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-    {
-        "id": "G.RLT.1",
-        "content": "\",\"3\n\"\n\""
-    },
-    {
-        "id": "G.RLT.2",
-        "content": "\",\"12\n\"\n\""
-    },
-    {
-        "id": "G.RLT.3",
-        "content": "\",\"17\n\"\n\"Triangles\n\",\"23\n\"\n\""
-    },
-    {
-        "id": "G.TR.1",
-        "content": "\",\"23\n\"\n\""
-    },
-    {
-        "id": "G.TR.2",
-        "content": "\",\"29\n\"\n\""
-    },
-    {
-        "id": "G.TR.3",
-        "content": "\",\"38\n\"\n\""
-    },
-    {
-        "id": "G.TR.4",
-        "content": "\",\"44\n\"\n\"Polygons and Circles.\n\",\"50\n\"\n\""
-    },
-    {
-        "id": "G.PC.1",
-        "content": "\",\"50\n\"\n\""
-    },
-    {
-        "id": "G.PC.2",
-        "content": "\",\"56\n\"\n\""
-    },
-    {
-        "id": "G.PC.3",
-        "content": "\",\"60\n\"\n\""
-    },
-    {
-        "id": "G.PC.4",
-        "content": "\",\"65\n\"\n\"Two- and Three-Dimensional Figures.\n\",\"70\n\"\n\""
-    },
-    {
-        "id": "G.DF.1",
-        "content": "\",\"70\n\"\n\""
-    },
-    {
-        "id": "G.DF.2",
-        "content": "\",\"76\n\" Table of Contents VIRGINIA DEPARTMENT OF EDUCATION | doe.virginia.gov Introduction The Mathematics Instructional Guide, a companion document to the 2023 Mathematics Standards of Learning, amplifies the Standards of Learning by defining the core knowledge and skills in practice, supporting teachers and their instruction, and serving to transition classroom instruction from the 2016 Mathematics Standards of Learning to the newly adopted 2023 Mathematics Standards of Learning. Instructional supports are accessible in #GoOpenVA and support the decisions local school divisions must make concerning local curriculum development and how best to help students meet the goals of the standards. The local curriculum should include a variety of information sources, readings, learning experiences, and forms of assessment selected at the local level to create a rigorous instructional program. For a complete list of the changes by standard, the 2023 Virginia Mathematics Standards of Learning - Overview of Revisions is available and delineates in greater specificity the changes for each grade level and course. The Instructional Guide is divided into three sections: Understanding the Standard, Skills in Practice, and Concepts and Connections aligned to the Standard. The purpose of each is explained below. Understanding the Standard This section includes mathematics understandings and key concepts that assist teachers in planning standards-focused instruction. The statements may provide definitions, explanations, or examples regarding information sources that support the content. They describe what students should know (core knowledge) as a result of the instruction specific to the course/grade level and include evidence-based practices to approaching the Standard. There are also possible misconceptions and common student errors for each standard to help teachers plan their instruction. Skills in Practice This section outlines supporting questions and skills that are specifically linked to the standard. They frame student inquiry, promote critical thinking, and assist in learning transfer. Curriculum writers and teachers should use them to plan instruction to deepen understanding of the broader unit and course objectives. This is not meant to be an exhaustive list of student expectations. Concepts and Connections This section outlines concepts that transcend grade levels and thread through the K through 12 mathematics program as appropriate at each level. Concept connections reflect connections to prior grade-level concepts as content and practices build within the discipline as well as potential connections across disciplines. VIRGINIA DEPARTMENT OF EDUCATION | doe.virginia.gov Geometry Reasoning, Lines, and Transformations Reasoning is the foundation for problem solving, higher-order thinking, understanding, and application. Students use reasoning skills in every branch of mathematics in addition to other courses. These skills unpack building blocks that are necessary for advanced mathematical thinking, understanding, and application. Also, mastery of contextual application of lines and transformations prepare students for engineering, calculus, and physical sciences. Throughout Geometry, students will translate, construct, and judge the validity of a logical argument and use and interpret Venn diagrams. Additionally, students will analyze the relationships of parallel lines cut by a transversal, and solve problems including contextual problems, involving symmetry and transformation."
