data/Gr9B_Mathematics_Learner_Eng.json

{
  "source_file": "Gr9B_Mathematics_Learner_Eng.txt",
  "title": "Grade 9B Mathematics",
  "table_of_contents": [
    {
      "section_id": "1",
      "title": "Functions"
    },
    {
      "section_id": "2",
      "title": "Algebraic expressions"
    },
    {
      "section_id": "3",
      "title": "Equations"
    },
    {
      "section_id": "4",
      "title": "Graphs"
    },
    {
      "section_id": "5",
      "title": "Surface area, volume and capacity of 3D objects"
    },
    {
      "section_id": "6",
      "title": "Transformation geometry"
    },
    {
      "section_id": "7",
      "title": "Geometry of 3D objects"
    },
    {
      "section_id": "8",
      "title": "Collect, organise and summarise data"
    },
    {
      "section_id": "9",
      "title": "Representing data"
    },
    {
      "section_id": "10",
      "title": "Interpret, analyse and report on data"
    },
    {
      "section_id": "11",
      "title": "Probability"
    }
  ],
  "front_matter": "MATHEMATICS\nGrade 9\nBook 2\nCAPS\nLearner Book\nDeveloped and funded as an ongoing project by the Sasol Inzalo \nFoundation in partnership with the Ukuqonda Institute.\nMaths2_Gr9_LB_Book.indb   1\n2014/09/08   09:06:03 AM\n\nPublished by The Ukuqonda Institute\n9 Neale Street, Rietondale 0084\nRegistered as a Title 21 company, registration number 2006/026363/08\nPublic Benefit Organisation, PBO Nr. 930035134\nWebsite: http://www.ukuqonda.org.za\nFirst published in 2014\n© 2014. Copyright in the work is vested in the publisher.\nCopyright in the text remains vested in the contributors.\nISBN: 978-1-920705-29-9\nThis book was developed with the participation of the Department of Basic  \nEducation of South Africa with funding from the Sasol Inzalo Foundation.\nContributors:\nPiet Human, Erna Lampen, Marthinus de Jager, Louise Keegan, Paul van Koersveld,  \nNathi Makae, Enoch Masemola, Therine van Niekerk, Alwyn Olivier, Cerenus Pfeiffer,  \nRenate Röhrs, Dirk Wessels, Herholdt Bezuidenhout\nAcknowledgements:\nFor the chapters on Data Handling, some valuable ideas and data sets were gleaned  \nfrom the following sources:\nhttp://www.statssa.gov.za/censusatschool/docs/Study_guide.pdf\nhttp://www.statssa.gov.za/censusatschool/docs/Census_At_School_2009_Report.pdf\nIllustrations and computer graphics:\nLeonora van Staden, Lisa Steyn Illustration, Ian Greenop\nZhandré Stark, Lebone Publishing Services\nComputer graphics for chapter frontispieces: Piet Human\nCover illustration: Leonora van Staden\nText design: Mike Schramm\nLayout and typesetting: Lebone Publishing Services\nPrinted by: [printer name and address]\nMaths2_Gr9_LB_Book.indb   2\n2014/09/08   09:06:03 AM\n\nCOPYRIGHT NOTICE\nYour freedom to legally copy this book\nThis book is published under a Creative Commons Attribution-NonCommercial 4.0  \nUnported License (CC BY-NC).\nYou are allowed and encouraged to freely copy this book. You can photocopy, print  \nand distribute it as often as you like. You may download it onto any electronic device, \ndistribute it via email, and upload it to your website, at no charge. You may also adapt \nthe text and illustrations, provided you acknowledge the copyright holders (‘attribute \nthe original work’).\nRestrictions: You may not make copies of this book for a profit-seeking purpose.  \nThis holds for printed, electronic and web-based copies of this book,  \nand any part of this book.\nFor more information about the Creative Commons Attribution-NonCommercial 4.0 \nUnported (CC BY-NC 4.0) license, see http://creativecommons.org/ \nlicenses/by-nc/4.0/\nAll reasonable efforts have been made to ensure that materials included are not already \ncopyrighted to other entities, or in a small number of cases, to acknowledge copyright \nholders. In some cases this may not have been possible. The publishers welcome the \nopportunity for redress with any unacknowledged copyright holders.\nExcept where otherwise noted, this work is licensed under  \nhttp://creativecommons.org/licenses/by-nc/4.0/\nMaths2_Gr9_LB_Book.indb   3\n2014/09/08   09:06:04 AM\n\nTable of contents\nTerm 3\nChapter 1:\nFunctions..................................................................................\t\n1\nChapter 2:\nAlgebraic expressions..............................................................\t\n13\nChapter 3:\nEquations..................................................................................\t\n31\nChapter 4:\nGraphs......................................................................................\t\n47\nChapter 5:\nSurface area, volume and capacity of 3D objects................\t\n75\nChapter 6:\nTransformation geometry.......................................................\t\n89\nChapter 7:\nGeometry of 3D objects.........................................................\t 109\nMaths2_Gr9_LB_Book.indb   4\n2014/09/08   09:06:04 AM\n\nTerm 4\nChapter 8:\nCollect, organise and summarise data..................................\t 127\nChapter 9:\nRepresenting data....................................................................\t 141\nChapter 10: \nInterpret, analyse and report on data....................................\t 161\nChapter 11:\nProbability................................................................................\t 177\nMaths2_Gr9_LB_Book.indb   5\n2014/09/08   09:06:04 AM\n\nMaths2_Gr9_LB_Book.indb   6\n2014/09/08   09:06:04 AM",
  "chapters": [
    {
      "title": "Functions",
      "content": "1\t Functions\n1.1\t\nFrom formulas to words, tables and graphs\nthe same instructions in words and in symbols\n1.\t Each of the formulas below indicates a relationship between two sets of numbers  \nthat may be called the input numbers and the output numbers. For each formula, \ncalculate the output numbers that correspond to the input numbers 0; 1; 2 and 10.\n\t\n(a)\t y = 3x + 5\t\n(b)\t y = 3(x + 5)\n\t\n(c)\t y = 3x + 5x\t\n(d)\t y = 3x2 + 5\n\t\n(e)\t y = 3x2 + 5x\t\n(f)\t y = 3x(x + 5)\n2.\t The information provided in the formula y = 5x2 − 3x can also be represented in \nwords, for example: To get the output number, you have to subtract 3 times the input \nnumber from 5 times the square of the input number.\n\t\nRepresent each of the formulas in question 1 in words:\n\t\n(a)\t y = 3x + 5  \n\t\n(b)\t y = 3(x + 5)\n\t\n(c)\t y = 3x + 5x          \n\t\n(d)\t y = 3x2 + 5       \n\t\n(e)\t y = 3x2 + 5x         \nMaths2_Gr9_LB_Book.indb   3\n2014/09/08   09:06:04 AM\n\n4\t\nMATHEMATICS Grade 9: Term 3\n\t\n(f)\t y = 3x(x + 5)         \n3.\t For each set of instructions write a formula that provides the same information:\n\t\n(a)\t multiply the input number by 10, then subtract 3  \n\t\nto get the output number \n     \n\t\n(b)\t subtract 3 from the square of the input number,  \n\t\nthen multiply by 10 to get the output number \n\t\n(c)\t multiply the square of the input number by 10, then add  \n\t\n5 times the input number to get the output number \n\t\n(d)\t subtract 7 times the square of the input number from 100, \n\t\nthen multiply by 3 to get the output number \n\t\n(e)\t add 4 to the input number, then subtract the answer from 50 \n\t\nto get the output number \n\t\n(f)\t multiply the input number by 3, then subtract the  \n\t\nanswer from 15 to get the output number \n4.\t To check your answers for question 3, use the table below. First apply the verbal \ninstructions for the input numbers 1, 5 and 10 in each case. Then choose another \ninput number and do the same thing. Next use the formula you have written to \ncalculate the output numbers. Do corrections where there are differences.\n1\n5\n10\n(a)\nverbal description\nformula\n(b)\nverbal description\nformula\n(c)\nverbal description\nformula\n(d)\nverbal description\nformula\n(e)\nverbal description\nformula\n(f)\nverbal description\nformula\nMaths2_Gr9_LB_Book.indb   4\n2014/09/08   09:06:04 AM\n\n\t\nCHAPTER 1: FUNCTIONS\t\n5\n5.\t In certain cases, flow diagrams can be used to provide instructions on how output \nnumbers can be calculated. For each flow diagram below, represent the information \nin a formula and also in words.\n\t\n(a)\t  \n× 3\n+ 17\n \n\t\n(b)\t\n \n× 3\n+ 2\n+ 5\n \n\t\n(c)\t\n \n× 3\n+ 23\n– 2\n \n\t\n(d)\t\n \n× 5\n+ 4\n+ 3\n× 2\n \n\t\n(e)\t\n \n+ 4\n× 5\n× 2\n+ 3\n \n\t\n(f)\t  \n× 10\n+ 19\n \n\t\n(g)\t  \n+ 5\n× 10\n \n6.\t (a)\t Complete the following table.\nx\n0\n1\n2\n3\ny according to your formula for 5(a)\ny according to your formula for 5(b)\ny according to your formula for 5(c)\n\t\n(b)\t If your output numbers for 5(a), 5(b) and 5(c) are not the same, you have made  \n\t\na mistake somewhere. If this is the case, find your mistake and correct it.\n7.\t (a)\t Complete the following table.\nx\n−3\n−2\n−1\n0\ny according to your formula for 5(d)\ny according to your formula for 5(e)\ny according to your formula for 5(f)\ny according to your formula for 5(g)\nMaths2_Gr9_LB_Book.indb   5\n2014/09/08   09:06:05 AM\n\n6\t\nMATHEMATICS Grade 9: Term 3\n\t\n(b)\t If your output numbers for 5(d) and 5(f) are not the same, you have made a  \n\t\nmistake somewhere. If this is the case, find your mistake and correct it.\n\t\n(c)\t If your output numbers for 5(e) and 5(g) are not the same, you have made a  \n\t\nmistake somewhere. If this is the case, find your mistake and correct it.\n8.\t Explain why the output numbers in 5(a), 5(b) and 5(c) are the same.\n1.2\t Tables and graphs\n1.\t Complete the table to show some of the input and output numbers of the \nrelationship described by the formula y = 2x − 3.\ninput numbers\n−5\n0\n2\n4\n6\n8\noutput numbers\nThe vertical blue line on this graph  \nrepresents the input number 6. \nThe heavy horizontal red line  \nrepresents the output number 9.\nThe black point where the blue and  \nred lines intersect indicates that the  \ninput number 6 is associated with the  \noutput number 9.\nWe also say the black point represents  \nthe ordered number pair (6; 9).\n2.\t (a)\t Which ordered number pair does the red point on the graph represent?\n\t\n(b)\t Which ordered number pair does the blue point on the graph represent?\n0\n–10\n10\n–10\n10\nMaths2_Gr9_LB_Book.indb   6\n2014/09/08   09:06:05 AM\n\n\t\nCHAPTER 1: FUNCTIONS\t\n7\nA relationship between two variables can be represented by a table of some values of the \nindependent and dependent variables (input and output numbers):\nvalues of the independent variable\n3\n4\n5\n6\n7\n8\nvalues of the dependent variable\n12\n14\n16\n18\n20\n22\nThe same information can also be shown on a graph:\n0\n10\n10\n20\n20\n0\n5\n10\n20\n10\n3.\t Do the two graphs show the same relationship, or different relationships between  \ntwo variables?\n4.\t How do the two graphs differ?\n5.\t Use one of the graphs to find out how many yellow squares there will be, in an \narrangement like those at the top, with 12 blue squares. \n6.\t Does the table below represent the same relationship as the table at the top of  \nthe page? Explain your answer.\nvalues of the independent variable\n0\n5\n10\n15\n20\n25\nvalues of the dependent variable\n8\n18\n28\n38\n48\n58\nMaths2_Gr9_LB_Book.indb   7\n2014/09/08   09:06:06 AM\n\n8\t\nMATHEMATICS Grade 9: Term 3\n7.\t (a)\t Complete the following table for the relationship described by y = x2.\nx\n−5\n−4\n−3\n−2\n−1\n0\n1\n2\n3\n4\n5\ny\n\t\n(b)\t Represent the ordered number pairs in the table on the graph sheet below.\n0\n5\n10\n15\n–5\n–10\n–15\n5\n10\n15\n20\n25\n30\n8.\t Complete the table for the relationship y = 15 + x. Represent the ordered number  \npairs on the graph sheet above.\nx\n−15\n−10\n−5\n0\n5\n10\n15\n15 + x\n9.\t Complete the table for the relationship y = 15 − x. Represent the ordered number  \npairs on the graph sheet above.\nx\n−15\n−10\n−5\n0\n5\n10\n15\n15 − x\nMaths2_Gr9_LB_Book.indb   8\n2014/09/08   09:06:06 AM\n\n\t\nCHAPTER 1: FUNCTIONS\t\n9\n10.\t(a)\t The output values for y = x2 and y = 15 + x show patterns. Describe in words how  \n\t\nthe patterns differ. Use the words increase and decrease in your description.\n\t\n(b)\t Describe in words how the graphs of y = x2 and y = 15 + x differ.\n11.\t(a)\t Describe in words how the patterns in the output values for  y = 15 + x and  \n\t\ny = 15 − x differ. Use the words increase and decrease in your description.\n\t\n(b)\t Describe in words how the graphs of y = 15 + x and y = 15 − x differ.\n12.\tComplete each of the following tables by extending the pattern in the output \nnumbers. Also represent the relationship on the graph sheets below.\n(a)\ninput numbers\n0\n5\n10\n15\n20\n25\n30\noutput numbers\n0\n4\n8\n12\n(b)\ninput numbers\n0\n5\n10\n15\n20\n25\n30\noutput numbers\n0\n2\n4\n6\n0\n10\n10\n20\n20\n30\n30\n0\n10\n10\n20\n20\n30\n30\nMaths2_Gr9_LB_Book.indb   9\n2014/09/08   09:06:06 AM\n\n10\t\nMATHEMATICS Grade 9: Term 3\n13.\tHow do the patterns in 12(a) and (b) differ, and how do the graphs differ?\n14.\tEach table below shows some values for a relationship represented by one of  \nthese rules:\n\t\n\t\ny = −2x + 3\t \t\n\t\ny = 2x − 5\t \t\n\t\n\t\ny = −3x + 5\t \t\n\t\n\t\ny = −3(x + 2)\n\t\n\t\ny = 3x + 2\t \t\n\t\ny = 5(x − 2)\t\t\n\t\n\t\ny = 2x + 3\t \t\n\t\n\t\ny = 2x + 5\n\t\n\t\ny = −3x + 6\t \t\n\t\ny = 5x + 10\t \t\n\t\n\t\ny = 5x − 10\t \t\n\t\n\t\ny = −x + 3\n\t\n(a)\t Complete the tables below by extending the patterns in the output values.\n\t\n(b)\t For each table, describe what you did to produce more output values. Also write \n\t\ndown the rule (formula) that corresponds to the table.\n\t\n\t\nA.\t\n\t\n\t\nB.\t\n\t\n\t\nC.\t\n\t\n\t\nD.\t\n\t\n\t\nE.\t\n\t\n\t\nF.\t\n\t\n\t\nG.\t\nA.\nx\n0\n1\n2\n3\n4\n5\n6\n7\ny\n2\n5\n8\nB.\nx\n0\n1\n2\n3\n4\n5\n6\n7\ny\n3\n1\n−1\n−3\nC.\nx\n0\n1\n2\n3\n4\n5\n6\n7\ny\n−10\n−5\n0\n5\nD.\nx\n0\n1\n2\n3\n4\n5\n6\n7\ny\n−5\n−3\n−1\nE.\nx\n0\n1\n2\n3\n4\n5\n6\n7\ny\n6\n3\n0\nF.\nx\n0\n1\n2\n3\n4\n5\n6\n7\ny\n3\n2\n1\n0\nG.\nx\n0\n1\n2\n3\n4\n5\n6\n7\ny\n3\n5\n7\nMaths2_Gr9_LB_Book.indb   10\n2014/09/08   09:06:06 AM\n\n\t\nCHAPTER 1: FUNCTIONS\t\n11\nAn investigation: patterns in differences\n1.\t Complete the tables for y = x2, z = x2 + 12, w = x2 + 22 and s = x2 + 32.\nx\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\ny\nz\nw\ns\n2.\t Complete the tables for y = x2, p = (x + 1)2, q = (x + 2)2 and r = (x + 3)2.\n(a)\nx\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\np\ny\np − y\n(b)\nx\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\nq\ny\nq − y\n(c)\nx\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\nr \ny\nr − y\n3.\t In each of the following cases, you should have different output values for the  \ntwo relationships. If your output values are the same, find your mistakes and  \ncorrect your work.\n\t\n(a)\t z = x2 + 12 and p = (x + 1)2\t\n\t\n(b)\t w = x2 + 22 and q = (x + 2)2\n\t\n(c)\t s = x2 + 32 and r = (x + 3)2\nMaths2_Gr9_LB_Book.indb   11\n2014/09/08   09:06:07 AM\n\n12\t\nMATHEMATICS Grade 9: Term 3\n4.\t Complete the tables, for y = x2, p = (x + 1)2, q = (x + 2)2 and r = (x + 3)2.\n(a)\nx\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\np − y\nq − y\nr − y\n(b)\nx\n10\n11\n12\n13\n14\n15\n16\n17\np − y\nq − y\nr − y\n5.\t (a)\t Complete the table.\nx\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n2x + 1\n4x + 4\n6x + 9\n\t\n(b)\t What are the constant differences in the sequences of values of 2x + 1, 4x + 4  \n\t\nand 6x + 9, for x = 1;  2;  3;  4 . . . . . . ? \n\t\n(c)\t Do you have an idea whether the corresponding sequence for 12x + 36 will also  \n\t\nhave a constant difference and what the constant difference may be?\n\t\n(d)\t There are certain patterns in the coefficients and constant terms in the  \n\t\nexpressions in the above table. Continue the patterns to make some more  \n\t\nsimilar expressions and complete the table below for your expressions.\nx\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n6.\t (a)\t If your answers for the tables in 4(a) and 5(a) are correct, they will be the same. \n\t\nTry to explain why they are the same.\n\t\n(b)\t What expressions, similar to those in question 5(a), may have the same values  \n\t\nas (x + 4)2 − x2 and (x + 5)2 − x2 respectively? \nMaths2_Gr9_LB_Book.indb   12\n2014/09/08   09:06:07 AM\n\nChapter 2\nAlgebraic expressions\n\t\nCHAPTER 2: ALGEBRAIC EXPRESSIONS\t\n13\n2.1\t Introduction............................................................................................................. 15\n2.2\t Factors of expressions of the form ab + ac................................................................. 16\n2.3\t Factors of expressions of the form x2 + (b + c)x + bc.................................................. 19\n2.4\t Factors of expressions of the form a2 – b2.................................................................. 22\n2.5\t Simplification of algebraic fractions........................................................................... 25\nMaths2_Gr9_LB_Book.indb   13\n2014/09/08   09:06:07 AM\n\n14\t\nMATHEMATICS Grade 9: Term 3\nMaths2_Gr9_LB_Book.indb   14\n2014/09/08   09:06:07 AM\n\n\t\nCHAPTER 2: ALGEBRAIC EXPRESSIONS\t\n15",
      "chapter_id": "1"
    },
    {
      "title": "Algebraic expressions",
      "content": "2\t Algebraic expressions\n2.1\t Introduction\nmanipulating expressions\nThe process of writing a polynomial as a product is  \ncalled factorisation. This is the inverse of expansion.\n\t\nfactorisation\n\t\nx2 + 5x + 6  =  (x + 2)(x + 3).\n\t\nexpansion\nEach part of a product is called a factor of the  \nexpression. If c = ab, then a and b are factors of c. x + 2 and x + 3 are the factors of  \n(x + 2)(x + 3). Since x2 + 5x + 6 = (x + 2)(x + 3), x + 2 and x + 3 are the factors of x2 + 5x + 6.\n1.\t Calculate the value of each of the following expressions for x = 12.\n\t\n(a)\t (  + 2)(  + 5)\n + 2\nx\nx\nx\n\t\n(b)\t (\n3)(\n4)\n4\nx\nx\nx\n−\n−\n−\n\t\n(c)\t\nx\nx\nx\n(\n)\n2\n1\n2\n1\n+\n+\n\t\n(d)\t ( + 5)(\n5)\n5\nx\nx\nx\n−\n−\n2.\t Check whether the following statements are  \nidentities by expanding the expressions on  \nthe right.\n\t\n(a)\t x2 − 9 = (x + 3)(x − 3)\t\n(b)\t x2 + x − 6 = (x + 3)(x − 2)\n\t\n(c)\t x2 + 4x + 3 = (x + 3)(x + 1)\t\n(d)\t x2 + 3x = x(x + 3)\nA numerical or algebraic  \nexpression that requires  \nmultiplication as a last step is \ncalled a product. For example, \n12(37 + 63), 2x(x – 5) and xyz \nare called products. A product is \na monomial.\nA statement like 2(x + 3) = 2x + 6, \nwhich is true for all values of x you \ncan think of, is called an identity.\nMaths2_Gr9_LB_Book.indb   15\n2014/09/08   09:06:10 AM\n\n16\t\nMATHEMATICS Grade 9: Term 3\n3.\t Write down the factors of each of the following expressions.\n\t\n(a)\t x2 + x − 6 \n\t\n(b)\t x2 + 3x \n\t\n(c)\t x2 + 4x + 3 \n\t\n(d)\t x2 − 9 \n4.\t Simplify the following quotients (algebraic fractions).\n\t\n(a) \t x\nx\n2\n9\n3\n−\n+\n\t\n(b)\t x\nx\nx\n2\n6\n3\n+\n+\n−\n\t\n(c) \t x\nx\nx\n2\n6\n2\n+\n−\n−\n\t\n(d)\t\nx\nx\nx\nx\n2\n4\n3\n3\n1\n+\n+\n+\n+\n(\n)(\n)\n5. \t (a)\t Suppose you have to find the value of the expression for x = 15. Which  \n\t\nexpression will be the least amount of work? x\nx\n2\n9\n3\n−\n+\n or x − 3? \n\t\n(b)\t Are you sure that you will get the same answers for the two expressions? \nIn the following sections you will learn how to factorise certain types of expressions.  \nThe following identities are useful for the purposes of factorisation:\n\t\na(b + c) = ab + ac    (x + a)(x + b) = x2 + (a + b)x + ab    (a + b)(a − b) = a2 − b2\n2.2\t Factors of expressions of the form ab + ac\nthe greatest common factor\n1. \t (a)\t Is 5 a factor of 20?\t\n(b)\t Is 5 a factor of 30?\n\t\n(c)\t Is 5 a factor of 30 + 20?\t\n(d)\t Is 5 a factor of 30 − 20?\n2.\t (a)\t Is a a factor of ab?\t\n(b)\t Is a a factor of ac?\n\t\n(c)\t Is a a factor of ab + ac?\t\n(d)\t Find another factor of ab + ac.\n\t\n(e)\t Now try and simplify: ab\nac\na\n+\n \nSuppose you have to factorise 4x3 + 2x2 − 6x: It is clear that 2x is a factor of every term, \nhence it is a factor of 4x3 + 2x2 − 6x. \nBy division we get 4\n2\n6\n2\n3\n2\nx\nx\nx\nx\n+\n−\n = 2x2 + x − 3. Hence 4x3 + 2x2 − 6x = 2x(2x2 + x − 3).\nIt is always a good idea to check factorisation by expanding the answer and making \nsure that the result is equal to the original expression.\nMaths2_Gr9_LB_Book.indb   16\n2014/09/08   09:06:12 AM\n\n\t\nCHAPTER 2: ALGEBRAIC EXPRESSIONS\t\n17\n3.\t Complete the table.\nFor each expression, \nfind:\n3x + 6y\n4a3 + 2a\n5x − 2x2\nax2 − bx3\n12a2b + 18ab2\nthe factors of the first \nterm\n3;  x\nthe factors of the \nsecond term\n2;  3;  y\nthe greatest common \nfactor of the two terms\n3\nWrite the expression \nin factor form\n3(x + 2y)\n4.\t Study the example and then factorise the expressions that follow.\n\t\n\t\n(a − b)x + (b − a)y\t = (a − b)x − (a − b)y\t\n\t\n\t\n\t\n= (a − b)(x − y)\n\t\n(a)\t (a − b)x + a − b\t\n(b)\t (a − b)x − a + b\n\t\n(c)\t (a + b)2 − (a + b)\t\n(d)\t (a + b)x − a − b\n\t\n(e)\t 3x(2x − 3) − (3 − 2x)\t\n(f)\t (y2 − 4y) + (3y − 12)\nsomething in between\n1.\t By completing the tables below you will learn something that will help you to find \nthe factors of expressions of the form x2 + (b + c) x + bc, for example x2 + 17x + 30.\nb\n1\n2\n3\n5\n−1\n−2\n−3\n−5\nc\n30\n15\n10\n6\n−30\n−15\n−10\n−6\nb + c\nbc\nb\n−1\n−2\n−3\n−5\n1\n2\n3\n5\nc\n30\n15\n10\n6\n−30\n−15\n−10\n−6\nb + c\nbc\nNote that: \nb − a = –a + b = – (a – b)\nMaths2_Gr9_LB_Book.indb   17\n2014/09/08   09:06:12 AM\n\n18\t\nMATHEMATICS Grade 9: Term 3\n2.\t For each case below find two numbers x and y so that their product xy is 30 and their \nsum x + y is the given number. \n\t\n(a)\t xy = 30 and x + y = 13\t\n(b)\t xy = 30 and x + y = −17\n\t\n(c)\t xy = 30 and x + y = −11\t\n(d)\t xy = 30 and x + y = 11\n3.\t Find x and y in each case.\n\t\n(a)\t xy = −30 and x + y = −13\n\t\n(b)\t xy = 30 and x + y = −13\t\n(c)\t xy = −30 and x + y = 13\t\n\t\n(d)\t xy = −30 and x + y = −1\t\n(e)\t xy = −30 and x + y = 1\t\n4.\t Find x and y in each case.\n\t\n(a)\t xy = 36 and x + y = 15\t\n(b)\t xy = 40 and x + y = 22\n\t\n(c)\t xy = 36 and x + y = 20\t\n(d)\t xy = −40 and x + y = 18\n\t\n(e)\t xy = 36 and x + y = −20\t\n(f)\t xy = −40 and x + y = −18\n5.\t Evaluate each expression for x = 2. Also expand each expression.\n\t\n(a)\t (x + 5)(x − 2) \n\t\n(b)\t (x + 5)(x + 2) \n\t\n(c)\t (x − 5)(x − 2) \n\t\n(d)\t (x − 5)(x + 2) \n6.\t Evaluate each polynomial you formed in question 5 for x = 2. Compare the answers \nwith the values of the corresponding product expressions in question 1. In cases \nwhere the values differ, you have made a mistake somewhere. Sort out any mistakes \ncompletely before you continue with question 7.\nYou may use the tables you \ncompleted in question 1 to \nfind the answers to some of \nthese questions.\nMaths2_Gr9_LB_Book.indb   18\n2014/09/08   09:06:12 AM\n\n\t\nCHAPTER 2: ALGEBRAIC EXPRESSIONS\t\n19\n7.\t Expand each product.\n\t\n(a)\t (x + 3)(x + 8) \n\t\n(b)\t (x + 2)(x + 12) \n\t\n(c)\t (x + 4)(x + 6) \n\t\n(d)\t (x + 1)(x + 24) \n\t\n(e)\t (x + 3)(x − 8) \n\t\n(f)\t (x + 2)(x − 12) \n\t\n(g)\t (x + 4)(x − 6) \n\t\n(h)\t (x + 1)(x − 24) \n2.3\t Factors of expressions of the form x2 + (b + c)x + bc \nThe expanded form of a product of two linear binomials like (x + 3)(x + 8) or  \n(x + 3)(x − 8) is a quadratic trinomial like x2 + 11x + 24 or x2 − 5x − 24 with \n• a term in x2, \n• a term in x that is called the middle term, which is +11x in x2 + 11x + 24 \n and −5x in x2 − 5x − 24, and\n• a constant term also called the last term, which is +24 in x2 + 11x + 24, \n and −24 in x2 − 5x − 24.\nTo factorise an expression like x2 + 5x + 6 means to reverse the process of expansion.  \nThis means that we have to find out which binomials will produce the trinomial when \nthe product of the binomials is expanded, for example:\nx2 + 5x + 6 = (? + ?)(? + ?)\n(x + 2)(x + 3)\n\t\nexpansion\t\nfactorisation\n= x2 + 5x + 6\ntry to find the factors\n1.\t Fill in the missing parts of the factors in each of the following cases.\n\t\n(a)\t (x + 3)(x \n) = x2 + 9x + 18\t\n(b)\t (x + 2)(x \n) = x2 + 11x + 18 \n\t\n(c)\t (x + 3)(x − \n) = x2 + 9x − 18\nMaths2_Gr9_LB_Book.indb   19\n2014/09/08   09:06:13 AM\n\n20\t\nMATHEMATICS Grade 9: Term 3\n\t\n(d)\t (\n + \n)(x + 2) = x2 + 5x + 6\t\n(e)\t x2 − x − 6\n2.\t Expand each product:\n\t\n(a)\t (x + p)(x + q)\t\n\t\n\t\n\t\n\t\n(b)\t (x + p)(x − q)\t\n\t\n\t\n\t\n\t\n(c)\t (x − p)(x + q)\t\n\t\n\t\n\t\n\t\n(d)\t (x − p)(x − q)\t\n\t\n\t\n\t\nThe product of the first terms of the factors must be equal to the x2 term of the trinomial.\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\nMeaning:  x × x = x2\n\t\nx2 + 5x + 6    =    (x + 2)(x + 3)\t \t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\nMeaning:  2 × 3 = 6\nThe product of the last terms of the factors must be equal to the last term (the constant \nterm) of the trinomial. The sum of the inner and outer products must be equal to the \nterm in x (the middle term) of the trinomial.\n\t\nx2 + 5x + 6    =    (x + 2)(x + 3)\t \t\n\t\nMeaning: 2x + 3x = (2 + 3)x = 5x\n(x + a)(x + b) = x × x + ax + bx + a × b = x2 + (a + b)x + ab\n3.\t Try to factorise the following trinomials.\n\t\n(a)\t x2 + 8x + 12\t\n(b)\t x2 − 8x + 12\nMaths2_Gr9_LB_Book.indb   20\n2014/09/08   09:06:13 AM\n\n\t\nCHAPTER 2: ALGEBRAIC EXPRESSIONS\t\n21\npractice makes perfect\n1.\t Factorise the following trinomials: (Remember to check your answer by expanding \nthe factors to test if you do get the original expression.)\n\t\n(a)\t a2 + 9a + 14\t\n(b)\t x2 + 3x − 18\n\t\n(c)\t x2 − 18x + 17\t\n(d)\t y2 + 17y + 30\n\t\n(e)\t y2 − 13y − 30 \t\n(f)\t y2 + 7y − 30\n\t\n(g)\t x2 + 2x − 15\t\n(h)\t m2 + 4m − 21\n\t\n(i)\t x2 − 6x + 9\t\n(j)\t b2 + 15b + 56\n\t\n(k)\t a2 − 2a − 63\t\n(l)\t a2 − ab − 30b2\n\t\n(m)\tx2 − 5xy − 24y2\t\n(n)\t x2 − 13x + 40\nAn alternative method\n2.\t Study the example and then factorise the expressions that follow.\n\t\nExample:\t\nFactorise ac + bc + bd + ad\n\t\nac + bc + bd + ad\t = (ac + bc) + (bd + ad)\t\nOrder and group terms with common factors\n\t\n\t\n\t\n= c(a + b) + d(b + a)\t\nTake out the common factor\n\t\n\t\n\t\n= (a + b)(c + d)\t\nWrite expression as a product\n\t\n(a)\t px + py + qx + qy\t\n(b)\t 9x3 − 27x2 + x − 3\nMaths2_Gr9_LB_Book.indb   21\n2014/09/08   09:06:13 AM\n\n22\t\nMATHEMATICS Grade 9: Term 3\n\t\n(c)\t 4a + 4b + 3ap + 3bp\t\n(d)\t a4 + a3 + 3a + 3\n\t\n(e)\t xy + x + y + 1\t\n(f)\t ac − ad − bc + bd\nYet another method\n\t\nExample 1:\t\nExample 2:\t\nAction:\n\t\nx2 + 4x + 3\t\nx2 + 3x − 4\t\n=\t x2 + x + 3x + 3\t\n= x2 − x + 4x − 4\t\nRe-writing middle term as sum of two terms.\n=\t (x2 + x) + (3x + 3)\t\n= (x2 − x) + (4x − 4)\t\nGrouping.\n=\t x(x + 1) + 3(x + 1)\t\n= x(x − 1) + 4(x − 1)\t\nTaking out the GCF of each group.\n=\t (x + 1)(x + 3)\t\n= (x − 1)(x + 4)\t\nWrite it as a product.\n3.\t Factorise:\n\t\n(a)\t x2 + 7x + 12\t\n(b)\t x2 − 7x + 12\nThe challenge is to re-write the middle term as the sum of two terms in a way that you \nare able to take out the common factor. \n2.4\t Factors of expressions of the form a2 – b2\npreliminary work \n1.\t Complete the following table and see if you can notice a pattern (rule) whereby you \ncan predict the answers to the first column’s calculations without squaring it:\n(a)\n32 − 22\n3 + 2\n3 − 2\n(3 + 2)(3 − 2)\n(b)\n42 − 32\n4 + 3\n4 − 3\n(4 + 3)(4 − 3)\n(c)\n62 − 42\n6 + 4\n6 − 4\n(6 + 4)(6 − 4)\n(d)\n92 − 32\n9 + 3\n9 − 3\n(9 + 3)(9 − 3)\nMaths2_Gr9_LB_Book.indb   22\n2014/09/08   09:06:13 AM\n\n\t\nCHAPTER 2: ALGEBRAIC EXPRESSIONS\t\n23\n2.\t Do you notice a pattern (rule) whereby you can predict the answers to such \ncalculations?\n3.\t Now predict the answers to each of the following without squaring. Check your \nanswers where necessary. Does the rule that you discovered in question 2 also hold \nfor the following cases?\n\t\n(a) \t 172 − 132\t\n(b)\t 542 − 462\t\n(c)\t 282 − 222\n4.\t Formulate your rule in symbols: \n\t\na2 − b2 = \n5.\t Can you explain why factors of a2 − b2 have this form?\nStated differently: If p and q are perfect squares, \nalso “algebraic squares”, then:\n  p  −  q\t\n= (\np + q )(\np − q )\n\t 9x4 − 4y2\t\n= ( 9\n4\nx  + 4\n2\ny )( 9\n4\nx  − 4\n2\ny )\n\t \t\n= (3x2 + 2y)(3x2 − 2y)\n(Note the operations within the brackets differ.)\nfactorising difference BETWEEN two squares expressions\n1.\t Use the skills you learnt in the previous exercises to factorise the following:\n\t\n(a)\t 4a2 − b2\t\n(b)\t m2 − 9n2\n\t\n(c)\t 25x2 − 36y2\t\n(d)\t 121x2 − 144y2\n\t\n(e)\t 16p2 − 49q2\t\n(f)\t 64a2 − 25b2c2\n\t\n(g)\t x2 − 4\t\n(h)\t 16x2 − 36y2\nAn expression of the form  \na2 – b2 is called the difference \nbetween two squares.\nTo factorise a difference \nbetween squares, we use the \nidentity: a2 – b2 = (a + b)(a − b)  \nwhere a and b represent numbers \nor algebraic expressions.\nMaths2_Gr9_LB_Book.indb   23\n2014/09/08   09:06:14 AM\n\n24\t\nMATHEMATICS Grade 9: Term 3\nAlways factorise completely.\nAlways take out the greatest common factor if there \nis one. \nOne is a perfect square: 1 = 12 and 1m = 1.\nThe exponential law: am.an = am + n.\n2.\t Factorise.\n\t\n(a)\t x4 − 1\t\n(b)\t 16a4 − 81\n\t\n(c)\t 1 − a2b2c2\t\n(d)\t 25x10 − 49y8\n\t\n(e)\t 2x2 − 18\t\n(f)\t 200 − 2b2\n\t\n(g)\t 3xy2 − 48xa2\t\n(h)\t 5a4 − 20b2\nIn each case calculate the area of the shaded part.\nUse the shortest possible method.\n(a)\t\n\t\n\t\n\t\n\t\n\t\n(b)\n25\n25\n9\n9\n36\n36\n4\n4\nThis is how factorisation can make calculation easy!\nMaths2_Gr9_LB_Book.indb   24\n2014/09/08   09:06:14 AM\n\n\t\nCHAPTER 2: ALGEBRAIC EXPRESSIONS\t\n25\n2.5\t Simplification of algebraic fractions\nworking with algebraic fractions\nLiza and Madodo have to determine the value of x\nx\nx\n2\n2\n3\n3\n−\n−\n−\n for x = 4,6.\nLiza’s solution:\nMadoda’s solution:\n x\nx\nx\n2\n2\n3\n3\n−\n−\n−\n= 4 6\n2 4 6\n3\n4 6\n3\n2\n,\n( , )\n,\n−\n−\n−\n         Substitute x = 4,6\n= 21 16\n9 2\n3\n4 6\n3\n,\n,\n,\n−\n−\n−\n \n= 8 96\n1 6\n,\n,\n= 5,6\n x\nx\nx\n2\n2\n3\n3\n−\n−\n−\n= (\n)(\n)\nx\nx\nx\n−\n−\n+\n3\n1\n3\n      Factorise the numerator\n= x + 1                    Simplify the expression\n= 4,6 + 1                Substitute x = 4,6\n= 5,6\n1.\t Which solution do you prefer? Why?\nIt is useful to manipulate quotient expressions like \nx\nx\nx\n2\n5\n6\n2\n+\n+\n+\n into  simpler but equivalent sum expressions, \nlike x + 3 in this case.  It makes substitution and the \nsolving of equations easier.\n2.\t Solve the following problems.\n\t\n(a)\t Evaluate x\nx\nx\n2\n5\n6\n2\n+\n+\n+\n if x = 23. \t\n(b)\t Solve x\nx\nx\n2\n5\n6\n2\n+\n+\n+\n = 19.\n3.\t Determine the value of each of the following expressions if x = 36.  \nSee if you can use the shortest possible method.\n\t\n (a)\t x\nx\n2\n9\n3\n−\n+\n\t\n(b)\t x\nx\nx\n2\n6\n3\n+\n+\n−\nMaths2_Gr9_LB_Book.indb   25\n2014/09/08   09:06:16 AM\n\n26\t\nMATHEMATICS Grade 9: Term 3\nHow is it possible that 2 = 1?\nWhat went wrong in the following argument? \n\t\nLet:\t\n\t\na\t= b\t\n(If: b ≠ 0)\n\t\n× a:\t\n\t\na2\t= ab\n\t\n− b2:\t\n\t\na2 − b2\t= ab − b2\n\t Factorise:\t\n\t\n(a + b)(a − b)\t= b(a − b)\n\t ÷ (a − b):\t\n\t\na + b\t= b\n\t But a = b:\t\n\t\nb + b\t= b\n\tAdd terms:\t\n\t\n2b\t= b\n\t\n÷ b:\t\n\t\n2\t= 1\nExplain what went wrong and why it is wrong?\ndividing by zero cannot be done\n1.\t Complete the following table by evaluating the value of the expression x\nx\n+\n−\n2\n2  for the  \nx-values given in the top row:\nx\n−2\n0\n2\n4\nx\nx\n+\n−\n2\n2\n2.\t If x = 2 then x\nx\n+\n−\n2\n2 will have the value 4\n0. What is the value of 4\n0?\n3.\t One way to determine the value of 4\n0, you can set it as 4\n0 = a. Then 4 = 0 × a.  \nWhich values of a will make this statement true?\n4.\t What is the result of the calculation of 4 ÷ 0 on your calculator? Can you explain  \nthe message on your calculator? \nDivision by 0 is not possible. The algebraic fraction\nx\nx\n+\n−\n2\n2 cannot have a value a value if the denominator\n(x − 2) is equal to 0. We may say the expression x\nx\n+\n−\n2\n2 is\nundefined for x − 2 = 0 i.e. for x = 2. We also say x = 2 is\nan excluded value of x for x\nx\n+\n−\n2\n2.\nMaths2_Gr9_LB_Book.indb   26\n2014/09/08   09:06:17 AM\n\n\t\nCHAPTER 2: ALGEBRAIC EXPRESSIONS\t\n27\ndefining the undefined\n1.\t Is the following statements true? If not, correct the statement.\n\t\n(a)\t x\nx  = 1 for all values of x.\n\t\n(b)\t x\nx\n3\n2  = x for all values of x.\n\t\n(c)\t\nx\nx\n−\n−\n3\n3  = 1 for all values of x.\n\t\n(d)\t\nx\nx\nx x\n2\n1\n+\n+\n(\n) = 1 for all values of x.\n2.\t For which values of the variables will each expression be undefined?\n\t\n(a)\t 7\n5\n2\n(\n)\ny\ny\n+\n+\n\t\t\n\t\n\t\n(b)\t 3\n2\n4\nx\nx\n+\n+\n\t \t\n\t\n\t\n(c)\t 2\n1\n1\n2\nx\nx\n+\n−\n\t \t\n\t\n\t\n(d)\t\n2\n1\n2\n3\n2\nx\nx\nx\n−\n−\n+\n(\n)(\n) \t\nsimplifying algebraic fractions\nTo simplify an algebraic fraction that contains a \npolynomial as numerator or denominator, the \npolynomial should be factorised first.\nTo prevent division by zero, the excluded values \nmust be stated.\nMaths2_Gr9_LB_Book.indb   27\n2014/09/08   09:06:19 AM\n\n28\t\nMATHEMATICS Grade 9: Term 3\n1.\t Simplify each of the following algebraic fractions by factorising the numerator and \nthen using the property ax\na  = x if a ≠ 0. Give excluded values.\n\t\n(a)\t 3\n3\n2\nxy\ny\nx\ny\n+\n+\n\t\n(b)\t a b\nab\na\nb\n2\n2\n+\n+\n\t\n(c)\t 3\n6\n3\n2\n2\n2\nx y\nx y\nxy\n−\n\t\n(d)\t 10\n15\n5\n4\n3\n2\nx\nx\nx\n+\n2.\t Simplify each of the following algebraic fractions by factorising the numerator and \nthen using the property ax\na  = x if a ≠ 0. (See if you can factorise the trinomials.)\n\t\n(a)\t x\nx\nx\n2\n5\n6\n2\n+\n+\n+\n\t\n(b)\t x\nx\nx\n2\n2\n8\n2\n+\n−\n−\n \n\t\n(c)\t\nx\nx\nx\n2\n5\n50\n5\n−\n−\n+\n\t\n(d)\t x\nx\nx\n2\n16\n15\n15\n−\n−\n+\n3.\t Simplify each of the following algebraic fractions by factorising the numerator and \nthen using the property ax\na  = x if a ≠ 0. \n\t\n(a)\t\nx\nx\n2\n4\n2\n−\n−\n\t\n(b)\t 4\n1\n2\n1\n2\nx\nx\n−\n+\nMaths2_Gr9_LB_Book.indb   28\n2014/09/08   09:06:22 AM\n\n\t\nCHAPTER 2: ALGEBRAIC EXPRESSIONS\t\n29\nFactorisation can reduce calculations\nIn each case, use the shortest possible method to get to your answer.\n(a)\tCalculate the shaded area. \n\t\n(Area = πr 2 and use π = 3.142)\n22\n18\n(b)\tCalculate the length of side a\n\t\n(Pythagoras: c2 = a2 + b2)\na = ?\nb = 11\nc = 61\nThis is how factorisation can save you time!\nmore practice\n1.\t Factorise the following expressions completely.\n\t\n(a)\t 4a + 6b\t\n(b)\t x2 + 8x + 7\n\t\n(c)\t c2 − 9\t\n(d)\t y2 − 8y + 15\n  \n\t\n(e)\t −3ab + b\t\n(f)\t −3a(b − 1) + (b − 1)\n\t\n(g)\t dfg2 + d2g − df 2g\t\n(h)\t x2 + 6x + 8\n\t\n(i)\t a2 + 5a + 6\t\n(j)\t x2 − 8x − 20\n\t\n(k)\t x5y3 − x3y5\t\n(l)\t x3y − xy3\n\t\n(m)\t4 − 4y + y2\t\n(n)\t 3a2 + 18a − 21\n\t\n(o)\t 6a2 − 54\t\n(p)\t −a2 − 11a − 30\nMaths2_Gr9_LB_Book.indb   29\n2014/09/08   09:06:22 AM\n\n30\t\nMATHEMATICS Grade 9: Term 3\n\t\n(q)\t 2a2 + 10a − 72\t\n(r)\t 5x3 − 15x2 − 200x\n\t\n(s)\t (x + 2)2 − y2\t\n(t)\t (x + y)2 − a2\n\t\n(u)\t (a2 − 2a + 1) − b2\t\n(v)\t 1 − (a2 − 2ab + b2)\n \n\t\n(w)\t (a – b)x + (b − a)y\t\n(x)\t a(2x − y) + (y − 2x)\n\t\n(y)\t 2x2y10 − 8x10y2\t\n(z)\t (a + b)3 – 4(a + b)\n\t\n(aa)\t(a + b)2 − a − b\t\n(ab)\t(x + y)(a − b) + (−x − y)(b − a)\n2.\t Simplify each of the following algebraic fractions as far as possible. \n\t\n(a)\t 16\n9\n4\n3\n2\n−\n+\nx\nx \t\n(b)\t 25\n36\n5\n6\n2\n2\nx\nx\nx\n−\n+\n\t\n(c)\t\nx\nx\nx\nx\n3\n2\n30\n6\n+\n+\n−\n\t\n(d)\t 2\n5\n3\n2\n3\n2\nx\nx\nx\n+\n+\n+\n\t \n\t\n(e)\t ab\nbc\nabc\n+\n\t\n(f)\t\npa\npb\na\nb\n+\n+\nMaths2_Gr9_LB_Book.indb   30\n2014/09/08   09:06:23 AM\n\nChapter 3\nEquations\n\t\nCHAPTER 3: EQUATIONS\t\n31\nYou have already solved equations by inspection and inverse operations in the first term. \nIn this chapter you will first revise this work. Then you will work with equations which \ncontain product expressions, like 2x(x – 2) and (x – 5)(x + 3). You will learn new methods \nto solve these equations, based on the fact that if the product of two expressions (or \nnumbers) equals zero, one or both of the expressions or numbers must be zero. You will \nuse factorisation to write equations in the form pq = 0 so that you can solve them.\n3.1\t Introduction............................................................................................................. 33\n3.2\t Solving by factorisation Part 1................................................................................... 35\n3.3\t Solving by factorisation Part 2................................................................................... 37\n3.4\t Solving by factorisation Part 3................................................................................... 39\n3.5\t Set up equations to solve problems.......................................................................... 41\n3.6\t Equations and ordered pairs..................................................................................... 44\nMaths2_Gr9_LB_Book.indb   31\n2014/09/08   09:06:23 AM\n\n32\t\nMATHEMATICS Grade 9: Term 3\n0\n5\n5\n5\n5\n10\n15\n10\n10\n10\n15\nMaths2_Gr9_LB_Book.indb   32\n2014/09/08   09:06:24 AM\n\n\t\nCHAPTER 3: EQUATIONS\t\n33",
