value": -59, + "explanation": "No matching terms; blank = 38 + 29 − 81 − 45 = -59." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000330", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "58 + 76 + ___ = 76 + 96 + 58", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 58 + 76 + ___ = 76 + 96 + 58", + "options": [ + "116", + "96", + "86", + "106" + ], + "answer": "96", + "correct_value": 96, + "explanation": "A and B appear on both sides and cancel; blank = C = 96." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "83 + 29 + ___ = 29 + 64 + 49", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 83 + 29 + ___ = 29 + 64 + 49", + "options": [ + "20", + "30", + "40", + "50" + ], + "answer": "30", + "correct_value": 30, + "explanation": "B cancels; blank = C + D − A = 64 + 49 − 83 = 30." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "36 + 93 + ___ = 27 + 84", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 36 + 93 + ___ = 27 + 84", + "options": [ + "-2", + "-32", + "-18", + "-12" + ], + "answer": "-18", + "correct_value": -18, + "explanation": "No matching terms; blank = 27 + 84 − 36 − 93 = -18." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000331", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "65 × 28 × 93 = ___ × 93 × 65", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 65 × 28 × 93 = ___ × 93 × 65", + "options": [ + "48", + "28", + "18", + "38" + ], + "answer": "28", + "correct_value": 28, + "explanation": "A and C appear on both sides; blank = B = 28." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "81 + 78 + ___ = 78 + 25 + 33", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 81 + 78 + ___ = 78 + 25 + 33", + "options": [ + "-17", + "-37", + "-23", + "-7" + ], + "answer": "-23", + "correct_value": -23, + "explanation": "B cancels; blank = C + D − A = 25 + 33 − 81 = -23." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "50 + 62 + ___ = 62 + 48", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 50 + 62 + ___ = 62 + 48", + "options": [ + "-2", + "2", + "12", + "-18" + ], + "answer": "-2", + "correct_value": -2, + "explanation": "No matching terms; blank = 62 + 48 − 50 − 62 = -2." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000332", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "34 × 23 × 26 = ___ × 26 × 34", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 34 × 23 × 26 = ___ × 26 × 34", + "options": [ + "43", + "13", + "23", + "33" + ], + "answer": "23", + "correct_value": 23, + "explanation": "A and C appear on both sides; blank = B = 23." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "87 + 82 + ___ = 82 + 88 + 39", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 87 + 82 + ___ = 82 + 88 + 39", + "options": [ + "30", + "60", + "40", + "50" + ], + "answer": "40", + "correct_value": 40, + "explanation": "B cancels; blank = C + D − A = 88 + 39 − 87 = 40." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "55 + 20 + ___ = 98 + 71", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 55 + 20 + ___ = 98 + 71", + "options": [ + "94", + "84", + "114", + "104" + ], + "answer": "94", + "correct_value": 94, + "explanation": "No matching terms; blank = 98 + 71 − 55 − 20 = 94." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000333", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "33 × (59 + 72) = 33 × ___ + 33 × 72", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 33 × (59 + 72) = 33 × ___ + 33 × 72", + "options": [ + "69", + "79", + "59", + "49" + ], + "answer": "59", + "correct_value": 59, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 59." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "34 + 38 + ___ = 38 + 14 + 62", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 34 + 38 + ___ = 38 + 14 + 62", + "options": [ + "32", + "62", + "52", + "42" + ], + "answer": "42", + "correct_value": 42, + "explanation": "B cancels; blank = C + D − A = 14 + 62 − 34 = 42." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "38 + 95 + ___ = 35 + 36", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 38 + 95 + ___ = 35 + 36", + "options": [ + "-48", + "-58", + "-78", + "-62" + ], + "answer": "-62", + "correct_value": -62, + "explanation": "No matching terms; blank = 35 + 36 − 38 − 95 = -62." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000334", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 + 51 = ___ + 68", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 98 + 51 = ___ + 68", + "options": [ + "81", + "101", + "71", + "91" + ], + "answer": "81", + "correct_value": 81, + "explanation": "Blank = A + B − C = 98 + 51 − 68 = 81." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "15 + 87 + ___ = 87 + 58 + 50", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 15 + 87 + ___ = 87 + 58 + 50", + "options": [ + "103", + "93", + "113", + "83" + ], + "answer": "93", + "correct_value": 93, + "explanation": "B cancels; blank = C + D − A = 58 + 50 − 15 = 93." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "76 + 23 + ___ = 67 + 68", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 76 + 23 + ___ = 67 + 68", + "options": [ + "56", + "46", + "26", + "36" + ], + "answer": "36", + "correct_value": 36, + "explanation": "No matching terms; blank = 67 + 68 − 76 − 23 = 36." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000335", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "81 + 59 + ___ = 59 + 76 + 81", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 81 + 59 + ___ = 59 + 76 + 81", + "options": [ + "96", + "86", + "66", + "76" + ], + "answer": "76", + "correct_value": 76, + "explanation": "A and B appear on both sides and cancel; blank = C = 76." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "66 + 60 + ___ = 60 + 48 + 46", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 66 + 60 + ___ = 60 + 48 + 46", + "options": [ + "48", + "38", + "28", + "18" + ], + "answer": "28", + "correct_value": 28, + "explanation": "B cancels; blank = C + D − A = 48 + 46 − 66 = 28." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "47 + 88 + ___ = 16 + 98", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 47 + 88 + ___ = 16 + 98", + "options": [ + "-21", + "-9", + "-39", + "-19" + ], + "answer": "-21", + "correct_value": -21, + "explanation": "No matching terms; blank = 16 + 98 − 47 − 88 = -21." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000336", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "74 + 24 + ___ = 24 + 33 + 74", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 74 + 24 + ___ = 24 + 33 + 74", + "options": [ + "33", + "53", + "23", + "43" + ], + "answer": "33", + "correct_value": 33, + "explanation": "A and B appear on both sides and cancel; blank = C = 33." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "51 + 67 + ___ = 67 + 13 + 32", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 51 + 67 + ___ = 67 + 13 + 32", + "options": [ + "-4", + "-6", + "6", + "-24" + ], + "answer": "-6", + "correct_value": -6, + "explanation": "B cancels; blank = C + D − A = 13 + 32 − 51 = -6." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "68 + 21 + ___ = 42 + 51", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 68 + 21 + ___ = 42 + 51", + "options": [ + "-6", + "4", + "24", + "14" + ], + "answer": "4", + "correct_value": 4, + "explanation": "No matching terms; blank = 42 + 51 − 68 − 21 = 4." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000337", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24 + 67 + ___ = 67 + 91 + 24", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 24 + 67 + ___ = 67 + 91 + 24", + "options": [ + "81", + "101", + "91", + "111" + ], + "answer": "91", + "correct_value": 91, + "explanation": "A and B appear on both sides and cancel; blank = C = 91." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "75 + 88 + ___ = 88 + 30 + 58", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 75 + 88 + ___ = 88 + 30 + 58", + "options": [ + "23", + "33", + "13", + "3" + ], + "answer": "13", + "correct_value": 13, + "explanation": "B cancels; blank = C + D − A = 30 + 58 − 75 = 13." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "52 + 82 + ___ = 94 + 29", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 52 + 82 + ___ = 94 + 29", + "options": [ + "-29", + "1", + "-11", + "-9" + ], + "answer": "-11", + "correct_value": -11, + "explanation": "No matching terms; blank = 94 + 29 − 52 − 82 = -11." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000338", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "63 + 49 + ___ = 49 + 49 + 63", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 63 + 49 + ___ = 49 + 49 + 63", + "options": [ + "69", + "49", + "59", + "39" + ], + "answer": "49", + "correct_value": 49, + "explanation": "A and B appear on both sides and cancel; blank = C = 49." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 + 26 + ___ = 26 + 35 + 10", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 25 + 26 + ___ = 26 + 35 + 10", + "options": [ + "10", + "30", + "20", + "40" + ], + "answer": "20", + "correct_value": 20, + "explanation": "B cancels; blank = C + D − A = 35 + 10 − 25 = 20." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "73 + 73 + ___ = 79 + 10", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 73 + 73 + ___ = 79 + 10", + "options": [ + "-53", + "-43", + "-57", + "-73" + ], + "answer": "-57", + "correct_value": -57, + "explanation": "No matching terms; blank = 79 + 10 − 73 − 73 = -57." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000339", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "68 + 62 = ___ + 66", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 68 + 62 = ___ + 66", + "options": [ + "74", + "54", + "84", + "64" + ], + "answer": "64", + "correct_value": 64, + "explanation": "Blank = A + B − C = 68 + 62 − 66 = 64." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "41 + 80 + ___ = 80 + 45 + 89", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 41 + 80 + ___ = 80 + 45 + 89", + "options": [ + "83", + "113", + "93", + "103" + ], + "answer": "93", + "correct_value": 93, + "explanation": "B cancels; blank = C + D − A = 45 + 89 − 41 = 93." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "28 + 17 + ___ = 93 + 54", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 28 + 17 + ___ = 93 + 54", + "options": [ + "112", + "122", + "92", + "102" + ], + "answer": "102", + "correct_value": 102, + "explanation": "No matching terms; blank = 93 + 54 − 28 − 17 = 102." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000340", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "52 × (21 + 16) = 52 × ___ + 52 × 16", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 52 × (21 + 16) = 52 × ___ + 52 × 16", + "options": [ + "41", + "11", + "31", + "21" + ], + "answer": "21", + "correct_value": 21, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 21." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "30 + 96 + ___ = 96 + 46 + 88", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 30 + 96 + ___ = 96 + 46 + 88", + "options": [ + "104", + "124", + "94", + "114" + ], + "answer": "104", + "correct_value": 104, + "explanation": "B cancels; blank = C + D − A = 46 + 88 − 30 = 104." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "14 + 21 + ___ = 48 + 51", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 14 + 21 + ___ = 48 + 51", + "options": [ + "54", + "64", + "74", + "84" + ], + "answer": "64", + "correct_value": 64, + "explanation": "No matching terms; blank = 48 + 51 − 14 − 21 = 64." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000341", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "80 + 18 = ___ + 47", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 80 + 18 = ___ + 47", + "options": [ + "61", + "71", + "51", + "41" + ], + "answer": "51", + "correct_value": 51, + "explanation": "Blank = A + B − C = 80 + 18 − 47 = 51." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "98 + 45 + ___ = 45 + 52 + 77", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 98 + 45 + ___ = 45 + 52 + 77", + "options": [ + "41", + "31", + "51", + "21" + ], + "answer": "31", + "correct_value": 31, + "explanation": "B cancels; blank = C + D − A = 52 + 77 − 98 = 31." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "99 + 36 + ___ = 88 + 17", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 99 + 36 + ___ = 88 + 17", + "options": [ + "-20", + "-40", + "-10", + "-30" + ], + "answer": "-30", + "correct_value": -30, + "explanation": "No matching terms; blank = 88 + 17 − 99 − 36 = -30." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000342", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 + 87 = ___ + 46", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 98 + 87 = ___ + 46", + "options": [ + "139", + "149", + "159", + "129" + ], + "answer": "139", + "correct_value": 139, + "explanation": "Blank = A + B − C = 98 + 87 − 46 = 139." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "43 + 77 + ___ = 77 + 83 + 89", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 43 + 77 + ___ = 77 + 83 + 89", + "options": [ + "119", + "139", + "129", + "149" + ], + "answer": "129", + "correct_value": 129, + "explanation": "B cancels; blank = C + D − A = 83 + 89 − 43 = 129." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "76 + 90 + ___ = 73 + 73", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 76 + 90 + ___ = 73 + 73", + "options": [ + "-30", + "-20", + "0", + "-10" + ], + "answer": "-20", + "correct_value": -20, + "explanation": "No matching terms; blank = 73 + 73 − 76 − 90 = -20." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000343", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "34 × (10 + 25) = 34 × ___ + 34 × 25", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 34 × (10 + 25) = 34 × ___ + 34 × 25", + "options": [ + "30", + "0", + "20", + "10" + ], + "answer": "10", + "correct_value": 10, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 10." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "36 + 88 + ___ = 88 + 63 + 62", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 36 + 88 + ___ = 88 + 63 + 62", + "options": [ + "79", + "109", + "99", + "89" + ], + "answer": "89", + "correct_value": 89, + "explanation": "B cancels; blank = C + D − A = 63 + 62 − 36 = 89." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "81 + 91 + ___ = 37 + 11", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 81 + 91 + ___ = 37 + 11", + "options": [ + "-124", + "-136", + "-106", + "-116" + ], + "answer": "-124", + "correct_value": -124, + "explanation": "No matching terms; blank = 37 + 11 − 81 − 91 = -124." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000344", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "18 × (49 + 48) = 18 × ___ + 18 × 48", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 18 × (49 + 48) = 18 × ___ + 18 × 48", + "options": [ + "59", + "39", + "49", + "69" + ], + "answer": "49", + "correct_value": 49, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 49." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "97 + 52 + ___ = 52 + 53 + 60", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 97 + 52 + ___ = 52 + 53 + 60", + "options": [ + "26", + "36", + "6", + "16" + ], + "answer": "16", + "correct_value": 16, + "explanation": "B cancels; blank = C + D − A = 53 + 60 − 97 = 16." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "52 + 53 + ___ = 96 + 71", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 52 + 53 + ___ = 96 + 71", + "options": [ + "72", + "82", + "62", + "52" + ], + "answer": "62", + "correct_value": 62, + "explanation": "No matching terms; blank = 96 + 71 − 52 − 53 = 62." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000345", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "29 + 90 = ___ + 68", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 29 + 90 = ___ + 68", + "options": [ + "51", + "41", + "61", + "71" + ], + "answer": "51", + "correct_value": 51, + "explanation": "Blank = A + B − C = 29 + 90 − 68 = 51." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "17 + 76 + ___ = 76 + 48 + 82", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 17 + 76 + ___ = 76 + 48 + 82", + "options": [ + "123", + "113", + "103", + "133" + ], + "answer": "113", + "correct_value": 113, + "explanation": "B cancels; blank = C + D − A = 48 + 82 − 17 = 113." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "36 + 83 + ___ = 97 + 52", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 36 + 83 + ___ = 97 + 52", + "options": [ + "40", + "20", + "30", + "50" + ], + "answer": "30", + "correct_value": 30, + "explanation": "No matching terms; blank = 97 + 52 − 36 − 83 = 30." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000346", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "13 × (53 + 58) = 13 × ___ + 13 × 58", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 13 × (53 + 58) = 13 × ___ + 13 × 58", + "options": [ + "53", + "43", + "73", + "63" + ], + "answer": "53", + "correct_value": 53, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 53." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "33 + 86 + ___ = 86 + 24 + 49", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 33 + 86 + ___ = 86 + 24 + 49", + "options": [ + "50", + "40", + "30", + "60" + ], + "answer": "40", + "correct_value": 40, + "explanation": "B cancels; blank = C + D − A = 24 + 49 − 33 = 40." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "44 + 62 + ___ = 21 + 73", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 44 + 62 + ___ = 21 + 73", + "options": [ + "-12", + "2", + "-28", + "-8" + ], + "answer": "-12", + "correct_value": -12, + "explanation": "No matching terms; blank = 21 + 73 − 44 − 62 = -12." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000347", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "37 × (77 + 48) = 37 × ___ + 37 × 48", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 37 × (77 + 48) = 37 × ___ + 37 × 48", + "options": [ + "97", + "67", + "77", + "87" + ], + "answer": "77", + "correct_value": 77, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 77." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "14 + 35 + ___ = 35 + 21 + 66", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 14 + 35 + ___ = 35 + 21 + 66", + "options": [ + "93", + "63", + "83", + "73" + ], + "answer": "73", + "correct_value": 73, + "explanation": "B cancels; blank = C + D − A = 21 + 66 − 14 = 73." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "34 + 53 + ___ = 71 + 77", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 34 + 53 + ___ = 71 + 77", + "options": [ + "51", + "61", + "71", + "81" + ], + "answer": "61", + "correct_value": 61, + "explanation": "No matching terms; blank = 71 + 77 − 34 − 53 = 61." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000348", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "76 + 96 + ___ = 96 + 26 + 76", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 76 + 96 + ___ = 96 + 26 + 76", + "options": [ + "16", + "26", + "46", + "36" + ], + "answer": "26", + "correct_value": 26, + "explanation": "A and B appear on both sides and cancel; blank = C = 26." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "68 + 59 + ___ = 59 + 76 + 24", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 68 + 59 + ___ = 59 + 76 + 24", + "options": [ + "52", + "22", + "42", + "32" + ], + "answer": "32", + "correct_value": 32, + "explanation": "B cancels; blank = C + D − A = 76 + 24 − 68 = 32." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "97 + 86 + ___ = 21 + 52", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 97 + 86 + ___ = 21 + 52", + "options": [ + "-100", + "-90", + "-110", + "-120" + ], + "answer": "-110", + "correct_value": -110, + "explanation": "No matching terms; blank = 21 + 52 − 97 − 86 = -110." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000349", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12 × (83 + 49) = 12 × ___ + 12 × 49", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 12 × (83 + 49) = 12 × ___ + 12 × 49", + "options": [ + "103", + "83", + "73", + "93" + ], + "answer": "83", + "correct_value": 83, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 83." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "34 + 46 + ___ = 46 + 45 + 78", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 34 + 46 + ___ = 46 + 45 + 78", + "options": [ + "79", + "89", + "99", + "109" + ], + "answer": "89", + "correct_value": 89, + "explanation": "B cancels; blank = C + D − A = 45 + 78 − 34 = 89." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "44 + 29 + ___ = 30 + 45", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 44 + 29 + ___ = 30 + 45", + "options": [ + "12", + "22", + "-8", + "2" + ], + "answer": "2", + "correct_value": 2, + "explanation": "No matching terms; blank = 30 + 45 − 44 − 29 = 2." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000350", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "79 × 54 × 80 = ___ × 80 × 79", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 79 × 54 × 80 = ___ × 80 × 79", + "options": [ + "44", + "54", + "64", + "74" + ], + "answer": "54", + "correct_value": 54, + "explanation": "A and C appear on both sides; blank = B = 54." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "46 + 83 + ___ = 83 + 41 + 31", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 46 + 83 + ___ = 83 + 41 + 31", + "options": [ + "16", + "46", + "36", + "26" + ], + "answer": "26", + "correct_value": 26, + "explanation": "B cancels; blank = C + D − A = 41 + 31 − 46 = 26." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "79 + 66 + ___ = 16 + 26", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 79 + 66 + ___ = 16 + 26", + "options": [ + "-87", + "-117", + "-97", + "-103" + ], + "answer": "-103", + "correct_value": -103, + "explanation": "No matching terms; blank = 16 + 26 − 79 − 66 = -103." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000351", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "70 × 16", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 70 × 16?", + "options": [ + "1123", + "1119", + "1121", + "1120" + ], + "answer": "1120", + "correct_value": 1120, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "20 × 98", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 20 × 98?", + "options": [ + "1959", + "1961", + "1980", + "1960" + ], + "answer": "1960", + "correct_value": 1960, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "33 × 87", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 33 × 87?", + "options": [ + "3031", + "2973", + "3187", + "2871" + ], + "answer": "2871", + "correct_value": 2871, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000352", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "54 × 30", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 54 × 30?", + "options": [ + "1619", + "1622", + "1620", + "1621" + ], + "answer": "1620", + "correct_value": 1620, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "85 × 24", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 85 × 24?", + "options": [ + "2041", + "2040", + "2020", + "2039" + ], + "answer": "2040", + "correct_value": 2040, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "57 × 61", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 57 × 61?", + "options": [ + "3117", + "3881", + "3477", + "3711" + ], + "answer": "3477", + "correct_value": 3477, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000353", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "55 × 87", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 55 × 87?", + "options": [ + "4788", + "4785", + "4784", + "4786" + ], + "answer": "4785", + "correct_value": 4785, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "95 × 30", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 95 × 30?", + "options": [ + "2851", + "2870", + "2850", + "2849" + ], + "answer": "2850", + "correct_value": 2850, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "61 × 23", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 61 × 23?", + "options": [ + "1547", + "1325", + "1403", + "1507" + ], + "answer": "1403", + "correct_value": 1403, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000354", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "98 × 85", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 98 × 85?", + "options": [ + "14994", + "8330", + "15827", + "16660" + ], + "answer": "8330", + "correct_value": 8330, + "explanation": "Product has 4 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "40 × 36", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 40 × 36?", + "options": [ + "1440", + "1420", + "1439", + "1441" + ], + "answer": "1440", + "correct_value": 1440, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "21 × 37", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 21 × 37?", + "options": [ + "777", + "719", + "701", + "757" + ], + "answer": "777", + "correct_value": 777, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000355", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "75 × 96", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 75 × 96?", + "options": [ + "7201", + "7202", + "7200", + "7199" + ], + "answer": "7200", + "correct_value": 7200, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 × 46", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 25 × 46?", + "options": [ + "1150", + "1151", + "1149", + "1140" + ], + "answer": "1150", + "correct_value": 1150, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "89 × 75", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 89 × 75?", + "options": [ + "6739", + "6807", + "7239", + "6675" + ], + "answer": "6675", + "correct_value": 6675, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000356", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "59 × 61", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 59 × 61?", + "options": [ + "3598", + "3599", + "3600", + "3602" + ], + "answer": "3599", + "correct_value": 3599, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "30 × 26", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 30 × 26?", + "options": [ + "770", + "779", + "781", + "780" + ], + "answer": "780", + "correct_value": 780, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "95 × 33", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 95 × 33?", + "options": [ + "2849", + "3207", + "3135", + "3321" + ], + "answer": "3135", + "correct_value": 3135, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000357", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "72 × 94", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 72 × 94?", + "options": [ + "676875", + "6768", + "675", + "14204" + ], + "answer": "6768", + "correct_value": 6768, + "explanation": "Product has 4 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "30 × 88", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 30 × 88?", + "options": [ + "2639", + "2640", + "2620", + "2641" + ], + "answer": "2640", + "correct_value": 2640, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "95 × 47", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 95 × 47?", + "options": [ + "3947", + "4551", + "4171", + "4465" + ], + "answer": "4465", + "correct_value": 4465, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000358", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "35 × 84", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 35 × 84?", + "options": [ + "2940", + "2943", + "2939", + "2941" + ], + "answer": "2940", + "correct_value": 2940, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "30 × 86", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 30 × 86?", + "options": [ + "2580", + "2581", + "2590", + "2579" + ], + "answer": "2580", + "correct_value": 2580, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "57 × 33", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 57 × 33?", + "options": [ + "1755", + "1675", + "2083", + "1881" + ], + "answer": "1881", + "correct_value": 1881, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000359", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "94 × 66", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 94 × 66?", + "options": [ + "615", + "6204", + "11784", + "13024" + ], + "answer": "6204", + "correct_value": 6204, + "explanation": "Product has 4 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "15 × 12", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 15 × 12?", + "options": [ + "170", + "179", + "181", + "180" + ], + "answer": "180", + "correct_value": 180, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "97 × 11", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 97 × 11?", + "options": [ + "1125", + "1067", + "1063", + "1093" + ], + "answer": "1067", + "correct_value": 1067, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000360", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25 × 20", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 25 × 20?", + "options": [ + "499", + "500", + "503", + "501" + ], + "answer": "500", + "correct_value": 500, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "60 × 42", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 60 × 42?", + "options": [ + "2521", + "2519", + "2530", + "2520" + ], + "answer": "2520", + "correct_value": 2520, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "25 × 33", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 25 × 33?", + "options": [ + "825", + "911", + "863", + "743" + ], + "answer": "825", + "correct_value": 825, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000361", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "45 × 39", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 45 × 39?", + "options": [ + "1757", + "1754", + "1756", + "1755" + ], + "answer": "1755", + "correct_value": 1755, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "80 × 42", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 80 × 42?", + "options": [ + "3350", + "3361", + "3359", + "3360" + ], + "answer": "3360", + "correct_value": 3360, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "33 × 67", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 33 × 67?", + "options": [ + "1959", + "2009", + "2211", + "2137" + ], + "answer": "2211", + "correct_value": 2211, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000362", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "67 × 13", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 67 × 13?", + "options": [ + "872", + "870", + "871", + "874" + ], + "answer": "871", + "correct_value": 871, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "85 × 64", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 85 × 64?", + "options": [ + "5460", + "5439", + "5441", + "5440" + ], + "answer": "5440", + "correct_value": 5440, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "15 × 95", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 15 × 95?", + "options": [ + "1573", + "1495", + "1339", + "1425" + ], + "answer": "1425", + "correct_value": 1425, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000363", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "70 × 98", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 70 × 98?", + "options": [ + "6859", + "6861", + "6860", + "6863" + ], + "answer": "6860", + "correct_value": 6860, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "55 × 92", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 55 × 92?", + "options": [ + "5059", + "5060", + "5061", + "5070" + ], + "answer": "5060", + "correct_value": 5060, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "53 × 97", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 53 × 97?", + "options": [ + "5689", + "5553", + "5141", + "5047" + ], + "answer": "5141", + "correct_value": 5141, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000364", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "72 × 27", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 72 × 27?", + "options": [ + "1944", + "1945", + "1946", + "1943" + ], + "answer": "1944", + "correct_value": 1944, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "45 × 72", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 45 × 72?", + "options": [ + "3239", + "3260", + "3241", + "3240" + ], + "answer": "3240", + "correct_value": 3240, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "23 × 91", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 23 × 91?", + "options": [ + "2343", + "1867", + "2021", + "2093" + ], + "answer": "2093", + "correct_value": 2093, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000365", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "60 × 99", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 60 × 99?", + "options": [ + "5941", + "5939", + "5940", + "5942" + ], + "answer": "5940", + "correct_value": 5940, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "20 × 14", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 20 × 14?", + "options": [ + "290", + "280", + "281", + "279" + ], + "answer": "280", + "correct_value": 280, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "53 × 17", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 53 × 17?", + "options": [ + "989", + "901", + "873", + "849" + ], + "answer": "901", + "correct_value": 901, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000366", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "96 × 96", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 96 × 96?", + "options": [ + "9217", + "9215", + "9218", + "9216" + ], + "answer": "9216", + "correct_value": 9216, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "85 × 86", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 85 × 86?", + "options": [ + "7311", + "7320", + "7309", + "7310" + ], + "answer": "7310", + "correct_value": 7310, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "11 × 33", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 11 × 33?", + "options": [ + "343", + "333", + "325", + "363" + ], + "answer": "363", + "correct_value": 363, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000367", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "97 × 61", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 97 × 61?", + "options": [ + "5918", + "5920", + "5917", + "5916" + ], + "answer": "5917", + "correct_value": 5917, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "35 × 98", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 35 × 98?", + "options": [ + "3431", + "3430", + "3429", + "3410" + ], + "answer": "3430", + "correct_value": 3430, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "51 × 37", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 51 × 37?", + "options": [ + "1855", + "1735", + "1887", + "1801" + ], + "answer": "1887", + "correct_value": 1887, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000368", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "30 × 22", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 30 × 22?", + "options": [ + "661", + "659", + "660", + "663" + ], + "answer": "660", + "correct_value": 660, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "40 × 28", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 40 × 28?", + "options": [ + "1121", + "1120", + "1100", + "1119" + ], + "answer": "1120", + "correct_value": 1120, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "73 × 21", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 73 × 21?", + "options": [ + "1531", + "1533", + "1509", + "1641" + ], + "answer": "1533", + "correct_value": 1533, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000369", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "88 × 98", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 88 × 98?", + "options": [ + "17244", + "16382", + "8624", + "18968" + ], + "answer": "8624", + "correct_value": 8624, + "explanation": "Product has 4 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "75 × 42", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 75 × 42?", + "options": [ + "3130", + "3151", + "3150", + "3149" + ], + "answer": "3150", + "correct_value": 3150, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "25 × 29", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 25 × 29?", + "options": [ + "717", + "725", + "789", + "703" + ], + "answer": "725", + "correct_value": 725, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000370", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "95 × 32", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 95 × 32?", + "options": [ + "3039", + "3043", + "3040", + "3041" + ], + "answer": "3040", + "correct_value": 3040, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "60 × 38", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 60 × 38?", + "options": [ + "2270", + "2279", + "2280", + "2281" + ], + "answer": "2280", + "correct_value": 2280, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "75 × 25", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 75 × 25?", + "options": [ + "2027", + "1875", + "1645", + "1689" + ], + "answer": "1875", + "correct_value": 1875, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000371", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "62 × 35", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 62 × 35?", + "options": [ + "21755", + "217062", + "2170", + "21758" + ], + "answer": "2170", + "correct_value": 2170, + "explanation": "Product has 4 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "55 × 30", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 