-    },
-    {
-        "id": "G.RLT.1",
-        "content": "The student will translate logic statements, identify conditional statements, and use and interpret Venn diagrams. Students will demonstrate the following Knowledge and Skills: a) Translate propositional statements and compound statements into symbolic form, including negations (~p, read \"not $p^{\\prime \\prime}$), conjunctions (pq, read \"p and $q^{\\prime \\prime}$, disjunctions (p V q, read \"p or $q^{\\prime \\prime}$), conditionals ( $p\rightarrow q,$ read \"if p then q\"), and biconditionals $(p\\leftrightarrow q$, read \"p if and only if q\"), including statements representing geometric relationships. b) Identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement, and recognize the connection between a biconditional statement and a true conditional statement with a true converse, including statements representing geometric relationships. c) Use Venn diagrams to represent set relationships, including union, intersection, subset, and negation. d) Interpret Venn diagrams, including those representing contextual situations. Understanding the Standard Symbolic notation is used to represent logical arguments, including the use of,,,, A, and V. VIRGINIA DEPARTMENT OF EDUCATION | doe.virginia.gov \"V\n\",\"or\n\"\n\"\u039b\n\",\"and\n\"\n\"\u2190\n\",\"read \"\"implies\"\", if... then...\n\"\n\"\u2194\n\",\"read \"\"if and only if\"\"\n\"\n\"iff\n\",\"read \"\"if and only if\"\"\n\"\n,\"not\n\"\n\"\u0b83\n\",\"therefore\n\" Logical arguments consist of a set of premises or hypotheses and a conclusion. A conditional statement is a logical argument consisting of a set of premises, hypothesis (p), and conclusion (q). O If p, then q or $p\rightarrow q$ O If an angle is a right angle, then its measure is $90^{\\circ}.$ A converse is formed by interchanging the hypothesis and conclusion of a conditional statement. If q, then p or $q\rightarrow p$) If an angle measures $90^{\\circ}$, then the angle is a right angle. An inverse is formed by negating the hypothesis and conclusion of a conditional statement. O If $\\sim p$ then $\\sim q$ qor $\\sim p\rightarrow\\sim q$ O If an angle is not a right angle, then its measure is not $90^{\\circ}$ A contrapositive is formed by interchanging and negating the hypothesis and conclusion of a conditional statement. If $\\sim q$ then $\\sim p$ por $\\sim q\rightarrow\\sim p$ If an angle does not measure $90^{\\circ}$ then the angle is not a right angle. When a conditional $(p\rightarrow q)$ and its converse $(q\rightarrow p)$ are true, the statements can be written as a biconditional: piff q op if and only if q $p\\leftrightarrow q$ The Pythagorean Theorem and its converse can be used as an example: If a triangle is a right triangle, then the sum of the squares of the legs is equal to the square of the hypotenuse $(a^{2}+b^{2}=c^{2})$. If the sum of the squares of the legs is equal to the square of the hypotenuse $(a^{2}+b^{2}=c^{2}),$ then the triangle is a right triangle. Therefore, a triangle is a right triangle if and only if the sum of the squares of the legs is equal to the square of the hypotenuse $(a^{2}+b^{2}=c^{2})$. VIRGINIA DEPARTMENT OF EDUCATION | doe.virginia.gov \u2022 Truth and validity are not synonymous. Valid logical arguments may be false. Validity requires only logical consistency between the statements, but it does not imply true statements. For example, the following argument is valid, but not true: If you are a happy person, then you like animals. If you like animals, then you like dogs. Therefore, if you are a happy person, then you like dogs. Formal proofs utilize symbols of formal logic to determine the validity of a logical argument. Inductive reasoning, deductive reasoning, and proofs are critical in establishing general claims. Inductive reasoning is the method of drawing conclusions from a limited set of observations. Deductive reasoning is the method that uses logic to draw conclusions based on definitions, postulates, and theorems. Valid forms of deductive reasoning include the law of syllogism, the law of contrapositive, the law of detachment, and the identification of a counterexample. Proof is a justification that is logically valid and based on initial assumptions, definitions, postulates, theorems, and/or properties. The law of detachment states that if $p\rightarrow q$ is true and p is true, then q is true. For