      "chapter_id": "2"
    },
    {
      "title": "Equations",
      "content": "3\t Equations\n3.1\t Introduction\nsolution by inspection\n1.\t Complete the following table. Substitute the  \ngiven x-values into the equation until you find  \nthe value that makes the equation true.\nEquation\nLHS if \n x = 4\nIs LHS  \n= RHS ?\nLHS if \n x = 5\nIs LHS  \n= RHS ?\nLHS if \n x = 6\nIs LHS  \n= RHS ?\nCorrect \nsolution\n(a)\n3x − 4 = 11\nx = \n(b)\n2x + 7 =19\nx = \n(c)\n13 − 5x = −7\nx = \n(LHS = Left-hand side and RHS = Right-hand side)\n2.\t In the following table, you are given equations  \nwith their solutions. Insert + or − or = signs  \nbetween each term to make the equations true  \nfor the solution given.\nEquation\nSolution\n(a)\n\t\n2x\t \t 7 = 15\nx = 4\n(b)\n\t\n3\t \t 2x = 11\nx = −4\n(c)\n\t\n−x\t \t 7 = 3\nx = 4\n(d)\n\t\n28\t \t 5x = 3\nx = 5\nStatements like 21 − x = 2x + 3 and (x − 3)(x − 5) = 0, \nwhich are true for only some values of x are called \nequations.\nA statement like 2(x + 3) = 2x + 6, which is true for \nall values of x you can think of, is called an identity.\nA statement like 2(x + 3) = 2x + 3, where there \nare no values of x for which it is true, is called an \nimpossibility.\nYou can read the solutions of \nan equation from a table.\nThe “searching” for the  \nsolution of an equation is \nreferred to as solving the \nequation by inspection.\nMaths2_Gr9_LB_Book.indb   33\n2014/09/08   09:06:24 AM\n\n34\t\nMATHEMATICS Grade 9: Term 3\nsolving equations through inverse operations \nIn this section you are going to explore a different way of solving equations.\n1.\t Complete the following calculations.\n\t\n(a)\t 3 − 3\t\n(b)\t −9 765 + 9 765\n\t\n(c)\t −a + a\t\n(d)\t 13a − 13a\n2.\t What do you notice? \n3.\t Complete the following calculations.\n\t\n(a)\t 3 ÷ 3\t\n(b)\t 3 × 1\n3  \n\t\n(c)\t\n1\nx  × x\t\n(d)\t\nx\n3  × 3\nx\n4.\t What do you notice? \nWe can start with a solution as an equation and \nthen apply some operations to it to turn it into an \nequivalent but more complicated equation.\nBuilding an equation\nSolving an equation\nAction on both \nsides \nEquivalent \nequations\nAction on both \nsides\nEquivalent \nequations\nSolution (1)\n× 3\n+ 2\nSolution (2)\n× 2\n+ 6\n+ x\nfactorise\nx = 3\n3x = 9\n3x + 2 = 11\nx = −9\n2x = −18\n2x + 6 = −12\n3x + 6 = x − 12\n3(x + 2) = x − 12\nEquation (1)\n− 2\n÷ 3\nEquation (2)\nremove brackets\n− x\n− 6\n÷ 2\n3x + 2 = 11\n3x = 9\nx = 3\n3(x + 2) = x − 12\n3x + 6 = x − 12\n2x + 6 = −12\n2x = −18\nx = −9\nTwo equations are equivalent \nif they have the same solution.\nMaths2_Gr9_LB_Book.indb   34\n2014/09/08   09:06:25 AM\n\n\t\nCHAPTER 3: EQUATIONS\t\n35\nBuilding an equation\nSolving an equation\nAction on both \nsides \nSolution (3)\n× −1\n+ 3\n+ 2x\n÷ 2\nEquivalent \nequations\nx = 1\n− x = −1\n− x + 3 = 2\n+ x + 3 = 2 + 2x\n(\n)\nx + 3\n2\n = 1 + x\nAction on both \nsides\nEquation (3)\n× 2\n− 2x\n− 3\n÷ −1\nEquivalent \nequations\n(\n)\nx + 3\n2\n = 1 + x\nx + 3 = 2 + 2x\n− x + 3 = 2\n− x = − 1\nx = 1\nTry making up your own equations and then solving them. Did you get the “solution” \nthat you started with?\nWhen you solve an equation, you actually reverse the \nmaking of the equation.\n5.\t Solve for x:\n\t\n(a)\t 2(x + 4) + 9 = 15\t\n(b)\t 5(x − 2) = 7(2 − x)\n \n\t\n(c)\t 2\n3\nx  − 2 = 12\t\n(d)\t 3\n3\n2\ny −\n + 5\n2  = 5\n3\ny \t\nUp to now you have only dealt with equations of the first degree. That means \nthey contained only first powers of the unknown (x), for example 3x − 2 = 5x + 7. In \nthe following sections you will solve equations of the second degree, where the \nexpressions contain second powers. This is an equation of the second degree:\nx2 + 1 = x + 13.\nWhen the expression part of the equation is written as the product of a monomial and \na binomial, e.g. x(x − 2) = 0;  or the product of two binomials, e.g. (x − 2)(x + 3) = 0 the \nresult is also an equation of the second degree.\nMaths2_Gr9_LB_Book.indb   35\n2014/09/08   09:06:26 AM\n\n36\t\nMATHEMATICS Grade 9: Term 3\n3.2\t Solving by factorisation (Part 1)\ndeveloping a strategy: multiplying by zero\n1.\t Can you find two numbers x and y so that if you  \nmultiply them the answer is 0, i.e. xy = 0?\n2.\t Complete the following table:\nEquation\nFactors\nProduct\nFirst \npossible \nsolution\nSecond \npossible \nsolution\nExample\nx(x − 2) = 0\nx \nand \n(x − 2)\n0\nx = 0\nx − 2 = 0\nx = 2\n(a)\nx(x + 5) = 0\n(b)\n2x(3x − 12) = 0\n(c)\n0 = (x + 2)(x − 2)\nYou can rewrite an equation so that it is in the form  \nexpression = 0; for example you can write \nx2 − 2x = 3x + 6 as x2 − 5x − 6 = 0. \nYou can factorise x2 − 5x − 6 and then use the zero-  \nproduct property to solve the equation.\nx2 − 5x − 6 = 0\n(x − 6)(x + 1) = 0\nx = 6 or x = −1\n  \nIn a later section you will solve equations like the above example. You have to write the \nequation in the form, expression = 0, then factorise the left-hand side and then use the \nzero-product property.\nEach part of a product is called  \na factor of the expression.\nIf c = ab, then a and b are  \nfactors of c.\nIf x2 + 5x + 6 = (x + 2)(x + 3), \nthen x + 2 and x + 3 are factors \nof x2 + 5x + 6.\nZero-Product Property\nIf: \t\na × b = 0 \nThen: \t a = 0 or \n\t\nb = 0 or \n\t\na = 0 and b = 0\nMaths2_Gr9_LB_Book.indb   36\n2014/09/08   09:06:26 AM\n\n\t\nCHAPTER 3: EQUATIONS\t\n37\ntaking out the highest common factor\nThe process of writing a sum expression (polynomial) \nas a product (monomial) is called factorisation. \nThis is the inverse of expansion.\nLook at the expression 2x2 − 6x.\n2x is a factor of both terms, therefore it is a factor  \nof 2x2 − 6x. \nBy division we get 2\n6\n2\n2\nx\nx\nx\n−\n = x − 3. \nHence 2x2 − 6x = 2x(x − 3).\nDetermine the values of x which will make the following statements true:\n1.\t x2 = −3x\t\n2.\t\nx2 + 2x2 = 6x\n3.\t 6\n3\nx  + x = −4x2\t\n4.\t\nx = x(2 − x)\nIt is unnecessary to write out \nthe division step of this method. \nAfter finding the common \nfactor, we write down the  \nproduct form directly. \n2x2 – 6x = 2x(                      )\nMaths2_Gr9_LB_Book.indb   37\n2014/09/08   09:06:27 AM\n\n38\t\nMATHEMATICS Grade 9: Term 3\n3.3\t Solving by factorisation (Part 2)\nsolving by factorising trinomials\nThe product of the first terms of the factors must be equal to the x2 term of the trinomial. \nThe product of the last terms of the factors must be equal to the last term (the constant \nterm) of the trinomial. \n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\nMeaning:  x.x = x2\n\t\nx2 + 5x + 6      =      (x + 2)(x + 3)\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\nMeaning:  2.3 = 6\nThe sum of the inner and outer products must be equal to the x term of the trinomial.\n\t\nx2 + 5x + 6      =      (x + 2)(x + 3)\t\n\t\nMeaning:  (2 + 3)x = 5x\n\t\n  The factors are of the form:  (x.x) + (a + b)x + (a.b) = (x + a)(x + b).\nDetermine the values of x which will make the following statements true.\nRemember to write the equation in the form expression = 0 so that you can use the zero-\nproduct property.\n1.\t x2 + 9x = −14\t\n2.\t\nx2 + 3x = 18\n3.\t x2 − 18x = −17\t\n4.\t\nx2 + 30 = 11x\n5.\t x2 = 13x + 30 \t\n6.\t\nx2 + 7x = 30\nMaths2_Gr9_LB_Book.indb   38\n2014/09/08   09:06:27 AM\n\n\t\nCHAPTER 3: EQUATIONS\t\n39\nsolving by factorising the difference between two squares\nRemember from the previous chapter: \nIf p and q are perfect squares, also “algebraic  \nsquares”, then:\n  p − q      = (\np  + \nq )(\np  − \nq )\n\t 9x4 − 4y2   = (\n9\n4\nx  + \n4\n2\ny )(\n9\n4\nx  − \n4\n2\ny )\n\t \t\n\t\n = (3x2 + 2y)(3x2 − 2y)\nDetermine the values of the unknown (x or a or n, etc.) which will make the following \nstatements true.\nRemember to write the equation in the form expression = 0 so that you can use the  \nzero-product property.\n1.\t x2 = 4\t\n2.\t\nx2 = 16\n3.\t 4a2 = 9\t\n4.\t\n81 = 9n2\n5.\t 25x2 = 36\t\n6.\t\n121x2 = 144\n7.\t 16p2 = 49\t\n8.\t\n64a2 = 25\n \nAn expression of the form  \na2 – b2 is called the difference \nbetween two squares.\nTo factorise a difference \nbetween squares, we use the \nidentity: a2 – b2 = (a + b)(a − b)  \nwhere a and b represent numbers \nor algebraic expressions.\nMaths2_Gr9_LB_Book.indb   39\n2014/09/08   09:06:28 AM\n\n40\t\nMATHEMATICS Grade 9: Term 3\n3.4\t Solving by factorisation (Part 3)\nsolving by using properties of exponents\n1.\t Write the following numbers as the product of  \ntheir prime factors.\n\t\n(a)\t 128\t\n(b)\t 243\n\t\n(c)\t 125\t\n(d)\t 2 401\n2.\t Determine the values of x which will make the  \nfollowing statements true.\n\t\n(a)\t 2x = 27\t\n(b)\t 3x = 35\n\t\n(c)\t 5x = 53\t\n(d)\t 7x = 74\n3.\t Determine the values of x which will make the following statements true.\n\t\n(a)\t 2x = 128\t\n(b)\t 3x = 243\n\t\n(c)\t 5x = 125\t\n(d)\t 7x = 2 401\n\t\n(e)\t 2x + 9 = 25\t\n(f)\t 27(3x) = 3\nIn the equation 2x = 16, the letter symbol (x) is the  \nexponent. Equations with the letter symbol as an  \nexponent are referred to as exponential  \nequations.\nAll numbers can be written \nas the product of their prime \nfactors:\n16 = 4 × 4 = 2 × 2 × 2 × 2 = 24\nFactorise the number until all \nthe factors are prime numbers.\nIf the base of the LHS is the \nsame as the base of the RHS, \nthen the exponent on the LHS \nmust be equal to the exponent \non the RHS.\nIf a x = a y, then x = y.\nMaths2_Gr9_LB_Book.indb   40\n2014/09/08   09:06:28 AM\n\n\t\nCHAPTER 3: EQUATIONS\t\n41\nmixed exercises for more practice \nDetermine the values of the unknown (x or m or b, etc.) which will make the following \nstatements true.\n1.\t 6\n3\nx  + x = −4x2\t\n2.\t\nx = x(2 − x)\n3.\t x2 + 2x = 15\t\n4.\t\nm2 + 4m = 21\n \n5.\t x2 + 3 = 4x\t\n6.\t\nb2 − 16b = −15\n7.\t 1 = a2\t\n8.\t\n25x2 = 49\n \n9.\t 2x − 25 = −9\t\n10.\t 81(3x) = 3\nMaths2_Gr9_LB_Book.indb   41\n2014/09/08   09:06:28 AM\n\n42\t\nMATHEMATICS Grade 9: Term 3\n3.5\t Set up equations to solve problems\nthe mathematical modelling process\nConsider this problem involving a practical situation.\nPrinting shop A charges 45c per page and R12 for binding a book. \nPrinting shop B charges 35c per page and R15 for binding a book. \nFor a book with how many pages will the two shops charge the same?\nYou can write an equation to describe the problem.\nLet the number of pages for which the work costs  \nthe same be x. Then \n45x + 1 200 = 35x + 1 500. \nNow solve the equation.\n\t45x + 1 200\t = 35x + 1 500\n\t 45x − 35x\t =  1 500 − 1 200\n\t\n10x\t = 300\n\t\nx\t = 30\nWe may now ask what the solution to the  \nmathematical problem (“x = 30”) means in terms of the  \npractical situation. When the equation was set up above,  \nthe symbol x was used as a placeholder for the number of  \npages in a book for which the two shops would charge  \nthe same. So – what does the solution tell you?\nNow check whether the two shops will charge the same for a book with 30 pages. \nAt shop A 30 pages will cost 30 × 45c = 1350c = R13,50. Binding is R12, total cost is R25,50.\nAt shop B 30 pages will cost 30 × 35c = 1050c = R10,50. Binding is R15, total cost is R25,50.\nThe solution to the mathematical problem is also a  \nsolution to the practical problem.\nThe equation represents a  \nmathematical problem that can \nbe solved without necessarily \nkeeping the practical situation  \nin mind. It is called a  \nmathematical model of the \npractical situation.\nWe describe this as analysing  \nthe mathematical model, to \nproduce a mathematical \nsolution.\nThe mathematical solution \nmay be interpreted to  \nestablish what it means in  \nthe practical situation.\nThe mathematical solution  \nshould be tested in the \npractical situation,  \nbecause mistakes may have \nbeen made.\nMaths2_Gr9_LB_Book.indb   42\n2014/09/08   09:06:28 AM\n\n\t\nCHAPTER 3: EQUATIONS\t\n43\nPractical \nproblem\nPractical \nsolution\nMathematical\nproblem\nMathematical\nsolution\nDoes the solution\nwork in practice?\nSolve the \nmathematical problem\nmathematical problem\nInterpret the \nmathematical terms\nDescribe the situation in \nWhen people work like this, we say they do mathematical modelling.\npractice your modelling skills\nFor each situation in questions 1 to 3, the mathematical model is outlined and some \nclues are provided. Fill in the missing information.\n1.\t Louis is 6 years older than Karin and Karin is 4 years older than Heidi. The sum of \ntheir ages is 53 years. How old is Heidi?\n\t\nModel:\t\nLet x be:  \t\nHeidi’s age \n\t\n\t\nThen:  \t\nKarin’s age will be \n\t\n\t\nAnd:  \t\n\t\n\t\nHence:  \t\n  = 53\n\t\nAnalysis:\t\nx + (x + 4) + (x + 10) = 53\n\t\n\t\n\t\nInterpretation:\t\nSo Heidi is: \n2.\t The sum of two numbers is 15. Three times the smaller number is 5 more than the \nlarger number. Calculate the two numbers. (Hint: Let the smaller number be x.)\n\t\nModel:\t\nLet x be:  \t\n\t\n\t\nThen:   \t\n  is the larger number\n\t\n\t\nHence: \t\n\t\nAnalysis:\t\n\t\nInterpretation:\t\nSo the smaller number is: \n\t\n\t\nAnd the larger number is: \nMaths2_Gr9_LB_Book.indb   43\n2014/09/08   09:06:28 AM\n\n44\t\nMATHEMATICS Grade 9: Term 3\n3.\t The sum of three consecutive even numbers is 108. What are the numbers?  \nHint: Consecutive numbers are numbers that follow on each other.  \nWe define an even number as a number of the form 2n where n is a counting number.\n\t\nModel:\t\nLet the first number be:  \t\n \n\t\n\t\nThen:  \t\n\t\n\t\n\t\n\t\n\t\nHence:  \t\n\t\nAnalysis:\t\n \n\t\n\t\n\t\nInterpretation:\t\nSo the first number is:\t\n\t\n\t\nAnd the second number is: \t\n\t\n\t\nAnd the third number is: \t\n4.\t Firm A calculates the cost of a job using the formula Cost = 500 + 30t, where t is the \nnumber of days it takes to complete the job.  \n\t\nFirm B calculates the cost of the same job using the formula Cost = 260 + 48t, where  \nt is the number of days needed to complete the job.\n\t\n(a)\t What would Firm A charge for a job that takes 10 days?\n\t\n(b)\t How long would Firm B take to complete a job for which their charge is R596?\n\t\n(c)\t Here is a specific job for which firms charge the same and take the same time to \n\t\ncomplete. How long does this job take?\nMaths2_Gr9_LB_Book.indb   44\n2014/09/08   09:06:28 AM\n\n\t\nCHAPTER 3: EQUATIONS\t\n45\n3.6\t Equations and ordered pairs\nwhen unknowns become variables \nIn the previous sections we dealt with equations which had fixed or limited solutions. \nThey only had one letter symbol, which in this case acted as a placeholder for the  \nvalue/s which will make the statement true.\nStudy the equation: y = 5x + 2 \n1.\t How many letter symbols does the equation have? (List them.)\n2.\t Is it possible to solve this “equation”? \n3.\t Complete the table.\nx\n12\n10\n20\n5\n6\n−5\n−10\n5x + 2\nfunctions as sets of ordered pairs\nA specific input number, for example 10, and the  \noutput number associated with it (52 in the case of  \nthe function described by y = 5x + 2) is called an  \nordered pair. Ordered pairs can be represented in a  \ntable like the one you completed in question 3  \nabove. \nOrdered pairs can also be written in brackets:  \n(input number; output number). \nFor example the ordered pairs you entered into  \nthe table in 3 can be written as  \n(12; 62), (10; 52), (20; 102), (5; 27), (6; 32),  \n(−5; −23), (−10; −48)\n1.\t Complete each table by writing the ordered pairs in brackets below the table, in the \ntable as shown in the example. Then choose two more input numbers and write \ndown two additional ordered pairs that belong to each given function.\n\t\nFor the function with the rule y = 4x + 5\nx\n−2\n0\n1\n2\n5\ny\n−3\n5\n9\n13\n25\n\t\n(−2; −3), (0; 5), (1; 9), (2; 13), (5; 25), and (10; 45) and (20; 85)\nIn the function indicated by  \ny = 5x + 2 the letter symbol in \nthe formula (x) represents the \ninput or independent variable \nwhile the other letter symbol \n(y) represents the output or \ndependent variable.\nIf there is precisely one value of \ny for each value of x, we say that\ny is a function of x.\nMaths2_Gr9_LB_Book.indb   45\n2014/09/08   09:06:28 AM\n\n46\t\nMATHEMATICS Grade 9: Term 3\n\t\n(a)\t For the function with the rule y = x2 + 9\t\nx\n5\n0\n−3\ny\n18\n34\n\t\n\t\n(5;34), (3; 18 ), (0; 9), (−3; 18), (−5; 34), and ( . . ; . . .), and ( . . ; . . .)\n\t\n(b)\t For the function with the rule y = 3x − 2\t\nx\n5\n1\n0\n−3\ny\n−17\n\t\n\t\n(5; 13), (1; 1), (0; −2), (−3; −11), (−5; −17), and (. . . ;  . . .) and ( . . ; . . .)\n\t\n(c)\t For the function with the rule y = 5x − 4\t\nx\n−5\n−3\n1\n2\ny\n21\n\t\n\t\n(−5; −29), (−3; −19 ), (1; 1), (2; 6), (5; 21), and (. . . ;  . . .) and ( . . ; . . .)\n\t\n(d)\t For the function with the rule y = 12 − 3x\t\nx\n1\n2\n3\n4\ny\n−3\n\t\n\t\n(1; 9), (2; 6 ), (3; 3), (4; 0), (5; −3), and (. . . ; . .) and ( . . ; . . .)\n\t\n(e)\t For the function with the rule y = x2 + 2\t\nx\n−12\n−7\n−2\n3\ny\n102\n\t\n\t\n(−12; 146), (−7; 51 ), (−2; 6), (3; 11), (10; 102), and (. . . ; . . .) and ( . . ; . . .)\n\t\n(f)\t For the function with the rule y = 2x + 2\t\nx\n0\n1\n2\n3\ny\n18\n\t\n\t\n(0; 3), (1; 4 ), (2; 6), (3; 10), (4; 18) and (. . . ; . . .) and ( . . ; . . .)\n3.\t (a)\t Which ordered pair belongs to both y = 3x − 2 and y = 5x − 4? \n\t\n(b)\t Which ordered pair belongs to both y = 12 − 3x and y = 5x − 4? \n4.\t Which ordered pair belongs to both y = 5x + 7 and y = 3x + 25?   \nMaths2_Gr9_LB_Book.indb   46\n2014/09/08   09:06:29 AM\n\nChapter 4\nGraphs\n\t\nCHAPTER 4: GRAPHS\t\n47\nIn this chapter you will learn more about making graphs to show how quantities change, \nand about interpreting graphs. Graphs can show how quantities increase and decrease, \nhow rapidly they increase and decrease, and where they have maximum and minimum \nvalues. You will pay special attention to graphs of quantities which change at a constant \nrates. These graphs are straight lines.\n4.1\t Global graphs........................................................................................................... 49\n4.2\t Changes at different rates......................................................................................... 58\n4.3\t Draw graphs from tables of ordered pairs................................................................. 61\n4.4\t Gradient................................................................................................................... 63\n4.5\t Finding the formula for a graph................................................................................ 66\n4.6\t x- and y-intercepts.................................................................................................... 71\n4.7\t Graphs of non-linear functions.................................................................................. 73\nMaths2_Gr9_LB_Book.indb   47\n2014/09/08   09:06:29 AM\n\n48\t\nMATHEMATICS Grade 9: Term 3\n5\n0\n10\n20\n30\n40\n50\n60\n70\n80\n90\n100\n110\n120\n130\n5\n10\n15\n10\n15\nMaths2_Gr9_LB_Book.indb   48\n2014/09/08   09:06:30 AM\n\n\t\nCHAPTER 4: GRAPHS\t\n49",
      "chapter_id": "3"
    },
    {
      "title": "Graphs",
      "content": "4\t Graphs\n4.1\t Global graphs\ndiscrete and continuous variables\nSibongile collects honey on his farm and puts it in large jars to sell. His business is \ndoing so well that he can no longer do all the work himself. He needs to get some help. \nSibongile knows that one person can normally fill 2 jars in 3 days. He sets up this table \nto help him determine how many full-time workers he should employ to fill different \nnumbers of jars in a five-day week.\nJars per week\n3\n1\n3\n6\n2\n3\n10\n13\n1\n3\n16\n2\n3\n20\n23\n1\n3\nWorkers\n1\n2\n3\n4\n5\n6\n7\n1.\t (a)\t If Sibongile needs to produce 40 jars a week, how many workers does he need?\n\t\n(b)\t How many jars can 9 workers fill in a week? \n\t\n(c)\t How many workers does Sibongile need to produce 15 jars per week?\n\t\n(d)\t What are the two variables in the above situation?\nIn a situation like the above, one can have any number of jars, as well as fractions of a jar. \nOne can have a whole number of jars (for example 4 jars) or a fractional quantity of jars \n(for example 62\n3 or 4,45 jars). The other variable in the above situation, the number of  \nfull-time employees, is different. Only whole numbers of people are possible.\nQuantities like the quantity of jars of honey, which can include any fraction, are  \nsometimes called “continuous quantities” or “continuous variables”. Quantities that  \ncan be counted, like a number of people or a number of motor cars or rivers or towns,  \nare sometimes called “discrete quantities” or “discrete variables”. \nWhen a graph of a discrete variable is drawn, it does not normally make sense to join \nthe dots with a line, but for some purposes it may be useful.\n2.\t Can you use the second graph on the next page to find out how many workers are \nneeded to fill 30 jars in a week, and how many to fill 40 jars? Check your answers by \ndoing calculations.\nMaths2_Gr9_LB_Book.indb   49\n2014/09/08   09:06:30 AM\n\n50\t\nMATHEMATICS Grade 9: Term 3\nHere is a graph of the information in  \nSibongile’s table.\n5\n0\n10\n15\n20\n25\n30\n0\n1\n2\n3\n4\n5\n6\n7\nNumber of jars filled\nNumber of workers\nHere is another graph of the same information.\n3.\t In what way are these two  \ngraphs different?\n10\n0\n20\n30\n40\n50\n60\n0\n2\n4\n6\n8\n10\n12\n14\nNumber of jars filled\nNumber of workers\nMaths2_Gr9_LB_Book.indb   50\n2014/09/08   09:06:31 AM\n\n\t\nCHAPTER 4: GRAPHS\t\n51\n4.\t In each case say whether the variables are “discrete” or “continuous”.\n\t\n(a)\t You order pizzas for a class party and you need 1 pizza for every 3 learners.\n\t\n(b)\t Your height measured at different stages as you grew up.\n\t\n(c)\t The speed the car is travelling as you drive to town.\n5.\t The line graph shows the number of cars that a company sold between July and \nDecember of 2014. \n0\n5\n10\n15\n20\n25\n30\n35\n40\n45\n50\nJuly\nAug\nSept\nOct\nNov\nDec\nMonth\nCar Sales 2014\nNumber of cars\n\t\n(a)\t Is the data shown in the graph discrete or continuous? Explain your answer\n\t\n(b)\t  How many cars were sold in August? \n\t\n(c)\t During which months were the maximum and minimum number of cars sold?\n\t\n(d)\t How many more cars were sold in November than in July? \n\t\n(e)\t During which months did the car sales decrease?\n\t\n(f)\t Would you say that the car sales generally improved over the 6 months? Explain \n\t\nyour answer.\nMaths2_Gr9_LB_Book.indb   51\n2014/09/08   09:06:31 AM\n\n52\t\nMATHEMATICS Grade 9: Term 3\n6.\t The graph below shows the population of elephants at a game reserve in South  \nAfrica between 1960 and 2010. Study the graph and answer the questions that  \nfollow. \n0\n5\n10\n15\n20\n25\n30\n35\n1960\n1970\n1980\n1990\n2000\n2010\nYear\nElephants in the game reserve \n    \nNumber of elephants\n\t\n(a)\t Did the elephant population increase or decrease between 1970 and 1990?\n\t\n(b)\t Between which years did the elephant population increase?\n\t\n(c)\t In which year were there the most elephants on the game farm?\n\t\n(d)\t Is the data in this graph discrete or continuous?\n\t\n(e)\t How many elephants do you think there were on the game reserve in 1995?\n\t\n(f)\t The following data shows the number of elephants at a different game reserve.   \n\t\nPlot this information on the grid above.\nYear\n1960\n1970\n1980\n1990\n2000\n2010\nElephants\n30\n25\n20\n15\n20\n35\n\t\n(g)\t Would you say that the second game reserve had more elephants than the first  \n\t\ngame reserve between 1960 and 2010? Explain your answer.\nMaths2_Gr9_LB_Book.indb   52\n2014/09/08   09:06:31 AM\n\n\t\nCHAPTER 4: GRAPHS\t\n53\nshowing increase and decrease on graphs\nThe graph below shows the temperature over a 24-hour period in a town in the Free State. \nThe graph was drawn by connecting the points that show actual temperature readings.\n05:00\n10\n15\n20\n25\n10:00\n15:00\n20:00\nTime of day\nTemperature in °C\n1.\t (a)\t Do you think the above temperatures were recorded on a summer day or a  \n\t\nwinter day?\n\t\n(b)\t At what time of the day was the highest temperature recorded, and what was  \n\t\nthis temperature?\n\t\n(c)\t During what part of the day did the temperature rise, and during what part did \n\t\nthe temperature drop?\n\t\n(d)\t During what part of the period when the temperature was rising did it rise  \n\t\nmost rapidly?\n\t\n(e)\t During what part of the day did the temperature drop most rapidly?\nMaths2_Gr9_LB_Book.indb   53\n2014/09/08   09:06:31 AM\n\n54\t\nMATHEMATICS Grade 9: Term 3\n2.\t Here are descriptions of the temperature changes on five different days.\n\tDay A: It is already warm early in the morning. The temperature does not change \nmuch during the day but late in the afternoon a breeze causes the temperature  \nto drop quite sharply.\n\tDay B: It is very cold early in the morning but it gets quite hot soon after the sun gets \nup. By midday a cold wind comes up and the temperature drops till late in the  \nafternoon. The wind then stops and it gets warmer again into the evening.\n\tDay C: It is warm in the early morning and the temperature remains about the same \ntill midday. then the temperature drops slowly during the afternoon.\n\tDay D: It is cold in the early morning and it remains cold for the whole day, except  \nfor a short time after lunch when the sun comes out for a while.\n\tDay E: It is warm early in the morning, but the temperature drops sharply soon after \nsunrise and remains low until mid-afternoon, when it slowly warms up a little.\nThe shapes of some temperature graphs for 24-hour periods, starting early in the \nmorning, are given below. Below each graph, write which of the above days is possibly \nrepresented by the graph.\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n \nMaths2_Gr9_LB_Book.indb   54\n2014/09/08   09:06:31 AM\n\n\t\nCHAPTER 4: GRAPHS\t\n55\nWater is supplied to a township from three reservoirs. The amount of water in each \nreservoir is measured each day at 08:00 am. The water level in reservoir A is represented \nin red on the graph below, and the water levels in reservoirs B and C are represented in \nblue and green respectively.\nMon\nTues\nWed\nThurs\nFri\nSat\nSun\n0\n0,5\n1,0\n1,5\n2,0\nWater level in millions of litres\nReservoir A\nReservoir B\nReservoir C\n\t\nThe daily water levels in the three reservoirs, in millions of litres, are also given in the \ntable below.\nMon.\nTue.\nWed.\nThu.\nFri.\nSat.\nSun.\nReservoir A\n2,2\n1,9\n1,6\n1,3\n1,2\n1,2\n1,2 \nReservoir B\n1,6\n1,4\n1,4\n1,5\n1,8\n1,9\n2,1\nReservoir C\n0,9\n1,1\n1,3\n1,5\n1,7\n1,9\n2,1\n3.\t You may use the graph or the table, or both, to find the answers to the questions below.\n\t\n(a)\t On which days does the water level in reservoir B increase from one day to the next? \n\t\n(b)\t On which of these days does the water level in reservoir B increase most, and by \n\t\nhow much does it increase from that day to the next?\n\t\n(c)\t By how much does the water level in reservoir B change each day?\n\t\n(d)\t By how much does the water level in reservoir C change each day?\n\t\n(e)\t Describe the water level situation from Friday to Sunday, in reservoir A.\nMaths2_Gr9_LB_Book.indb   55\n2014/09/08   09:06:32 AM\n\n56\t\nMATHEMATICS Grade 9: Term 3\n4.\t During a certain day, these changes occur in the temperature at a certain place.\n\t\nBetween 00:00 and 03:00, the temperature drops by 2°C.\t \t\n\t\nBetween 03:00 and 06:00, the temperature drops by 3°C.\t \t\n\t\nBetween 06:00 and 10:00, the temperature remains constant.\t\n\t\nBetween 10:00 and 12:00, the temperature rises by 3°C.\t\n\t\n\t\nBetween 12:00 and 16:00, the temperature remains constant.\t\n\t\nBetween 16:00 and 18:00, the temperature drops by 4°C.\t \t\n\t\nBetween 18:00 and 22:00, the temperature drops by 5°C.\t \t\n\t\nBetween 22:00 and 24:00, the temperature remains constant.  \t\n\t\nWhich of the graphs below show the above temperature changes? \n05:00\n10:00\n15:00\nTime of day\n20:00\nTemperature in °C\n\t\nGraph A\n05:00\n10:00\n15:00\nTime of the day\n20:00\nTemperature in °C\n\t\nGraph B\n05:00\n10:00\n15:00\nTime of the day\n20:00\nTemperature in °C\n\t\nGraph C\nMaths2_Gr9_LB_Book.indb   56\n2014/09/08   09:06:32 AM\n\n\t\nCHAPTER 4: GRAPHS\t\n57\n5.\t Write a verbal description, like in question 4, of the temperature changes shown in \ngraph A in question 4.\n\t\nBetween \n and \n, the temperature \n\t\nBetween \n and \n, the temperature \n\t\nBetween \n and \n, the temperature \n\t\nBetween \n and \n, the temperature \n\t\nBetween \n and \n, the temperature \n\t\nBetween \n and \n, the temperature \n\t\nBetween \n and \n, the temperature \n\t\nBetween \n and \n, the temperature \n6.\t Write a verbal description, like in question 4, of the temperature changes shown in \ngraph C in question 4.\n\t\nBetween \n and \n, the temperature \n\t\nBetween \n and \n, the temperature \n\t\nBetween \n and \n, the temperature \n\t\nBetween \n and \n, the temperature \n\t\nBetween \n and \n, the temperature \n\t\nBetween \n and \n, the temperature \n\t\nBetween \n and \n, the temperature \n\t\nBetween \n and \n, the temperature \n7.\t Look at graph A in question 4.\n\t\n(a)\t By how much does the temperature drop from 13:00 to 19:00? \n\t\n(b)\t By how much does the temperature drop from 19:00 to 21:00? \n\t\n(c)\t When does the temperature drop most rapidly, from 13:00 to 19:00 or from  \n\t\n19:00 to 21:00? Explain your answer.\n8.\t Look at graph C in question 4.\n\t\n(a)\t By how much does the temperature increase from 07:00 to 09:00? \n\t\n(b)\t By how much does the temperature increase from 09:00 to 13:00? \n\t\n(c)\t When does the temperature increase more rapidly, from 07:00 to 09:00 or from \n\t\n09:00 to 13:00? Explain your answer.\nMaths2_Gr9_LB_Book.indb   57\n2014/09/08   09:06:33 AM\n\n58\t\nMATHEMATICS Grade 9: Term 3\n9.\t Look at graph B in question 4.