55 × 30?", + "options": [ + "1649", + "1651", + "1670", + "1650" + ], + "answer": "1650", + "correct_value": 1650, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "81 × 11", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 81 × 11?", + "options": [ + "845", + "919", + "923", + "891" + ], + "answer": "891", + "correct_value": 891, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000372", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "61 × 33", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 61 × 33?", + "options": [ + "201323", + "20152", + "2013", + "201310" + ], + "answer": "2013", + "correct_value": 2013, + "explanation": "Product has 4 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10 × 46", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 10 × 46?", + "options": [ + "470", + "461", + "459", + "460" + ], + "answer": "460", + "correct_value": 460, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "73 × 29", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 73 × 29?", + "options": [ + "2117", + "1873", + "2105", + "2331" + ], + "answer": "2117", + "correct_value": 2117, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000373", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "63 × 85", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 63 × 85?", + "options": [ + "5355", + "5356", + "5358", + "5354" + ], + "answer": "5355", + "correct_value": 5355, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "85 × 56", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 85 × 56?", + "options": [ + "4761", + "4740", + "4759", + "4760" + ], + "answer": "4760", + "correct_value": 4760, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "73 × 85", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 73 × 85?", + "options": [ + "6787", + "6205", + "5891", + "6675" + ], + "answer": "6205", + "correct_value": 6205, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000374", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "69 × 31", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 69 × 31?", + "options": [ + "2139", + "2142", + "2138", + "2140" + ], + "answer": "2139", + "correct_value": 2139, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "65 × 32", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 65 × 32?", + "options": [ + "2079", + "2090", + "2081", + "2080" + ], + "answer": "2080", + "correct_value": 2080, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "33 × 33", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 33 × 33?", + "options": [ + "1089", + "1059", + "1007", + "1047" + ], + "answer": "1089", + "correct_value": 1089, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000375", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "29 × 57", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 29 × 57?", + "options": [ + "1654", + "1653", + "1652", + "1656" + ], + "answer": "1653", + "correct_value": 1653, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "30 × 10", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 30 × 10?", + "options": [ + "300", + "299", + "280", + "301" + ], + "answer": "300", + "correct_value": 300, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "79 × 49", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 79 × 49?", + "options": [ + "3871", + "3663", + "4265", + "3399" + ], + "answer": "3871", + "correct_value": 3871, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000376", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "82 × 66", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 82 × 66?", + "options": [ + "11363", + "5412", + "10822", + "10281" + ], + "answer": "5412", + "correct_value": 5412, + "explanation": "Product has 4 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "75 × 32", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 75 × 32?", + "options": [ + "2399", + "2401", + "2400", + "2420" + ], + "answer": "2400", + "correct_value": 2400, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "29 × 37", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 29 × 37?", + "options": [ + "1097", + "1091", + "1195", + "1073" + ], + "answer": "1073", + "correct_value": 1073, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000377", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "36 × 60", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 36 × 60?", + "options": [ + "2159", + "2161", + "2160", + "2162" + ], + "answer": "2160", + "correct_value": 2160, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25 × 92", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 25 × 92?", + "options": [ + "2301", + "2280", + "2300", + "2299" + ], + "answer": "2300", + "correct_value": 2300, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "85 × 49", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 85 × 49?", + "options": [ + "4165", + "4007", + "4641", + "4137" + ], + "answer": "4165", + "correct_value": 4165, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000378", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "13 × 25", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 13 × 25?", + "options": [ + "325", + "3265", + "28", + "3253" + ], + "answer": "325", + "correct_value": 325, + "explanation": "Product has 3 digits; only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "30 × 98", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 30 × 98?", + "options": [ + "2950", + "2939", + "2940", + "2941" + ], + "answer": "2940", + "correct_value": 2940, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "57 × 27", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 57 × 27?", + "options": [ + "1693", + "1585", + "1547", + "1539" + ], + "answer": "1539", + "correct_value": 1539, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000379", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "20 × 64", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 20 × 64?", + "options": [ + "256", + "128", + "1280", + "128016" + ], + "answer": "1280", + "correct_value": 1280, + "explanation": "Product has 4 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "55 × 96", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 55 × 96?", + "options": [ + "5280", + "5279", + "5281", + "5300" + ], + "answer": "5280", + "correct_value": 5280, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "73 × 57", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 73 × 57?", + "options": [ + "4161", + "3825", + "4203", + "4281" + ], + "answer": "4161", + "correct_value": 4161, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000380", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "65 × 35", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 65 × 35?", + "options": [ + "2278", + "2274", + "2276", + "2275" + ], + "answer": "2275", + "correct_value": 2275, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "70 × 60", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 70 × 60?", + "options": [ + "4201", + "4199", + "4200", + "4190" + ], + "answer": "4200", + "correct_value": 4200, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "33 × 11", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 33 × 11?", + "options": [ + "407", + "337", + "397", + "363" + ], + "answer": "363", + "correct_value": 363, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000381", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "53 × 35", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 53 × 35?", + "options": [ + "185556", + "1855", + "367", + "18590" + ], + "answer": "1855", + "correct_value": 1855, + "explanation": "Product has 4 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "85 × 84", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 85 × 84?", + "options": [ + "7140", + "7139", + "7160", + "7141" + ], + "answer": "7140", + "correct_value": 7140, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "83 × 97", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 83 × 97?", + "options": [ + "7531", + "8539", + "8727", + "8051" + ], + "answer": "8051", + "correct_value": 8051, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000382", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10 × 84", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 10 × 84?", + "options": [ + "839", + "841", + "843", + "840" + ], + "answer": "840", + "correct_value": 840, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "15 × 94", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 15 × 94?", + "options": [ + "1411", + "1409", + "1410", + "1420" + ], + "answer": "1410", + "correct_value": 1410, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "55 × 85", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 55 × 85?", + "options": [ + "4675", + "5029", + "4325", + "4775" + ], + "answer": "4675", + "correct_value": 4675, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000383", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "20 × 48", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 20 × 48?", + "options": [ + "959", + "963", + "961", + "960" + ], + "answer": "960", + "correct_value": 960, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "85 × 84", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 85 × 84?", + "options": [ + "7140", + "7139", + "7130", + "7141" + ], + "answer": "7140", + "correct_value": 7140, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "13 × 79", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 13 × 79?", + "options": [ + "1113", + "1027", + "1033", + "1101" + ], + "answer": "1027", + "correct_value": 1027, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000384", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "83 × 23", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 83 × 23?", + "options": [ + "1910", + "1908", + "1909", + "1912" + ], + "answer": "1909", + "correct_value": 1909, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "95 × 36", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 95 × 36?", + "options": [ + "3419", + "3400", + "3420", + "3421" + ], + "answer": "3420", + "correct_value": 3420, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "53 × 73", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 53 × 73?", + "options": [ + "3955", + "3869", + "4057", + "3973" + ], + "answer": "3869", + "correct_value": 3869, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000385", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "84 × 30", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 84 × 30?", + "options": [ + "2521", + "2522", + "2519", + "2520" + ], + "answer": "2520", + "correct_value": 2520, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "90 × 48", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 90 × 48?", + "options": [ + "4319", + "4340", + "4320", + "4321" + ], + "answer": "4320", + "correct_value": 4320, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "85 × 53", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 85 × 53?", + "options": [ + "4149", + "4575", + "4943", + "4505" + ], + "answer": "4505", + "correct_value": 4505, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000386", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "33 × 66", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 33 × 66?", + "options": [ + "2179", + "2178", + "2177", + "2180" + ], + "answer": "2178", + "correct_value": 2178, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "90 × 34", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 90 × 34?", + "options": [ + "3040", + "3061", + "3060", + "3059" + ], + "answer": "3060", + "correct_value": 3060, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "13 × 91", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 13 × 91?", + "options": [ + "1055", + "1183", + "1063", + "1321" + ], + "answer": "1183", + "correct_value": 1183, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000387", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "97 × 48", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 97 × 48?", + "options": [ + "928", + "462", + "10236", + "4656" + ], + "answer": "4656", + "correct_value": 4656, + "explanation": "Product has 4 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "75 × 36", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 75 × 36?", + "options": [ + "2710", + "2701", + "2699", + "2700" + ], + "answer": "2700", + "correct_value": 2700, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "73 × 21", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 73 × 21?", + "options": [ + "1379", + "1533", + "1451", + "1683" + ], + "answer": "1533", + "correct_value": 1533, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000388", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "67 × 63", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 67 × 63?", + "options": [ + "4222", + "4221", + "4224", + "4220" + ], + "answer": "4221", + "correct_value": 4221, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "65 × 70", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 65 × 70?", + "options": [ + "4530", + "4549", + "4550", + "4551" + ], + "answer": "4550", + "correct_value": 4550, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "73 × 79", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 73 × 79?", + "options": [ + "5441", + "5767", + "6255", + "6485" + ], + "answer": "5767", + "correct_value": 5767, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000389", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "45 × 42", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 45 × 42?", + "options": [ + "1890", + "1889", + "1891", + "1893" + ], + "answer": "1890", + "correct_value": 1890, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "55 × 36", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 55 × 36?", + "options": [ + "1980", + "1990", + "1981", + "1979" + ], + "answer": "1980", + "correct_value": 1980, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "97 × 25", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 97 × 25?", + "options": [ + "2197", + "2557", + "2425", + "2431" + ], + "answer": "2425", + "correct_value": 2425, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000390", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "84 × 12", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 84 × 12?", + "options": [ + "1010", + "1009", + "1008", + "1007" + ], + "answer": "1008", + "correct_value": 1008, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "95 × 50", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 95 × 50?", + "options": [ + "4770", + "4751", + "4749", + "4750" + ], + "answer": "4750", + "correct_value": 4750, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "71 × 63", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 71 × 63?", + "options": [ + "4993", + "4307", + "4907", + "4473" + ], + "answer": "4473", + "correct_value": 4473, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000391", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "84 × 69", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 84 × 69?", + "options": [ + "5795", + "5798", + "5796", + "5797" + ], + "answer": "5796", + "correct_value": 5796, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "60 × 66", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 60 × 66?", + "options": [ + "3961", + "3960", + "3959", + "3980" + ], + "answer": "3960", + "correct_value": 3960, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "35 × 91", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 35 × 91?", + "options": [ + "3185", + "3057", + "3127", + "3347" + ], + "answer": "3185", + "correct_value": 3185, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000392", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "47 × 35", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 47 × 35?", + "options": [ + "1648", + "1644", + "1646", + "1645" + ], + "answer": "1645", + "correct_value": 1645, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "40 × 42", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 40 × 42?", + "options": [ + "1679", + "1681", + "1690", + "1680" + ], + "answer": "1680", + "correct_value": 1680, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "33 × 61", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 33 × 61?", + "options": [ + "2007", + "1775", + "2013", + "2255" + ], + "answer": "2013", + "correct_value": 2013, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000393", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "29 × 21", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 29 × 21?", + "options": [ + "610", + "612", + "609", + "608" + ], + "answer": "609", + "correct_value": 609, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "40 × 44", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 40 × 44?", + "options": [ + "1761", + "1760", + "1740", + "1759" + ], + "answer": "1760", + "correct_value": 1760, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "11 × 77", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 11 × 77?", + "options": [ + "863", + "847", + "935", + "947" + ], + "answer": "847", + "correct_value": 847, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000394", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "90 × 27", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 90 × 27?", + "options": [ + "2429", + "2430", + "2432", + "2431" + ], + "answer": "2430", + "correct_value": 2430, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "80 × 20", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 80 × 20?", + "options": [ + "1601", + "1590", + "1600", + "1599" + ], + "answer": "1600", + "correct_value": 1600, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "63 × 33", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 63 × 33?", + "options": [ + "1929", + "2155", + "2079", + "2201" + ], + "answer": "2079", + "correct_value": 2079, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000395", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "72 × 27", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 72 × 27?", + "options": [ + "1944", + "1945", + "1943", + "1946" + ], + "answer": "1944", + "correct_value": 1944, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10 × 42", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 10 × 42?", + "options": [ + "420", + "419", + "440", + "421" + ], + "answer": "420", + "correct_value": 420, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "87 × 43", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 87 × 43?", + "options": [ + "3713", + "3741", + "3691", + "3863" + ], + "answer": "3741", + "correct_value": 3741, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000396", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "81 × 15", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 81 × 15?", + "options": [ + "121573", + "1215", + "239", + "117" + ], + "answer": "1215", + "correct_value": 1215, + "explanation": "Product has 4 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "55 × 96", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 55 × 96?", + "options": [ + "5281", + "5280", + "5270", + "5279" + ], + "answer": "5280", + "correct_value": 5280, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "77 × 83", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 77 × 83?", + "options": [ + "6137", + "7085", + "5963", + "6391" + ], + "answer": "6391", + "correct_value": 6391, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000397", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "29 × 13", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 29 × 13?", + "options": [ + "380", + "376", + "378", + "377" + ], + "answer": "377", + "correct_value": 377, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "40 × 16", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 40 × 16?", + "options": [ + "620", + "639", + "640", + "641" + ], + "answer": "640", + "correct_value": 640, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "69 × 91", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 69 × 91?", + "options": [ + "5941", + "6279", + "6889", + "5607" + ], + "answer": "6279", + "correct_value": 6279, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000398", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "15 × 45", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 15 × 45?", + "options": [ + "676", + "677", + "675", + "674" + ], + "answer": "675", + "correct_value": 675, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "85 × 64", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 85 × 64?", + "options": [ + "5441", + "5440", + "5439", + "5420" + ], + "answer": "5440", + "correct_value": 5440, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "61 × 21", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 61 × 21?", + "options": [ + "1181", + "1355", + "1281", + "1227" + ], + "answer": "1281", + "correct_value": 1281, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000399", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "75 × 35", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 75 × 35?", + "options": [ + "2625", + "2624", + "2626", + "2628" + ], + "answer": "2625", + "correct_value": 2625, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "90 × 52", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 90 × 52?", + "options": [ + "4679", + "4680", + "4681", + "4700" + ], + "answer": "4680", + "correct_value": 4680, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "71 × 45", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 71 × 45?", + "options": [ + "3243", + "3385", + "3195", + "2971" + ], + "answer": "3195", + "correct_value": 3195, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000400", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "57 × 69", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 57 × 69?", + "options": [ + "3935", + "3932", + "3934", + "3933" + ], + "answer": "3933", + "correct_value": 3933, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "95 × 18", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 95 × 18?", + "options": [ + "1709", + "1700", + "1711", + "1710" + ], + "answer": "1710", + "correct_value": 1710, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "65 × 21", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 65 × 21?", + "options": [ + "1331", + "1365", + "1443", + "1429" + ], + "answer": "1365", + "correct_value": 1365, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + } +] \ No newline at end of file