example, if two angles are vertical, then they are congruent. $\u0007ngle A$ and $\u0007ngle B$ are vertical, therefore $\u0007ngle A\\cong\u0007ngle B$. The law of syllogism states that if $p\rightarrow q$ is true and $q\rightarrow r$ is true, then $p\rightarrow r$ is true. For example, if two angles are vertical, then they are congruent. If two angles are congruent, then they have the same measure. Thus, if two angles are vertical, then they have the same measure. The law of contrapositive states that if a conditional statement $(p\rightarrow q)$ is true, then its contrapositive $(\\sim q\rightarrow\\sim p)$ is also true. For example, if two angles are vertical, then they are congruent. $\u0007ngle A\\cong\u0007ngle B$, therefore $\u0007ngle A$ and $\u0007ngle B$ are not vertical. A counterexample is used to show an argument is false. For example, the argument \"All rectangles are squares,\" is proven false with the following counterexample since quadrilateral ABCD is a rectangle but not a square. feet D A feet feet B feet C A counterexample of a statement confirms the hypothesis but negates the conclusion. Exploration of the representation of conditional statements using Venn diagrams may assist in deepening student understanding. VIRGINIA DEPARTMENT OF EDUCATION | doe.virginia.gov Venn diagrams can be interpreted within contextual situations. Venn diagrams can be used to support the understanding of special quadrilateral relationships or problems involving probability. Surveys can provide opportunities for discussion of experimental probability. The Venn diagram below shows the results of a survey of students to determine who likes comedy movies (C) and/or horror movies (H). Eight students like comedy movies, but not horror movies; five students like horror movies, but not comedy movies; and two students like both comedy movies and horror movies. C H 5 Skills in Practice While the five process goals are expected to be embedded in each standard, the Skills in Practice highlight the most prevalent process goals in relation to the content presented. Mathematical Representations: Students will benefit from additional practice using and interpreting logic symbols, A and V. Let m represent: Angle A is obtuse. Let n represent: Angle B is obtuse. Which is a symbolic representation of the following argument? Angle A is obtuse if and only if Angle B is obtuse. Angle A is obtuse or Angle B is obtuse. Therefore, Angle A is obtuse and Angle B is obtuse. A. $m\rightarrow n$ $m\\wedge n$ B. $m\rightarrow n$ $m\u000bee n$ C. $m\\leftrightarrow n$ mAn D. $m\\leftrightarrow n$ $m\u000bee n$ mVn mAn mVn mAn VIRGINIA DEPARTMENT OF EDUCATION | doe.virginia.gov Misconceptions occur when students do not fully understand the meaning of each logic symbol. They tend to make mistakes when converting between symbolic form and written statements. Common errors include using in place of or confusing A and V. The correct answer to this problem is D. Mathematical Reasoning: Students may need practice judging the validity of a logical argument and using valid forms of deductive reasoning, including the law of syllogism, the law of contrapositive, the law of detachment, and counterexamples. Let $p=$ a dog eats bread\" Let $q=$ \"the dog gains weight\" $p\rightarrow q$, \"If a dog eats bread, then the dog gains weight\" is a true statement. John's dog eats bread. What can be concluded? Students may incorrectly apply geometric rules or properties to situations where they don't apply, leading to errors in problem- solving. Since John has a dog who eats bread and the conditional stated \"If a dog eats bread, then the dog gains weight,\" it can be concluded John's dog gains weight. Using the Law of Detachment, p, John's dog eats bread, is true, thus q, John's dog gains weight is true. Mathematical Connections: Students may have difficulty using Venn diagrams to represent set relationships and contextual situations. Venn diagrams provide a visual representation of two or more sets. The following are examples of a two set and a three set Venn diagram with the following features A B A B C VIRGINIA DEPARTMENT OF EDUCATION | doe.virginia.gov The universal set is a rectangle outlining the space in which all values within the smaller sets are found. The set A, shown using a circle and labeled A. The set B, shown using a circle and labeled B. The set C, shown using a circle and labeled C (reference