\n\t\n(a)\t By how much does the temperature drop from 16:00 to 18:00? \n\t\n(b)\t By how much does the temperature drop from 18:00 to 22:00? \n\t\n(c)\t When does the temperature drop more rapidly, from 16:00 to 18:00 or from  \n\t\n18:00 to 22:00? Explain your answer.\n4.2\t Change at different rates\nThe water levels in kilolitres (kl) in different water storage  \ntanks over a period of 30 hours are represented on the  \ngraphs below and on the next page. \n1.\t (a)\t In which  tanks does the water level rise during the 30-hour period? \n\t\n(b)\t In which  tanks does the water level drop during the 30-hour period? \n2.\t How much water is there at the start of the 30-hour period, in each of the tanks?\n3.\t (a)\t Which tank is losing water most rapidly? Explain your answer.\n\t\n(b)\t Which tank is gaining water most slowly? Explain your answer.\n5\n0\n0\n5\n10\n15\n20\n25\n30\n10\n15\nHours\nWater level in kilolitres\nTank A\n20\n25\n30\n5\n0\n0\n5\n10\n15\n20\n25\n30\n10\n15\nHours\nWater level in kilolitres\nTank B\n20\n25\n30\n1 kilolitre = 1 000 litre\nMaths2_Gr9_LB_Book.indb   58\n2014/09/08   09:06:33 AM\n\n\t\nCHAPTER 4: GRAPHS\t\n59\n5\n0\n0\n5\n10\n15\n20\n25\n30\n10\n15\nHours\nWater level in kilolitres\nTank C\n20\n25\n30\n5\n0\n0\n5\n10\n15\n20\n25\n30\n10\n15\nHours\nWater level in kilolitres\nTank D\n20\n25\n30\n5\n0\n0\n5\n10\n15\n20\n25\n30\n10\n15\nHours\nWater level in kilolitres\nTank E\n20\n25\n30\n5\n0\n0\n5\n10\n15\n20\n25\n30\n10\n15\nHours\nWater level in kilolitres\nTank F\n20\n25\n30\n3.\t Complete the table. Use negative numbers for decreases.\nChange over each hour\nChange over any period of 5 hours\nTank A\nTank B\nTank C\nTank D\nTank E\nTank F\nMaths2_Gr9_LB_Book.indb   59\n2014/09/08   09:06:34 AM\n\n60\t\nMATHEMATICS Grade 9: Term 3\nIf a constant stream of water is pumped into a tank so  \nthat the water level is increased by 3 kilolitre in each  \nhour, we say:\nWater is pumped into the tank at a constant rate of  \n3 kilolitres per hour.\n4.\t (a)\t Tank G contains 12 kilolitres at the beginning of a 30-hour period. Water is then \n\t\npumped into it at a constant rate of 3 kilolitres per hour. Draw a dotted line  \n\t\ngraph to show the water level in Tank G on the graph sheet below. \n\t\n(b)\t Tank H also contains 12 kilolitres at the beginning of a 30-hour period. Water is \n\t\nthen pumped into at a constant rate of 1,5 kilolitres per hour. Draw a solid line \n\t\ngraph to show the water level in Tank H on the graph sheet below.\n5\n0\n0\n10\n20\n30\n40\n50\n60\n10\n15\nHours\nWater level in kilolitres\n20\n25\n30\n5.\t Complete the table for tanks G and H over the 30-hour period.\nHours\n0\n5\n10\n15\n20\n25\n30\nKilolitres in tank G\n12\nKilolitres in tank H\n12\nMaths2_Gr9_LB_Book.indb   60\n2014/09/08   09:06:35 AM\n\n\t\nCHAPTER 4: GRAPHS\t\n61\n4.3\t Draw graphs from tables of ordered pairs\nA “coordinate” graph shows the relationship between two variables, the dependent and \nindependent variable in a function. The value of the dependent variable depends on the \nvalue given to the independent variable, hence its name. Sometimes there is no pattern \nto the relationship between the two variables and sometimes there is. In Grade 9 we will \nfocus on graphs where there is a pattern to the relationship. Specifically, we will focus on \ngraphs of linear functions. The graph of a linear function is a straight line.\ngraphs of functions with constant differences\n1.\t Complete the table. \nx\n0\n1\n2\n3\n4\n5\n6\n7\n8\n9\nFunction A\n8\n8\n1\n2  \n9\n9\n1\n2\nFunction B\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\nFunction C\n0\n1\n1\n2\n3\n4\n1\n2\nFunction D\n−4\n−2\n0\n2\n2.\t Represent each of the functions in question 1 with a graph by plotting the points on \nthe grids below. You may join the points in each case and write down the constant \ndifference between the function values.\n\t\nFunction A\t\nFunction B\n\t\nConstant difference = \n   \t\nConstant difference = \n0\n10\n—10\n—10\n10\n0\n10\n—10\n—10\n10\nMaths2_Gr9_LB_Book.indb   61\n2014/09/08   09:06:36 AM\n\n62\t\nMATHEMATICS Grade 9: Term 3\n\t\nFunction C\t\nFunction D\n\t\nConstant difference = \n   \t\nConstant difference = \n0\n10\n—10\n—10\n10\n0\n10\n—10\n—10\n10\n3.\t Some of the graphs you have drawn “go upwards” (or downwards) quickly, like a steep \nhill or mountain; others “go up” (or down) slowly. \n\t\n(a)\t Is there a link between the constant difference and the “steepness” of the graph?\n\t\n(b)\t Try to explain why this is the case.\n4.\t (a)\t Complete the following tables.\nx\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n2x + 3\n5x + 4\n3x + 3\n\t\n(b)\t Determine the difference between consecutive terms in each of the above three \n\t\nnumber sequences. What do you notice about this difference?\n\t\n(c)\t What difference between consecutive terms would you expect in the output \n\t\nnumbers for 4x + 5, if the input numbers are the natural numbers 1; 2; 3;  . . . . . ?\nMaths2_Gr9_LB_Book.indb   62\n2014/09/08   09:06:36 AM\n\n\t\nCHAPTER 4: GRAPHS\t\n63\n4.4\t Gradient\nThe “steepness” or slope of a line can be indicated by a number, as described below. This \nnumber is called the gradient of the line.\nThe gradient is the vertical change divided by the  \nhorizontal change as you move from left to right  \non the line.\nGradient = \nvertical change\nhorizontal change\n5\n10\n5\n10\n15\n5\n10\n0\n5\n10\n15\nHorizontal change\nHorizontal change\nVertical change\nVertical change\nThe gradient of the blue line above is 9\n10  = 0,9.\nThe gradient of the green line is −13\n10  = −1,3.\nMaths2_Gr9_LB_Book.indb   63\n2014/09/08   09:06:37 AM\n\n64\t\nMATHEMATICS Grade 9: Term 3\nNote that the horizontal change is always taken to be positive (moving to the right), but \nthe vertical change can be positive (if it is upwards) or negative (if it is downwards).\n1.\t A certain line passes through the points (2; 3) and (8; 15). A straight line is drawn \nthrough the two points.\n\t\n(a)\t Try to think of a way in which you can work out the gradient of the line that  \n\t\npasses through the two points. \n\t\n(b)\t Plot the two points on the graph  \n\t\nsheet on the right.\n\t\n(c)\t What horizontal change and  \n\t\nvertical change is needed to move  \n\t\nfrom the point (2; 3) to the point \n\t\n(8; 15)? You may draw arrows on  \n\t\nyour graph to help you to think  \n\t\nclearly about this.\n\t\n(d)\t Work out the gradient of the line that passes through the two points.\n2.\t Complete the table and plot graphs of  \ny = 2x + 3 and y = −2x + 5 on the given  \ngraph sheet.\nx\n−3\n1\n3\n5\n2x + 3\n−2x + 5\n10\n10\n–10\n0\n–10\n10\n10\n–10\n0\n–10\nMaths2_Gr9_LB_Book.indb   64\n2014/09/08   09:06:38 AM\n\n\t\nCHAPTER 4: GRAPHS\t\n65\n3.\t Work out the gradients of the  graphs of y = 2x + 3 and y = −2x + 5. You may use the \ncoordinates of any of the points you have plotted.\nSuppose the coordinates of point A are (xA; yA) and the coordinates of B are (xB; yB).\nThe gradient of line AB is: mAB = \nVertical change\nHorizontal change  = y\ny\nx\nx\nA\nB\nA\nB\n−\n−\n \nIn summary:\nIf you have two points A (xA; yA) and B (xB; yB) then the\nformula for the gradient is: m = y\ny\nx\nx\nA\nB\nA\nB\n−\n−\nExamples of finding the gradient between two points:\nCalculate the gradient of the line that goes through the points:\n(a)\tA(2; 5) and B(4 ; 1)\n\t\nm\t = y\ny\nx\nx\nA\nB\nA\nB\n−\n−\n\t\t\n\t\n\t\n= 5\n1\n2\n4\n−\n−\n\t  \n\t\n\t\n= 4\n2\n−\n\t\n\t\n= −2\n(b)\tC(2; 2) and D(−6; 0)\n\t\nm\t = y\ny\nx\nx\nC\nD\nC\nD\n−\n−\n\t\n\t\n= \n2\n0\n2\n6\n−\n−−\n(\n)\n\t\n\t\n= 2\n8\n\t\n\t\n= 1\n4\n(c)\tA(0; −1) and B(1; 1)\n\t\nm\t = y\ny\nx\nx\nA\nB\nA\nB\n−\n−\n\t\n\t\n= −−\n−\n1\n1\n0\n1\n\t\n\t\n= −\n−\n2\n1\n\t\n\t\n= 2\nx\ny\nHorizontal change (x – x )\nB\nA\nVertical change (y  – y )\nA\nB\nB(x  ; y  )\nB\nB\nA(x  ; y )\nA\nA\n0\nMaths2_Gr9_LB_Book.indb   65\n2014/09/08   09:06:40 AM\n\n66\t\nMATHEMATICS Grade 9: Term 3\nThe gradient of a straight line is the same everywhere, \nso it doesn’t matter which 2 points you use to \ndetermine the gradient.\ndetermine the gradient\nDo the following task in your exercise book.\n1.\t Determine the gradient of the lines that go through the following points:\n\t\n(a)\t A(2; 10) and B(6; 12)\t\n(b)\t C(1; 3) and D(−2; −3)\t\n(c)\t E(0; 3) and F(4; −1)\n\t\n(d)\t G(5; 2), H(4; 4) and I(2; 8)\n2.\t Determine the gradient of the following lines:\n\t\n(a)\t \t\n(b)\t\n\t\n(c)\t\n\t\nx\ny\n1\n2\n3\n1\n–1\n–1\n–2\n–3\n–2\n–3\n0\n2\n3\nx\ny\n1\n2\n3\n1\n–1\n–1\n–2\n–3\n0\n2\nA(2; –1)\nA(–1; 2)\n3\nx\ny\n1\n2\n3\n1\n–1\n–1\n–2\n–3\n–2\n–3\n0\n4.5\t Finding the formula for a graph\ntables and formulas\n1.\t Each table on the next page shows values for a relationship represented by one of \nthese rules:\n\t\ny = −2x + 3\t \t\n\t\ny = 2x − 5\t \t\n\t\n\t\ny = −3x + 5\t \t\n\t\n\t\ny = −3(x + 2)\n\t\ny = 3x + 2\t\n\t\n\t\ny = 5(x − 2)\t\t\n\t\n\t\ny = 2x + 3\t \t\n\t\n\t\ny = 2x + 5\n\t\ny = −3x + 6\t \t\n\t\ny = 5x + 10\t \t\n\t\n\t\ny = 5x − 10\t \t\n\t\n\t\ny = −x + 3\nMaths2_Gr9_LB_Book.indb   66\n2014/09/08   09:06:40 AM\n\n\t\nCHAPTER 4: GRAPHS\t\n67\n\t\n(a)\t Complete the tables below by extending the patterns in the output values.\nA.\nx\n0\n1\n2\n3\n4\n5\n6\n7\ny\n2\n5\n8\nB.\nx\n0\n1\n2\n3\n4\n5\n6\n7\ny\n3\n1\n−1\n−3\nC.\nx\n0\n1\n2\n3\n4\n5\n6\n7\ny\n−10\n−5\n0\n5\nD.\nx\n0\n1\n2\n3\n4\n5\n6\n7\ny\n−5\n−3\n−1\nE.\nx\n0\n1\n2\n3\n4\n5\n6\n7\ny\n6\n3\n0\nF.\nx\n0\n1\n2\n3\n4\n5\n6\n7\ny\n3\n2\n1\n0\nG.\nx\n0\n1\n2\n3\n4\n5\n6\n7\ny\n3\n5\n7\n\t\n(b)\t For each table, describe what you did to produce more output values. Also write \n\t\ndown the rule (formula) that corresponds to the table.\nYou may have noticed that the equations of  \nstraight lines look similar.\nThe equation of a straight line is: y = mx + c \nm tells us the gradient of the line. \nc tells us where the line crosses the y-axis.  \nThis is called the y-intercept and it has the  \ncoordinates (0; c).\nGradient\nGradient means the steepness or \nslope of the line. \nIntercept\nThe point where a line crosses \none of the axes.\nMaths2_Gr9_LB_Book.indb   67\n2014/09/08   09:06:40 AM\n\n68\t\nMATHEMATICS Grade 9: Term 3\n• the line y = 3x + 4 has a gradient of 3 and the y-intercept is  (0 ; 4).\n• the equation of a line with a gradient of −2 and y-intercept of (0 ;10) is \t\n\t\ny = −2x + 10.\n• the line y = 2x has a gradient of 2 and the y-intercept is (0; 0).\n• the line y = 5 has a gradient of 0 and the y-intercept is (0; 5).\n• What is the gradient and y-intercept of the line 2y = 6x + 10? \nIf you said m = 6 and c = 10 you would be wrong. The equation is not in standard form. \nThe equation must be written in standard form before you can read off the values of the \ngradient and the y-intercept.\n\t\n\t\n2y = 6x + 10\t ➝ Divide both sides by 2\n\t\n\t\n  y = 3x + 5 \nTherefore the gradient is 3 and the y-intercept is  \n(0; 5).\nIf m > 0 the line will be increasing.\nIf m < 0 the line will be decreasing.\nIf the line is horizontal m = 0.\nIf the line is vertical m is undefined.\n2.\t Complete the following table:\nEquation\nGradient\ny-intercept\ny = 3x + 5\ny = x\n2  − 7\ny = 2 − 3x \n−y = 5x − 10\ny = 3\n1\n(0; 0)\n−2\n(0; −7)\n3.\t Write each of the following equations in standard form and then determine the \ngradient and y-intercept.\n\t\n(a)\t 2y + 4x = 10\t\n(b)\t −3x = y + 4\t\n(c)\t 3x − 4 = y\nStandard form\nThe standard form of a straight \nline graph is y = mx + c.\nOn one side there should only \nbe a “y” (with a coefficient of 1).\nMaths2_Gr9_LB_Book.indb   68\n2014/09/08   09:06:41 AM\n\n\t\nCHAPTER 4: GRAPHS\t\n69\n\t\n(d)\t 3y + 6 = x\t\n(e)\t y = −3x + 4y − 12\t\n(f)\t y = 3x − 2\n\t\n(g)\t y = 1\n4 x + 6\t\n(h)\t y = −12\t\n(i)\t x = 15\nDetermine the equation of a straight line\nThe equation of a straight line is y = mx + c. If you need to determine the equation of a \nstraight line then all you need to know are the values of m and c.\nIf you know the values of two points on the graph then you can determine the  \ngradient using the formula: m = y\ny\nx\nx\nA\nB\nA\nB\n−\n−\n \nOnce you know the gradient you can calculate the value of the y-intercept using \nsubstitution. \nExample 1: Determine the equation of the straight line that goes through (1; 1) and (5; 13).\nStep 1:\t\nCalculate the gradient.\n\t\n\t\n\t\n\t\n\t\nm = y\ny\nx\nx\nA\nB\nA\nB\n−\n−\n = 1\n13\n1\n5\n−\n−\n = −\n−\n12\n4  = 3\nStep 2:\t\nSince you now know m = 3 you can substitute it into the equation y = mx + c.\n\t\n\t\n\t\n\t\n\t\nTherefore y = 3x + c.\nStep 3:\t\nTo determine c you need to substitute the coordinates of a point on the line \n\t\n\t\ninto the equation. (It can be either of the points that were given, so choose \t\t\n\t\n\t\nthe easier one.)\n\t\n\t\n\t\nSubstitute (5; 13) into y = 3x + c\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n (13) = 3(5) + c\n\t\n\t\n\t\n\t\n \t\n\t\n\t\n\t\n   13 = 15 + c\n\t\n\t\n\t\n\t\n  \t\n\t\n\t\n 13 − 15 =  c\n\t\n\t\n\t\n\t\n    \t \t\n\t\n  \t\n   −2 = c\nStep 4:\t\nWrite down the equation: y = 3x − 2\nMaths2_Gr9_LB_Book.indb   69\n2014/09/08   09:06:41 AM\n\n70\t\nMATHEMATICS Grade 9: Term 3\nExample 2: Determine the equation of the line that passes through (4; −1) and (7; 5).\nInformation\nm (Gradient)\nc (y-intercept)\ny = mx + c \n(Equation)\n(4; −1)\n(7; 5)\nm = y\ny\nx\nx\nA\nB\nA\nB\n−\n−\n  = −−\n−\n1\n5\n4\n7\n \n= −\n−\n6\n3\n                    = 2\nSubstitute m = 2 and (7; 5) \ny = mx + c\ny = 2x + c\n(5) = 2(7) + c\n5 = 14 + c\n−9 = c\ny = 2x − 9\nExample 3: Determine the equation of the line with a gradient of 4 passing through (2 ; 6).\nInformation\nm (Gradient)\nc (y-intercept)\ny = mx + c\n(Equation)\nm = 4\n(2; 6)\nm = 4 \nSubstitute m = 4 and (2; 6) \ny = mx + c\ny = 4x + c\n6 = 4(2) + c\n−2 = c\ny = 4x − 2\nYou may want to set your work out as shown in Examples 2 and 3 above.\n1.\t Determine the equation of the each of the straight lines passing through the points \ngiven.  \n\t\n(a)\t (3; 10) and (2; 5)\t\n(b)\t (−4; 5) and (2; 5)\t\n(c)\t (0; 0) and (4; −8)\n \n\t\n(d)\t (11\n2; 4) and (−1\n2; 12)\t\n(e)\t (3; 4) and (−7; −1)\t\n(f)\t (0; 3) and (−14; −4)\n2.\t Determine the equation of the straight line with:\n\t\n(a)\t a gradient of 5 and passing through the point (1; −3) \t\n\t\n(b)\t a gradient of −2 passing through the point (0; 0)\t \t\n\t\n(c)\t a y-intercept of 7 passing through the point (1; −3)\t\nMaths2_Gr9_LB_Book.indb   70\n2014/09/08   09:06:42 AM\n\n\t\nCHAPTER 4: GRAPHS\t\n71\n3.\t Determine the equations of the straight lines. Question (d) is a challenge.\nx\ny\n3\n0\ny\nA\n11\n2\nx\ny\n5\nB\n5\n0\nx\ny\n0\nC\n(1; 4)\nx\ny\n2\n0\n4\nD\n4.6\t x- and y-intercepts\nx\ny\n1\n2\nA\nD\nB\nC\n3\n1\n1\n1\n2\n3\n2\n3\n4\n0\n2\n3\n4\n1.\t Write down the coordinates of the points  \nwhere each line cuts the 2 axes:\nx-intercept\ny-intercept\nA\nB\nC\nD\n2.\t What do all the x-intercepts have in common?\n(a)\n(b)\n(c)\n(d)\nMaths2_Gr9_LB_Book.indb   71\n2014/09/08   09:06:43 AM\n\n72\t\nMATHEMATICS Grade 9: Term 3\n3.\t What do all the y-intercepts have in common?\n4.\t Determine the coordinates of the intercepts of the following straight line graphs.\n\t\n(a)\t y = 3x + 12\t\n(b)\t y = x − 3\n\t\n(c)\t y = −2x − 4\t\n(d)\t 2y = 6x + 12\n\t\n(e)\t 4x + 2y = 20\t\n(f)\t 13 − y = −26x\nvertical and horizontal lines\nSome special lines are so easy that you don’t need any  \nfancy methods to draw them or get their equation; you  \ncan just look at them. \n1.\t What do the following coordinate pairs have  \nin common?\n\t\n(2; 3), (2; −2), (2; 0) and (2; −3)\n2.\t Write down two more points that have an \nx-coordinate of 2.\nIf you plot these points on a set of axes you will see that they form a vertical line. \nThe equation of the line is x = 2. \n3.\t Will the two extra points you wrote down (question 2) also be on the line?\n4.\t Write down five coordinate pairs with x = −1. \n1\n0\n1\n2 1\n1\n2\n3\n4\n3\n4\n2\n3\n4\n2\n3\n4 y\nx\nMaths2_Gr9_LB_Book.indb   72\n2014/09/08   09:06:43 AM\n\n\t\nCHAPTER 4: GRAPHS\t\n73\n4.7\t Graphs of non-linear functions\nSome of the following relationships are represented by graphs on the next page. Identify \nwhich of the relationships are represented by which set of points on the graph. You may \nuse the tables below to help you to answer this question. For example, you may calculate \nsome output numbers by using the formulas and record this in the tables.\ny = −x2\t \t\n\t\n\t\ny = (−x)2\t\n\t\n\t\ny = x2 + 130\t\t\n\t\n\t\ny = (x − 5)2 + 10\ny = x2\t\n\t\n\t\n\t\ny = −x2 + 130\t\n\t\ny = 130 − x2\t\t\n\t\n\t\ny = x2 − 10x + 35\nWrite your answers here:\nSet of points in yellow \n \nSet of points in blue\t\n \nSet of points in red\t\n \nMaths2_Gr9_LB_Book.indb   73\n2014/09/08   09:06:43 AM\n\n74\t\nMATHEMATICS Grade 9: Term 3\n5\n0\n10\n20\n30\n40\n50\n60\n70\n80\n90\n100\n110\n120\n130\n5\n10\n15\n10\n15\nMaths2_Gr9_LB_Book.indb   74\n2014/09/08   09:06:44 AM\n\nChapter 5\nSurface area, volume and\ncapacity of 3D objects\n\t\nCHAPTER 5: SURFACE AREA, VOLUME AND CAPACITY OF 3D OBJECTS\t\n75\nBy now you should know how to calculate the surface area and volume of cubes, \nrectangular prisms and triangular prisms. In this chapter, you will revise how to do this, \npractise converting between equivalent units used for volume, and revise the difference \nbetween volume and capacity. You will investigate how to calculate the surface area and \nvolume of cylinders, and explore how the volumes of a prism and cylinder are affected \nwhen one or more of their dimensions is doubled.\n5.1\t Surface area.............................................................................................................. 77\n5.2\t Volume..................................................................................................................... 81\n5.3\t Capacity................................................................................................................... 85\n5.4\t Doubling dimensions and the effect on volume........................................................ 86\nMaths2_Gr9_LB_Book.indb   75\n2014/09/08   09:06:44 AM\n\n76\t\nMATHEMATICS Grade 9: Term 3\nMaths2_Gr9_LB_Book.indb   76\n2014/09/08   09:06:45 AM\n\n\t\nCHAPTER 5: SURFACE AREA, VOLUME AND CAPACITY OF 3D OBJECTS\t\n77\n5\t Surface area, volume and \n\t\ncapacity of 3D objects\n5.1\t Surface area\nsurface area of prisms\nThe surface area of an object is the total area of all of its faces added together. You \nlearnt the following formula in previous grades:\nSurface area of a prism = Sum of the areas of all its faces\nCalculate the surface area of the following objects to revise what you should already know.\n1.\t \t\n\t\n\t\n2.\t\n10 cm\n10 cm\n10 cm\n12 cm\n5 cm\n8 cm\n[We use SA for surface area.]\n3.\t \t\n\t\n\t\n4.\t\n25 mm\n25 mm\n100 mm\n60 mm\nh = 40 mm\n70 mm\n50 mm\nMaths2_Gr9_LB_Book.indb   77\n2014/09/08   09:06:46 AM\n\n78\t\nMATHEMATICS Grade 9: Term 3\n5.\t \t\n\t\n\t\n6.\n10 m\nx m\n10 m\n3 m\n4 m\nx m\n1,5 m\n2 m\n7.\t \t\n\t\n\t\n8.\n6 cm\n5 cm\n14 cm\n20 cm\n7 cm\n10 cm\n12 cm\n25 cm\nMaths2_Gr9_LB_Book.indb   78\n2014/09/08   09:06:46 AM\n\n\t\nCHAPTER 5: SURFACE AREA, VOLUME AND CAPACITY OF 3D OBJECTS\t\n79\ninvestigating the surface area of cylinders\nIn order to calculate the surface area of a cylinder, you need to know what shape the \nsurfaces of the cylinder are.\nThe surfaces of the top and base of a cylinder are made up of circles. The curved surface \nbetween the top and base of a cylinder can be unrolled to create a rectangle.\nSo the net of a cylinder looks like this:\nArea = πr2\nl = circumference of circle = 2πr\nArea = πr2\nb = h of cylinder\nSurface area of a cylinder\t = Area of all its surfaces\n\t\n= Area of top + Area of base + Area of curved surface \n\t\n= πr2 + πr2 + (l × b)\n\t\n= 2πr2 + (2πr × h)\n\t\n= 2πr(r + h)\nCan you explain why the \nlength of the rectangle is \nequal to the circumference \nof the top or base of the \ncylinder?\nMaths2_Gr9_LB_Book.indb   79\n2014/09/08   09:06:46 AM\n\n80\t\nMATHEMATICS Grade 9: Term 3\ncalculating the surface area of cylinders\nFrom the formula on the previous page, you can see that we need only know the radius \n(r) and the height (h) of a cylinder in order to work out its surface area.\n1.\t Calculate the surface areas of the following objects. Use π = 3,14 and round off all  \nyour answers to two decimal places.\n\t\nA.\t \t\n\t\nB.\n6 cm\nr = 6 cm\n8 m\nr = 4 m\n2.\t Calculate the surface area of a \ncylinder if its height is 60 cm and the \ncircumference of its base is 25,12 cm. \n3.\t Calculate the surface area of a \ncylinder if its height is 5 m and the \ncircumference of its base is 12,56 m. \n4.\t The outside of a cylindrical structure at a factory must be painted. Its radius is 3,5 m \nand its height is 8 m. How many litres of paint must be bought if 1 litre covers 10 m2? \n(The bottom of the structure will not be painted.)\nMaths2_Gr9_LB_Book.indb   80\n2014/09/08   09:06:47 AM\n\n\t\nCHAPTER 5: SURFACE AREA, VOLUME AND CAPACITY OF 3D OBJECTS\t\n81\n5.2\t Volume\nThe volume of an object is the amount of space it  \noccupies. We usually measure volume in cubic units,  \nsuch as mm3, cm3 and m3.\nformulas for volume of prisms\nThe general formula for the volume of a prism is:\nVolume of a prism  =  Area of base × height.\nTherefore, the formulas to work out the volumes of the following prisms are:\nRectangular prism\nCube\nTriangular prism\nb\nh\ns\ns\ns\nl\nh of prism\nbase \nof triangle\nh\nV = (l × b) × h\t \t\n\t\n  V = (s × s) × s\t\t\n\t\nV = (1\n2 base × hb) × hp \n\t\n\t\n\t\n\t\n      = s3\ncalculating the volume of prisms\n1.\t Calculate the volumes of the following prisms. \n\t\nA.\t \t\n\t\nB.\n12 cm\n5 cm\n8 cm\n4 m\n1,5 m\n2 m\nTo convert between cubic units, \nremember:\n1 cm3 = 1 000 mm3\n1 m3 = 1 000 000 cm3\nIn case of a triangular prism do \nnot confuse the height of the \nbase of the triangle (hb) with the \nheight of the prism (hp).\nMaths2_Gr9_LB_Book.indb   81\n2014/09/08   09:06:47 AM\n\n82\t\nMATHEMATICS Grade 9: Term 3\n\t\nC.\t \t\n\t\nD.\n15 m\n15 m\n15 m\n70 mm\n50 mm\nh = 40 mm\n60 mm\n2.\t (a)\t The area of the base of a rectangular prism is 32 m2 and its height is 12 m. What  \n\t\nis its volume?\n\t\n(b)\t The volume of a cube is 216 m3. What is the length of one of its edges?\n3.\t Calculate the volume of the following objects.\n\t\nA.\t \t\n\t\nB.\n6 cm\n5 cm\n14 cm\n20 cm\n7 cm\n10 cm\n12 cm\n25 cm\nMaths2_Gr9_LB_Book.indb   82\n2014/09/08   09:06:48 AM\n\n\t\nCHAPTER 5: SURFACE AREA, VOLUME AND CAPACITY OF 3D OBJECTS\t\n83\nvolume of cylinders\nYou also calculate the volume of a cylinder by multiplying the  \narea of the base by the height of the cylinder. The base of a  \ncylinder is circular, therefore:\nVolume of a cylinder = Area of base × h\n\t \t\n\t\n \t\n   = πr 2 × h\n1.\t Calculate the volume of the following cylinders. Use π = 3,14 and round off all \nanswers to two decimal places.\n\t\nA.\t \t\n\t\nB.\n15 cm\nr = 5 cm\n12 cm\nr = 3 cm\n\t\nC.\t \t\n\t\nD.\n10 m\ndiameter = 15 m\n9 m\ndiameter  \n= 14 m\nh\nr\nArea of circle = πr2 \nMaths2_Gr9_LB_Book.indb   83\n2014/09/08   09:06:48 AM\n\n84\t\nMATHEMATICS Grade 9: Term 3\n2.\t Without using a calculator, calculate the volume of cylinders with the following \nmeasurements. Use π = 22\n7 .\n\t\n(a)  r = 14 cm, h = 20 cm\n(b)  r = 7 cm, h = 35 cm\n(c)  diameter = 28 cm, h = 50 cm\n(d)  diameter = 7 cm, h = 10 cm\n3.\t Calculate the volume of the following object. Use a calculator and round off all \nanswers to two decimal places.\n20 cm\n30 cm\n16 cm\n10 cm\ndiameter = 8 cm\nMaths2_Gr9_LB_Book.indb   84\n2014/09/08   09:06:49 AM\n\n\t\nCHAPTER 5: SURFACE AREA, VOLUME AND CAPACITY OF 3D OBJECTS\t\n85\n5.3\t Capacity\nRemember that the capacity of an object is the  \namount of space inside the object. You can think of  \nthe capacity of an object as the amount of liquid that  \nthe object can hold.\nThe volume of a solid block of wood is 10 cm × 6 cm × 4 cm = 240 cm3.\nThe same block of wood is carved out to make a hollow container with inside  \nmeasurements of 8 cm × 4 cm × 3 cm. (Its walls are 1 cm thick.) The amount of space \ninside the container must be calculated using the inside measurements. So the capacity \nof the container is 8 cm × 4 cm × 3 cm = 96 cm3.\nA. Solid block with outside\nmeasurements\nB. Hollowed block with inside\nmeasurements\n10 cm\n6 cm\n4 cm\n10 cm\n6 cm\n4 cm\n8 cm\n3 cm\n4 cm\n1.\t Write, in ml, the volume of water that would fill container B.\n2.\t If the walls and bottom of container B were 0,5 cm thick, what  \nwould its capacity be? Write the answer in ml.\n3.\t The inside measurements of a swimming pool are 9 m × 4 m × 2 m. What is the \ncapacity of the pool in kl?\nThe volume of an object is \nthe amount of space that the \nobject itself takes up.\nRemember:\n1 cm3 = 1 ml\n1 m3 = 1 kl\nMaths2_Gr9_LB_Book.indb   85\n2014/09/08   09:06:49 AM\n\n86\t\nMATHEMATICS Grade 9: Term 3\n5.4\t Doubling dimensions and the effect on volume\ndoubling the dimensions of a prism\nThe first prism below measures 5 cm × 3 cm × 2 cm. The other diagrams show the prism \nwith one or more of its dimensions doubled.\n1.\t Work out the volume of each prism.\n3 cm\n2 cm\n5 cm\nOne dimension doubled\n3 cm\n2 cm\n10 cm\nTwo dimensions doubled\n6 cm\n2 cm\n10 cm\nThree dimensions doubled\n6 cm\n4 cm\n10 cm\n2.\t Complete the following:\n\t\n(a)\t When one dimension of a prism is doubled, the volume \n.\n\t\n(b)\t When two dimensions of a prism are doubled, the volume increases by \n times.\n\t\n(c)\t When all three dimensions of a prism are doubled, the volume increases by \n \n\t\ntimes.\nMaths2_Gr9_LB_Book.indb   86\n2014/09/08   09:06:49 AM\n\n\t\nCHAPTER 5: SURFACE AREA, VOLUME AND CAPACITY OF 3D OBJECTS\t\n87\n3.\t The volume of a prism is 80 cm3. What is its volume if:\n\t\n(a)\t its length is doubled? \t\t\n\t\n\t\n\t\n\t\n\t\n\t\n(b)\t its length and breadth are doubled? \t \t\n\t\n\t\n(c)\t its length, breadth and height are doubled?\t\ndoubling the dimensions of a cylinder\nThe first cylinder below has a radius of 7 cm and a height of 2 cm. The other diagrams \nshow the cylinder with one or more of its dimensions doubled.\n1.\t Work out the volume of each cylinder.\nr = 7 cm\nh = 2 cm\nOnly height doubled\nr = 7 cm\nh = 4 cm\nOnly radius doubled\nr = 14 cm\nh = 2 cm\nRadius and height doubled\nr = 14 cm\nh = 4 cm\nMaths2_Gr9_LB_Book.indb   87\n2014/09/08   09:06:50 AM\n\n88\t\nMATHEMATICS Grade 9: Term 3\n2.\t Complete the following:\n\t\n(a)\t When the height of a cylinder is doubled, the volume \n.\n\t\n(b)\t When the radius of a cylinder is doubled, the volume increases by \n times.\n\t\n(c)\t When height and radius of cylinder are doubled, the volume increases by \n times.\n3.\t The volume of a cylinder is 462 cm3. What is its volume if:\n\t\n(a)\t its height is doubled?\t \t\n\t\n\t\n\t\n\t\n(b)\t its radius is doubled?\t \t\n\t\n\t\n\t\n\t\n(c)\t its height and radius are doubled?\t\n4.\t (a)\t Study the following tables. Without using the formulas to calculate volume,  \n\t\ncomplete the last column in each table. (Hint: Identify which dimensions are \n\t\ndoubled each time, then work out the volume accordingly.)\nRectangular prism\nLength (l) \nin m\nBreadth (b) \nin m\nHeight (h) \nin m\nVolume (V) \nin m3\n4\n2\n1\n4\n4\n1\n8\n2\n1\n8\n2\n2\n8\n4\n2\nCylinder\nRadius (r) \nin m\nHeight (h) \nin m\nVolume (V) \nin m3\n3,5\n4\n7\n4\n3,5\n8\n7\n8\n\t\n(b)\t Explain how you worked out the answers in the tables.\nMaths2_Gr9_LB_Book.indb   88\n2014/09/08   09:06:50 AM\n\nChapter 6\nTransformation geometry\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n89\nIn Grade 8, you learnt how to describe and perform translations, reflections and rotations \non a coordinate system. In such transformations, the original figure and its image are \nalways congruent. In this grade, you will explore in more detail the change in the \ncoordinates of original figures and their images after different types of reflections and \ntranslations.\nYou will then revise how a figure is enlarged or reduced when its sides are multiplied \nby the same number, called a scale factor, and how the scale factor affects the area and \nperimeter of an image. In enlargements and reductions, the corresponding sides of the \noriginal figure and its image are in proportion, which makes the figures similar. You will also \nperform enlargement and reduction of figures on the coordinate system, and investigate \nthe coordinates of the vertices of such figures.\n6.1\t Points on a coordinate system.................................................................................. 91\n6.2\t Reflection (flip)......................................................................................................... 92\n6.3\t Translation (slide)...................................................................................................... 96\n6.4\t Enlargement (expansion) and reduction (shrinking)................................................ 100\nMaths2_Gr9_LB_Book.indb   89\n2014/09/08   09:06:50 AM\n\n90\t\nMATHEMATICS Grade 9: Term 3\nMaths2_Gr9_LB_Book.indb   90\n2014/09/08   09:06:51 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n91\n6\t Transformation geometry\n6.1\t Points on a coordinate system\nA rectangular coordinate system is  \nalso called a Cartesian coordinate  \nsystem. It consists of a horizontal  \nx-axis and a vertical y-axis. \nThe intersection of the axes is called  \nthe origin, and represents the point  \n(0; 0). \nAny point can be represented on a  \ncoordinate system using an x-value  \nand a y-value. These numbers are  \ncalled coordinates, and describe the  \nposition of the point with reference  \nto the two axes.\nThe coordinates of a point are always written in a certain order: \nThe horizontal distance from the origin  \n(x-coordinate) is written first.\nThe vertical distance from the origin  \n(y-coordinate) is written second.\nThese numbers, called an ordered pair, are  \nseparated by a semi-colon (;) and are placed  \nbetween brackets. Here is an example of an  \nordered pair: (4; 3) (see on the coordinate  \nsystem above).\nThe x-axis and y-axis divide the coordinate  \nsystem into four sections called quadrants.  \nThe diagram alongside shows how the  \nquadrants are numbered, and also whether the  \nx- and y-coordinates are negative or positive in each quadrant.\n1.\t In which quadrant will the following points be plotted?\n\t\n(a)\t (−4; 1)\t\n\t (b)\t (−1; −5)\t\n\t\n(c)\t (4; −3)\t\n\t (d)\t (5; 2)\t\n\t\n2.\t Plot the points in question 1 on the coordinate system above.\n4\n1\n2\n3\n–4\n–5\n–6\n–1\n–2\n–3\n5\n1\n2\n3\n–4\n–5\n–1\n–2\n–3\ny-axis\nx-axis\n(4; 3)\norigin\n0\nfirst quadrant\nsecond quadrant\nthird quadrant\nfourth quadrant\n(+; –)\n(+; +)\n(–; –)\n(–; +)\n–\n–\n+\n+\nx\ny\n0\nMaths2_Gr9_LB_Book.indb   91\n2014/09/08   09:06:51 AM\n\n92\t\nMATHEMATICS Grade 9: Term 3\nWhen a point is translated to a different position on a coordinate system, the new \nposition is called the image of the point. We use the prime symbol (') to indicate \nan image. For example, the image of A is indicated by A' (read as “A prime”). If the \ncoordinates of A are labelled as (x; y), the coordinates of A' can be labelled as (x'; y'). \nWe write A ➝ A' and (x; y) ➝ (x'; y') to indicate that A is mapped to A'.\n6.2\t Reflection (flip)\nThe mirror image or reflection of a point is on \nthe opposite side of a line of reflection.\nThe original point and its mirror image are the same  \ndistance away from the line of reflection, and the line that joins the point and its image \nis perpendicular to the line of reflection.\nAny line on the coordinate system can be a line of reflection, including the x-axis, the \ny-axis and the line y = x.\nreflecting points in the x-axis, y-axis and the line y = x\n1.\t The points A(5; 4) and B(−3; −2) are plotted on a coordinate system.\n\t\n(a)\t Reflect points A and B in the x-axis and write down the coordinates of the  \n\t\nimages.\n\t\n(b)\t Reflect points A and B in the y-axis and write down the coordinates of the  \n\t\nimages.\n\t\n(c)\t Compare the coordinates of the original points with those of its images. What  \n\t\ndo you notice?\n“Reflecting a point in the x-axis” \nmeans that the x-axis is the line \nof reflection. \n4\n1\n2\n3\n–4\n–5\n–6\n–1\n–2\n–3\n5\n1\n2\n3\n–4\n–5\n–1\n–2\n–3\ny\nx\nA(5; 4)\nB(–3; –2)\n0\n4\nMaths2_Gr9_LB_Book.indb   92\n2014/09/08   09:06:51 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n93\n2.\t Write down the coordinates of the images of the following reflected points.\nPoint\nReflection in the x-axis\nReflection in the y-axis\n(−131; 24)\n(−459; −795)\n(x; y)\n3.\t The points J(−1; 5), K(−2; −4) and L(1; −2) are plotted on the coordinate system.  \nK' is the reflection of point K in the line y = x. This means that the line y = x is the  \nline of reflection.\n\t\n(a)\t Reflect J and L in the line y = x. \n\t\n(b)\t Write down the coordinates  \n\t\nof the images of the points.\n\t\n(c)\t What do you notice about  \n\t\nthe coordinates of the images  \n\t\nof the points in (b) above?\n\t\n(d)\t Use your observation in (c)  \n\t\nabove to complete this table.\nPoint\nCoordinates of the image of  \nthe point reflected in y = x\n(−1 001; −402)\n(459; −795)\n(−342; 31)\n(21; 67)\n(x; y)\n1\n–1\n–2\n–3\n–4\n–5\n–6\n–7\n–8\n–9\n2\n3\n4\n5\n6\n7\n8\n9\ny\nx\nL\n0\n0\n–2\n1\n2\n3\n4\n5\n6\n7\n–3\n–4\n–5\n–1\nK\nK'\nJ\ny = x\nMaths2_Gr9_LB_Book.indb   93\n2014/09/08   09:06:51 AM\n\n94\t\nMATHEMATICS Grade 9: Term 3\nWhile doing the previous activity, you may have noticed the following.\n• For a reflection in the y-axis, the sign of the x-coordinate changes and the  \ny-coordinate stays the same:  (x; y) → (−x; y) or x' = −x and y' = y,  \nfor example: (−3; 4)→ (3; 4)\n• For a reflection in the x-axis, the sign of the y-coordinate changes and the  \nx-coordinate stays the same:  (x; y) → (x; −y) or x' = x and y' = −y,  \nfor example: (−3; 4) → (−3; −4)\n• For a reflection in the line y = x, the values of the x- and y-coordinates are \ninterchanged:   \n(x; y) → (y; x) or x' = y and y' = x, for example: (−3; 4) → (4; −3).\n4.\t Investigate the effect of reflection in the line y = −x on the coordinates of a point.\n5.\t A is the point (5; −2). Write the coordinates of the mirror images of A if the point is \nreflected in:\n\t\n(a)\t the y-axis             \t\n\t\n\t\n\t\n\t\n\t\n(b)\t the line y = −x\n\t\n\t\n\t\n(c)\t the line y = x       \t\n\t\n\t\n\t\n\t\n\t\n(d)\t the x-axis                  \n\t\n                 \nreflecting geometric figures \nThe same principles as above apply when reflecting geometric figures.\n1.\t (a)\t Reflect ∆PQR in the x-axis, in the y-axis and in the line y = x in the coordinate  \n\t\nsystem (first reflect the vertices and then join the reflected points).\n1\n–1\n–2\n–3\n–4\n–5\n–6\n–7\n–8\n–9\n2\n3\n4\n5\n6\n7\n8\n9\ny\nx\n0\n0\n–2\n1\n2\n3\n4\n5\n6\n–3\n–4\n–5\n–6\n–1\nQ\nP\ny = x\nR\nMaths2_Gr9_LB_Book.indb   94\n2014/09/08   09:06:52 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n95\n\t\n(b)\t Look at your completed reflections in question 1(a), and write down the  \n\t\ncoordinates of the image points in the following table.\nVertices of triangle\nReflection in the \nx-axis\nReflection in the \ny-axis\nReflection in the \nline y = x\nP(−6; 3)\nQ(−5; −2)\nR(−2; 1)\n\t\n(c)\t What do you notice about ∆PQR, ∆P'Q'R', ∆P''Q''R'' and ∆P'''Q'''R'''?\n2.\t Reflect ∆DEF in the x-axis, in the y-axis and in the line y = x. \n1\n–1\n–2\n–3\n–4\n–5\n–6\n–7\n–8\n–9\n2\n3\n4\n5\n6\n7\n8\n9\ny\nx\n0\n0\n–2\n1\n2\n3\n4\n5\n6\n–3\n–4\n–5\n–6\n–1\nE\nD\nF\n3.\t A quadrilateral has the following vertices: A(1; 4), B(−6; 1), C(−2; −1) and D(7; 2). \nWithout performing the actual reflections, write down the coordinates of the  \nvertices of the image when the quadrilateral is:\n\t\n(a)\t reflected in the x-axis\n\t\n(b)\t reflected in the y-axis\n\t\n(c)\t reflected in the line y = x\nMaths2_Gr9_LB_Book.indb   95\n2014/09/08   09:06:52 AM\n\n96\t\nMATHEMATICS Grade 9: Term 3\n4.\t In each case state around which line the point was reflected.\n\t\n(a)\t (−4; 5) → (−4; −5)\t \t\n\t\n\t\n(b)\t (2; −3) → (−2; −3)\t \t\n\t\n\t\n(c)\t (−13; −3) → (−3; −13)\t\n\t\n\t\n(d)\t (1; 16) → (16; 1)\t\n\t\n\t\n\t\n(e)\t (12; −8) → (−12; −8)\t\n\t\n\t\n(f)\t (−7; −5) → (−5; −7)\t\t\n\t\n\t\n(g)\t (2; −3) → (−2; −3)\t \t\n\t\n6.3\t Translation (slide)\nRemember: A translation of a point or geometric figure on a coordinate system means \nmoving or sliding the point in a vertical direction, in a horizontal direction, or in both a \nvertical and horizontal direction.\ntranslating points horizontally or vertically on a \ncoordinate system\n1.\t Points R and W are plotted  \n\t\non a coordinate system.\n\t\n(a)\t Plot the image of point R  \n\t\nafter a translation of:\n• 5 units to the right\n• 5 units to the left\n• 2 units up\n• 2 units down\n\t\n(b)\t Plot the image of point W  \n\t\nafter a translation of:\n• 4 units to the right\n• 4 units to the left\n• 3 units up\n• 3 units down\n1\n–1\n–2\n–3\n–4\n–5\n–6\n–7\n–8\n–9\n2\n3\n4\n5\n6\n7\n8\n9\ny\nx\n0\n–2\n1\n2\n3\n4\n5\n6\n–3\n–4\n–5\n–6\n–1\nW\nR\nMaths2_Gr9_LB_Book.indb   96\n2014/09/08   09:06:52 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n97\n\t\n(c)\t Look at your completed translations in (a) and (b) on the previous page.  \n\t\nComplete the following table by writing down the coordinates of the original \n\t\npoints and their images after each translation.\n\t\nCoordinates of original points\nR(−1; 4)\nW(−2; −3)\nCoordinates of image after a translation to the right\nCoordinates of image after a translation to the left\nCoordinates of image after a translation up\nCoordinates of image after a translation down\n\t\n\t\n(d)\t Look at your completed table in (c) above. Choose the correct answers below to \t\n\t\nmake each statement true:\n• For translations to the right or left, the (x-value/y-value) changes and the \n(x-value/y-value) stays the same.\n• For translations up or down, the (x-value/y-value) changes and the  \n(x-value/y-value) stays the same.\n• For translations to the right, (add/subtract) the number of translated units \n(to/from) the x-value.\n• For translations to the left, (add/subtract) the number of translated units  \n(to/from) the x-value.\n• For translations up, (add/subtract) the number of translated units (to/from) \nthe y-value.\n• For translations down, (add/subtract) the number of translated units  \n(to/from) the y-value.\n2.\t Write down the coordinates of each image after the following translations.\nPoint\n3 units to the \nright\n4 units to the \nleft\n2 units up\n5 units down\n(3; 5)\n(−13; 42)\n(−59; −95)\n(x; y)\nMaths2_Gr9_LB_Book.indb   97\n2014/09/08   09:06:52 AM\n\n98\t\nMATHEMATICS Grade 9: Term 3\n3.\t Write down the coordinates of each image after the following translations:\nPoint\n4 units to the \nright and  \n3 units up\n2 units to the \nleft and \n1 unit up\n1 unit to the \nright and \n5 units down\n6 units to the \nleft and \n2 units down\n(4; 2)\n(−32; 21)\n(−68; −57)\n(x; y)\n\t\nWhile doing the previous activity, you may have noticed the following:\n• For a horizontal translation through the distance p, the x-coordinate increases  \nby the distance p if the slide is to the right, and decreases by the distance p if the \nslide is to the left. We may write x' = x + p, with p > 0 for a translation to the right, \nand p < 0 for a translation to the left. The y-coordinate remains the same, so  \n(x; y) → (x + p; y).\n• For a vertical translation through the distance q, the y-coordinate increases by \nthe distance q if the slide is upwards, and decreases by the distance q if the slide is \ndownwards. We may write y' = y + q, with q > 0 for a translation vertically upwards, \nand q < 0 for a translation vertically downwards. The x-coordinate remains the \nsame, so (x; y) → (x; y + q).\ntranslation of geometric figures on a coordinate system\n1.\t (a)\t Translate ∆PQR  \n\t\n5 units to the  \n\t\nright and 3 units  \n\t\ndown.\n\t\n(b)\t Translate ∆PQR  \n\t\n2 units to the left  \n\t\nand 3 units up. \n\t\n(c)\t Are all the triangles  \n\t\ncongruent? \n–2\n–4\n–6\n–8\n2\n4\n1\n3\ny\nx\n0\n–2\n2\n4\n6\n8\n–4\nQ\nP\nR\n1\n3\n5\n7\n–1\n–3\n5\n–1\n–3\n–5\n–7\n–9\nMaths2_Gr9_LB_Book.indb   98\n2014/09/08   09:06:52 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n99\n2.\t (a)\t Translate ∆DEF 4 units to the left and 2 units down.\n\t\n(b)\t Translate ∆DEF 3 units to the right and 4 units up.\n\t\n(c)\t Are all the triangles congruent? \n–2\n2\n4\n6\n8\n10\n12\n14\n–4\n–6\n–8\n–10\ny\nx\n0\n0\n–2\n2\n4\n6\n–4\n–6\n–8\nE\nD\nF\n\t\n3.\t The vertices of a quadrilateral have the following coordinates: K(−5; 2), L(−4; −2), \nM(1; −3) and N(4; 3). Write down the coordinates of the image of the quadrilateral \nafter the following translations.\n\t\n(a)\t 7 units to the right and 2 units up\n\t\n(b)\t 5 units to the right and 2 units down\n\t\n(c)\t 4 units to the left and 3 units down\n\t\n(d)\t 2 units to the left and 7 units up\n4.\t Describe the translation if the coordinates of the original point and the image point are:\n\t\n(a)\t (−2; −3) → (−2; −5)\t\t\n\t\n(b)\t (4; −7) → (−6; 0) \t\n\t\n\t\n(c)\t (3; 11) → (16; 20) \t \t\n\t\n(d)\t (−1; −2) → (5; −4)\t \t\n\t\n(e)\t (8; −11) → (−2; −3)\t \t\nMaths2_Gr9_LB_Book.indb   99\n2014/09/08   09:06:53 AM\n\n100\t MATHEMATICS Grade 9: Term 3\n6.4\t Enlargement (expansion) and reduction (shrinking)\nwhat are enlargements and reductions?\nYou will remember the following from Grade 8.\n• An image is an enlargement or reduction of the original figure only if all the \ncorresponding sides between the two figures are in proportion. This means  \nthat all the sides of the original figure are multiplied by the same number (the \nscale factor) to produce the image.\n• Scale factor = \nside length of image\nlength of corresponding side of original figure\nºº\nIf the scale factor is > 1, the image is an enlargement.\nºº\nIf the scale factor is < 1, the image is a reduction.\n• The original figure and its enlarged or reduced image are similar.\n• Perimeter of image = Perimeter of original figure × scale factor\n• Area of image = Area of original figure × (scale factor)2\n6\nA = 24 units2\nP = 24 units\nA = 96 units2\nP = 48 units\nA = 6 units2\nP = 12 units\n8\n10\n16\n12\n20\n3\n4\n5\nEnlargement by \na factor of 2\nReduction by \na factor of 2\nDivide each side by 2 \n(or multiply each side by    )\nMultiply each side by 2\nP = perimeter\nA = area\n1\n2\nMaths2_Gr9_LB_Book.indb   100\n2014/09/08   09:06:53 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n101\nSometimes the terminology used for enlargements and reductions can be confusing. \nMake sure you understand the following examples. Refer to the diagram on the  \nprevious page.\n“Enlarge a figure by a scale factor of 2” means: \n• \nside length of image\nlength of corresponding side of original figure  = 2\n• Each side of the original figure must be multiplied by 2.\n• Each side of the image will be 2 times longer than its corresponding side in the \noriginal figure.\n• The perimeter of the image will be 2 times longer than the perimeter of the  \noriginal figure.\n• The area of the image will be 22 times (2 × 2 = 4 times) bigger than the area of the \noriginal figure.\n“Reduce a figure by a scale factor of 2” means:\n• \nside length of image\nlength of corresponding side of original figure  = 0,5\n• Each side of the original figure must be multiplied by 1\n2  (or divided by 2).\n• Each side of the image will be 2 times  \nshorter than its corresponding side in the  \noriginal figure.\n• The perimeter of the image will be 2 times  \nshorter than the perimeter of the original figure.\n• The area of the image will be 22 times (2 × 2 = 4 times) smaller than the area of \nthe original figure. (Or area of image = (1\n2)2 = 1\n4 of the area of the original figure.)\npractise working with enlargements and reductions\n1.\t Work out the scale factor of each original figure and its image.\n\t\n(a)\nA\nB\nC\nA'\nB'\nC'\n6 cm\n5 cm\n7,5 cm\n12 cm\n10 cm\n15 cm\nA = 56 cm2\nP = 37 cm\nNote that the multiplicative\ninverse of 2 is 1\n2 .\nMaths2_Gr9_LB_Book.indb   101\n2014/09/08   09:06:54 AM\n\n102\t MATHEMATICS Grade 9: Term 3\n\t\n(b)\nD\nE\nF\nD'\nE'\nF'\n4 cm\n6 cm\n20 cm\n30 cm\nA = 550 cm2\nP = 100 cm\nG\nG'\n\t\n(c)\t\nJ\nK\nL\nJ'\nK'\nL'\n15 cm\n20 cm\n8 cm\nA = 48 cm2\nP = 28 cm\nM\nM'\n6 cm\n2.\t For each set of figures in question 1, write down by how many times the perimeter of \neach image is longer or shorter than the perimeter of the original image. Also write \ndown the perimeter of each image.\n\t\n(a)\t\n\t\n(b)\t\n\t\n(c)\t\n3.\t For each set of figures in question 1, write down by how many times the area of each \nimage is bigger or smaller than the area of the original image. Also write down the \narea of each image.\n\t\n(a)\t\n\t\n\t\n(b)\t\n\t\n(c)\t\n\t\n4.\t The perimeter of rectangle DEFG = 20 cm. Write down the perimeter of the rectangle \t\n\t\nD'E'F'G' if the scale factor is 3.\t \n5.\t The perimeter of quadrilateral PQRS = 30 cm and its area is 50 cm2.\n\t\n(a)\t Find the perimeter of P'Q'R'S' if the scale factor is 1\n5 . \n\t\n(b)\t Determine the area of quadrilateral P'Q'R'S' if the scale factor is 1\n5. \nMaths2_Gr9_LB_Book.indb   102\n2014/09/08   09:06:54 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n103\n6.\t The perimeter of ∆DEF = 17 cm and the perimeter of ∆D'E'F' = 25,5 cm.\n\t\n(a)\t What is the scale factor of enlargement? \n \n\t\n(b)\t What is the area of ∆D'E'F' if the area of ∆DEF = 14 cm2? \n7.\t The area of ∆ABC = 20 cm2 and the area of ∆A'B'C' = 5 cm2.\n\t\n(a)\t What is the scale factor of reduction? \n\t\n(b)\t What is the perimeter of the image if the perimeter of ∆ABC = 22 cm? \ninvestigating enlargement and reduction\nWhen we do enlargements or reductions on a coordinate system, we use one point from \nwhich to perform the enlargement or reduction. This point is known as the centre of \nenlargement or reduction.\nThe centre of enlargement or reduction can be any point on the coordinate system.  \nIn this chapter, we will always use the origin as the centre of enlargement or reduction.\nRectangle ABCD, rectangle A'B'C'D' and rectangle A''B''C''D'' are plotted on a  \ncoordinate system as shown below. \ny\nx\n14\n12\n10\n8\n6\n4\n2\n–2\n0\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\nC'\nA'\nB'\nD'\nC\nA\nB\nD\nC''\nA''\nB''\nD''\n1.\t (a)\t Is rectangle A''B''C''D'' an enlargement of rectangle ABCD? Explain your answer.\nMaths2_Gr9_LB_Book.indb   103\n2014/09/08   09:06:54 AM\n\n104\t MATHEMATICS Grade 9: Term 3\n\t\n(b)\t Is rectangle A'B'C'D' a reduction of rectangle ABCD? Explain your answer.\n2.\t (a)\t The origin is the centre of enlargement and reduction. Draw four line segments \t\n\t\nto join the origin with A'', B'', C'' and D''.\n\t\n(b)\t What do you notice about these line segments?\n3.\t (a)\t List the coordinates of the images to complete the following table\nVertices of ABCD\nVertices of A'B'C'D'\nVertices of A''B''C''D''\nA(6; 6)\nB(6; 4)\nC(10; 4)\nD(10; 6)\n\t\n(b)\t What do you notice about the coordinates of the vertices of the original  \n\t\nrectangle and the coordinates of the vertices of the image?\nFrom the previous activity, you should have found the following:\nOn a coordinate system, the line that joins the \ncentre of an enlargement or reduction to a vertex \nof the original figure also passes through the \ncorresponding vertex of the enlarged or reduced \nimage.\nThe coordinates of a vertex of the enlarged or \nreduced image are equal to the scale factor × the \ncoordinates of the corresponding vertex of the \noriginal figure.\nMaths2_Gr9_LB_Book.indb   104\n2014/09/08   09:06:54 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n105\nFor example:\nB(6; 4) → B' (3; 2): The coordinates of B' are 1\n2  the coordinates of B. Note that the scale \nfactor is 1\n2 .\nB(6; 4) → B'' (12; 8): The coordinates of B'' are 2 times the coordinates of B. Note that \nthe scale factor is 2.\nIn general, we therefore use the following notation to describe the enlargement or  \nreduction with respect to the origin: \n(x; y) → (kx; ky) or (x'; y') = (kx; ky) where k is the \nscale factor.\nIf 0 < k < 1, the image is a reduction.\nIf k > 1, the image is an enlargement.\npractise\n1.\t Draw the enlarged or reduced images of the following figures according to the scale \nfactor given. In each case, use the origin as the centre of enlargement or \nreduction.\n\t\n(a)\t Scale factor = 2\n–2\n–4\n–6\n–8\n2\n4\n6\n8\ny\nx\n0\n0\n–2\n2\n4\n6\n8\n–4\n–6\n–8\nT\nS\nR\n–10\nMaths2_Gr9_LB_Book.indb   105\n2014/09/08   09:06:55 AM\n\n106\t MATHEMATICS Grade 9: Term 3\n\t\n(b)\t Scale factor = 1\n2    \n–2\n–4\n–6\n–8\n2\n4\n6\n8\ny\nx\n0\n0\n–2\n2\n4\n6\n8\n–4\n–6\n–8\nK\nJ\nH\n\t\n\t\n(c)\t Scale factor = 1\n2\n–4\n–6\n–8\n–10\n–2\n2\n4\n6\ny\nx\n0\n0\n6\n8\n10\n–2\n–4\nQ\nS\nR\n2\n4\n–12\nP\nMaths2_Gr9_LB_Book.indb   106\n2014/09/08   09:06:55 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n107\n\t\n(d)\t Scale factor = 1\n3  \n–2\n–4\n–6\n–8\n–10\n2\n4\n6\ny\nx\n0\n0\n–2\n2\n4\n6\n8\n–4\n–6\n–8\nL\nM\nK\nN\nO\n2.\t A quadrilateral has the following vertices: A(−2; 4), B(−4; −2), C(4; −3) and D(2; 1). \t\nDetermine the coordinates of the enlarged image if the scale factor = 2.\n3.\t A quadrilateral has the following vertices: P(−4; 0), Q(2,5; 4,5), R(6; −2,25) and  \nS(2; −4). Determine the coordinates of the enlarged image if the scale factor = 4.\n4.\t A quadrilateral has the following vertices: D(6; −4 ), E(4; −6), F(−4; 2) and G(−2; −2). \nDetermine the coordinates of the reduced image if the scale factor = 1\n2 .\n5.\t A quadrilateral has the following vertices: K(8; −2), L(4; −6), M(−8; −4) and  \nN(−6; 10). Determine the coordinates of the reduced image if the scale factor = 1\n4 .\n6.\t Describe the following transformations:\n\t\n(a)\t A(7; −5) → A'(9; 0)\nMaths2_Gr9_LB_Book.indb   107\n2014/09/08   09:06:56 AM\n\n108\t MATHEMATICS Grade 9: Term 3\n\t\n(b)\t A(−4; 6) → A'(4; 6)\n\t\n(c)\t A(−3; −2) → A'(−2; −3)\n\t\n(d)\t A(8; 1) → A'(8; −1)\n\t\n(e)\t A(4; −2) → A'(8; −4)\n\t\n(f)\t A(12; −16) → A'(3; −4)\n\t\n(g)\t A(2; −1) → A'(−3; −5)\n7.\t Describe each of the following transformations.\n\t\n(a)\t \t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(b)\n1\n–1\n–2\n–3\n–4\n–5\n2\n3\n4\n5\n6\ny\nx\n0\n0\n–2\n1\n2\n3\n4\n5\n6\n–1\nA\nB\nC\nD\nA'\nB'\nC'\nD'\n–2\n2\n4\n6\n8\n10\ny\nx\n0\n0\n–2\n2\n4\nC\nB\nA\nD\nC'\nB'\nA'\nD'\nMaths2_Gr9_LB_Book.indb   108\n2014/09/08   09:06:56 AM\n\nChapter 7\nGeometry of 3D objects\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n109\nIn this chapter, you will revise the properties of prisms and pyramids, which you \ninvestigated in previous grades. This includes using nets to construct models of these \nobjects as a further means of consolidating your understanding of polyhedra. You will also \nrevise the properties and definitions of the five Platonic solids, which you first learnt about \nin Grade 8, as well as how Euler’s formula describes a relation between the number of \nvertices, faces and edges of any polyhedron.\nNew to this grade are investigations of the properties of cylinders and spheres. Although \nyou should be able to recognise these 3D objects by now, you will examine some of their \nproperties in more detail, and learn how to construct a net and model of a cylinder.\n7.1\t Classifying 3D objects............................................................................................. 111\n7.2\t Nets and models of prisms and pyramids................................................................ 113\n7.3\t Platonic solids......................................................................................................... 115\n7.4\t Euler’s formula........................................................................................................ 119\n7.5\t Cylinders................................................................................................................ 121\n7.6\t Spheres................................................................................................................... 124\nMaths2_Gr9_LB_Book.indb   109\n2014/09/08   09:06:56 AM\n\n110\t MATHEMATICS Grade 9: Term 3\nMaths2_Gr9_LB_Book.indb   110\n2014/09/08   09:06:57 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n111\n7\t Geometry of 3D objects\n7.1\t\nClassifying 3D objects\n3D objects with flat faces which are called  \npolyhedra. Prisms and pyramids are two types of  \npolyhedra. \nTriangular prism\nRectangular prism\nRectangular-based pyramid\nCube\nExamples of 3D objects that have at least one curved surface are cylinders, spheres  \nand cones.\nCylinders\nSpheres\nCones\nWhen we study the properties of a 3D  \nobject, we investigate the shapes of its  \nfaces, its number of faces, its number of  \nvertices and its number of edges. For  \nexample, the pyramid alongside has  \n1 square face and 4 triangular faces,  \n5 vertices and 8 edges.\nEdges\nFaces\nVertices\nSquare-based pyramid\nA polyhedron is a 3D object \nwith only flat faces.\nMaths2_Gr9_LB_Book.indb   111\n2014/09/08   09:06:57 AM\n\n112\t MATHEMATICS Grade 9: Term 3\nClassifying and describing 3d objects\n1.\t Label parts (a) to (c) on the prism correctly.\n\t\n(a)\t\n\t\n(b)\t\n\t\n(c)\t\n2.\t Complete the table.\n3D object\nName of the object\nNumber of faces \nand shape of faces\nNumber of \nvertices\n(a)\nTriangular prism\n2 triangles and 3 \nrectangles\n6\n(b)\n(c)\n6 squares\n8\n(d)\n1 rectangle and 4 \ntriangles\n5\n(e)\n(f)\n(g)\n(a)\n(b)\n(c)\nMaths2_Gr9_LB_Book.indb   112\n2014/09/08   09:06:58 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n113\n3.\t Say whether each statement below is true or false.\n\t\n(a)\t A cylinder is a polyhedron.\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(b)\t A triangular-based pyramid has 4 triangular faces.\t\n\t\n\t\n\t\n\t\n(c)\t A cube is also known as a hexahedron.\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(d)\t A triangular-based pyramid has 6 vertices.\t\n\t\n\t\n\t\n\t\n\t\n\t\n(e)\t A pyramid is a 3D object.\t \t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n7.2\t\nNets and models of prisms and pyramids\nA net is a flat pattern that can be used to represent a 3D  \nobject. The net can be folded up to create a model of the  \n3D object.\n1.\t Name each object below and draw an arrow to match it with its net.\n(a)\n(b)\n(c)\n(d)\nNet of a cube\nMaths2_Gr9_LB_Book.indb   113\n2014/09/08   09:06:59 AM\n\n114\t MATHEMATICS Grade 9: Term 3\n2.\t Construct an accurate net for each of the following 3D objects.\n\t\n(a)\n\t\n\t\n1 cm\n4 cm\n2 cm\n\t\n(b)\n\t\n\t\n3 cm\n4 cm\n5 cm\nMaths2_Gr9_LB_Book.indb   114\n2014/09/08   09:06:59 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n115\n\t\n(c)\n\t\n\t\n4 cm\n2 cm\n3.\t Construct models of  \nthe objects in question 2  \nbut double all the  \nmeasurements.\n7.3\t\nPlatonic solids\nA Platonic solid is a 3D object which has identical faces, and all of the faces are \nidentical regular polygons. This means that all its faces are the same shape and size and \nall the vertices are identical.\n1.\t Which of the following objects are Platonic solids?\n\t\nA.\t \t\n\t\n\t\n\t\nB.\t \t\n\t\n\t\n\t\nC.\t \t\n\t\n\t\n   D.\t\n\t\n\t\n\t\nE.\n\t\n\t\nF.\t\n\t\n\t\n\t\nG.\t \t\n\t\n\t\n   H.\t\n\t\n\t\n\t\n  I.\t \t\n\t\n        J.\n\t\n\t\n\t\n \n2.\t How many Platonic solids are there in question 1? \nMaths2_Gr9_LB_Book.indb   115\n2014/09/08   09:07:00 AM\n\n116\t MATHEMATICS Grade 9: Term 3\nOnly five platonic solids?\nYou can use your knowledge about angles to prove that the five Platonic solids are  \nthe only 3D objects that can be made from identical regular polygons. Keep the \nfollowing facts in mind:\n• A 3D object has at least three faces that meet at each vertex.\n• The sum of the angles that meet at a vertex must be less than  \n360°. If it is equal to 360°, it will form a flat surface. If it is  \ngreater than 360°, the faces will overlap.\n• Each Platonic solid is made up of one type of regular polygon only.\nWhat 3D objects can you make from equilateral triangles?\nWe use the following reasoning:\nsize of each interior angle = 180° ÷ 3 = 60°\n∴\t 3 triangles = 3 × 60° = 180°\t\n[< 360°]\n\t\n4 triangles = 4 × 60° = 240°\t\n[< 360°]\n\t\n5 triangles = 5 × 60° = 300°\t\n[< 360°]\n\t\n6 triangles = 6 × 60° = 360°\nAny more than 5 triangles will be equal to or more than 360° and will therefore form a flat \nsurface or overlap.\nThis means that we can make three 3D objects from equilateral triangles:\nIf 3 triangles are at each vertex, it will form a tetrahedron.\nIf 4 triangles are at each vertex, it will form an octahedron.\nIf 5 triangles are at each vertex, it will form an icosahedron.\nTetrahedron\nOctahedron\nIcosahedron\n1\n2\n3\nMaths2_Gr9_LB_Book.indb   116\n2014/09/08   09:07:00 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n117\nWhat 3D objects can you make from squares?\nComplete the statements: size of each interior angle \n\t\n∴ \t\n3 squares = 3 × \n  \n\t\n\t\n4 squares = 4 × \n \nTherefore we can make only one 3D object using squares. This 3D  \nobject is called a hexahedron (or cube).\nWhat 3D objects can you make from regular pentagons?\nComplete the statements:\n\t\nSize of each interior angle \n\t\n∴ \t\n3 pentagons = \n  \n\t\n\t\n4 pentagons = \n \nTherefore we can make only one 3D object using regular  \npentagons. This 3D object is called a dodecahedron.\nWhat 3D objects can you make from regular hexagons?\nComplete the statements: \n\t\nSize of each interior angle \n\t\n∴  \t 3 hexagons = \nThree hexagons will already form a flat surface. Therefore it is  \nimpossible to make a 3D object from regular hexagons. \nAlso, the interior angles of polygons with more than 6 sides are bigger than those of  \na hexagon, so it is not possible to make 3D objects from any other regular polygons.\nTherefore the five Platonic solids already \nmentioned (tetrahedron, octahedron, \nicosahedron, hexahedron and dodecahedron) \nare the only ones that can be made of identical \nregular polygons. Each of these solids is named \nafter the number of faces it has.\nHexahedron (cube)\nDodecahedron\nMaths2_Gr9_LB_Book.indb   117\n2014/09/08   09:07:00 AM\n\n118\t MATHEMATICS Grade 9: Term 3\nproperties of the platonic solids\nComplete the information about each of the following Platonic solids.\n1.\nName: \nShape of the faces: \n \nNumber of faces: \nNumber of edges: \nNumber of vertices: \n2.\nName: \n \nShape of the faces: \n \nNumber of faces: \nNumber of edges: \nNumber of vertices: \n3.\nName: \n \nShape of the faces: \n \nNumber of faces: \n \nNumber of edges: \nNumber of vertices: \n4.\nName: \nShape of the faces: \n \nNumber of faces: \nEdges: \nVertices: \n5.\nName:  \n \nShape of the faces: \nNumber of faces: \nEdges: \nVertices: \nMaths2_Gr9_LB_Book.indb   118\n2014/09/08   09:07:02 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n119\n7.4\t\nEuler’s formula\nEuler’s formula and platonic solids\n1.\t You learnt about Euler’s formula in Grade 8. Complete the following table to \ninvestigate whether or not Euler’s formula holds true for Platonic solids.\nName\nShape of \nfaces\nNo. of \nfaces (F)\nNo. of \nvertices \n(V)\nNo. of \nedges  \n(E)\nF + V − E\n2.\t Complete Euler’s formula for polyhedra:\n\t\nF + \n    \n3.\t Apply Euler’s formula to each of the following:\n\t\n(a)\t A polyhedron has 25 faces and 13 vertices. How many edges will it have? \t\n\t\n(b)\t A polyhedron has 11 vertices and 23 edges. How many faces does it have?\t\n\t\n(c)\t A polyhedron has 8 faces and 12 edges. How many vertices does it have?\t\nMaths2_Gr9_LB_Book.indb   119\n2014/09/08   09:07:02 AM\n\n120\t MATHEMATICS Grade 9: Term 3\neuler’s formula and other polyhedra\n1.\t Is each of the following statements true or false?\n\t\n(a)\t A polyhedron with 10 vertices and 15 edges must have 7 faces.\t\n\t\n\t\n\t\n(b)\t A polyhedron will always have more edges than either faces or vertices.\t\n\t\n(c)\t A polyhedron with 5 faces must have 6 edges.\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(d)\t A pyramid will always have the same number of faces and vertices.\t \t\n2.\t Complete the following table.\n\t\nNo. of \nfaces (F)\nNo. of \nvertices (V)\nNo. of \nedges (E)\nName of \npolyhedron\nShapes of faces\n(a)\n6\n12\nRectangles\n(b)\n7\nHexagonal \npyramid\n(c)\n4\n4\n(d)\n5\n6\n9\nTriangles and \nrectangles\n3.\t A soccer ball consists of pentagons and hexagons. \n\t\n(a)\t How many pentagons does it consist of? \n\t\n(b)\t How many hexagons does it consist of? \n\t\n(c)\t How many edges does it have? \n\t\n(d)\t How many vertices does it have? \n\t\n(e)\t Does Euler’s formula apply to soccer balls too?\n\t\n\t\nNet of a soccer ball\n \nMaths2_Gr9_LB_Book.indb   120\n2014/09/08   09:07:02 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n121\n7.5\t\nCylinders\nProperties of cylinders\n1.\t Which of the following 3D objects are cylinders? \nA\nB\nC\nD\nE\nF\n2.\t Tick the statement or statements below that are true only for cylinders and not for \nthe other objects shown in question 1:\n\t\n\t It is a 3D object.\n\t\n\t It has a curved surface.\n\t\n\t It has two circular bases that are parallel to each other.\n\t\n\t It has two flat circular bases and a curved surface.\n\t\n\t The radius of its curved surface is equal from the top to the bottom between the \t\n\t\nbases. \n\t\n\t It has two circular bases opposite each other, joined by a curved surface whose \t \t\n\t\nradius is equal from the top to the bottom between the bases.\n3.\t Look at the cylinder alongside and complete the following:\n\t\n(a)\t Number and shape of faces: \n \n\t\n(b)\t Number of vertices: \n \n\t\n(c)\t Number of edges: \n \nNets of cylinders\nIn Chapter 5, you learnt about the net of a cylinder. If you cut the  \ncurved surface of a cylinder vertically and flatten it, it will be the  \nshape of a rectangle.\n1.\t Explain why the length of the rectangular face is  \nequal to the circumference of the base.\nMaths2_Gr9_LB_Book.indb   121\n2014/09/08   09:07:03 AM\n\n122\t MATHEMATICS Grade 9: Term 3\n2.\t Will each of the following nets form a cylinder?\n\t\nA.\t \t\nB.\t\n\t\nC.\n\t\n\t\n\t\nD.\t \t\nE.\t\n\t\nF.\n\t\n3.\t In each of the following questions, use π = 22\n7  and round off your answer to two \ndecimal places to do the calculations.\n\t\n(a)\t If the radius of a cylinder is 3 cm, what is the length of the rectangular surface of \t\n\t\nthe cylinder?\n\t\n(b)\t If the radius of a cylinder is 5 cm, what is the length of the rectangular surface of \t\n\t\nthe cylinder?\n\t\n(c)\t If the diameter of a cylinder is 8 cm, what is the length of the rectangular surface \t\n\t\nof the cylinder?\n\t\n(d)\t If the diameter of a cylinder is 9 cm, what is the length of the rectangular surface \t\n\t\nof the cylinder?\nMaths2_Gr9_LB_Book.indb   122\n2014/09/08   09:07:03 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n123\n4.\t Use a ruler and a set of compasses to construct the following nets as accurately as \npossible. Show the measurements on each net.\n\t\n(a)\t Net of a cylinder with a radius of 1 cm and a height of 4 cm.\n\t\n(b)\t Net of a cylinder with a radius of 1,5 cm and a height of 3 cm\n5.\t Construct models of the cylinders in question 5 but double the measurements.\nMaths2_Gr9_LB_Book.indb   123\n2014/09/08   09:07:04 AM\n\n124\t MATHEMATICS Grade 9: Term 3\n7.6\t\nSpheres\n1.\t Which of the following 3D objects are spheres?\nA\nB.\nC.\nD.\nE.\nF.\nG.\n.\n2.\t Tick the property or properties below that are true for spheres only and not for the \nother objects shown in question 1.\n\t\n\t It is a 3D object.\n\t\n\t It has one curved surface.\n\t\n\t It has no bases.\n\t\n\t It has no vertices.\n\t\n\t It has no edges.\n\t\n\t The distance from its centre to any point on its surface is always equal.\n3.\t Complete the following information for a sphere:\n\t\n(a)\t Number and shape of faces: \n\t\n(b)\t Number of vertices:  \n \n\t\n(c)\t Number of edges:  \n \nFrom your study of spheres in the above activity, you  \nshould have found the following:\nA sphere is a round 3D object with only one \ncurved surface and the distance from its centre  \nto any point on its surface is always equal. It has \nno vertices or edges.\nr\nr\nr\nr\nMaths2_Gr9_LB_Book.indb   124\n2014/09/08   09:07:04 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n125\n4.\t In the sphere alongside, write down the length of:\n\t\n(a)\t the radius: \n\t\n(b)\t the diameter: \n\t\n(c)\t MD: \n\t\n(d)\t CD: \n5.\t The drawing alongside shows part of a sphere  \nwith a diameter of 100 km. Imagine that you  \nare at point M, at the centre inside the sphere.  \nPeople A, B and C are all at different places on  \nthe surface of the sphere.\n\t\n(a)\t Which of the people – A, B or C – is closest to \t\n\t\nyou?\n\t\n(b)\t How far away is person C from you?\nNet of a sphere\nIt is impossible to make a perfect sphere (ball or globe) from a flat sheet of paper. Paper can curve in \none direction, but cannot curve in two directions at the same time. So all spheres made from paper or \ncard will be approximations. This is the best net we can make of a sphere. \nCan you make your own \npaper model of a sphere?\nA\nB\nC\nD\n5 cm\nM\nA\nB\nC\nM\nMaths2_Gr9_LB_Book.indb   125\n2014/09/08   09:07:04 AM\n\nWorksheet\nGEOMETRY OF 3D OBJECTS\n1.\t Grade 9 learners were asked to represent a 3D object and give the class clues as \t\n\t\nto which polyhedron they represent. Name their objects:\n\t\n(a)\tAmy: I have 6 faces and they are all the same size.\t\n\t\n(b)\tJohn: I have 6 faces and 12 edges. I am not a cube.\t\n\t\n\t\n(c)\tOnke: I have 3 faces. I also have two edges.\t\n\t\n\t\n(d)\tTessa: I have 8 edges and I have 5 vertices.\t\n\t\n\t\n(e)\t Mandlakazi: I have 6 edges and 4 vertices.\t\n\t\n\t\n(f)\t Chiquita: I have 8 faces and am a Platonic solid.\t\n\t\n\t\n(g)\tSeni: I do not have any edges.\t\n\t\n\t\n(h)\tMpu: My faces are made only of regular pentagons.\t\n\t\n2.\t Write down the required information about each object below.\n\t\nA.\t\n\t\nB.\t\n \n\t\nObject A\nObject B\nName\nNumber of faces\nShape/s of faces\nNumber of edges\nNumber of vertices\nDoes Euler’s formula work?\nIs it a Platonic solid?\n3.\t (a)\tOn a separate sheet of paper, construct a net of a cylinder with a diameter of \n\t\n\t\n7 cm and a height of 10 cm.\n\t\n(b)\tFold your net to make a model of the cylinder.\nMaths2_Gr9_LB_Book.indb   126\n2014/09/08   09:07:05 AM\n\nChapter 8\nCollect, organise and\nsummarise data\n\t CHAPTER 8: Collect, organise and summarise data\t\n127\nYou have learnt how to collect, organise and summarise data in previous grades. In Grade 9, \nyou need to decide which methods are best in certain situations and you need to be able to \njustify your choices.