the three set Venn diagram). Set A and set B (and set C) overlap, showing the shared parts of set A, set B, (and set C). This is called the intersection. To analyze data using Venn diagrams, all of the values within each set must be correctly allocated into the correct part of the Venn diagram. The number of sets is usually outlined or logically deduced from the information provided. When constructing a Venn diagram, draw a region containing two or more overlapping circles (or ellipses), each representing a set. Then, either fill in the information that is given or what can be logically deduced. For example, the Venn diagram below shows the set of numbers {1, 2, 3, 4, 5, 6, 7, 8, 9, 10 that have been sorted into factors of 10 (set F) and even numbers (set E). F E 10 All numbers 1 through 10 are represented on the Venn diagram. The factors of 10 appear within set F; and even numbers appear within set E. The numbers that are both factors of 10 and even numbers appear in the intersection of sets F and E. The numbers that are not factors of 10 or even numbers are outside of set F and set E but remain part of the universal set of numbers 1-10. To describe a subset, students need to understand key symbols and set notation for different sets including the intersection of sets, the union of sets and the absolute complement of sets. The following are examples of set notation with appropriate symbols and their meanings - VIRGINIA DEPARTMENT OF EDUCATION | doe.virginia.gov VIRGINIADEPARTMENT OF EDUCATION | doe.virginia.gov9 \u2022\nTo calculate the number of items in a subset of a Venn diagram, add together the frequencies of the required subset. The following\nVenn diagram shows the number of people who own a cat (C) or a dog (D). VIRGINIADEPARTMENT OF EDUCATION | doe.virginia.gov10\n\u2022\nHow many people own a dog? Some students will say that only 12 people own a dog. These students have neglected to\ninclude 6, the intersection of people owning both a dog and a cat. Another common misconception is for students to add 12 + + 3, including those who own neither a dog nor cat. Teachers may suggest the strategy of shading set D (dog owners) to help\nstudents see what numbers are included. See the illustration below. \u2022\nHow many people own a dog or a cat? A common misconception is for students to add all the numbers 12 + 6 + 9 + 3, thinking\nthe entire diagram represents only dog and cat owners. This shows a misunderstanding of the universal set, and that 3 people\nare not dog or cat owners. Concepts and Connections\nCONCEPTS\nLogic and reasoning provide the foundation of how we use explanations and justifications. Proofs are developed through using higher\norder thinking and logical statements. VIRGINIADEPARTMENT OF EDUCATION | doe.virginia.gov11 CONNECTIONS\n\u2022\nWithin the grade level/course:\no Throughout each Geometry SOL, students will apply logic and reasoning skills to prove, justify, or confirm answers using\npostulates, theorems, definitions, and other appropriate justification statements.\n\u2022\nVertical Progression:\no Critical thinking skills have been embedded in each Mathematics SOL prior to Geometry. Beyond Geometry, students will\ncontinue to apply logic and reasoning skills.\no A.EO.1 The student will represent verbal quantitative situations algebraically and evaluate these expressions for given\nreplacement values of the variables. Resources to Support Local Curriculum\n\u2022\nA list of approved textbooks and instructional materials is posted on the VDOE website.\n\u2022\nEOC Geometry Formula Sheet (PDF) VIRGINIADEPARTMENT OF EDUCATION | doe.virginia.gov12"
-    },
-    {
-        "id": "G.RLT.2",
-        "content": "The student will analyze, prove, and justify the relationships of parallel lines cut by a transversal.\nStudents will demonstrate the following Knowledge and Skills:\na) Prove and justify angle pair relationships formed by two parallel lines and a transversal, including:\ni) corresponding angles;\nii) alternate interior angles;\niii) alternate exterior angles;\niv) same-side (consecutive) interior angles; and\nv) same-side (consecutive) exterior angles.\nb) Prove two or more lines are parallel given angle measurements expressed numerically or algebraically.\nc) Solve problems by using the relationships between pairs of angles formed by the intersection of two parallel lines and a\ntransversal.