\n8.1\t Collecting data....................................................................................................... 129\n8.2\t Organising data...................................................................................................... 133\n8.3\t Summarising data................................................................................................... 136\nMaths2_Gr9_LB_Book.indb   127\n2014/09/08   09:07:05 AM\n\n128\t MATHEMATICS Grade 9: Term 4\nMaths2_Gr9_LB_Book.indb   128\n2014/09/08   09:07:05 AM\n\n\t CHAPTER 8: Collect, organise and summarise data\t\n129\n\t CHAPTER 8: Collect, organise and summarise data\t\n129\n8\t Collect, organise and summarise data\n8.1\t Collecting data\nAvoiding bias when selecting a sample\nThe methods that we use to collect data must help us to make sure that the data is \nreliable. This means that it is data that we can trust. \nData cannot be trusted unless it has been collected in a way that makes sure that every \nmember of the population had the same chance of being selected in the sample. \nIt is not practical to taste all the oranges on a tree to know whether the oranges are \nsweet. Only a small number of oranges can be tested, otherwise the farmer would have \ntoo few oranges to sell. The oranges that are tested are called a sample, and all the  \noranges harvested from the tree are called the population.\nSample bias occurs when the particular section of the population from which the \nsample is drawn does not represent that population. The way to avoid sample bias is to \ntake a random sample. A sample is random if every member of the population \nhas the same chance of being selected. A random sample of the orange trees means \nthat every tree should have a chance of being selected for the sample. Every person in \nthe country should have a chance of being selected for the housing survey in a random \nsample.\nAn example of sample bias would be to survey only the people in Limpopo  \nabout their views on housing provision when you want to know the views of the whole \ncountry. For the sample to provide information on the population as a whole, each \nperson in the country should have the same chance of being part of the survey. \nData can be collected through questionnaires, through observation and through \naccess to databases. \nHow to develop a good questionnaire\nThe questionnaire also has an important role in making sure that the information you \ncollect is reliable. You should aim to get a high number of respondents and accurate \ninformation. If not enough people fill in the questionnaire, then you don’t know \nwhether the information you get reflects the real situation. Sampling techniques and \nrules developed by statisticians determine the numbers needed.\nMaths2_Gr9_LB_Book.indb   129\n2014/09/08   09:07:05 AM\n\n130\t MATHEMATICS Grade 9: Term 4\nThere are some important points to consider when designing a questionnaire. Two \nof the most important points are that the questions are clear and accurate and that \npeople find the questionnaire relatively easy to complete.\n1.\t Keep in mind the length of the questionnaire and the time that it takes to complete. \nYour participants will more likely complete a short questionnaire that is quick and \neasy to complete. Exclude information that is not needed. \n2.\t Write down a selection of questions that you think will provide the information that \nyou want. \n3.\t Check the wording for each question. \n4.\t Order the items so that they are in a logical sequence. It might make sense to have the \neasiest questions first but in some cases the more general questions should come first \nand the more specific questions towards the end of the questionnaire.\n5.\t Then try the questionnaire out on a partner. Ask the following questions: \n• Is this question necessary? What information will be provided by the answer?\n• How easy will it be for the respondent to answer this question? How much time \t\nwill it take to answer the question? \n• Do the questions ask for sensitive information? Will people want to answer the \t\nquestion? Will the respondent answer the question honestly?\n• Can the question be answered quickly?\n6.\t Decide how the answers should be provided. Questions may require open-ended \nresponses or closed-ended responses, as described blow.\nIn an open-ended question, the person responds in his or her own words. Through his, \nor her, own words important information can be gained; the person is free to write what \nthey like. A disadvantage is that you might not get the information you want and that it \nmight take a long time to answer.\nIn a closed-ended question the respondents are given some options to choose from. \nThey tick the box which most closely represents their response. These options can be \nconstructed in categories. For example age may be categorised as follows:\nUnder 10 \n    From 10 to 14 \n    From 15 to 19 \n    20 and older \nMaths2_Gr9_LB_Book.indb   130\n2014/09/08   09:07:05 AM\n\n\t CHAPTER 8: Collect, organise and summarise data\t\n131\nthink about data collection and develop a questionnaire\n1.\t Which method for collecting data would be most appropriate for each of the cases \nbelow? Give reasons for your choice.\n\t\n(a)\t The number of learners who bring lunch to schools. What are the contents of the \t\n\t\nschool lunch? \n\t\n(b)\t Whether the tellers at a supermarket chain are happy with their conditions  \n\t\nof work. \n\t\n(c)\t Whether the clients of a clinic are satisfied with the professional conduct of the \t\n\t\nmedical staff.\n\t\n(d) \t The types of activities preschool children choose during their free time.\n\t\n(e)\t The number of Grade 9 learners in the Gauteng North district. \n2.\t You are doing some market research for a new fast food shop near the high \nschool. The owners of the shop want to find out what kind of food and music the \ntarget market likes. The target market is learners from the high school. Develop a \nquestionnaire to collect this information, on the next page. \nMaths2_Gr9_LB_Book.indb   131\n2014/09/08   09:07:05 AM\n\n132\t MATHEMATICS Grade 9: Term 4\nMaths2_Gr9_LB_Book.indb   132\n2014/09/08   09:07:05 AM\n\n\t CHAPTER 8: Collect, organise and summarise data\t\n133\n8.2\t Organising data\nThere is a difference between data and information. Data is unorganised facts.  \nWhen data is organised and analysed so that people can make decisions, it may be  \ncalled information. Data can be organised in many different ways. Some methods are \ndescribed below.\nData can be organised by making a tally table. Here is an example of a tally table \nshowing the numbers of learners in a class that participate in different sports.\nSport\nTally marks\nSoccer\n////    ////   ////    ////  ////          \nAthletics\n////   ///\nNetball\n////    ////   ////    ////  /\nChess\n////   /\nThe above data can also be organized in a frequency table:\nSport\nFrequency\nSoccer\n25\nAthletics\n8\nNetball\n21\nChess\n6\nNumerical data sets with many items are often grouped into equal class intervals and \nrepresented in a table of frequencies for the different class intervals. This is very useful \nsince it makes it easy to see how the data is spread out.\nHere is an example of grouped data showing the heights of all the learners in a school. \nTo make a frequency table for numerical data, the data has to be arranged \nfrom smallest to biggest first.\nHeight in m\nNumber of learners \n(Frequency)\n< 1,20 m\n13\n1,20 m – 1,30 m\n28\n1,30 m – 1,40 m\n57\n1,40 m – 1,50 m\n164\n1,50 m – 1,60 m\n274\n1,60 m – 1,70 m\n198\n1,70 m – 1,80 m\n73\n> 1,80 m\n13\nA value equal to the lower \nboundary of a class interval \nis counted in that interval. For \nexample a length of 1,60 m is \ncounted in the interval 1,60 \n– 1,70, and not in the interval \n1,50 – 1,60 m.\nHowever, 1,599 m is less than \n1,60 m, so it belongs in the \ninterval 1,50 m – 1,60 m.\nMaths2_Gr9_LB_Book.indb   133\n2014/09/08   09:07:06 AM\n\n134\t MATHEMATICS Grade 9: Term 4\nA stem-and-leaf display is a useful way to organise numerical data. It also shows you \nwhat the “shape” of the data is like. Here is an example of a stem-and-leaf display. \n\t\nKey: 35 | 4 means 354\n34 0  4\n35 4  8  8\n36 0  1  6  8\n37 1  3  5  8  8  8  9\n38 2  4  9\n39 0  3  4  4  5  6  9\n40 0  3  7\n41 1\nThe above stem-and-leaf display represents the following data about the masses in grams \nof the chickens in a sample of 6-week-old chickens on a chicken farm.\n399\n378\n382\n360\n396\n389\n344\n411\n378\n394\n394\n354\n375\n378\n400\n371\n379\n358\n366\n403\n358\n395\n390\n340\n393\n384\n361\n407\n373\n368\nTo make a stem-and-leaf display, it helps to first arrange the data from smallest to largest, \nas shown here for the above data set.\n340\n344\n354\n358\n358\n360\n361\n366\n368\n371\n373\n375\n378\n378\n378\n379\n382\n384\n389\n390\n393\n394\n394\n395\n396\n399\n400\n403\n407\n411\nThe same data set is displayed in two slightly different ways below.\n384\n382\n399\n379\n399\n379\n396\n378\n396\n368\n378\n395\n378\n395\n366\n378\n394\n368\n378\n394\n361\n378\n394\n411\n358\n366\n375\n389\n394\n407\n354\n360\n375\n393\n407\n344\n358\n361\n373\n384\n393\n403\n344\n358\n373\n390\n403\n340\n354\n360\n371\n382\n390\n400\n411\n340\n358\n371\n389\n400\nIn this display the width of each class interval\t\nIn this display the width of \nis 10, as in the stem-and-leaf display above.\t\neach class interval is 15.\nMaths2_Gr9_LB_Book.indb   134\n2014/09/08   09:07:06 AM\n\n\t CHAPTER 8: Collect, organise and summarise data\t\n135\nworking with grouped data\n1.\t An organisation called Auto Rescue recorded the following numbers of calls from \nmotorists each day for roadside service during March 2014. \n28\n122\n217\n130\n120\n86\n80\n90\n120\n140\n70\n40\n145\n187\n113\n90\n68\n174\n194\n170\nl00\n75\n104\n97\n75\n123\n100\n82\n109\n120\n81\n\t\nSet up a tally and frequency table for this set of data values, in intervals of width 40.\n2.\t When geologists go out into the field they make sure they have their rulers and \nmeasurement instruments in their bags. They also have their “inbuilt rulers”, for \nexample their handspans. A handspan is the distance from the tip of the thumb to \nthe tip of the fifth finger on an outstretched hand. Measure your handspan against \nthe ruler! This frequency table shows the handspans of different Grade 9 learners, in cm.\nHandspan of Grade 9 learners in cm\nFrequency\n15–18\n7\n18–21\n9\n21–24\n10\n24 and greater\n4\n\t\n(a)\t How many learner handspans were measured altogether?\n\t\n(b)\t How many learner handspans are less than 21cm wide?\nMaths2_Gr9_LB_Book.indb   135\n2014/09/08   09:07:06 AM\n\n136\t MATHEMATICS Grade 9: Term 4\n\t\n(c)\t How many handspans are 18 cm or wider?\n\t\n(d)\t In which interval would you place a handspan of 18 cm?\n8.3\t Summarising data\nThe mean, median, mode and range are single numbers that provide some information \nabout a data set, without listing all the data values. \nThe mode is the value that occurs most frequently. \nTo find the mode, look for the number or category \nthat is listed in the data set most often. Some data sets \nhave more than one mode, and some may have none.\nThe median is the number that separates the set \nof values into an upper half and a lower half. The \nmedian can be found by arranging the values from \nsmall to big or big to small. If the data set consists of \nan even number of items, the median is the sum of \nthe two middle values divided by 2.\nThe mean (average) of a set of numerical data is the \nsum of the values divided by the number of values in \nthe data set. \nMean = the sum of the values ÷ the number of values. \nThe range is a number that tells us how spread out \nthe data values are. It is the difference between the \nlargest and smallest values.\nThe mean, median and mode don’t work equally well for all sets of data. It depends on \nthe kind of data, and also on whether the data is evenly spread out or not.\norganise, summarise and compare some data\n1.\t A researcher analyses data about the people who are suffering from three different \ntypes of the flu virus: A, B and C. The ages of the people in the different groups are:\n\t\nType A: 60, 80, 75, 87, 88, 49, 94, 84, 59, 86, 82, 62, 79, 89 and 78.\n\t\nType B: 27, 39, 43, 29, 36, 70, 56, 25, 54, 36, 66, 45, 33, 46, 14 and 41.\n\t\nType C: 33, 48, 64, 15, 31, 20, 70, 21, 18, 49, 21, 19, 57, 23, 29 and 20.\nMaths2_Gr9_LB_Book.indb   136\n2014/09/08   09:07:06 AM\n\n\t CHAPTER 8: Collect, organise and summarise data\t\n137\n\t\nFor each group:\n• Draw a stem-and-leaf plot.\n• Calculate the range, mean and median of the ages.\n• Look at the shape of the stem-and-leaf displays as well as the summary measures. \nDiscuss the spread of the data in each case, and compare the three different  \ngroups.\nWork and report on your work below and on the next page.\nType A: \n  \nMaths2_Gr9_LB_Book.indb   137\n2014/09/08   09:07:06 AM\n\n138\t MATHEMATICS Grade 9: Term 4\nType B: \nType C: \nMaths2_Gr9_LB_Book.indb   138\n2014/09/08   09:07:06 AM\n\n\t CHAPTER 8: Collect, organise and summarise data\t\n139\n2.\t Fill in the statistic (mode, mean or median) that would best summarise each data set, \nand indicate the central tendency of the data. \nData set\nBest measure of central tendency\nThe shoe sizes of boys in Grade 9\nAn evenly-spread set of measurement values, \nsuch as the heights of learners in a class\nA set of measurement values with a few very \nlow values and mostly high values\nThe number of siblings each person in your \nclass has\nThe sizes of properties in a town, where \nmost people live in small apartments or RDP \nhouses, and a few live on large properties\nextreme values and outliers \nAn extreme value or outlier is a data value that lies an abnormal distance from other  \nvalues in a random sample from a population. Sometimes there are reasons why this  \ndata value is so different to the others. It is important to comment on the possible \nreasons. \nWhen you are summarising data (and also when you analyse data), you need to decide \nwhether an outlier makes sense in the context you are looking at. \nIt is possible that an outlier does not make sense, as it lies too far away and is an  \nunreasonable measurement. Then you need to think about the fact that this data value \nmay be an error. For example:\n11 12 13 14 15 16 17 18 19 20 21 22 23 24 25\nage\nIn this case, the value of 24 years old could be an unreasonable value. This depends on \nthe context of the survey.\nYou will learn more about extreme values and outliers in Chapter 10. \nMaths2_Gr9_LB_Book.indb   139\n2014/09/08   09:07:06 AM\n\n140\t MATHEMATICS Grade 9: Term 4\nUse this information about 14 countries to answer the questions that follow.\nCountry\nTotal population  \n(in 1 000s)\nTotal annual \nnational income \nper person (US$)\nPercentage of \nincome spent on \nhealth\nAngola\n18 498\n4 830\n4,6\nBotswana\n1 950\n13 310\n10,3\nDRC \n66 020\n280\n2,0\nLesotho\n2 067\n1 970\n8,2\nMalawi\n15 263\n810\n6,2\nMauritius\n1 288\n12 580\n5,7\nMozambique\n22 894\n770\n5,7\nNamibia\n2 171\n6 250\n5,9\nSeychelles\n84\n19 650\n4,0\nSouth Africa\n50 110\n9 790\n8,5\nSwaziland\n1 185\n5 000\n6,3\nTanzania\n43 739\n1 260\n5,1\nZambia\n12 935\n1 230\n4,8\nZimbabwe\n12 523\n170\nNot available\n1.\t Look at the total population for each country. \n\t\n(a)\t Calculate the mean of the data. \n\t\n(b)\t Draw a dot plot on the number line below to show the data.\nPopulation (thousands)\n0\n10 000\n20 000\n30 000\n40 000\n50 000\n60 000\n\t\n(c)\t Find the median of the data.\n\t\n(d)\t What is the range of the data?\n\t\n(e)\t Which measure of central tendency do you think represents the data more \t\t\n\t\n    accurately? Explain. \n \n2.\t Look at the Total annual national income per person in US dollars. Comment on the \nspread of the data.  \n \nMaths2_Gr9_LB_Book.indb   140\n2014/09/08   09:07:06 AM\n\nChapter 9\nRepresenting data\n\t\nCHAPTER 9: representing data\t\n141\nIn the previous chapter, you focused on methods of collecting, organising and summarising \ndata. Now we focus on representing data in bar graphs, double bar graphs, histograms, pie \ncharts and broken-line graphs. You will practise drawing these graphs. You will also decide \nwhy a certain kind of graph is useful in a particular context.\n9.1\t Bar graphs and double bar graphs.......................................................................... 143\n9.2\t Histograms............................................................................................................. 146\n9.3 \t Pie charts................................................................................................................ 149\n9.4\t Broken-line graphs.................................................................................................. 151\n9.5\t Scatter plots........................................................................................................... 154\nMaths2_Gr9_LB_Book.indb   141\n2014/09/08   09:07:06 AM\n\n142\t MATHEMATICS Grade 9: Term 4\n0\n2 000 000\n4 000 000\n2011 Mid-year estimates\nSouth Africa’s population by province\n6 000 000\n8 000 000\n10 000 000\n12 000 000\nEastern Cape\nFree State\nGauteng\nKwaZulu-Natal\nLimpopo\nMpumalanga\nNorthern Cape\nNorth West\nWestern Cape\n0\nJanuary\nFebruary\nMarch\nApril\nMay\nJune\nJuly\nAugust\nSeptember\nOctober\nNovember\nDecember\n20\n40\n60\n80\n100\n120\n140\n160\n180\nRainfall (mm)\nRainfall for Ceres, Mahikeng and Amatole\nCeres, WC\nMahikeng, NW\nAmatole, KZN\nMaths2_Gr9_LB_Book.indb   142\n2014/09/08   09:07:07 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n143\n\t\nCHAPTER 9: representing data\t\n143\n9\t Representing data\n9.1\t Bar graphs and double bar graphs\nrevising bar graphs and double bar graphs\nA bar graph shows categories of data along the horizontal axis, and the frequency of \neach category along the vertical axis. An example is given below.\nGraphs need a title to tell you what they are about.\nThis shows the frequency \nof each category.\n8 learners chose orange juice.\nThis axis gives all the categories of data.\nNumber of learners\nFruit juice\n0\n2\n4\n6\n8\n10\n12\nApple\nApricot\nGrenadilla\nMango\nOrange\nPeach     \nPineapple\nStrawberry\nFavourite fruit juice in a Grade 9 class\nA double bar graph shows two sets of data in the same categories on the same set of \naxes. This is useful when we need to show two groups within each category.\nNumber of learners\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\nGrade 8\nBoys\nGrade 9\nGrade 10\nYear\nGrade 11\nGrade 12\nNumber of boys and girls in each grade\nat Malbongwe High School\nGirls\nThis is a key to show \nwhat the two colours \nrepresent\nThe two bars show \nthe differences \nin numbers of boys \nand girls.\nMaths2_Gr9_LB_Book.indb   143\n2014/09/08   09:07:07 AM\n\n144\t MATHEMATICS Grade 9: Term 4\ndrawing bar graphs and double bar graphs\n1.\t Obese (very overweight) people have many health problems. It is a concern all \naround the world. Health researchers analysed the change over 28 years in the \nnumbers of people who are overweight and obese in different areas of the world. This \ntable summarises some of the data.\n\t\nPercentage of population that is overweight and obese\n1980\n2008\nSub-Saharan Africa\n12%\n23%\nNorth Africa and Middle East\n33%\n58%\nLatin America\n30%\n57%\nEast Asia (low income countries)\n13%\n25%\nEurope\n45%\n58%\nNorth America (high income \ncountries)\n43%\n70%\n\t\n(a)\t The table summarises “some” of the data. What would some other important \t \t\n\t\ndata be? Think of as many things as you can. \n\t\n(b)\t Which data stands out the most for you in the table above? Give your personal \t\t\n\t\nopinion. \n\t\n(c)\t Plot a double bar graph to compare the data for the areas, and for the two years. \t\n\t\nUse the grid on the next page. Remember to give your graph a key.\nMaths2_Gr9_LB_Book.indb   144\n2014/09/08   09:07:07 AM\n\n\t\nCHAPTER 9: representing data\t\n145\n\t\n(d)\t Look carefully at the comparisons that the graph makes. Has your opinion of  \n\t\nthe most interesting differences changed, now that you see the double bar graph? \n\t\nExplain.\n\t\n(e)\t In some countries the obesity problem has been labelled “Obesity in the face of \t\n\t\npoverty”. Write a short report on the data and your double bar graph to support \t\n\t\nthis argument.\nMaths2_Gr9_LB_Book.indb   145\n2014/09/08   09:07:07 AM\n\n146\t MATHEMATICS Grade 9: Term 4\n9.2\t Histograms\nrevising histograms\nA histogram is a graph of the frequencies of data in different class intervals, as \ndemonstrated in the example below. Each class interval is used for a range of values.  \nThe different class intervals are consecutive and cannot have values that overlap. The \ndata may result from counting or from measurement.\nA histogram looks somewhat like a bar graph, but is normally drawn without gaps  \nbetween the bars. \nrepresenting data in histograms\n1.\t (a)\t A fruit farmer wants to know which of his trees are producing good plums, and \n\t\nwhich trees need to be replaced. \n\t\n\t\nHe collects 100 plums each from two trees and measures their masses. \n\t\n\t\nThe data below gives the mass of plums from the first tree.\nMass of plums (g)\n20–29\n30–39\n40–49\n50–59\n60–69\nFrequency\n6\n18\n34\n30\n12\n\t\n\t\nRepresent the data in a histogram on the grid below. \nMaths2_Gr9_LB_Book.indb   146\n2014/09/08   09:07:07 AM\n\n\t\nCHAPTER 9: representing data\t\n147\n\t\n(b)\t Now draw another histogram to represent the following data giving the mass of \t\n\t\nthe same type of plums from another tree in the orchard.\nMass of plums (g)\n20–29\n30–39\n40–49\n50–59\n60–69\nFrequency\n3\n14\n26\n36\n21\n\t\n(c)\t Study the two histograms and then comment on the number of plums produced \t\n\t\nby the two trees. \nMaths2_Gr9_LB_Book.indb   147\n2014/09/08   09:07:07 AM\n\n148\t MATHEMATICS Grade 9: Term 4\n2.\t (a)\t Draw a histogram to represent the data in the table below. Group the data in  \n\t\nintervals of 0,5 kg.\n\t\n\t\nBirth weights (kg) of 28 babies at a clinic\n3,3\n1,34\n2,88\n2,54\n1,87\n2,06\n2,72\n1,89\n0,85\n1,99\n2,43\n1,66\n2,45\n1,62\n1,91\n1,20\n2,45\n1,38\n0,9\n2,65\n2,88\n1,75\n2,11\n3,2\n1,74\n0,6\n3,1\n1,86\n\t\n(b)\t Calculate the mean and median of the data.\n\t\n(c)\t Records from the whole country show that the birth weight of babies ranges \t\n\t\n\t\nfrom 0,5 kg to 4,5 kg, and the mean birth weight is 3,18 kg. Use the graph and  \n\t\nthe mean and median to write a short report on the data from the clinic.  \nMaths2_Gr9_LB_Book.indb   148\n2014/09/08   09:07:08 AM\n\n\t\nCHAPTER 9: representing data\t\n149\n9.3\t Pie charts\nA pie chart consists of a circle divided into sectors (slices). Each sector shows one \ncategory of data. Bigger categories of data have bigger slices of the circle. \nHere is an example of a pie chart:\nThis pie chart shows 5 \ncategories of data.\nThe size of each \nslice is the fraction \nor percentage of \nthe whole that the \ncategory forms.\nThe key (or legend) \nshows the category \nthat each colour \nstands for\n30% or\n40% or\n15% or\n10% or\nVery good\nCustomer opinion on service at Fishy Fun \nrestaurant as reflected in survey\nGood\nNeutral\nPoor\nVery poor\n2\n5\n3\n20\n1\n10\n1\n20\n3\n10\n5% or\ndrawing pie charts\n1.\t The following bar graph shows the population of South Africa by province.\n0\n2 000 000\n4 000 000\n2011 Mid-year population estimates\nSouth Africa’s population by province\n6 000 000\n8 000 000\n10 000 000\n12 000 000\nEastern Cape\nFree State\nGauteng\nKwaZulu-Natal\nLimpopo\nMpumalanga\nNorthern Cape\nNorth West\nWestern Cape\n\t\n(a)\t Write the figures in the graph correct to the nearest 500 000.\nProvince\nE Cape\nFS\nGau\nKZN\nLim\nMpum\nNC\nNW\nWC\nPopulation\n(× 1 000)\n\t\n(b)\t What is the total of the rounded off numbers? \nMaths2_Gr9_LB_Book.indb   149\n2014/09/08   09:07:08 AM\n\n150\t MATHEMATICS Grade 9: Term 4\n\t\n(c)\t Work out the percentage of the whole for each province.\nProvince\nE Cape\nFS\nGau\nKZN\nLim\nMpum\nNC\nNW\nWC\nPercentage \nof total\n\t\n(d)\t Draw a pie chart showing the data in the completed table. (Estimate the sizes of \t\n\t\nthe slices.)\n\t\n(e)\t Write a short report explaining the difference in the way the data is represented \t\n\t\nin the pie chart and the bar graph. Which do you think is a better method to \t\n\t\n\t\nshow this data?\nMaths2_Gr9_LB_Book.indb   150\n2014/09/08   09:07:08 AM\n\n\t\nCHAPTER 9: representing data\t\n151\n9.4\t Broken-line graphs \nBroken-line graphs\nBroken-line graphs are used to represent data that changes continuously over time. \nFor example, the rainfall for a whole month is captured as one data point, even though \nthe rain is spread out over the month, and it rains on some days and not on others. \nBroken line graphs are useful to identify and display trends.\nHere is some data that can be represented with broken-line graphs.\nRainfall at three locations in South Africa in 2012\nAmatole, KZN\nMahikeng, NW\nCeres, WC\nRainfall (mm)\nRainfall (mm)\nRainfall (mm)\nJanuary\n101\n118\n27\nFebruary\n108\n90\n23\nMarch\n117\n86\n41\nApril\n77\n61\n60\nMay\n46\n14\n130\nJune\n27\n6\n168\nJuly\n32\n3\n152\nAugust\n48\n7\n162\nSeptember\n76\n18\n88\nOctober\n112\n46\n60\nNovember\n115\n75\n41\nDecember\n100\n86\n36\nHere is a broken line graph for the Amatole rainfall data.\n0\nJanuary\nFebruary\nMarch\nApril\nMay\nJune\nJuly\nAugust\nSeptember\nOctober\nNovember\nDecember\n20\n40\n60\n80\n100\n120\n140\nRainfall (mm)\nRainfall for Amatole, KZN\nMaths2_Gr9_LB_Book.indb   151\n2014/09/08   09:07:08 AM\n\n152\t MATHEMATICS Grade 9: Term 4\n1.\t During which four months does Amatole have the least rain?\n2.\t During which six months does Amatole have the most rain?\n3.\t During which months would you plan a hike if you were only considering the rainfall \npatterns?\n4.\t What other factors should you consider when planning a hike in this region?\n5.\t Make a broken-line graph for the Mahikeng rainfall data on the grid below.\n \nMaths2_Gr9_LB_Book.indb   152\n2014/09/08   09:07:08 AM\n\n\t\nCHAPTER 9: representing data\t\n153\n6.\t Make a broken-line graph for the Ceres-rainfall data on the grid below.\n7.\t Write a few lines on the difference in rainfall patterns between Ceres and Mahikeng.  \n8.\t Draw a combined broken-line graph with the information from all three regions on \none graph.\nMaths2_Gr9_LB_Book.indb   153\n2014/09/08   09:07:09 AM\n\n154\t MATHEMATICS Grade 9: Term 4\n9.4\t Scatter plots\nunderstanding and constructing scatter plots\nScatter plots show how two sets of numerical data are related. Matching pairs of \nnumbers are treated as coordinates and are plotted as a single point. All the points,  \nmade up of two data items each, show a scattering across the graph.\n1.\t This table shows a data set  \nwith two variables. Study the  \ninformation in the table.\n2.\t Make a dot for each learner’s  \nmark for each subject on the  \nnumber lines below. \nLearners\nMaths marks\nNatural \nScience marks\nZinzi\n25\n26\nJohn\n23\n25\nPalesa\n22\n25\nSiza\n21\n23\nEric\n20\n23\nChokocha\n19\n21\nGabriel\n17\n20\nSimon\n16\n19\nMiriam\n15\n18\nFrederik\n15\n16\nSibusiso\n12\n15\nMeshack\n11\n13\nDuma\n11\n12\nSamuel\n10\n12\nLola\n10\n11\nThandile\n9\n10\nJabulani\n8\n10\nManare\n7\n9\nMarlene\n7\n7\nMary\n5\n7\n0\n2\n4\n6\n8\n10 12 14 16\nNatural Sciences marks\n18 20 22 24 26 28 30\n0\n2\n4\n6\n8\n10 12 14 16\nMathematics marks\n18 20 22 24 26 28 30\nMaths2_Gr9_LB_Book.indb   154\n2014/09/08   09:07:09 AM\n\n\t\nCHAPTER 9: representing data\t\n155\n3.\t What if you were to show both sets of marks on the same graph, instead of a separate \nnumber line for each set? The graph below shows a scatter plot that represents both \nsets of data. Each dot represents one learner. \n\t\nNatural Sciences marks\nMathematics marks\n0\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\n24\n26\n28\n30\n2\n4\n6\n8 10 12 14 16 18 20 22 24 26 28 30\nCorrelation between Mathematics\nand Natural Sciences\n\t\nThe scatter plot shows the relationship between the Natural Sciences mark and the  \nMathematics mark. \n4.\t Find the dot for Sibusiso in the data set. He obtained a mark of 12 for the  \nMathematics test and a mark of 15 for Natural Sciences. Find 12 on the  \nhorizontal axis. Follow the vertical line up until you reach a blue dot. Find 15 on  \nthe vertical axis. Follow the line horizontally until you reach the same blue dot.  \nThis blue dot represents the two marks that belong to Sibusiso. Circle the blue dot \nand label it S.\n5.\t Find the data points for Zinzi, Palesa, Jabulani and Mary. Circle them and label them \nZ, P, J and M.\nIn the above example, a higher Mathematics mark \ncorresponds to a higher Science mark. We say there is \na positive correlation between the Mathematics \nmarks and the Science marks.\nMaths2_Gr9_LB_Book.indb   155\n2014/09/08   09:07:09 AM\n\n156\t MATHEMATICS Grade 9: Term 4\n6.\t Study this data set and the scatter plot of the data given on the next page.\nLearner\nMaths marks\nArt marks\nZinzi\n25\n7\nJohn\n23\n7\nJabulani\n22\n9\nSiza\n21\n10\nEric\n20\n10\nChokocha\n19\n11\nGabriel\n17\n12\nSimon\n16\n12\nMiriam\n15\n15\nFrederik\n15\n15\nSibusiso\n12\n16\nMishack\n11\n17\nDuma\n11\n19\nSamuel\n10\n20\nLola\n10\n21\nThandile\n9\n23\nPalesa\n8\n23\nManare\n7\n25\nMarlene\n7\n25\nMary\n5\n26\n7.\t Find Eric in the table. Note his marks for Mathematics and Art. Find the dot that \nrepresents his marks on the scatter plot. Encircle it and label it E.\n8.\t Find Samuel in the table. Note his marks for Mathematics and Art. Find the dot that \nrepresents his marks. Encircle it and label it S.\n9.\t Compare the two sets of marks for Eric and for Samuel. What do you notice about  \nthe marks?\nMaths2_Gr9_LB_Book.indb   156\n2014/09/08   09:07:09 AM\n\n\t\nCHAPTER 9: representing data\t\n157\n10.\tFind the data points on the scatter plot for Zinzi, Eric, Miriam, Frederik, Samuel and \nMary. Circle the points and label them Z, E, M, F, S and Ma\n11.\tWhat do you notice about the pattern of marks in Mathematics and Art for this  \ndata set?\nArt marks\nMathematics marks\n0\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\n24\n26\n28\n30\n2\n4\n6\n8 10 12 14 16 18 20 22 24 26 28 30\nCorrelation between Mathematics\nand Art\nA negative correlation is a correlation in which an increase in the value of one piece \nof data tends to be matched by the decrease in the other set of data. Learners who obtain \na high mark for Mathematics appear to obtain a low mark for Art. We say there is a \nnegative correlation between the Mathematics and the Art scores for this data set.\nA correlation is an assessment of how strongly two sets of data appear to be connected. \nTwo sets of data may be correlated or may show no correlation. \nMaths2_Gr9_LB_Book.indb   157\n2014/09/08   09:07:09 AM\n\n158\t MATHEMATICS Grade 9: Term 4\nHere is the scatter plot for the Mathematics and Life Skills marks of the same group of \nlearners. The table for this data is given on the next page.\nLife Skills marks\nMathematics marks\n0\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\n24\n26\n28\n30\n2\n4\n6\n8 10 12 14 16 18 20 22 24 26 28 30\nCorrelation between Mathematics \nand Life Skills marks\n12. Study the scatter plot and the data table on the next page.\n13. Find the data points on the scatter plot for Zinzi, Eric, Miriam, Lola, and Mary. Circle \nthe points and label them Z, E, M, L and Ma.\n14. What do you notice about the pattern of marks in Mathematics and Life Skills for this \ndata set?\nMaths2_Gr9_LB_Book.indb   158\n2014/09/08   09:07:10 AM\n\n\t\nCHAPTER 9: representing data\t\n159\nLearner\nMaths\nLife Skills\nZinzi\n25\n5\nJohn\n23\n14\nJabulani\n22\n23\nSiza\n21\n16\nEric\n20\n5\nChokocha\n19\n4\nGabriel\n17\n9\nSimon\n16\n6\nMiriam\n15\n25\nFrederik\n15\n27\nSibusiso\n12\n29\nMeshack\n11\n17\nDuma\n11\n11\nSamuel\n10\n1\nLola\n10\n25\nThandile\n9\n5\nPalesa\n8\n28\nManare\n7\n26\nMarlene\n7\n2\nMary\n5\n15\nthe relationship between arm span and height\nThe idea that a person’s arm span (the distance from the tip of the middle finger on one \nhand to the tip of the middle finger on the other hand when the arms are stretched out \nsideways) is the same as one’s height has been explored many times. \nA data set for 13 people is given on the next page.\n1.\t Make a scatter plot of this data on the given grid.\n\t\nFor example, take Cilla’s arm span. Find 156 on the horizontal axis. Follow a vertical \nline up. Then on the vertical axis find 162. Follow a horizontal line across. Where the \ntwo points meet, draw a dot.\nMaths2_Gr9_LB_Book.indb   159\n2014/09/08   09:07:10 AM\n\n160\t MATHEMATICS Grade 9: Term 4\nPerson\nArm span\nHeight\nCilla\n156\n162\nMeshack\n159\n162\nTony\n161\n160\nEllen\n162\n170\nKarin\n170\n170\nSibongile\n173\n185\nGabriel\n177\n173\nAlpheus   \n178\n178\nMfiki\n188\n188\nNathi\n188\n182\nManare\n188\n192\nKhanyi\n196\n184\n2.\t What would you say about the correlation between the arm span and the height?\nMaths2_Gr9_LB_Book.indb   160\n2014/09/08   09:07:10 AM\n\nChapter 10\nInterpret, analyse and\nreport on data\n\tCHAPTER 10: INTERPRET, ANALYSE AND REPORT ON DATA\t\n161\nIn this chapter, you will develop and practise some critical data analysis skills. This means \nlooking at reported data and analysing the whole data handling cycle for this data. You \nneed to decide which way of representing data is best in a given situation. In summarising \ndata, some measures are more appropriate for different types of data. You also need to \nrecognise some ways in which bias can appear in data, including methods of collecting, \nrepresenting and summarising data.\n10.1\t Which graph is best?.............................................................................................. 163\n10.2\t The effects of summary statistics on how data is reported....................................... 167\n10.3\t Misleading graphs.................................................................................................. 168\n10.4\t Analysing extreme values and outliers..................................................................... 172\nMaths2_Gr9_LB_Book.indb   161\n2014/09/08   09:07:10 AM\n\n162\t MATHEMATICS Grade 8: Term 4\nCollect\ndata\nPose a \nquestion\nPresent \nthe data\nOrganise\nthe data\nInterpret \nand analyse \nthe data\nReport on \nthe data\nThe data\ncycle\nMaths2_Gr9_LB_Book.indb   162\n2014/09/08   09:07:10 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n163\n\tCHAPTER 10: INTERPRET, ANALYSE AND REPORT ON DATA\t\n163\n10\tInterpret, analyse and report on data\n10.1\t Which graph is best?\nYou have learnt that certain types of graphs are best for displaying certain kinds of \ninformation. The type of graph depends mostly on the type of data that needs to be \nrepresented. Here is a summary of the advantages of different types of graphs:\nTables show more information than graphs but  \nthe patterns are not as easy to see. They do not give a \nvisual impression of particular trends. \nPie charts show a whole divided into parts. They \nshow how the parts relate to each other and how the \nparts relate to a whole. They do not show the  \nquantities involved.\nBar graphs show the amounts or quantities  \ninvolved but do not show the relationship as  \neffectively as pie charts. They are useful for showing \nquantitative data. Bar charts allow us to compare \nthe quantities of different categories, for example, \nthe sales of different items.\nA double-bar graph is used to compare two or \nmore things for each category. For example, we could \nuse a double-bar graph to compare the differences \nbetween males and females. \nHistograms are used to represent numerical data \nthat is grouped into equal class intervals. Histograms \nare useful to show the way the data is spread out.\nBroken-line graphs show trends or changes in \nquantities over time. \nMaths2_Gr9_LB_Book.indb   163\n2014/09/08   09:07:10 AM\n\n164\t MATHEMATICS Grade 9: Term 4\nchoose the best representation\n1.\t Which kind of graph is best to represent each of the following? Explain your answers.\n\t\n(a)\t Showing the value of the rand against the US dollar over several years\n\t\n(b)\t Comparing the monthly sales of six different makes of car in 2014 and 2015\n\t\n(c)\t The proportion of people of different age groups in a town\n\t\n(d)\t The quantities of different crops produced on a farm\n\t\n(e)\t The percentages of different goods sold to make up the total sales for a shop\n\t\n(f)\t The change in HIV infection rates over time\n2.\t This graph was published by Statistics South Africa to show the assets owned by \nSouth Africans. The blue bar shows the Census 2011 results and the yellow bar shows \nthe General Household Survey 2012 results.\n\t\nPercent\nVehicle\nComputer\nTV\nRefrigerator\nCensus 2011\nGeneral\nhousehold\nsurvey 2012\n0\n10\n20\n30\n40\n50\n60\n70\nPercentage of household assets owned\nWashing\nmachine\nElectric\nstove\n90\n80\nMaths2_Gr9_LB_Book.indb   164\n2014/09/08   09:07:10 AM\n\n\tCHAPTER 10: INTERPRET, ANALYSE AND REPORT ON DATA\t\n165\n\t\nGive reasons for your answers to the questions below.\n\t\n(a)\t Is it useful to show the differences in the results of Census 2011 and the General \t\n\t\nHousehold Survey 2012?\n\t\n(b)\t Is it useful to collect data on assets that people own? \n\t\n(c)\t Is it useful to show that lower percentages of people own certain assets?\n\t\n(d)\t The different coloured bars represent the two different surveys. Draw up a table \t\n\t\nto show the data in table form. (Read the percentages as accurately as you can  \n\t\nfrom the graph and round off the data to the nearest whole number for the table.)\n\t\n(e)\t Does the table show the data as effectively as the double bar chart? Give your \t \t\n\t\nown opinion.\nMaths2_Gr9_LB_Book.indb   165\n2014/09/08   09:07:10 AM\n\n166\t MATHEMATICS Grade 9: Term 4\n3.\t The table below shows the employment status of people ages 15–64 years in  \nSouth Africa. Discuss some ways of representing the data (e.g. graphs). Justify your \nanswers.\nJul–Sept 2012\nApr–June 2013\nJul–Sep 2013\nNumber of people (thousands)\nPopulation 15–64 years old\n33 017\n33 352\n33 464\nLabour force\n18 313\n18 444\n18 638\nEmployed\n13 645\n13 720\n14 028\n  Formal sector  \n  (non-agricultural)\n9 663\n9 694\n10 008\n  Informal sector  \n  (non-agricultural)\n2 197\n2 221\n2 182\n  Agriculture\n661\n712\n706\n  Private households\n1 124\n1 093\n1 132\nUnemployed\n4 668\n4 723\n4 609\nNot economically active\n14 705\n14 908\n14 826\n  Discouraged work-seekers\n2 170\n2 365\n2 240\n  Other (not economically \n  active)\n12 535\n12 543\n12 586\nUnemployment rate (%) \n25,5\n25,6\n24,7\n\t\n(a)\t The percentages of the employed, unemployed, and not economically active \t\n\t\n\t\npeople in July–September 2013.\n\t\n(b)\t The change in the employment rates over three time periods\n\t\n(c)\t The proportions of employed people who work in the formal sector, informal  \n\t\nsector, agriculture and private households.\n\t\n(d)\t The numbers of the employed and unemployed over the three time periods.\nMaths2_Gr9_LB_Book.indb   166\n2014/09/08   09:07:10 AM\n\n\tCHAPTER 10: INTERPRET, ANALYSE AND REPORT ON DATA\t\n167\n10.2\tThe effects of summary statistics on how data is\n\t\nreported\nInformation articles often use averages to report information. The articles might not use \nthe exact terms for average that you have learnt about: the mean, median and mode. \nInstead, they may use terms such as ‘most’. However, it is important to be sure which \nkind of average a report refers to, because they give us different information.\n• Remember that the mean is useful for describing a set of measurement values, but \ncan also be used for other numerical data sets. The word ‘average’ usually refers to \nthe ‘mean’ if it is not explained further. The mean is not reliable if a data set is too \nspread out. \n• The median is the value in the middle of a data set when it is arranged in order. \nHalf the values in the data set are lower than the median and half of them are \nhigher than the median. The median is often the average used when data values \nare not uniformly distributed, because the mean is affected by extreme values in \nthe data set, while the median is not. For example, house prices vary widely, so \nthe median would be a better description of the data than the mean. When the \nmedian is given in a report, the writer should state that they are using the median \nor middle value.  \n• The mode is the number that occurs most often in a set of data. For example, if we \ncollect data about people’s favourite colours, the data set would be a list of colours, \nand the mode would be the colour that comes up most often. The mode can also \nbe used for numbers. Not all data sets have a mode, because sometimes none of the \nnumbers occurs more than once. \nExample\nThe standard way of reporting house prices in South Africa and internationally is the \nmedian house price, which is used by economists in financial reports. The median is \nregarded as more useful than the mean house price because the sale of a few expensive \nhouses would increase the mean, but would not affect the median.\nIf a bank gives bonds for eight houses to the value of R100 000, and for two houses  \nto the value of R1 million, then the mean would be R280 000. This does not seem to  \nbe an accurate reflection of the value of the houses, because it is distorted by the higher \nvalues. The median house price would be R100 000, which is an accurate reflection of  \nthe prices. \nRemember that the median is the middle point, and half of the values fall below the \nmedian, and half above. If the median is lower than the mean, this shows us that there \nare high values that are distorting the mean.\nMaths2_Gr9_LB_Book.indb   167\n2014/09/08   09:07:10 AM\n\n168\t MATHEMATICS Grade 9: Term 4\nusing different summary statistics\n1.\t What kind of average is used in each of these statements?\n\t\n(a)\t The average family has 2,6 children. \n\t\n(b)\t Most families have 3 children. \n    \n\t\n(c)\t Most people prefer red cars. \n   \n\t\n(d)\t The average height for women is 1,62 m.  \n   \n\t\n(e)\t More people shop after work than at any other time during the day. \n  \n2.\t The mean monthly salary of all the staff at company ABC is R8 000 per month, but \nthe median salary is R5 000. \n\t\n(a)\t Explain why the two summary statistics are so different. \n\t\n(b)\t Which summary statistic gives a better idea of the salaries at the company? Give \t\n\t\nreasons for your answer.\n10.3\tMisleading graphs\nThe media (newspapers, magazines, television), regularly use graphs to show \ninformation. Unfortunately, the information is often manipulated to emphasise \na particular result. This may be because the writer simply wants to make his or her \nargument more obvious to the reader. \nChanging the scale of the axis \nIf you change the scale of the vertical axis on bar graphs and line graphs, you will change \nthe way the graphs look. For a bar graph, the larger the spaces between the numbers \non the vertical axis, the bigger the difference between the bars. The smaller the spaces \nbetween the numbers on the axis, the smaller the difference in the height of the bars. \nThe same is true for a line graph which will either have sharp points or be much flatter \ndepending on how you have changed the scale.\nMaths2_Gr9_LB_Book.indb   168\n2014/09/08   09:07:10 AM\n\n\tCHAPTER 10: INTERPRET, ANALYSE AND REPORT ON DATA\t\n169\nExample \nThe two broken-line graphs below show the same sales data for a business over a period \nof six months. Which graph gives the more accurate impression? \nQuantity\nSales (July to December)\n0\n100\n200\n300\n400\n500\n600\n700\n800\nA\nJ\nO\nS\nD\nN\nMonth\nGraph A\nQuantity\nSales (July to December)\n0\n350\n400\n450\n500\n550\n600\n650\n700\nA\nJ\nO\nS\nD\nN\nMonth\nGraph B\nGraph B has a different scale on the vertical axis. The vertical axis does not start at 0 and \nso two blocks on the vertical axis represent 100 items instead of only one block, as in \nGraph A. This makes it look as if the sales increased rapidly over the six months. \nNote that it is not necessarily wrong to change the scale on the axes or not to start  \nat 0. For example, graphs showing stock exchange fluctuations rarely show the origin on \nthe graph and stockbrokers are taught to interpret the graphs in that form. Sometimes \nsmall changes in data values have important effects and in these cases, it may be valid to \nchange the scale to show these. \nMaths2_Gr9_LB_Book.indb   169\n2014/09/08   09:07:11 AM\n\n170\t MATHEMATICS Grade 9: Term 4\nanalysing graphs \n1.\t This graph from Statistics South Africa shows the increase in the percentage of \nhouseholds that had access to piped water over a ten-year period. \nPercentage\nHousehold access to piped water\n82,5\n85\n87,5\n90\n92,5\nYear\n2002\n2004\n2006\n2008\n2010\n2012\n\t\n(a)\t Comment on the scale used on the vertical axis. Is this a misleading graph?\n\t\n(b)\t How could you redraw the graph so that the differences on the graph are more \t\t\n\t\nnoticeable?\n\t\n(c)\t How could you draw the graph so that the differences are less noticeable?\nMaths2_Gr9_LB_Book.indb   170\n2014/09/08   09:07:11 AM\n\n\tCHAPTER 10: INTERPRET, ANALYSE AND REPORT ON DATA\t\n171\n2.\t In this graph the height of the houses represents the number of sales. \nHome Sales\n2010\t                       2014\n\t\nDo you think that this graph is misleading? Give reason(s) for your answer.\n3.\t Look at the two graphs below: \n\t\nGraph A\t\nGraph B\n\t\nWhich graph do you think is drawn correctly? Explain your answer.\nMaths2_Gr9_LB_Book.indb   171\n2014/09/08   09:07:11 AM\n\n172\t MATHEMATICS Grade 9: Term 4\n10.4\tAnalysing extreme values and outliers\nA data item that is very different from all (or most) of \nthe other items in a data set is called an outlier.\nIt is sometimes difficult to notice outliers in numerical data. However, outliers often \nbecome clearly noticeable when data is displayed with graphs.\nMathematics marks\nHistory marks\nRaphael\nRolene\nThuni\nTebogo\nBussi\nRallai\nJoamiah\nSara\nDikgang\nBen\nSipho\nMary\nMichel\nBongilel\n \n1.\t The above scatter plot shows the performance of a group of learners in Mathematics \nand History. Which of the points on the scatter plot can be regarded as outliers? Give \nreasons for your answer.\nMaths2_Gr9_LB_Book.indb   172\n2014/09/08   09:07:11 AM\n\n\tCHAPTER 10: INTERPRET, ANALYSE AND REPORT ON DATA\t\n173\nOutliers in data sets can be very important. We need to decide whether there is a \nparticular reason for the value being so different to the others. Sometimes it gives us \nimportant information. In some cases, the data collected for that point could be wrong.\nThe scatter plot below is for data collected by a transport company. \nLoad weight (kg)\nFuel consumption (l/100 km)\n1 000\n2 000\n3 000\n4 000\n5 000\n6 000\n7 000\n40\n45\n50\n55\n60\n65\n70\nThe company uses just one type of truck. Before each transport job, the company has to \nspecify the price for the job. In order to specify a price before a job, the company needs to \nestimate how much their costs will be for doing the job. One of the main costs is the cost \nof fuel, and the main factor influencing the amount of fuel used is the distance. The load \nweight also plays a role: the greater the load weight, the higher the fuel consumption \n(litres/100 km). \nThe table on the next page gives information that was recorded for previous transport \njobs. The jobs are numbered from 1 to 16 and for each job the values of the four variables \ndistance, load weight, amount of fuel used and fuel consumption rate are given. \n2.\t (a)\t Which of the four variables are represented on the scatter plot given above?\n\t\n(b)\t What are the values of these two variables for the point indicated by the blue  \n\t\narrow on the scatter plot?\nMaths2_Gr9_LB_Book.indb   173\n2014/09/08   09:07:11 AM\n\n174\t MATHEMATICS Grade 9: Term 4\nJob number\nDistance (km)\nLoad weight  \n(kg)\nFuel used \n(litres)\nFuel \nconsumption \n(litres/100 km)\n1\n1 304\n5 445\n879\n67.4\n2\n1 320\n2 954\n639\n48.4\n3\n1 151\n4 705\n698\n60.6\n4\n1 371\n4 378\n787\n57.4\n5\n325\n3 673\n176\n54.2\n6\n1 630\n5 995\n1 113\n68.3\n7\n1 023\n5 357\n600\n58.7\n8\n620\n4 988\n382\n61.6\n9\n73\n1 992\n35\n47.9\n10\n1 071\n5 529\n680\n63.5\n11\n370\n4 140\n218\n58.9\n12\n1 423\n4 062\n843\n59.2\n13\n394\n4 068\n221\n56.1\n14\n1 536\n1 678\n682\n44.4\n15\n1 633\n3 736\n887\n54.3\n16\n435\n3 644\n241\n55.4\n3.\t (a)\t Consider the scatter plot and the data set. What is the effect of load weight on \t \t\n\t\nfuel consumption?\n\t\n(b)\t Is job 7 an exception in this respect? Explain your answer.\n4.\t Further investigations revealed that the driver for jobs 2 and 7 was the same person, \nand that he was not the driver for any other jobs. What may this indicate?\nMaths2_Gr9_LB_Book.indb   174\n2014/09/08   09:07:11 AM\n\n\tCHAPTER 10: INTERPRET, ANALYSE AND REPORT ON DATA\t\n175\nfind outliers\nResearchers collected data on the population of some African countries plus the \nSeychelles, the income per person, and the percentage of the income spent on health.\nCountry\nTotal population \n(in 1 000s)\nTotal annual national \nincome per person \n(US$)\nPercentage of \nincome spent on \nhealth\nAngola\n18 498\n4 830\n4,6\nBotswana\n1 950\n13 310\n10,3\nDRC \n66 020\n280\n2,0\nLesotho\n2 067\n1 970\n8,2\nMalawi\n15 263\n810\n6,2\nMauritius\n1 288\n12 580\n5,7\nMozambique\n22 894\n770\n5,7\nNamibia\n2 171\n6 250\n5,9\nSeychelles\n84\n19 650\n4,0\nSouth Africa\n50 110\n9 790\n8,5\nSwaziland\n1 185\n5 000\n6,3\nTanzania\n43 739\n1 260\n5,1\nZambia\n12 935\n1 230\n4,8\n1.\t What are the three variables in this table?\n2.\t Why do you think it is important to look at income per person in this case, rather \nthan the total income?\nMaths2_Gr9_LB_Book.indb   175\n2014/09/08   09:07:12 AM\n\n176\t MATHEMATICS Grade 9: Term 4\n3.\t Plot the points for the national income per person and the percentage spent on \nhealth care for each country. \n4.\t Write a short report on the data in the table and what the scatter plot shows you \nabout the data. Comment on the general trend and any outliers. \nMaths2_Gr9_LB_Book.indb   176\n2014/09/08   09:07:12 AM\n\nChapter 11\nProbability\n\t\nCHAPTER 11: PROBABILITY\t\n177\nIn this chapter you will learn about the idea of probability, and what information  \nprobabilities provide about what may happen in future. You will also learn about  \ncompound events.\n11.1\t Simple events......................................................................................................... 179\n11.2\t Compound events.................................................................................................. 184\nMaths2_Gr9_LB_Book.indb   177\n2014/09/08   09:07:12 AM\n\n178\t MATHEMATICS Grade 9: Term 4\nMaths2_Gr9_LB_Book.indb   178\n2014/09/08   09:07:13 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n179\n\t\nCHAPTER 11: PROBABILITY\t\n179\n11\tProbability\n11.1\t Simple events\nrevision\nyellow\ngreen\npink\nblue\nred\nbrown\ngrey\nblack\n1.\t (a)\t Suppose the 8 coloured buttons above are in a bag and you draw one button from \t\n\t\nthe bag without looking. Can you tell what colour you will draw? \n\t\n(b)\t Suppose you repeatedly draw a button from the bag, note its colour, then put it \t\n\t\nback. Can you tell in approximately what fraction of all the trials the button will \t\n\t\nbe yellow?  \nArchie has a theory. Because the 8 possible outcomes  \nare equally likely, he believes that if you perform 8  \ntrials in a situation like the above you will draw each  \ncolour once. \n2.\t If Archie’s theory is correct, how many times will  \neach colour be drawn if 40 trials are performed?\n3.\t If Archie’s theory is correct, in what fraction of the  \ntotal number of trials will each colour be drawn?\nEach time you draw a button \nfrom the bag without looking \nyou perform a trial. If you do \nthis and put the button back, \nand repeat the same actions 8 \ntimes, you have performed 8 \ntrials.\nThe number of times an event \noccurs during a set of trials is \ncalled the frequency of the \nevent.\nWhen the frequency of an \nevent is expressed as a fraction \nof the total number of trials, it is \ncalled the relative frequency.\nMaths2_Gr9_LB_Book.indb   179\n2014/09/08   09:07:13 AM\n\n180\t MATHEMATICS Grade 9: Term 4\n4.\t If Archie’s theory is correct, how many times will each of the colours be drawn if \na total of 40 trials is performed? Write your answers in the second row of the  \ntable below. Write the predicted relative frequencies in row 3 as fortieths, and \nin row 4 as twohundredths. \ncolour\nyellow\ngreen\npink\nblue\nred\nbrown\ngrey\nblack\nfrequencies predicted \nby Archie\nrelative frequencies \npredicted by Archie \nexpressed in \n40ths\nrelative frequencies \npredicted by Archie \nexpressed in 200ths\nThe relative frequency for each colour that Archie predicted is called the probability  \nof drawing that colour. If all the outcomes are equally likely, then\nprobability of an outcome = \n1\nthe total number of equally-likely outcomes\nYou will now investigate whether Archie’s theory is correct.\n5.\t (a)\t Make 8 small cards and write the name of one of the above colours on each card, \t\n\t\nso that you have cards with the eight colour names. Perform 8 trials to check \n\t\nwhether Archie’s theory is correct. Record your results (your tally marks 1 and  \n\t\nyour frequencies 1) in the relevant row of the table below.\n\t\n(b)\t Find out what any four of your classmates found when they did the experiment. \t\n\t\nEnter their results in your table too (Friend 1, 2, 3, 4 frequencies).\nTable for the results of the experiments\ncolour\nyellow\ngreen\npink\nblue\nred\nbrown\ngrey\nblack\nyour tally marks (1)\nyour frequencies (1)\nFriend 1 frequencies\nFriend 2 frequencies\nFriend 3 frequencies\nFriend 4 frequencies\nTotal frequencies for \n5 experiments \nMaths2_Gr9_LB_Book.indb   180\n2014/09/08   09:07:13 AM\n\n\t\nCHAPTER 11: PROBABILITY\t\n181\n6.\t (a)\t What was the total number of trials in the five experiments you recorded in the \t\n\t\nabove table? \n\t\n(b)\t What is the total of the frequencies for the different colours, in the last row of \t \t\n\t\nyour table? \n \n7.\t Is Archie’s theory correct? \n \nBettina has a different theory to Archie’s. She believes that if one does many trials with \nthe eight buttons in a bag, each colour will be drawn in approximately one-eighth of \nthe cases. In other words Bettina believes that the relative frequency of each outcome \nwill be close to the probability of that outcome, but may not be equal to it. \n8.\t (a)\t You and your four classmates performed 40 trials in total. Enter the results in the \t\n\t\nsecond row of the table below. Also express each frequency as a fraction of 40, in \t\n\t\nfortieths and in twohundredths.\ncolour\nyellow\ngreen\npink\nblue\nred\nbrown\ngrey\nblack\nactual frequencies \nobtained in your \nexperiments  \n(40 trials) \nrelative frequencies \nas 40ths\nrelative frequencies \nas 200ths\nprobability as \n200ths\n\t\n(b)\t Do your experiments show that Bettina’s theory is correct or not?\nJayden believes that when more trials are performed, the relative frequencies \nwill get closer to the probabilities. \nYou will now do an investigation to investigate whether Jayden’s theory is true.\nMaths2_Gr9_LB_Book.indb   181\n2014/09/08   09:07:13 AM\n\n182\t MATHEMATICS Grade 9: Term 4\ninvestigate what happens when more trials are done\n1.\t Perform 40 trials by drawing one card at a time from eight small cards with the names \nof the colours written on them, and enter your results in the second and third rows of \nthe table below.\ncolour\nyellow\ngreen\npink\nblue\nred\nbrown\ngrey\nblack\ntally marks\nfrequencies\nrelative frequencies \nas 40ths\nrelative frequencies \nas 200ths\nprobabilities as \n200ths\n2.\t Make a copy of the above table, without the row for tally marks, and without the  \nrow for the relative frequencies as fourtieths and the row for the probabilities, on  \na loose sheet of paper. Exchange it with a classmate. Enter the results of your \nclassmate on table 1 and 2 on the next page. Also enter your own results for  \nquestion 1 on the tables.\n3.\t Get hold of the data reports of three other classmates, and enter these on the tables \non the next page too.\n4.\t Add the frequencies of the various colours in the five sets of data for 40 trials each, \nand calculate the relative frequencies expressed as twohundredths.\n5.\t Is the range of relative frequencies for 200 trials smaller than the ranges for the \nfive different sets of 40 trials each? What does this indicate with respect to Jayden’s \ntheory?\nWhen only a small number of trials are done, the \nactual relative frequencies for different outcomes \nmay differ a lot from the probabilities of the \noutcomes.\nWhen many trials are done, the actual relative \nfrequencies of the different outcomes are quite close \nto the probabilities of the outcomes.\nMaths2_Gr9_LB_Book.indb   182\n2014/09/08   09:07:13 AM\n\n\t\nCHAPTER 11: PROBABILITY\t\n183\nTable 1: Frequencies for 5 sets of 40 trials each\ncolour\nyellow\ngreen\npink\nblue\nred\nbrown\ngrey\nblack\nfrequencies for your \nown 40 trials in \nquestion 1\nfrequencies for 40 trials \nby classmate 1\nfrequencies for 40 trials \nby classmate 2\nfrequencies for 40 trials \nby classmate 3\nfrequencies for 40 trials \nby classmate 4\ntotal frequencies for \n200 trials\nrelative frequencies for \n200 trials as \n200ths\nTable 2: Relative frequencies for each of the 5 sets of 40 trials each\n(expressed as 200ths)\ncolour\nyellow\ngreen\npink\nblue\nred\nbrown\ngrey\nblack\nrelative frequencies for \nyour own 40 trials\nrelative frequencies for \n40 trials by classmate 1\nrelative frequencies for \n40 trials by classmate 2\nrelative frequencies for \n40 trials by classmate 3\nrelative frequencies for \n40 trials by classmate 4\n6.\t How many different three-digit numbers can be formed with the symbols 3 and 5, \nif no other symbols are used? You may use one, two or three of the symbols in each \nnumber, and you may repeat the same symbol. \nMaths2_Gr9_LB_Book.indb   183\n2014/09/08   09:07:13 AM\n\n184\t MATHEMATICS Grade 9: Term 4\n11.2\t Compound events\ntossing a coin and giving birth\n1.\t Simon threw a coin and the outcome was heads. He will now throw the coin again.\n\t\n(a)\t What are the possible outcomes? \n\t\n(b)\t What is the probability of each of the possible outcomes? \n \n\t\n(c)\t What are the possible outcomes if Simon throws the coin for a third time? \n\t\n(d)\t What is the probability of each of the possible outcomes for the third throw?\nWhat happens when a coin is thrown for a second \ntime has nothing to do with what happened when it \nwas thrown the first time.\nThe first throw and the second throws are called \nindependent events: what happened on the first \nthrow cannot influence what will happen on the \nsecond throw.\n2.\t (a)\t If an event has four different equally-likely outcomes, what is the probability of \t\n\t\neach of the four outcomes? \n   \n\t\n(b)\t Does that mean that if the event is repeated 4 times, each of the four outcomes \t\t\n\t\nwill happen once? \n  \n\t\n(c)\t Does your answer in (a) means that if the event is repeated 100 times, each of \t \t\n\t\nthe four outcomes will happen 25 times? \n  \n3.\t (a)\t What are the possible outcomes when two coins are thrown? Use the two-way \t\n\t\ntable below to answer this question. One possible outcome is already given.\nHeads\nTails\nHeads\nH T\nTails\n\t\n(b)\t Do you think these four outcomes are equally likely? \n  \n\t\n(c)\t What is the probability of each of the four outcomes? \n  \n\t\n(d)\t What is the probability of getting a head and a tail? \nMaths2_Gr9_LB_Book.indb   184\n2014/09/08   09:07:13 AM\n\n\t\nCHAPTER 11: PROBABILITY\t\n185\n4.\t Let us consider the possible outcomes if three coins are thrown. \n\t\nBelow is a tree diagram that can help you figure out what the different possible \t \t\n\t\noutcomes are. Complete the diagram by filling in the missing information.\nFirst coin\nSecond coin\nThird coin\nOutcome\nheads\ntails\nheads\ntails\nheads\ntails\nheads\nHHH\nHHT\nHTH\nHTT\ntails\n5.\t (a)\t Do you think the eight different outcomes in question 4 are equally likely?  \n\t\n(b)\t What is the probability of each of the eight outcomes? \n  \n\t\n(c)\t What is the probability of throwing two heads and one tail? \n\t\n\t\n6.\t In question 6 on page 183 you were asked to write down the various numbers that \ncan be formed by using symbols 3 and 5. Think of all the four-letter codes that you \ncan form by using only two letters, P and Q. Any letter can be used more than once \nin one code. First think about how you will go about finding all the possibilities in a \nsystematic way and then try to set up a tree diagram to help you.\n\t\n(a)\t Draw a tree diagram in your exercise book to help you to solve this problem. List \t\n\t\nall the outcomes.\nMaths2_Gr9_LB_Book.indb   185\n2014/09/08   09:07:14 AM\n\n186\t MATHEMATICS Grade 9: Term 4\n\t\n(b)\t If the codes are formed by randomly choosing the letters, what is the probability \t\n\t\nthat the code will consist of the using the same letter four times? \n\t\n(c)\t What is the probability that the code will consist of two P’s and two Q’s? \n \nWhen a woman is pregnant, the baby can be a boy or a girl. Suppose we make the \nassumption that the two possibilities are equally likely, so the probability of a boy is 1\n2  \nand the probability of a girl is 1\n2 . \n7.\t (a)\t Complete this two-way table to show the possible outcomes of the gender of the \t\n\t\ntwo children in a family \nBoy\nGirl\n\t\n(b)\t List the possible outcomes.\n\t\n(c)\t What is the probability that the two children in the family will be of the same \t \t\n\t\ngender?\n\t\n(d)\t What is the probability that the eldest child will be a boy and then they will have \t\n\t\na girl?\n8.\t A certain woman already has one child, which is a  \nboy. She now expects a second child. What is the  \nprobability of it being a boy again, if we make the  \nassumption that a baby being a boy or a girl are  \nequally likely events? \n9.\t (a)\t A woman gets married and plans to have a baby in one year and another \t\n\t\n\t\n\t\nbaby in the next year. What is the probability that both babies will be girls? \n   \n\t\n(b)\t A woman gets married and plans to have a baby in each of the first three years \t \t\n\t\nof the marriage. What is the probability that she will have a boy in the first year,\n \t\n\t\nand girls in the second and third years? \n   \nThe assumption that a boy or a \ngirl being born are equally likely \nevents may not actually be true. \nHowever, probabilities can only \nbe calculated and used to make \npredictions if it is assumed that \noutcomes are equally likely.\nMaths2_Gr9_LB_Book.indb   186\n2014/09/08   09:07:14 AM",
      "chapter_id": "4"
    },
    {
      "title": "Surface area, volume and capacity of 3D objects",
      "content": "",
      "chapter_id": "5"
    },
    {
      "title": "Transformation geometry",