\nUnderstanding the Standard\n\u2022\nParallel lines are coplanar lines that do not intersect. Parallel lines have the same slope.\n\u2022\nA transversal is a line that intersects at least two other lines.\n\u2022\nParallel lines intersected by a transversal form angles with specific relationships.\no Corresponding angles are in matching positions when a transversal crosses at least two lines. In the model below \u22202 and\n\u22206 are corresponding angles.\no Alternate interior angles are inside the parallel lines and on opposite sides of the transversal. In the model below \u22203 and\n\u22206 are alternate interior angles.\no Alternate exterior angles are outside the two parallel lines and on opposite sides of the transversal. In the model below \u22202\nand \u22207 are alternate exterior angles.\no Consecutive interior angles are inside the two parallel lines and on the same side of the transversal. In the model below\n\u22204 and \u22206 are consecutive interior angles.\no Consecutive exterior angles are outside the parallel lines and on the same side of the transversal. In the model below \u22201\nand \u22207 are consecutive exterior angles. \u2022\nIf two parallel lines are intersected by a transversal, then:\no corresponding angles are congruent\no alternate interior angles are congruent;\no alternate exterior angles are congruent;\no consecutive (same-side) interior angles are supplementary; and\no consecutive (same-side) exterior angles are supplementary.\n\u2022\nTransformations of vertical and linear angle pairs can be used to explore relationships of alternate interior, alternate exterior,\ncorresponding, and same-side interior angles.\n\u2022\nTo prove two or more lines parallel, one of the angle pairs listed above must be shown to be true. The angles must be on the same\ntransversal that intersects both or all of the lines.\n\u2022\nThe parallel line construction uses the Converse of the Corresponding Angles Theorem which states, \u201cIf two lines (\ud835\udc38\ud835\udc38\ud835\udc38\ud835\udc38\n\u20d6\u20d7\nand\n\ud835\udc3a\ud835\udc3a\ud835\udc3a\ud835\udc3a\n\u20d6\u20d7\n) and a transversal (\ud835\udc38\ud835\udc38\n\u20d6\u20d7\n) form corresponding angles (\u2220\ud835\udc3b\ud835\udc3b \ud835\udc3b\ud835\udc3b and \u2220\ud835\udc3e\ud835\udc3e \ud835\udc3e\ud835\udc3e) that are congruent, then the lines (\ud835\udc38\ud835\udc38\ud835\udc38\ud835\udc38\n\u20d6\u20d7\nand \ud835\udc3a\ud835\udc3a\ud835\udc3a\ud835\udc3a\n\u20d6\u20d7\n) are parallel. VIRGINIADEPARTMENT OF EDUCATION | doe.virginia.gov14\nSkills in Practice Mathematical Problem Solving: Students may benefit from additional practice identifying which parts of a figure can be used to\ndetermine whether lines are parallel when more than one statement may be true.\n\u2022\nGiven: Lines a and b intersect lines c and d. \u2022\nWhich of these statements could be used to prove a\u2551b and c\u2551d?\nA. \u22201 and \u22202 are supplementary and \u22205 \u2245 \u22206\nB. \u22201 \u2245 \u22203 and \u22203 \u2245 \u22205\nC. \u22203 and \u22205 are supplementary, and \u22205 and \u22206 are supplementary\nD. \u22203 \u2245 \u22204 and \u22202 \u2245 \u22206 A common mistake is when students do not select a statement that applies to appropriate angle pairs from both sets of lines that\nneed to be proven parallel. They must select appropriate pairs of angles from both lines \ud835\udc4e\ud835\udc4e and \ud835\udc4f\ud835\udc4f and \ud835\udc50\ud835\udc50 and \ud835\udc51\ud835\udc51 with appropriate\njustifications. Some students may select option B. This indicates they understand that \u22201 and \u22203 and \u22205 do have a relationship,\nbut have confused consecutive exterior angles with corresponding angles or have mistakenly thought they were congruent and\nnot supplementary. The correct answer to this problem is: C, \u22203 and \u22205 are supplementary and they are consecutive exterior\nangles which proves c\u2551d; \u22205 and \u22206 are supplementary and they are consecutive exterior angles which proves a\u2551b. VIRGINIADEPARTMENT OF EDUCATION | doe.virginia.gov15\nMathematical Reasoning: Students may need more practice determining parallelism when given complex figures. The example shows a\ncomplex figure with more than one transversal. Given: a \u01c1 c \u2022\nWhat is the value of x?