      "content": "6\t Transformation geometry\n6.1\t Points on a coordinate system\nA rectangular coordinate system is  \nalso called a Cartesian coordinate  \nsystem. It consists of a horizontal  \nx-axis and a vertical y-axis. \nThe intersection of the axes is called  \nthe origin, and represents the point  \n(0; 0). \nAny point can be represented on a  \ncoordinate system using an x-value  \nand a y-value. These numbers are  \ncalled coordinates, and describe the  \nposition of the point with reference  \nto the two axes.\nThe coordinates of a point are always written in a certain order: \nThe horizontal distance from the origin  \n(x-coordinate) is written first.\nThe vertical distance from the origin  \n(y-coordinate) is written second.\nThese numbers, called an ordered pair, are  \nseparated by a semi-colon (;) and are placed  \nbetween brackets. Here is an example of an  \nordered pair: (4; 3) (see on the coordinate  \nsystem above).\nThe x-axis and y-axis divide the coordinate  \nsystem into four sections called quadrants.  \nThe diagram alongside shows how the  \nquadrants are numbered, and also whether the  \nx- and y-coordinates are negative or positive in each quadrant.\n1.\t In which quadrant will the following points be plotted?\n\t\n(a)\t (−4; 1)\t\n\t (b)\t (−1; −5)\t\n\t\n(c)\t (4; −3)\t\n\t (d)\t (5; 2)\t\n\t\n2.\t Plot the points in question 1 on the coordinate system above.\n4\n1\n2\n3\n–4\n–5\n–6\n–1\n–2\n–3\n5\n1\n2\n3\n–4\n–5\n–1\n–2\n–3\ny-axis\nx-axis\n(4; 3)\norigin\n0\nfirst quadrant\nsecond quadrant\nthird quadrant\nfourth quadrant\n(+; –)\n(+; +)\n(–; –)\n(–; +)\n–\n–\n+\n+\nx\ny\n0\nMaths2_Gr9_LB_Book.indb   91\n2014/09/08   09:06:51 AM\n\n92\t\nMATHEMATICS Grade 9: Term 3\nWhen a point is translated to a different position on a coordinate system, the new \nposition is called the image of the point. We use the prime symbol (') to indicate \nan image. For example, the image of A is indicated by A' (read as “A prime”). If the \ncoordinates of A are labelled as (x; y), the coordinates of A' can be labelled as (x'; y'). \nWe write A ➝ A' and (x; y) ➝ (x'; y') to indicate that A is mapped to A'.\n6.2\t Reflection (flip)\nThe mirror image or reflection of a point is on \nthe opposite side of a line of reflection.\nThe original point and its mirror image are the same  \ndistance away from the line of reflection, and the line that joins the point and its image \nis perpendicular to the line of reflection.\nAny line on the coordinate system can be a line of reflection, including the x-axis, the \ny-axis and the line y = x.\nreflecting points in the x-axis, y-axis and the line y = x\n1.\t The points A(5; 4) and B(−3; −2) are plotted on a coordinate system.\n\t\n(a)\t Reflect points A and B in the x-axis and write down the coordinates of the  \n\t\nimages.\n\t\n(b)\t Reflect points A and B in the y-axis and write down the coordinates of the  \n\t\nimages.\n\t\n(c)\t Compare the coordinates of the original points with those of its images. What  \n\t\ndo you notice?\n“Reflecting a point in the x-axis” \nmeans that the x-axis is the line \nof reflection. \n4\n1\n2\n3\n–4\n–5\n–6\n–1\n–2\n–3\n5\n1\n2\n3\n–4\n–5\n–1\n–2\n–3\ny\nx\nA(5; 4)\nB(–3; –2)\n0\n4\nMaths2_Gr9_LB_Book.indb   92\n2014/09/08   09:06:51 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n93\n2.\t Write down the coordinates of the images of the following reflected points.\nPoint\nReflection in the x-axis\nReflection in the y-axis\n(−131; 24)\n(−459; −795)\n(x; y)\n3.\t The points J(−1; 5), K(−2; −4) and L(1; −2) are plotted on the coordinate system.  \nK' is the reflection of point K in the line y = x. This means that the line y = x is the  \nline of reflection.\n\t\n(a)\t Reflect J and L in the line y = x. \n\t\n(b)\t Write down the coordinates  \n\t\nof the images of the points.\n\t\n(c)\t What do you notice about  \n\t\nthe coordinates of the images  \n\t\nof the points in (b) above?\n\t\n(d)\t Use your observation in (c)  \n\t\nabove to complete this table.\nPoint\nCoordinates of the image of  \nthe point reflected in y = x\n(−1 001; −402)\n(459; −795)\n(−342; 31)\n(21; 67)\n(x; y)\n1\n–1\n–2\n–3\n–4\n–5\n–6\n–7\n–8\n–9\n2\n3\n4\n5\n6\n7\n8\n9\ny\nx\nL\n0\n0\n–2\n1\n2\n3\n4\n5\n6\n7\n–3\n–4\n–5\n–1\nK\nK'\nJ\ny = x\nMaths2_Gr9_LB_Book.indb   93\n2014/09/08   09:06:51 AM\n\n94\t\nMATHEMATICS Grade 9: Term 3\nWhile doing the previous activity, you may have noticed the following.\n• For a reflection in the y-axis, the sign of the x-coordinate changes and the  \ny-coordinate stays the same:  (x; y) → (−x; y) or x' = −x and y' = y,  \nfor example: (−3; 4)→ (3; 4)\n• For a reflection in the x-axis, the sign of the y-coordinate changes and the  \nx-coordinate stays the same:  (x; y) → (x; −y) or x' = x and y' = −y,  \nfor example: (−3; 4) → (−3; −4)\n• For a reflection in the line y = x, the values of the x- and y-coordinates are \ninterchanged:   \n(x; y) → (y; x) or x' = y and y' = x, for example: (−3; 4) → (4; −3).\n4.\t Investigate the effect of reflection in the line y = −x on the coordinates of a point.\n5.\t A is the point (5; −2). Write the coordinates of the mirror images of A if the point is \nreflected in:\n\t\n(a)\t the y-axis             \t\n\t\n\t\n\t\n\t\n\t\n(b)\t the line y = −x\n\t\n\t\n\t\n(c)\t the line y = x       \t\n\t\n\t\n\t\n\t\n\t\n(d)\t the x-axis                  \n\t\n                 \nreflecting geometric figures \nThe same principles as above apply when reflecting geometric figures.\n1.\t (a)\t Reflect ∆PQR in the x-axis, in the y-axis and in the line y = x in the coordinate  \n\t\nsystem (first reflect the vertices and then join the reflected points).\n1\n–1\n–2\n–3\n–4\n–5\n–6\n–7\n–8\n–9\n2\n3\n4\n5\n6\n7\n8\n9\ny\nx\n0\n0\n–2\n1\n2\n3\n4\n5\n6\n–3\n–4\n–5\n–6\n–1\nQ\nP\ny = x\nR\nMaths2_Gr9_LB_Book.indb   94\n2014/09/08   09:06:52 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n95\n\t\n(b)\t Look at your completed reflections in question 1(a), and write down the  \n\t\ncoordinates of the image points in the following table.\nVertices of triangle\nReflection in the \nx-axis\nReflection in the \ny-axis\nReflection in the \nline y = x\nP(−6; 3)\nQ(−5; −2)\nR(−2; 1)\n\t\n(c)\t What do you notice about ∆PQR, ∆P'Q'R', ∆P''Q''R'' and ∆P'''Q'''R'''?\n2.\t Reflect ∆DEF in the x-axis, in the y-axis and in the line y = x. \n1\n–1\n–2\n–3\n–4\n–5\n–6\n–7\n–8\n–9\n2\n3\n4\n5\n6\n7\n8\n9\ny\nx\n0\n0\n–2\n1\n2\n3\n4\n5\n6\n–3\n–4\n–5\n–6\n–1\nE\nD\nF\n3.\t A quadrilateral has the following vertices: A(1; 4), B(−6; 1), C(−2; −1) and D(7; 2). \nWithout performing the actual reflections, write down the coordinates of the  \nvertices of the image when the quadrilateral is:\n\t\n(a)\t reflected in the x-axis\n\t\n(b)\t reflected in the y-axis\n\t\n(c)\t reflected in the line y = x\nMaths2_Gr9_LB_Book.indb   95\n2014/09/08   09:06:52 AM\n\n96\t\nMATHEMATICS Grade 9: Term 3\n4.\t In each case state around which line the point was reflected.\n\t\n(a)\t (−4; 5) → (−4; −5)\t \t\n\t\n\t\n(b)\t (2; −3) → (−2; −3)\t \t\n\t\n\t\n(c)\t (−13; −3) → (−3; −13)\t\n\t\n\t\n(d)\t (1; 16) → (16; 1)\t\n\t\n\t\n\t\n(e)\t (12; −8) → (−12; −8)\t\n\t\n\t\n(f)\t (−7; −5) → (−5; −7)\t\t\n\t\n\t\n(g)\t (2; −3) → (−2; −3)\t \t\n\t\n6.3\t Translation (slide)\nRemember: A translation of a point or geometric figure on a coordinate system means \nmoving or sliding the point in a vertical direction, in a horizontal direction, or in both a \nvertical and horizontal direction.\ntranslating points horizontally or vertically on a \ncoordinate system\n1.\t Points R and W are plotted  \n\t\non a coordinate system.\n\t\n(a)\t Plot the image of point R  \n\t\nafter a translation of:\n• 5 units to the right\n• 5 units to the left\n• 2 units up\n• 2 units down\n\t\n(b)\t Plot the image of point W  \n\t\nafter a translation of:\n• 4 units to the right\n• 4 units to the left\n• 3 units up\n• 3 units down\n1\n–1\n–2\n–3\n–4\n–5\n–6\n–7\n–8\n–9\n2\n3\n4\n5\n6\n7\n8\n9\ny\nx\n0\n–2\n1\n2\n3\n4\n5\n6\n–3\n–4\n–5\n–6\n–1\nW\nR\nMaths2_Gr9_LB_Book.indb   96\n2014/09/08   09:06:52 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n97\n\t\n(c)\t Look at your completed translations in (a) and (b) on the previous page.  \n\t\nComplete the following table by writing down the coordinates of the original \n\t\npoints and their images after each translation.\n\t\nCoordinates of original points\nR(−1; 4)\nW(−2; −3)\nCoordinates of image after a translation to the right\nCoordinates of image after a translation to the left\nCoordinates of image after a translation up\nCoordinates of image after a translation down\n\t\n\t\n(d)\t Look at your completed table in (c) above. Choose the correct answers below to \t\n\t\nmake each statement true:\n• For translations to the right or left, the (x-value/y-value) changes and the \n(x-value/y-value) stays the same.\n• For translations up or down, the (x-value/y-value) changes and the  \n(x-value/y-value) stays the same.\n• For translations to the right, (add/subtract) the number of translated units \n(to/from) the x-value.\n• For translations to the left, (add/subtract) the number of translated units  \n(to/from) the x-value.\n• For translations up, (add/subtract) the number of translated units (to/from) \nthe y-value.\n• For translations down, (add/subtract) the number of translated units  \n(to/from) the y-value.\n2.\t Write down the coordinates of each image after the following translations.\nPoint\n3 units to the \nright\n4 units to the \nleft\n2 units up\n5 units down\n(3; 5)\n(−13; 42)\n(−59; −95)\n(x; y)\nMaths2_Gr9_LB_Book.indb   97\n2014/09/08   09:06:52 AM\n\n98\t\nMATHEMATICS Grade 9: Term 3\n3.\t Write down the coordinates of each image after the following translations:\nPoint\n4 units to the \nright and  \n3 units up\n2 units to the \nleft and \n1 unit up\n1 unit to the \nright and \n5 units down\n6 units to the \nleft and \n2 units down\n(4; 2)\n(−32; 21)\n(−68; −57)\n(x; y)\n\t\nWhile doing the previous activity, you may have noticed the following:\n• For a horizontal translation through the distance p, the x-coordinate increases  \nby the distance p if the slide is to the right, and decreases by the distance p if the \nslide is to the left. We may write x' = x + p, with p > 0 for a translation to the right, \nand p < 0 for a translation to the left. The y-coordinate remains the same, so  \n(x; y) → (x + p; y).\n• For a vertical translation through the distance q, the y-coordinate increases by \nthe distance q if the slide is upwards, and decreases by the distance q if the slide is \ndownwards. We may write y' = y + q, with q > 0 for a translation vertically upwards, \nand q < 0 for a translation vertically downwards. The x-coordinate remains the \nsame, so (x; y) → (x; y + q).\ntranslation of geometric figures on a coordinate system\n1.\t (a)\t Translate ∆PQR  \n\t\n5 units to the  \n\t\nright and 3 units  \n\t\ndown.\n\t\n(b)\t Translate ∆PQR  \n\t\n2 units to the left  \n\t\nand 3 units up. \n\t\n(c)\t Are all the triangles  \n\t\ncongruent? \n–2\n–4\n–6\n–8\n2\n4\n1\n3\ny\nx\n0\n–2\n2\n4\n6\n8\n–4\nQ\nP\nR\n1\n3\n5\n7\n–1\n–3\n5\n–1\n–3\n–5\n–7\n–9\nMaths2_Gr9_LB_Book.indb   98\n2014/09/08   09:06:52 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n99\n2.\t (a)\t Translate ∆DEF 4 units to the left and 2 units down.\n\t\n(b)\t Translate ∆DEF 3 units to the right and 4 units up.\n\t\n(c)\t Are all the triangles congruent? \n–2\n2\n4\n6\n8\n10\n12\n14\n–4\n–6\n–8\n–10\ny\nx\n0\n0\n–2\n2\n4\n6\n–4\n–6\n–8\nE\nD\nF\n\t\n3.\t The vertices of a quadrilateral have the following coordinates: K(−5; 2), L(−4; −2), \nM(1; −3) and N(4; 3). Write down the coordinates of the image of the quadrilateral \nafter the following translations.\n\t\n(a)\t 7 units to the right and 2 units up\n\t\n(b)\t 5 units to the right and 2 units down\n\t\n(c)\t 4 units to the left and 3 units down\n\t\n(d)\t 2 units to the left and 7 units up\n4.\t Describe the translation if the coordinates of the original point and the image point are:\n\t\n(a)\t (−2; −3) → (−2; −5)\t\t\n\t\n(b)\t (4; −7) → (−6; 0) \t\n\t\n\t\n(c)\t (3; 11) → (16; 20) \t \t\n\t\n(d)\t (−1; −2) → (5; −4)\t \t\n\t\n(e)\t (8; −11) → (−2; −3)\t \t\nMaths2_Gr9_LB_Book.indb   99\n2014/09/08   09:06:53 AM\n\n100\t MATHEMATICS Grade 9: Term 3\n6.4\t Enlargement (expansion) and reduction (shrinking)\nwhat are enlargements and reductions?\nYou will remember the following from Grade 8.\n• An image is an enlargement or reduction of the original figure only if all the \ncorresponding sides between the two figures are in proportion. This means  \nthat all the sides of the original figure are multiplied by the same number (the \nscale factor) to produce the image.\n• Scale factor = \nside length of image\nlength of corresponding side of original figure\nºº\nIf the scale factor is > 1, the image is an enlargement.\nºº\nIf the scale factor is < 1, the image is a reduction.\n• The original figure and its enlarged or reduced image are similar.\n• Perimeter of image = Perimeter of original figure × scale factor\n• Area of image = Area of original figure × (scale factor)2\n6\nA = 24 units2\nP = 24 units\nA = 96 units2\nP = 48 units\nA = 6 units2\nP = 12 units\n8\n10\n16\n12\n20\n3\n4\n5\nEnlargement by \na factor of 2\nReduction by \na factor of 2\nDivide each side by 2 \n(or multiply each side by    )\nMultiply each side by 2\nP = perimeter\nA = area\n1\n2\nMaths2_Gr9_LB_Book.indb   100\n2014/09/08   09:06:53 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n101\nSometimes the terminology used for enlargements and reductions can be confusing. \nMake sure you understand the following examples. Refer to the diagram on the  \nprevious page.\n“Enlarge a figure by a scale factor of 2” means: \n• \nside length of image\nlength of corresponding side of original figure  = 2\n• Each side of the original figure must be multiplied by 2.\n• Each side of the image will be 2 times longer than its corresponding side in the \noriginal figure.\n• The perimeter of the image will be 2 times longer than the perimeter of the  \noriginal figure.\n• The area of the image will be 22 times (2 × 2 = 4 times) bigger than the area of the \noriginal figure.\n“Reduce a figure by a scale factor of 2” means:\n• \nside length of image\nlength of corresponding side of original figure  = 0,5\n• Each side of the original figure must be multiplied by 1\n2  (or divided by 2).\n• Each side of the image will be 2 times  \nshorter than its corresponding side in the  \noriginal figure.\n• The perimeter of the image will be 2 times  \nshorter than the perimeter of the original figure.\n• The area of the image will be 22 times (2 × 2 = 4 times) smaller than the area of \nthe original figure. (Or area of image = (1\n2)2 = 1\n4 of the area of the original figure.)\npractise working with enlargements and reductions\n1.\t Work out the scale factor of each original figure and its image.\n\t\n(a)\nA\nB\nC\nA'\nB'\nC'\n6 cm\n5 cm\n7,5 cm\n12 cm\n10 cm\n15 cm\nA = 56 cm2\nP = 37 cm\nNote that the multiplicative\ninverse of 2 is 1\n2 .\nMaths2_Gr9_LB_Book.indb   101\n2014/09/08   09:06:54 AM\n\n102\t MATHEMATICS Grade 9: Term 3\n\t\n(b)\nD\nE\nF\nD'\nE'\nF'\n4 cm\n6 cm\n20 cm\n30 cm\nA = 550 cm2\nP = 100 cm\nG\nG'\n\t\n(c)\t\nJ\nK\nL\nJ'\nK'\nL'\n15 cm\n20 cm\n8 cm\nA = 48 cm2\nP = 28 cm\nM\nM'\n6 cm\n2.\t For each set of figures in question 1, write down by how many times the perimeter of \neach image is longer or shorter than the perimeter of the original image. Also write \ndown the perimeter of each image.\n\t\n(a)\t\n\t\n(b)\t\n\t\n(c)\t\n3.\t For each set of figures in question 1, write down by how many times the area of each \nimage is bigger or smaller than the area of the original image. Also write down the \narea of each image.\n\t\n(a)\t\n\t\n\t\n(b)\t\n\t\n(c)\t\n\t\n4.\t The perimeter of rectangle DEFG = 20 cm. Write down the perimeter of the rectangle \t\n\t\nD'E'F'G' if the scale factor is 3.\t \n5.\t The perimeter of quadrilateral PQRS = 30 cm and its area is 50 cm2.\n\t\n(a)\t Find the perimeter of P'Q'R'S' if the scale factor is 1\n5 . \n\t\n(b)\t Determine the area of quadrilateral P'Q'R'S' if the scale factor is 1\n5. \nMaths2_Gr9_LB_Book.indb   102\n2014/09/08   09:06:54 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n103\n6.\t The perimeter of ∆DEF = 17 cm and the perimeter of ∆D'E'F' = 25,5 cm.\n\t\n(a)\t What is the scale factor of enlargement? \n \n\t\n(b)\t What is the area of ∆D'E'F' if the area of ∆DEF = 14 cm2? \n7.\t The area of ∆ABC = 20 cm2 and the area of ∆A'B'C' = 5 cm2.\n\t\n(a)\t What is the scale factor of reduction? \n\t\n(b)\t What is the perimeter of the image if the perimeter of ∆ABC = 22 cm? \ninvestigating enlargement and reduction\nWhen we do enlargements or reductions on a coordinate system, we use one point from \nwhich to perform the enlargement or reduction. This point is known as the centre of \nenlargement or reduction.\nThe centre of enlargement or reduction can be any point on the coordinate system.  \nIn this chapter, we will always use the origin as the centre of enlargement or reduction.\nRectangle ABCD, rectangle A'B'C'D' and rectangle A''B''C''D'' are plotted on a  \ncoordinate system as shown below. \ny\nx\n14\n12\n10\n8\n6\n4\n2\n–2\n0\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\nC'\nA'\nB'\nD'\nC\nA\nB\nD\nC''\nA''\nB''\nD''\n1.\t (a)\t Is rectangle A''B''C''D'' an enlargement of rectangle ABCD? Explain your answer.\nMaths2_Gr9_LB_Book.indb   103\n2014/09/08   09:06:54 AM\n\n104\t MATHEMATICS Grade 9: Term 3\n\t\n(b)\t Is rectangle A'B'C'D' a reduction of rectangle ABCD? Explain your answer.\n2.\t (a)\t The origin is the centre of enlargement and reduction. Draw four line segments \t\n\t\nto join the origin with A'', B'', C'' and D''.\n\t\n(b)\t What do you notice about these line segments?\n3.\t (a)\t List the coordinates of the images to complete the following table\nVertices of ABCD\nVertices of A'B'C'D'\nVertices of A''B''C''D''\nA(6; 6)\nB(6; 4)\nC(10; 4)\nD(10; 6)\n\t\n(b)\t What do you notice about the coordinates of the vertices of the original  \n\t\nrectangle and the coordinates of the vertices of the image?\nFrom the previous activity, you should have found the following:\nOn a coordinate system, the line that joins the \ncentre of an enlargement or reduction to a vertex \nof the original figure also passes through the \ncorresponding vertex of the enlarged or reduced \nimage.\nThe coordinates of a vertex of the enlarged or \nreduced image are equal to the scale factor × the \ncoordinates of the corresponding vertex of the \noriginal figure.\nMaths2_Gr9_LB_Book.indb   104\n2014/09/08   09:06:54 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n105\nFor example:\nB(6; 4) → B' (3; 2): The coordinates of B' are 1\n2  the coordinates of B. Note that the scale \nfactor is 1\n2 .\nB(6; 4) → B'' (12; 8): The coordinates of B'' are 2 times the coordinates of B. Note that \nthe scale factor is 2.\nIn general, we therefore use the following notation to describe the enlargement or  \nreduction with respect to the origin: \n(x; y) → (kx; ky) or (x'; y') = (kx; ky) where k is the \nscale factor.\nIf 0 < k < 1, the image is a reduction.\nIf k > 1, the image is an enlargement.\npractise\n1.\t Draw the enlarged or reduced images of the following figures according to the scale \nfactor given. In each case, use the origin as the centre of enlargement or \nreduction.\n\t\n(a)\t Scale factor = 2\n–2\n–4\n–6\n–8\n2\n4\n6\n8\ny\nx\n0\n0\n–2\n2\n4\n6\n8\n–4\n–6\n–8\nT\nS\nR\n–10\nMaths2_Gr9_LB_Book.indb   105\n2014/09/08   09:06:55 AM\n\n106\t MATHEMATICS Grade 9: Term 3\n\t\n(b)\t Scale factor = 1\n2    \n–2\n–4\n–6\n–8\n2\n4\n6\n8\ny\nx\n0\n0\n–2\n2\n4\n6\n8\n–4\n–6\n–8\nK\nJ\nH\n\t\n\t\n(c)\t Scale factor = 1\n2\n–4\n–6\n–8\n–10\n–2\n2\n4\n6\ny\nx\n0\n0\n6\n8\n10\n–2\n–4\nQ\nS\nR\n2\n4\n–12\nP\nMaths2_Gr9_LB_Book.indb   106\n2014/09/08   09:06:55 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n107\n\t\n(d)\t Scale factor = 1\n3  \n–2\n–4\n–6\n–8\n–10\n2\n4\n6\ny\nx\n0\n0\n–2\n2\n4\n6\n8\n–4\n–6\n–8\nL\nM\nK\nN\nO\n2.\t A quadrilateral has the following vertices: A(−2; 4), B(−4; −2), C(4; −3) and D(2; 1). \t\nDetermine the coordinates of the enlarged image if the scale factor = 2.\n3.\t A quadrilateral has the following vertices: P(−4; 0), Q(2,5; 4,5), R(6; −2,25) and  \nS(2; −4). Determine the coordinates of the enlarged image if the scale factor = 4.\n4.\t A quadrilateral has the following vertices: D(6; −4 ), E(4; −6), F(−4; 2) and G(−2; −2). \nDetermine the coordinates of the reduced image if the scale factor = 1\n2 .\n5.\t A quadrilateral has the following vertices: K(8; −2), L(4; −6), M(−8; −4) and  \nN(−6; 10). Determine the coordinates of the reduced image if the scale factor = 1\n4 .\n6.\t Describe the following transformations:\n\t\n(a)\t A(7; −5) → A'(9; 0)\nMaths2_Gr9_LB_Book.indb   107\n2014/09/08   09:06:56 AM\n\n108\t MATHEMATICS Grade 9: Term 3\n\t\n(b)\t A(−4; 6) → A'(4; 6)\n\t\n(c)\t A(−3; −2) → A'(−2; −3)\n\t\n(d)\t A(8; 1) → A'(8; −1)\n\t\n(e)\t A(4; −2) → A'(8; −4)\n\t\n(f)\t A(12; −16) → A'(3; −4)\n\t\n(g)\t A(2; −1) → A'(−3; −5)\n7.\t Describe each of the following transformations.\n\t\n(a)\t \t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(b)\n1\n–1\n–2\n–3\n–4\n–5\n2\n3\n4\n5\n6\ny\nx\n0\n0\n–2\n1\n2\n3\n4\n5\n6\n–1\nA\nB\nC\nD\nA'\nB'\nC'\nD'\n–2\n2\n4\n6\n8\n10\ny\nx\n0\n0\n–2\n2\n4\nC\nB\nA\nD\nC'\nB'\nA'\nD'\nMaths2_Gr9_LB_Book.indb   108\n2014/09/08   09:06:56 AM\n\nChapter 7\nGeometry of 3D objects\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n109\nIn this chapter, you will revise the properties of prisms and pyramids, which you \ninvestigated in previous grades. This includes using nets to construct models of these \nobjects as a further means of consolidating your understanding of polyhedra. You will also \nrevise the properties and definitions of the five Platonic solids, which you first learnt about \nin Grade 8, as well as how Euler’s formula describes a relation between the number of \nvertices, faces and edges of any polyhedron.\nNew to this grade are investigations of the properties of cylinders and spheres. Although \nyou should be able to recognise these 3D objects by now, you will examine some of their \nproperties in more detail, and learn how to construct a net and model of a cylinder.\n7.1\t Classifying 3D objects............................................................................................. 111\n7.2\t Nets and models of prisms and pyramids................................................................ 113\n7.3\t Platonic solids......................................................................................................... 115\n7.4\t Euler’s formula........................................................................................................ 119\n7.5\t Cylinders................................................................................................................ 121\n7.6\t Spheres................................................................................................................... 124\nMaths2_Gr9_LB_Book.indb   109\n2014/09/08   09:06:56 AM\n\n110\t MATHEMATICS Grade 9: Term 3\nMaths2_Gr9_LB_Book.indb   110\n2014/09/08   09:06:57 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n111",
      "chapter_id": "6"
    },
    {
      "title": "Geometry of 3D objects",
      "content": "7\t Geometry of 3D objects\n7.1\t\nClassifying 3D objects\n3D objects with flat faces which are called  \npolyhedra. Prisms and pyramids are two types of  \npolyhedra. \nTriangular prism\nRectangular prism\nRectangular-based pyramid\nCube\nExamples of 3D objects that have at least one curved surface are cylinders, spheres  \nand cones.\nCylinders\nSpheres\nCones\nWhen we study the properties of a 3D  \nobject, we investigate the shapes of its  \nfaces, its number of faces, its number of  \nvertices and its number of edges. For  \nexample, the pyramid alongside has  \n1 square face and 4 triangular faces,  \n5 vertices and 8 edges.\nEdges\nFaces\nVertices\nSquare-based pyramid\nA polyhedron is a 3D object \nwith only flat faces.\nMaths2_Gr9_LB_Book.indb   111\n2014/09/08   09:06:57 AM\n\n112\t MATHEMATICS Grade 9: Term 3\nClassifying and describing 3d objects\n1.\t Label parts (a) to (c) on the prism correctly.\n\t\n(a)\t\n\t\n(b)\t\n\t\n(c)\t\n2.\t Complete the table.\n3D object\nName of the object\nNumber of faces \nand shape of faces\nNumber of \nvertices\n(a)\nTriangular prism\n2 triangles and 3 \nrectangles\n6\n(b)\n(c)\n6 squares\n8\n(d)\n1 rectangle and 4 \ntriangles\n5\n(e)\n(f)\n(g)\n(a)\n(b)\n(c)\nMaths2_Gr9_LB_Book.indb   112\n2014/09/08   09:06:58 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n113\n3.\t Say whether each statement below is true or false.\n\t\n(a)\t A cylinder is a polyhedron.\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(b)\t A triangular-based pyramid has 4 triangular faces.\t\n\t\n\t\n\t\n\t\n(c)\t A cube is also known as a hexahedron.\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(d)\t A triangular-based pyramid has 6 vertices.\t\n\t\n\t\n\t\n\t\n\t\n\t\n(e)\t A pyramid is a 3D object.\t \t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n7.2\t\nNets and models of prisms and pyramids\nA net is a flat pattern that can be used to represent a 3D  \nobject. The net can be folded up to create a model of the  \n3D object.\n1.\t Name each object below and draw an arrow to match it with its net.\n(a)\n(b)\n(c)\n(d)\nNet of a cube\nMaths2_Gr9_LB_Book.indb   113\n2014/09/08   09:06:59 AM\n\n114\t MATHEMATICS Grade 9: Term 3\n2.\t Construct an accurate net for each of the following 3D objects.\n\t\n(a)\n\t\n\t\n1 cm\n4 cm\n2 cm\n\t\n(b)\n\t\n\t\n3 cm\n4 cm\n5 cm\nMaths2_Gr9_LB_Book.indb   114\n2014/09/08   09:06:59 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n115\n\t\n(c)\n\t\n\t\n4 cm\n2 cm\n3.\t Construct models of  \nthe objects in question 2  \nbut double all the  \nmeasurements.\n7.3\t\nPlatonic solids\nA Platonic solid is a 3D object which has identical faces, and all of the faces are \nidentical regular polygons. This means that all its faces are the same shape and size and \nall the vertices are identical.\n1.\t Which of the following objects are Platonic solids?\n\t\nA.\t \t\n\t\n\t\n\t\nB.\t \t\n\t\n\t\n\t\nC.\t \t\n\t\n\t\n   D.\t\n\t\n\t\n\t\nE.\n\t\n\t\nF.\t\n\t\n\t\n\t\nG.\t \t\n\t\n\t\n   H.\t\n\t\n\t\n\t\n  I.\t \t\n\t\n        J.\n\t\n\t\n\t\n \n2.\t How many Platonic solids are there in question 1? \nMaths2_Gr9_LB_Book.indb   115\n2014/09/08   09:07:00 AM\n\n116\t MATHEMATICS Grade 9: Term 3\nOnly five platonic solids?\nYou can use your knowledge about angles to prove that the five Platonic solids are  \nthe only 3D objects that can be made from identical regular polygons. Keep the \nfollowing facts in mind:\n• A 3D object has at least three faces that meet at each vertex.\n• The sum of the angles that meet at a vertex must be less than  \n360°. If it is equal to 360°, it will form a flat surface. If it is  \ngreater than 360°, the faces will overlap.\n• Each Platonic solid is made up of one type of regular polygon only.\nWhat 3D objects can you make from equilateral triangles?\nWe use the following reasoning:\nsize of each interior angle = 180° ÷ 3 = 60°\n∴\t 3 triangles = 3 × 60° = 180°\t\n[< 360°]\n\t\n4 triangles = 4 × 60° = 240°\t\n[< 360°]\n\t\n5 triangles = 5 × 60° = 300°\t\n[< 360°]\n\t\n6 triangles = 6 × 60° = 360°\nAny more than 5 triangles will be equal to or more than 360° and will therefore form a flat \nsurface or overlap.\nThis means that we can make three 3D objects from equilateral triangles:\nIf 3 triangles are at each vertex, it will form a tetrahedron.\nIf 4 triangles are at each vertex, it will form an octahedron.\nIf 5 triangles are at each vertex, it will form an icosahedron.\nTetrahedron\nOctahedron\nIcosahedron\n1\n2\n3\nMaths2_Gr9_LB_Book.indb   116\n2014/09/08   09:07:00 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n117\nWhat 3D objects can you make from squares?\nComplete the statements: size of each interior angle \n\t\n∴ \t\n3 squares = 3 × \n  \n\t\n\t\n4 squares = 4 × \n \nTherefore we can make only one 3D object using squares. This 3D  \nobject is called a hexahedron (or cube).\nWhat 3D objects can you make from regular pentagons?\nComplete the statements:\n\t\nSize of each interior angle \n\t\n∴ \t\n3 pentagons = \n  \n\t\n\t\n4 pentagons = \n \nTherefore we can make only one 3D object using regular  \npentagons. This 3D object is called a dodecahedron.\nWhat 3D objects can you make from regular hexagons?\nComplete the statements: \n\t\nSize of each interior angle \n\t\n∴  \t 3 hexagons = \nThree hexagons will already form a flat surface. Therefore it is  \nimpossible to make a 3D object from regular hexagons. \nAlso, the interior angles of polygons with more than 6 sides are bigger than those of  \na hexagon, so it is not possible to make 3D objects from any other regular polygons.\nTherefore the five Platonic solids already \nmentioned (tetrahedron, octahedron, \nicosahedron, hexahedron and dodecahedron) \nare the only ones that can be made of identical \nregular polygons. Each of these solids is named \nafter the number of faces it has.\nHexahedron (cube)\nDodecahedron\nMaths2_Gr9_LB_Book.indb   117\n2014/09/08   09:07:00 AM\n\n118\t MATHEMATICS Grade 9: Term 3\nproperties of the platonic solids\nComplete the information about each of the following Platonic solids.\n1.\nName: \nShape of the faces: \n \nNumber of faces: \nNumber of edges: \nNumber of vertices: \n2.\nName: \n \nShape of the faces: \n \nNumber of faces: \nNumber of edges: \nNumber of vertices: \n3.\nName: \n \nShape of the faces: \n \nNumber of faces: \n \nNumber of edges: \nNumber of vertices: \n4.\nName: \nShape of the faces: \n \nNumber of faces: \nEdges: \nVertices: \n5.\nName:  \n \nShape of the faces: \nNumber of faces: \nEdges: \nVertices: \nMaths2_Gr9_LB_Book.indb   118\n2014/09/08   09:07:02 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n119\n7.4\t\nEuler’s formula\nEuler’s formula and platonic solids\n1.\t You learnt about Euler’s formula in Grade 8. Complete the following table to \ninvestigate whether or not Euler’s formula holds true for Platonic solids.\nName\nShape of \nfaces\nNo. of \nfaces (F)\nNo. of \nvertices \n(V)\nNo. of \nedges  \n(E)\nF + V − E\n2.\t Complete Euler’s formula for polyhedra:\n\t\nF + \n    \n3.\t Apply Euler’s formula to each of the following:\n\t\n(a)\t A polyhedron has 25 faces and 13 vertices. How many edges will it have? \t\n\t\n(b)\t A polyhedron has 11 vertices and 23 edges. How many faces does it have?\t\n\t\n(c)\t A polyhedron has 8 faces and 12 edges. How many vertices does it have?\t\nMaths2_Gr9_LB_Book.indb   119\n2014/09/08   09:07:02 AM\n\n120\t MATHEMATICS Grade 9: Term 3\neuler’s formula and other polyhedra\n1.\t Is each of the following statements true or false?\n\t\n(a)\t A polyhedron with 10 vertices and 15 edges must have 7 faces.\t\n\t\n\t\n\t\n(b)\t A polyhedron will always have more edges than either faces or vertices.\t\n\t\n(c)\t A polyhedron with 5 faces must have 6 edges.\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(d)\t A pyramid will always have the same number of faces and vertices.\t \t\n2.\t Complete the following table.\n\t\nNo. of \nfaces (F)\nNo. of \nvertices (V)\nNo. of \nedges (E)\nName of \npolyhedron\nShapes of faces\n(a)\n6\n12\nRectangles\n(b)\n7\nHexagonal \npyramid\n(c)\n4\n4\n(d)\n5\n6\n9\nTriangles and \nrectangles\n3.\t A soccer ball consists of pentagons and hexagons. \n\t\n(a)\t How many pentagons does it consist of? \n\t\n(b)\t How many hexagons does it consist of? \n\t\n(c)\t How many edges does it have? \n\t\n(d)\t How many vertices does it have? \n\t\n(e)\t Does Euler’s formula apply to soccer balls too?\n\t\n\t\nNet of a soccer ball\n \nMaths2_Gr9_LB_Book.indb   120\n2014/09/08   09:07:02 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n121\n7.5\t\nCylinders\nProperties of cylinders\n1.\t Which of the following 3D objects are cylinders? \nA\nB\nC\nD\nE\nF\n2.\t Tick the statement or statements below that are true only for cylinders and not for \nthe other objects shown in question 1:\n\t\n\t It is a 3D object.\n\t\n\t It has a curved surface.\n\t\n\t It has two circular bases that are parallel to each other.\n\t\n\t It has two flat circular bases and a curved surface.\n\t\n\t The radius of its curved surface is equal from the top to the bottom between the \t\n\t\nbases. \n\t\n\t It has two circular bases opposite each other, joined by a curved surface whose \t \t\n\t\nradius is equal from the top to the bottom between the bases.\n3.\t Look at the cylinder alongside and complete the following:\n\t\n(a)\t Number and shape of faces: \n \n\t\n(b)\t Number of vertices: \n \n\t\n(c)\t Number of edges: \n \nNets of cylinders\nIn Chapter 5, you learnt about the net of a cylinder. If you cut the  \ncurved surface of a cylinder vertically and flatten it, it will be the  \nshape of a rectangle.\n1.\t Explain why the length of the rectangular face is  \nequal to the circumference of the base.\nMaths2_Gr9_LB_Book.indb   121\n2014/09/08   09:07:03 AM\n\n122\t MATHEMATICS Grade 9: Term 3\n2.\t Will each of the following nets form a cylinder?\n\t\nA.\t \t\nB.\t\n\t\nC.\n\t\n\t\n\t\nD.\t \t\nE.\t\n\t\nF.\n\t\n3.\t In each of the following questions, use π = 22\n7  and round off your answer to two \ndecimal places to do the calculations.\n\t\n(a)\t If the radius of a cylinder is 3 cm, what is the length of the rectangular surface of \t\n\t\nthe cylinder?\n\t\n(b)\t If the radius of a cylinder is 5 cm, what is the length of the rectangular surface of \t\n\t\nthe cylinder?\n\t\n(c)\t If the diameter of a cylinder is 8 cm, what is the length of the rectangular surface \t\n\t\nof the cylinder?\n\t\n(d)\t If the diameter of a cylinder is 9 cm, what is the length of the rectangular surface \t\n\t\nof the cylinder?\nMaths2_Gr9_LB_Book.indb   122\n2014/09/08   09:07:03 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n123\n4.\t Use a ruler and a set of compasses to construct the following nets as accurately as \npossible. Show the measurements on each net.\n\t\n(a)\t Net of a cylinder with a radius of 1 cm and a height of 4 cm.\n\t\n(b)\t Net of a cylinder with a radius of 1,5 cm and a height of 3 cm\n5.\t Construct models of the cylinders in question 5 but double the measurements.\nMaths2_Gr9_LB_Book.indb   123\n2014/09/08   09:07:04 AM\n\n124\t MATHEMATICS Grade 9: Term 3\n7.6\t\nSpheres\n1.\t Which of the following 3D objects are spheres?\nA\nB.\nC.\nD.\nE.\nF.\nG.\n.\n2.\t Tick the property or properties below that are true for spheres only and not for the \nother objects shown in question 1.\n\t\n\t It is a 3D object.\n\t\n\t It has one curved surface.\n\t\n\t It has no bases.\n\t\n\t It has no vertices.\n\t\n\t It has no edges.\n\t\n\t The distance from its centre to any point on its surface is always equal.\n3.\t Complete the following information for a sphere:\n\t\n(a)\t Number and shape of faces: \n\t\n(b)\t Number of vertices:  \n \n\t\n(c)\t Number of edges:  \n \nFrom your study of spheres in the above activity, you  \nshould have found the following:\nA sphere is a round 3D object with only one \ncurved surface and the distance from its centre  \nto any point on its surface is always equal. It has \nno vertices or edges.\nr\nr\nr\nr\nMaths2_Gr9_LB_Book.indb   124\n2014/09/08   09:07:04 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n125\n4.\t In the sphere alongside, write down the length of:\n\t\n(a)\t the radius: \n\t\n(b)\t the diameter: \n\t\n(c)\t MD: \n\t\n(d)\t CD: \n5.\t The drawing alongside shows part of a sphere  \nwith a diameter of 100 km. Imagine that you  \nare at point M, at the centre inside the sphere.  \nPeople A, B and C are all at different places on  \nthe surface of the sphere.\n\t\n(a)\t Which of the people – A, B or C – is closest to \t\n\t\nyou?\n\t\n(b)\t How far away is person C from you?\nNet of a sphere\nIt is impossible to make a perfect sphere (ball or globe) from a flat sheet of paper. Paper can curve in \none direction, but cannot curve in two directions at the same time. So all spheres made from paper or \ncard will be approximations. This is the best net we can make of a sphere. \nCan you make your own \npaper model of a sphere?\nA\nB\nC\nD\n5 cm\nM\nA\nB\nC\nM\nMaths2_Gr9_LB_Book.indb   125\n2014/09/08   09:07:04 AM\n\nWorksheet\nGEOMETRY OF 3D OBJECTS\n1.\t Grade 9 learners were asked to represent a 3D object and give the class clues as \t\n\t\nto which polyhedron they represent. Name their objects:\n\t\n(a)\tAmy: I have 6 faces and they are all the same size.\t\n\t\n(b)\tJohn: I have 6 faces and 12 edges. I am not a cube.\t\n\t\n\t\n(c)\tOnke: I have 3 faces. I also have two edges.\t\n\t\n\t\n(d)\tTessa: I have 8 edges and I have 5 vertices.\t\n\t\n\t\n(e)\t Mandlakazi: I have 6 edges and 4 vertices.\t\n\t\n\t\n(f)\t Chiquita: I have 8 faces and am a Platonic solid.\t\n\t\n\t\n(g)\tSeni: I do not have any edges.\t\n\t\n\t\n(h)\tMpu: My faces are made only of regular pentagons.\t\n\t\n2.\t Write down the required information about each object below.\n\t\nA.\t\n\t\nB.\t\n \n\t\nObject A\nObject B\nName\nNumber of faces\nShape/s of faces\nNumber of edges\nNumber of vertices\nDoes Euler’s formula work?\nIs it a Platonic solid?\n3.\t (a)\tOn a separate sheet of paper, construct a net of a cylinder with a diameter of \n\t\n\t\n7 cm and a height of 10 cm.\n\t\n(b)\tFold your net to make a model of the cylinder.\nMaths2_Gr9_LB_Book.indb   126\n2014/09/08   09:07:05 AM\n\nChapter 8\nCollect, organise and\nsummarise data\n\t CHAPTER 8: Collect, organise and summarise data\t\n127\nYou have learnt how to collect, organise and summarise data in previous grades. In Grade 9, \nyou need to decide which methods are best in certain situations and you need to be able to \njustify your choices.\n8.1\t Collecting data....................................................................................................... 129\n8.2\t Organising data...................................................................................................... 133\n8.3\t Summarising data................................................................................................... 136\nMaths2_Gr9_LB_Book.indb   127\n2014/09/08   09:07:05 AM\n\n128\t MATHEMATICS Grade 9: Term 4\nMaths2_Gr9_LB_Book.indb   128\n2014/09/08   09:07:05 AM\n\n\t CHAPTER 8: Collect, organise and summarise data\t\n129\n\t CHAPTER 8: Collect, organise and summarise data\t\n129",