\nSome students may think x = 85 degrees, incorrectly assuming b \u01c1 c. It is important for students to understand they must\nuse the given information and not make assumptions about a diagram based on how it looks. Teachers may wish to have\nstudents use colored pencils to highlight the given parallel lines and transversal that intersects those lines. When parallel\nlines a and c are highlighted one colored and transversal line d is highlighted a different color, students are able to see the\nrelationship between 105 degrees and x. \u2022\nCan you prove b \u01c1 c? Explain your answer.\nSome students will provide a \u201cyes\u201d or \u201cno\u201d without an explanation. Having students explain or justify their answer will help\nthem analyze the information given and think critically about what is and is not provided in the figure. Lines b and c are not\nparallel because the only angles given relative to these lines are the two 85 degree angles. There is not an angle\nrelationship that exists between them. Students should be reminded to rely on postulates and theorems learned to make judgements regarding parallelism. Concepts and Connections\nCONCEPTS\nLines transcend mathematical content areas. Lines and angles are embedded in every area of mathematics. CONNECTIONS\n\u2022\nWithin the grade level/course:\no"
-    },
-    {
-        "id": "G.RLT.1",
-        "content": "\u2013 The student will translate logic statements, identify conditional statements, and use and interpret Venn\ndiagrams.\no"
-    },
-    {
-        "id": "G.TR.2",
-        "content": "\u2013 The student will, given information in the form of a figure or statement, prove and justify two triangles are\ncongruent using direct and indirect proofs, and solve problems involving measured attributes of congruent triangles.\no"
-    },
-    {
-        "id": "G.TR.3",
-        "content": "\u2013 The student will, given information in the form of a figure or statement, prove and justify two triangles are similar\nusing direct and indirect proofs, and solve problems, including those in context, involving measured attributes of similar\ntriangles.\no"
-    },
-    {
-        "id": "G.PC.1",
-        "content": "\u2013 The student will prove and justify theorems and properties of quadrilaterals and verify and use properties of\nquadrilaterals to solve problems, including the relationships between the sides, angles, and diagonals.\n\u2022\nVertical Progression:\no 8.MG.1 \u2013 The student will use the relationships among pairs of angles that are vertical angles, adjacent angles,\nsupplementary angles, and complementary angles to determine the measure of unknown angles.\no A.F.1 \u2013 The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear\nrelationships. Resources to Support Local Curriculum\n\u2022\nA list of approved textbooks and instructional materials is posted on the VDOE website.\n\u2022\nEOC Geometry Formula Sheet (PDF)\n\u2022\nParallel Lines and Angle Relationships Exemplar Mathematics Instructional Plan (Word | PDF)"
-    },
-    {
-        "id": "G.RLT.3",
-        "content": "The student will solve problems, including contextual problems, involving symmetry and transformation.\nStudents will demonstrate the following Knowledge and Skills:\na) Locate, count, and draw lines of symmetry given a figure, including figures in context.\nb) Determine whether a figure has point symmetry, line symmetry, both, or neither, including figures in context.\nc) Given an image or preimage, identify the transformation or combination of transformations that has/have occurred.\nTransformations include:\ni) translations;\nii) reflections over any horizontal or vertical line or the lines y = x or y = -x;\niii) clockwise or counterclockwise rotations of 90\u00b0, 180\u00b0, 270\u00b0, or 360\u00b0 on a coordinate grid where the center of rotation is limited\nto the origin; and\niv) dilations, from a fixed point on a coordinate grid.\nUnderstanding the Standard\n\u2022\nA transformation of a figure, called a preimage, changes the size, shape, and/or position of the figure to a new figure called the\nimage.\n\u2022\nA rigid transformation (or isometry) is a transformation that does not change the size or shape of a geometric figure. This is a special\nkind of transformation that does not change the size or shape of a figure.\n\u2022\nThe image of an object or function graph after a rigid transformation is congruent to the preimage of the object.\n\u2022\nCongruent figures can be shown through a series of rigid transformations.\n\u2022\nA translation is a rigid transformation in which an image is formed by moving every point on the preimage the same distance in the\nsame direction. \u2022\nA reflection is a rigid transformation in which an image is formed by reflecting the preimage over a line called the line of reflection. All\ncorresponding points in the image are equidistant from the line of reflection. The figure in the image below has been reflected over the\ny-axis. \u2022\nThe midpoint between any set of reflected points lies on the line of reflection.\n\u2022\nThe line of reflection can be determined by finding the midpoint (or balance point) between any set of two reflected points.\n\u2022\nA rotation is a rigid transformation in which an image is formed by rotating the preimage about a point called the center of rotation.\nThe center of rotation may or may not be on the preimage. The figure in the image below has been transformed by a 90 degree\nclockwise rotation about the origin. \u2022\nA dilation is a transformation in which an image is formed by enlarging or reducing the preimage proportionally by a scale factor from\nthe center of dilation. The center of dilation may or may not be on the preimage. The image is similar to the preimage. The image below\nhas been dilated by a scale factor of 2. \u2022\nSymmetry and transformations can be explored with coordinate methods.\n\u2022\nTransformations and combinations of transformations can be used to define and describe the movement of objects in a plane or\ncoordinate system.\n\u2022\nThe rules for transformations can be described using coordinates and/or verbal descriptions.\nCoordinate Transformations\nTranslation (x, y)\u2192(x + a, y + b)\nReflection across the x-axis (x, y)\u2192(x, \u2212y)\nReflection across the y-axis (x, y)\u2192(\u2212x, y)\nReflection across the line y=x (x, y)\u2192(y, x)\nReflection across the line y=-x (x, y)\u2192(\u2212y, \u2212x)\nRotation 90\u00b0 (counterclockwise) about the origin (x, y)\u2192(\u2212y, x)\nRotation 180\u00b0 about the origin (x, y)\u2192(\u2212x, \u2212y)\nRotation 270\u00b0 (counterclockwise) about the origin (x, y)\u2192(y, \u2212x)\nDilation with respect to the origin and scale factor of k (x, y)\u2192(kx, ky) \u2022\nA set of points has line symmetry if and only if there is a line, l, such that the reflection through l of each point in the set is also a\npoint in the set.\n\u2022 Point symmetry exists when a figure is built around a single point called the center of the figure. For every point in the figure, there\nis another point found directly opposite it on the other side of the center, at the same distance from the center. A figure with point\nsymmetry will appear the same after a 180\u00b0 rotation. In point symmetry, the center point is the midpoint of every segment formed\nby joining a point to its image. \u2022\nThe perpendicular bisector construction creates the perpendicular bisector as the line of reflection of the provided line segment. Skills in Practice\nWhile the five process goals are expected to be embedded in each standard, the Skills in Practice highlight the most prevalent process\ngoals in relation to the content presented. Mathematical Problem Solving: Students may need practice identifying the result of a combination of transformations and completing a\ncombination of transformations to determine the new coordinates of a given figure. Example: Which sequence of two transformations maps \u25b3ABC to \u25b3A\u2032B\u2032C\u2032? (The vertices of the triangles have integral coordinates.) \u2022\nA rotation 90\u00b0 counterclockwise about the origin, then a reflection over the line \ud835\udc66\ud835\udc66 = \u2212\ud835\udc65\ud835\udc65\n\u2022\nA rotation 90\u00b0 clockwise about the origin, then a reflection over the x-axis\n\u2022\nA translation 7 units to the right, then a reflection over the x-axis\n\u2022\nA reflection over the y-axis, then a translation 8 units down Common errors include confusing the x-axis with the y-axis; confusing clockwise and counterclockwise; confusing horizontal with\nvertical; and incorrectly locating the line \ud835\udc66\ud835\udc66 = \ud835\udc65\ud835\udc65 or \ud835\udc66\ud835\udc66 = \u2212\ud835\udc65\ud835\udc65. The answer is: a rotation 90\u00b0 clockwise about the origin, then a reflection over\nthe x-axis. Mathematical Connections: Students may be"
-    }
-]