      "chapter_id": "7"
    },
    {
      "title": "Collect, organise and summarise data",
      "content": "8\t Collect, organise and summarise data\n8.1\t Collecting data\nAvoiding bias when selecting a sample\nThe methods that we use to collect data must help us to make sure that the data is \nreliable. This means that it is data that we can trust. \nData cannot be trusted unless it has been collected in a way that makes sure that every \nmember of the population had the same chance of being selected in the sample. \nIt is not practical to taste all the oranges on a tree to know whether the oranges are \nsweet. Only a small number of oranges can be tested, otherwise the farmer would have \ntoo few oranges to sell. The oranges that are tested are called a sample, and all the  \noranges harvested from the tree are called the population.\nSample bias occurs when the particular section of the population from which the \nsample is drawn does not represent that population. The way to avoid sample bias is to \ntake a random sample. A sample is random if every member of the population \nhas the same chance of being selected. A random sample of the orange trees means \nthat every tree should have a chance of being selected for the sample. Every person in \nthe country should have a chance of being selected for the housing survey in a random \nsample.\nAn example of sample bias would be to survey only the people in Limpopo  \nabout their views on housing provision when you want to know the views of the whole \ncountry. For the sample to provide information on the population as a whole, each \nperson in the country should have the same chance of being part of the survey. \nData can be collected through questionnaires, through observation and through \naccess to databases. \nHow to develop a good questionnaire\nThe questionnaire also has an important role in making sure that the information you \ncollect is reliable. You should aim to get a high number of respondents and accurate \ninformation. If not enough people fill in the questionnaire, then you don’t know \nwhether the information you get reflects the real situation. Sampling techniques and \nrules developed by statisticians determine the numbers needed.\nMaths2_Gr9_LB_Book.indb   129\n2014/09/08   09:07:05 AM\n\n130\t MATHEMATICS Grade 9: Term 4\nThere are some important points to consider when designing a questionnaire. Two \nof the most important points are that the questions are clear and accurate and that \npeople find the questionnaire relatively easy to complete.\n1.\t Keep in mind the length of the questionnaire and the time that it takes to complete. \nYour participants will more likely complete a short questionnaire that is quick and \neasy to complete. Exclude information that is not needed. \n2.\t Write down a selection of questions that you think will provide the information that \nyou want. \n3.\t Check the wording for each question. \n4.\t Order the items so that they are in a logical sequence. It might make sense to have the \neasiest questions first but in some cases the more general questions should come first \nand the more specific questions towards the end of the questionnaire.\n5.\t Then try the questionnaire out on a partner. Ask the following questions: \n• Is this question necessary? What information will be provided by the answer?\n• How easy will it be for the respondent to answer this question? How much time \t\nwill it take to answer the question? \n• Do the questions ask for sensitive information? Will people want to answer the \t\nquestion? Will the respondent answer the question honestly?\n• Can the question be answered quickly?\n6.\t Decide how the answers should be provided. Questions may require open-ended \nresponses or closed-ended responses, as described blow.\nIn an open-ended question, the person responds in his or her own words. Through his, \nor her, own words important information can be gained; the person is free to write what \nthey like. A disadvantage is that you might not get the information you want and that it \nmight take a long time to answer.\nIn a closed-ended question the respondents are given some options to choose from. \nThey tick the box which most closely represents their response. These options can be \nconstructed in categories. For example age may be categorised as follows:\nUnder 10 \n    From 10 to 14 \n    From 15 to 19 \n    20 and older \nMaths2_Gr9_LB_Book.indb   130\n2014/09/08   09:07:05 AM\n\n\t CHAPTER 8: Collect, organise and summarise data\t\n131\nthink about data collection and develop a questionnaire\n1.\t Which method for collecting data would be most appropriate for each of the cases \nbelow? Give reasons for your choice.\n\t\n(a)\t The number of learners who bring lunch to schools. What are the contents of the \t\n\t\nschool lunch? \n\t\n(b)\t Whether the tellers at a supermarket chain are happy with their conditions  \n\t\nof work. \n\t\n(c)\t Whether the clients of a clinic are satisfied with the professional conduct of the \t\n\t\nmedical staff.\n\t\n(d) \t The types of activities preschool children choose during their free time.\n\t\n(e)\t The number of Grade 9 learners in the Gauteng North district. \n2.\t You are doing some market research for a new fast food shop near the high \nschool. The owners of the shop want to find out what kind of food and music the \ntarget market likes. The target market is learners from the high school. Develop a \nquestionnaire to collect this information, on the next page. \nMaths2_Gr9_LB_Book.indb   131\n2014/09/08   09:07:05 AM\n\n132\t MATHEMATICS Grade 9: Term 4\nMaths2_Gr9_LB_Book.indb   132\n2014/09/08   09:07:05 AM\n\n\t CHAPTER 8: Collect, organise and summarise data\t\n133\n8.2\t Organising data\nThere is a difference between data and information. Data is unorganised facts.  \nWhen data is organised and analysed so that people can make decisions, it may be  \ncalled information. Data can be organised in many different ways. Some methods are \ndescribed below.\nData can be organised by making a tally table. Here is an example of a tally table \nshowing the numbers of learners in a class that participate in different sports.\nSport\nTally marks\nSoccer\n////    ////   ////    ////  ////          \nAthletics\n////   ///\nNetball\n////    ////   ////    ////  /\nChess\n////   /\nThe above data can also be organized in a frequency table:\nSport\nFrequency\nSoccer\n25\nAthletics\n8\nNetball\n21\nChess\n6\nNumerical data sets with many items are often grouped into equal class intervals and \nrepresented in a table of frequencies for the different class intervals. This is very useful \nsince it makes it easy to see how the data is spread out.\nHere is an example of grouped data showing the heights of all the learners in a school. \nTo make a frequency table for numerical data, the data has to be arranged \nfrom smallest to biggest first.\nHeight in m\nNumber of learners \n(Frequency)\n< 1,20 m\n13\n1,20 m – 1,30 m\n28\n1,30 m – 1,40 m\n57\n1,40 m – 1,50 m\n164\n1,50 m – 1,60 m\n274\n1,60 m – 1,70 m\n198\n1,70 m – 1,80 m\n73\n> 1,80 m\n13\nA value equal to the lower \nboundary of a class interval \nis counted in that interval. For \nexample a length of 1,60 m is \ncounted in the interval 1,60 \n– 1,70, and not in the interval \n1,50 – 1,60 m.\nHowever, 1,599 m is less than \n1,60 m, so it belongs in the \ninterval 1,50 m – 1,60 m.\nMaths2_Gr9_LB_Book.indb   133\n2014/09/08   09:07:06 AM\n\n134\t MATHEMATICS Grade 9: Term 4\nA stem-and-leaf display is a useful way to organise numerical data. It also shows you \nwhat the “shape” of the data is like. Here is an example of a stem-and-leaf display. \n\t\nKey: 35 | 4 means 354\n34 0  4\n35 4  8  8\n36 0  1  6  8\n37 1  3  5  8  8  8  9\n38 2  4  9\n39 0  3  4  4  5  6  9\n40 0  3  7\n41 1\nThe above stem-and-leaf display represents the following data about the masses in grams \nof the chickens in a sample of 6-week-old chickens on a chicken farm.\n399\n378\n382\n360\n396\n389\n344\n411\n378\n394\n394\n354\n375\n378\n400\n371\n379\n358\n366\n403\n358\n395\n390\n340\n393\n384\n361\n407\n373\n368\nTo make a stem-and-leaf display, it helps to first arrange the data from smallest to largest, \nas shown here for the above data set.\n340\n344\n354\n358\n358\n360\n361\n366\n368\n371\n373\n375\n378\n378\n378\n379\n382\n384\n389\n390\n393\n394\n394\n395\n396\n399\n400\n403\n407\n411\nThe same data set is displayed in two slightly different ways below.\n384\n382\n399\n379\n399\n379\n396\n378\n396\n368\n378\n395\n378\n395\n366\n378\n394\n368\n378\n394\n361\n378\n394\n411\n358\n366\n375\n389\n394\n407\n354\n360\n375\n393\n407\n344\n358\n361\n373\n384\n393\n403\n344\n358\n373\n390\n403\n340\n354\n360\n371\n382\n390\n400\n411\n340\n358\n371\n389\n400\nIn this display the width of each class interval\t\nIn this display the width of \nis 10, as in the stem-and-leaf display above.\t\neach class interval is 15.\nMaths2_Gr9_LB_Book.indb   134\n2014/09/08   09:07:06 AM\n\n\t CHAPTER 8: Collect, organise and summarise data\t\n135\nworking with grouped data\n1.\t An organisation called Auto Rescue recorded the following numbers of calls from \nmotorists each day for roadside service during March 2014. \n28\n122\n217\n130\n120\n86\n80\n90\n120\n140\n70\n40\n145\n187\n113\n90\n68\n174\n194\n170\nl00\n75\n104\n97\n75\n123\n100\n82\n109\n120\n81\n\t\nSet up a tally and frequency table for this set of data values, in intervals of width 40.\n2.\t When geologists go out into the field they make sure they have their rulers and \nmeasurement instruments in their bags. They also have their “inbuilt rulers”, for \nexample their handspans. A handspan is the distance from the tip of the thumb to \nthe tip of the fifth finger on an outstretched hand. Measure your handspan against \nthe ruler! This frequency table shows the handspans of different Grade 9 learners, in cm.\nHandspan of Grade 9 learners in cm\nFrequency\n15–18\n7\n18–21\n9\n21–24\n10\n24 and greater\n4\n\t\n(a)\t How many learner handspans were measured altogether?\n\t\n(b)\t How many learner handspans are less than 21cm wide?\nMaths2_Gr9_LB_Book.indb   135\n2014/09/08   09:07:06 AM\n\n136\t MATHEMATICS Grade 9: Term 4\n\t\n(c)\t How many handspans are 18 cm or wider?\n\t\n(d)\t In which interval would you place a handspan of 18 cm?\n8.3\t Summarising data\nThe mean, median, mode and range are single numbers that provide some information \nabout a data set, without listing all the data values. \nThe mode is the value that occurs most frequently. \nTo find the mode, look for the number or category \nthat is listed in the data set most often. Some data sets \nhave more than one mode, and some may have none.\nThe median is the number that separates the set \nof values into an upper half and a lower half. The \nmedian can be found by arranging the values from \nsmall to big or big to small. If the data set consists of \nan even number of items, the median is the sum of \nthe two middle values divided by 2.\nThe mean (average) of a set of numerical data is the \nsum of the values divided by the number of values in \nthe data set. \nMean = the sum of the values ÷ the number of values. \nThe range is a number that tells us how spread out \nthe data values are. It is the difference between the \nlargest and smallest values.\nThe mean, median and mode don’t work equally well for all sets of data. It depends on \nthe kind of data, and also on whether the data is evenly spread out or not.\norganise, summarise and compare some data\n1.\t A researcher analyses data about the people who are suffering from three different \ntypes of the flu virus: A, B and C. The ages of the people in the different groups are:\n\t\nType A: 60, 80, 75, 87, 88, 49, 94, 84, 59, 86, 82, 62, 79, 89 and 78.\n\t\nType B: 27, 39, 43, 29, 36, 70, 56, 25, 54, 36, 66, 45, 33, 46, 14 and 41.\n\t\nType C: 33, 48, 64, 15, 31, 20, 70, 21, 18, 49, 21, 19, 57, 23, 29 and 20.\nMaths2_Gr9_LB_Book.indb   136\n2014/09/08   09:07:06 AM\n\n\t CHAPTER 8: Collect, organise and summarise data\t\n137\n\t\nFor each group:\n• Draw a stem-and-leaf plot.\n• Calculate the range, mean and median of the ages.\n• Look at the shape of the stem-and-leaf displays as well as the summary measures. \nDiscuss the spread of the data in each case, and compare the three different  \ngroups.\nWork and report on your work below and on the next page.\nType A: \n  \nMaths2_Gr9_LB_Book.indb   137\n2014/09/08   09:07:06 AM\n\n138\t MATHEMATICS Grade 9: Term 4\nType B: \nType C: \nMaths2_Gr9_LB_Book.indb   138\n2014/09/08   09:07:06 AM\n\n\t CHAPTER 8: Collect, organise and summarise data\t\n139\n2.\t Fill in the statistic (mode, mean or median) that would best summarise each data set, \nand indicate the central tendency of the data. \nData set\nBest measure of central tendency\nThe shoe sizes of boys in Grade 9\nAn evenly-spread set of measurement values, \nsuch as the heights of learners in a class\nA set of measurement values with a few very \nlow values and mostly high values\nThe number of siblings each person in your \nclass has\nThe sizes of properties in a town, where \nmost people live in small apartments or RDP \nhouses, and a few live on large properties\nextreme values and outliers \nAn extreme value or outlier is a data value that lies an abnormal distance from other  \nvalues in a random sample from a population. Sometimes there are reasons why this  \ndata value is so different to the others. It is important to comment on the possible \nreasons. \nWhen you are summarising data (and also when you analyse data), you need to decide \nwhether an outlier makes sense in the context you are looking at. \nIt is possible that an outlier does not make sense, as it lies too far away and is an  \nunreasonable measurement. Then you need to think about the fact that this data value \nmay be an error. For example:\n11 12 13 14 15 16 17 18 19 20 21 22 23 24 25\nage\nIn this case, the value of 24 years old could be an unreasonable value. This depends on \nthe context of the survey.\nYou will learn more about extreme values and outliers in Chapter 10. \nMaths2_Gr9_LB_Book.indb   139\n2014/09/08   09:07:06 AM\n\n140\t MATHEMATICS Grade 9: Term 4\nUse this information about 14 countries to answer the questions that follow.\nCountry\nTotal population  \n(in 1 000s)\nTotal annual \nnational income \nper person (US$)\nPercentage of \nincome spent on \nhealth\nAngola\n18 498\n4 830\n4,6\nBotswana\n1 950\n13 310\n10,3\nDRC \n66 020\n280\n2,0\nLesotho\n2 067\n1 970\n8,2\nMalawi\n15 263\n810\n6,2\nMauritius\n1 288\n12 580\n5,7\nMozambique\n22 894\n770\n5,7\nNamibia\n2 171\n6 250\n5,9\nSeychelles\n84\n19 650\n4,0\nSouth Africa\n50 110\n9 790\n8,5\nSwaziland\n1 185\n5 000\n6,3\nTanzania\n43 739\n1 260\n5,1\nZambia\n12 935\n1 230\n4,8\nZimbabwe\n12 523\n170\nNot available\n1.\t Look at the total population for each country. \n\t\n(a)\t Calculate the mean of the data. \n\t\n(b)\t Draw a dot plot on the number line below to show the data.\nPopulation (thousands)\n0\n10 000\n20 000\n30 000\n40 000\n50 000\n60 000\n\t\n(c)\t Find the median of the data.\n\t\n(d)\t What is the range of the data?\n\t\n(e)\t Which measure of central tendency do you think represents the data more \t\t\n\t\n    accurately? Explain. \n \n2.\t Look at the Total annual national income per person in US dollars. Comment on the \nspread of the data.  \n \nMaths2_Gr9_LB_Book.indb   140\n2014/09/08   09:07:06 AM\n\nChapter 9\nRepresenting data\n\t\nCHAPTER 9: representing data\t\n141\nIn the previous chapter, you focused on methods of collecting, organising and summarising \ndata. Now we focus on representing data in bar graphs, double bar graphs, histograms, pie \ncharts and broken-line graphs. You will practise drawing these graphs. You will also decide \nwhy a certain kind of graph is useful in a particular context.\n9.1\t Bar graphs and double bar graphs.......................................................................... 143\n9.2\t Histograms............................................................................................................. 146\n9.3 \t Pie charts................................................................................................................ 149\n9.4\t Broken-line graphs.................................................................................................. 151\n9.5\t Scatter plots........................................................................................................... 154\nMaths2_Gr9_LB_Book.indb   141\n2014/09/08   09:07:06 AM\n\n142\t MATHEMATICS Grade 9: Term 4\n0\n2 000 000\n4 000 000\n2011 Mid-year estimates\nSouth Africa’s population by province\n6 000 000\n8 000 000\n10 000 000\n12 000 000\nEastern Cape\nFree State\nGauteng\nKwaZulu-Natal\nLimpopo\nMpumalanga\nNorthern Cape\nNorth West\nWestern Cape\n0\nJanuary\nFebruary\nMarch\nApril\nMay\nJune\nJuly\nAugust\nSeptember\nOctober\nNovember\nDecember\n20\n40\n60\n80\n100\n120\n140\n160\n180\nRainfall (mm)\nRainfall for Ceres, Mahikeng and Amatole\nCeres, WC\nMahikeng, NW\nAmatole, KZN\nMaths2_Gr9_LB_Book.indb   142\n2014/09/08   09:07:07 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n143\n\t\nCHAPTER 9: representing data\t\n143",
      "chapter_id": "8"
    },
    {
      "title": "Representing data",
      "content": "9\t Representing data\n9.1\t Bar graphs and double bar graphs\nrevising bar graphs and double bar graphs\nA bar graph shows categories of data along the horizontal axis, and the frequency of \neach category along the vertical axis. An example is given below.\nGraphs need a title to tell you what they are about.\nThis shows the frequency \nof each category.\n8 learners chose orange juice.\nThis axis gives all the categories of data.\nNumber of learners\nFruit juice\n0\n2\n4\n6\n8\n10\n12\nApple\nApricot\nGrenadilla\nMango\nOrange\nPeach     \nPineapple\nStrawberry\nFavourite fruit juice in a Grade 9 class\nA double bar graph shows two sets of data in the same categories on the same set of \naxes. This is useful when we need to show two groups within each category.\nNumber of learners\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\nGrade 8\nBoys\nGrade 9\nGrade 10\nYear\nGrade 11\nGrade 12\nNumber of boys and girls in each grade\nat Malbongwe High School\nGirls\nThis is a key to show \nwhat the two colours \nrepresent\nThe two bars show \nthe differences \nin numbers of boys \nand girls.\nMaths2_Gr9_LB_Book.indb   143\n2014/09/08   09:07:07 AM\n\n144\t MATHEMATICS Grade 9: Term 4\ndrawing bar graphs and double bar graphs\n1.\t Obese (very overweight) people have many health problems. It is a concern all \naround the world. Health researchers analysed the change over 28 years in the \nnumbers of people who are overweight and obese in different areas of the world. This \ntable summarises some of the data.\n\t\nPercentage of population that is overweight and obese\n1980\n2008\nSub-Saharan Africa\n12%\n23%\nNorth Africa and Middle East\n33%\n58%\nLatin America\n30%\n57%\nEast Asia (low income countries)\n13%\n25%\nEurope\n45%\n58%\nNorth America (high income \ncountries)\n43%\n70%\n\t\n(a)\t The table summarises “some” of the data. What would some other important \t \t\n\t\ndata be? Think of as many things as you can. \n\t\n(b)\t Which data stands out the most for you in the table above? Give your personal \t\t\n\t\nopinion. \n\t\n(c)\t Plot a double bar graph to compare the data for the areas, and for the two years. \t\n\t\nUse the grid on the next page. Remember to give your graph a key.\nMaths2_Gr9_LB_Book.indb   144\n2014/09/08   09:07:07 AM\n\n\t\nCHAPTER 9: representing data\t\n145\n\t\n(d)\t Look carefully at the comparisons that the graph makes. Has your opinion of  \n\t\nthe most interesting differences changed, now that you see the double bar graph? \n\t\nExplain.\n\t\n(e)\t In some countries the obesity problem has been labelled “Obesity in the face of \t\n\t\npoverty”. Write a short report on the data and your double bar graph to support \t\n\t\nthis argument.\nMaths2_Gr9_LB_Book.indb   145\n2014/09/08   09:07:07 AM\n\n146\t MATHEMATICS Grade 9: Term 4\n9.2\t Histograms\nrevising histograms\nA histogram is a graph of the frequencies of data in different class intervals, as \ndemonstrated in the example below. Each class interval is used for a range of values.  \nThe different class intervals are consecutive and cannot have values that overlap. The \ndata may result from counting or from measurement.\nA histogram looks somewhat like a bar graph, but is normally drawn without gaps  \nbetween the bars. \nrepresenting data in histograms\n1.\t (a)\t A fruit farmer wants to know which of his trees are producing good plums, and \n\t\nwhich trees need to be replaced. \n\t\n\t\nHe collects 100 plums each from two trees and measures their masses. \n\t\n\t\nThe data below gives the mass of plums from the first tree.\nMass of plums (g)\n20–29\n30–39\n40–49\n50–59\n60–69\nFrequency\n6\n18\n34\n30\n12\n\t\n\t\nRepresent the data in a histogram on the grid below. \nMaths2_Gr9_LB_Book.indb   146\n2014/09/08   09:07:07 AM\n\n\t\nCHAPTER 9: representing data\t\n147\n\t\n(b)\t Now draw another histogram to represent the following data giving the mass of \t\n\t\nthe same type of plums from another tree in the orchard.\nMass of plums (g)\n20–29\n30–39\n40–49\n50–59\n60–69\nFrequency\n3\n14\n26\n36\n21\n\t\n(c)\t Study the two histograms and then comment on the number of plums produced \t\n\t\nby the two trees. \nMaths2_Gr9_LB_Book.indb   147\n2014/09/08   09:07:07 AM\n\n148\t MATHEMATICS Grade 9: Term 4\n2.\t (a)\t Draw a histogram to represent the data in the table below. Group the data in  \n\t\nintervals of 0,5 kg.\n\t\n\t\nBirth weights (kg) of 28 babies at a clinic\n3,3\n1,34\n2,88\n2,54\n1,87\n2,06\n2,72\n1,89\n0,85\n1,99\n2,43\n1,66\n2,45\n1,62\n1,91\n1,20\n2,45\n1,38\n0,9\n2,65\n2,88\n1,75\n2,11\n3,2\n1,74\n0,6\n3,1\n1,86\n\t\n(b)\t Calculate the mean and median of the data.\n\t\n(c)\t Records from the whole country show that the birth weight of babies ranges \t\n\t\n\t\nfrom 0,5 kg to 4,5 kg, and the mean birth weight is 3,18 kg. Use the graph and  \n\t\nthe mean and median to write a short report on the data from the clinic.  \nMaths2_Gr9_LB_Book.indb   148\n2014/09/08   09:07:08 AM\n\n\t\nCHAPTER 9: representing data\t\n149\n9.3\t Pie charts\nA pie chart consists of a circle divided into sectors (slices). Each sector shows one \ncategory of data. Bigger categories of data have bigger slices of the circle. \nHere is an example of a pie chart:\nThis pie chart shows 5 \ncategories of data.\nThe size of each \nslice is the fraction \nor percentage of \nthe whole that the \ncategory forms.\nThe key (or legend) \nshows the category \nthat each colour \nstands for\n30% or\n40% or\n15% or\n10% or\nVery good\nCustomer opinion on service at Fishy Fun \nrestaurant as reflected in survey\nGood\nNeutral\nPoor\nVery poor\n2\n5\n3\n20\n1\n10\n1\n20\n3\n10\n5% or\ndrawing pie charts\n1.\t The following bar graph shows the population of South Africa by province.\n0\n2 000 000\n4 000 000\n2011 Mid-year population estimates\nSouth Africa’s population by province\n6 000 000\n8 000 000\n10 000 000\n12 000 000\nEastern Cape\nFree State\nGauteng\nKwaZulu-Natal\nLimpopo\nMpumalanga\nNorthern Cape\nNorth West\nWestern Cape\n\t\n(a)\t Write the figures in the graph correct to the nearest 500 000.\nProvince\nE Cape\nFS\nGau\nKZN\nLim\nMpum\nNC\nNW\nWC\nPopulation\n(× 1 000)\n\t\n(b)\t What is the total of the rounded off numbers? \nMaths2_Gr9_LB_Book.indb   149\n2014/09/08   09:07:08 AM\n\n150\t MATHEMATICS Grade 9: Term 4\n\t\n(c)\t Work out the percentage of the whole for each province.\nProvince\nE Cape\nFS\nGau\nKZN\nLim\nMpum\nNC\nNW\nWC\nPercentage \nof total\n\t\n(d)\t Draw a pie chart showing the data in the completed table. (Estimate the sizes of \t\n\t\nthe slices.)\n\t\n(e)\t Write a short report explaining the difference in the way the data is represented \t\n\t\nin the pie chart and the bar graph. Which do you think is a better method to \t\n\t\n\t\nshow this data?\nMaths2_Gr9_LB_Book.indb   150\n2014/09/08   09:07:08 AM\n\n\t\nCHAPTER 9: representing data\t\n151\n9.4\t Broken-line graphs \nBroken-line graphs\nBroken-line graphs are used to represent data that changes continuously over time. \nFor example, the rainfall for a whole month is captured as one data point, even though \nthe rain is spread out over the month, and it rains on some days and not on others. \nBroken line graphs are useful to identify and display trends.\nHere is some data that can be represented with broken-line graphs.\nRainfall at three locations in South Africa in 2012\nAmatole, KZN\nMahikeng, NW\nCeres, WC\nRainfall (mm)\nRainfall (mm)\nRainfall (mm)\nJanuary\n101\n118\n27\nFebruary\n108\n90\n23\nMarch\n117\n86\n41\nApril\n77\n61\n60\nMay\n46\n14\n130\nJune\n27\n6\n168\nJuly\n32\n3\n152\nAugust\n48\n7\n162\nSeptember\n76\n18\n88\nOctober\n112\n46\n60\nNovember\n115\n75\n41\nDecember\n100\n86\n36\nHere is a broken line graph for the Amatole rainfall data.\n0\nJanuary\nFebruary\nMarch\nApril\nMay\nJune\nJuly\nAugust\nSeptember\nOctober\nNovember\nDecember\n20\n40\n60\n80\n100\n120\n140\nRainfall (mm)\nRainfall for Amatole, KZN\nMaths2_Gr9_LB_Book.indb   151\n2014/09/08   09:07:08 AM\n\n152\t MATHEMATICS Grade 9: Term 4\n1.\t During which four months does Amatole have the least rain?\n2.\t During which six months does Amatole have the most rain?\n3.\t During which months would you plan a hike if you were only considering the rainfall \npatterns?\n4.\t What other factors should you consider when planning a hike in this region?\n5.\t Make a broken-line graph for the Mahikeng rainfall data on the grid below.\n \nMaths2_Gr9_LB_Book.indb   152\n2014/09/08   09:07:08 AM\n\n\t\nCHAPTER 9: representing data\t\n153\n6.\t Make a broken-line graph for the Ceres-rainfall data on the grid below.\n7.\t Write a few lines on the difference in rainfall patterns between Ceres and Mahikeng.  \n8.\t Draw a combined broken-line graph with the information from all three regions on \none graph.\nMaths2_Gr9_LB_Book.indb   153\n2014/09/08   09:07:09 AM\n\n154\t MATHEMATICS Grade 9: Term 4\n9.4\t Scatter plots\nunderstanding and constructing scatter plots\nScatter plots show how two sets of numerical data are related. Matching pairs of \nnumbers are treated as coordinates and are plotted as a single point. All the points,  \nmade up of two data items each, show a scattering across the graph.\n1.\t This table shows a data set  \nwith two variables. Study the  \ninformation in the table.\n2.\t Make a dot for each learner’s  \nmark for each subject on the  \nnumber lines below. \nLearners\nMaths marks\nNatural \nScience marks\nZinzi\n25\n26\nJohn\n23\n25\nPalesa\n22\n25\nSiza\n21\n23\nEric\n20\n23\nChokocha\n19\n21\nGabriel\n17\n20\nSimon\n16\n19\nMiriam\n15\n18\nFrederik\n15\n16\nSibusiso\n12\n15\nMeshack\n11\n13\nDuma\n11\n12\nSamuel\n10\n12\nLola\n10\n11\nThandile\n9\n10\nJabulani\n8\n10\nManare\n7\n9\nMarlene\n7\n7\nMary\n5\n7\n0\n2\n4\n6\n8\n10 12 14 16\nNatural Sciences marks\n18 20 22 24 26 28 30\n0\n2\n4\n6\n8\n10 12 14 16\nMathematics marks\n18 20 22 24 26 28 30\nMaths2_Gr9_LB_Book.indb   154\n2014/09/08   09:07:09 AM\n\n\t\nCHAPTER 9: representing data\t\n155\n3.\t What if you were to show both sets of marks on the same graph, instead of a separate \nnumber line for each set? The graph below shows a scatter plot that represents both \nsets of data. Each dot represents one learner. \n\t\nNatural Sciences marks\nMathematics marks\n0\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\n24\n26\n28\n30\n2\n4\n6\n8 10 12 14 16 18 20 22 24 26 28 30\nCorrelation between Mathematics\nand Natural Sciences\n\t\nThe scatter plot shows the relationship between the Natural Sciences mark and the  \nMathematics mark. \n4.\t Find the dot for Sibusiso in the data set. He obtained a mark of 12 for the  \nMathematics test and a mark of 15 for Natural Sciences. Find 12 on the  \nhorizontal axis. Follow the vertical line up until you reach a blue dot. Find 15 on  \nthe vertical axis. Follow the line horizontally until you reach the same blue dot.  \nThis blue dot represents the two marks that belong to Sibusiso. Circle the blue dot \nand label it S.\n5.\t Find the data points for Zinzi, Palesa, Jabulani and Mary. Circle them and label them \nZ, P, J and M.\nIn the above example, a higher Mathematics mark \ncorresponds to a higher Science mark. We say there is \na positive correlation between the Mathematics \nmarks and the Science marks.\nMaths2_Gr9_LB_Book.indb   155\n2014/09/08   09:07:09 AM\n\n156\t MATHEMATICS Grade 9: Term 4\n6.\t Study this data set and the scatter plot of the data given on the next page.\nLearner\nMaths marks\nArt marks\nZinzi\n25\n7\nJohn\n23\n7\nJabulani\n22\n9\nSiza\n21\n10\nEric\n20\n10\nChokocha\n19\n11\nGabriel\n17\n12\nSimon\n16\n12\nMiriam\n15\n15\nFrederik\n15\n15\nSibusiso\n12\n16\nMishack\n11\n17\nDuma\n11\n19\nSamuel\n10\n20\nLola\n10\n21\nThandile\n9\n23\nPalesa\n8\n23\nManare\n7\n25\nMarlene\n7\n25\nMary\n5\n26\n7.\t Find Eric in the table. Note his marks for Mathematics and Art. Find the dot that \nrepresents his marks on the scatter plot. Encircle it and label it E.\n8.\t Find Samuel in the table. Note his marks for Mathematics and Art. Find the dot that \nrepresents his marks. Encircle it and label it S.\n9.\t Compare the two sets of marks for Eric and for Samuel. What do you notice about  \nthe marks?\nMaths2_Gr9_LB_Book.indb   156\n2014/09/08   09:07:09 AM\n\n\t\nCHAPTER 9: representing data\t\n157\n10.\tFind the data points on the scatter plot for Zinzi, Eric, Miriam, Frederik, Samuel and \nMary. Circle the points and label them Z, E, M, F, S and Ma\n11.\tWhat do you notice about the pattern of marks in Mathematics and Art for this  \ndata set?\nArt marks\nMathematics marks\n0\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\n24\n26\n28\n30\n2\n4\n6\n8 10 12 14 16 18 20 22 24 26 28 30\nCorrelation between Mathematics\nand Art\nA negative correlation is a correlation in which an increase in the value of one piece \nof data tends to be matched by the decrease in the other set of data. Learners who obtain \na high mark for Mathematics appear to obtain a low mark for Art. We say there is a \nnegative correlation between the Mathematics and the Art scores for this data set.\nA correlation is an assessment of how strongly two sets of data appear to be connected. \nTwo sets of data may be correlated or may show no correlation. \nMaths2_Gr9_LB_Book.indb   157\n2014/09/08   09:07:09 AM\n\n158\t MATHEMATICS Grade 9: Term 4\nHere is the scatter plot for the Mathematics and Life Skills marks of the same group of \nlearners. The table for this data is given on the next page.\nLife Skills marks\nMathematics marks\n0\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\n24\n26\n28\n30\n2\n4\n6\n8 10 12 14 16 18 20 22 24 26 28 30\nCorrelation between Mathematics \nand Life Skills marks\n12. Study the scatter plot and the data table on the next page.\n13. Find the data points on the scatter plot for Zinzi, Eric, Miriam, Lola, and Mary. Circle \nthe points and label them Z, E, M, L and Ma.\n14. What do you notice about the pattern of marks in Mathematics and Life Skills for this \ndata set?\nMaths2_Gr9_LB_Book.indb   158\n2014/09/08   09:07:10 AM\n\n\t\nCHAPTER 9: representing data\t\n159\nLearner\nMaths\nLife Skills\nZinzi\n25\n5\nJohn\n23\n14\nJabulani\n22\n23\nSiza\n21\n16\nEric\n20\n5\nChokocha\n19\n4\nGabriel\n17\n9\nSimon\n16\n6\nMiriam\n15\n25\nFrederik\n15\n27\nSibusiso\n12\n29\nMeshack\n11\n17\nDuma\n11\n11\nSamuel\n10\n1\nLola\n10\n25\nThandile\n9\n5\nPalesa\n8\n28\nManare\n7\n26\nMarlene\n7\n2\nMary\n5\n15\nthe relationship between arm span and height\nThe idea that a person’s arm span (the distance from the tip of the middle finger on one \nhand to the tip of the middle finger on the other hand when the arms are stretched out \nsideways) is the same as one’s height has been explored many times. \nA data set for 13 people is given on the next page.\n1.\t Make a scatter plot of this data on the given grid.\n\t\nFor example, take Cilla’s arm span. Find 156 on the horizontal axis. Follow a vertical \nline up. Then on the vertical axis find 162. Follow a horizontal line across. Where the \ntwo points meet, draw a dot.\nMaths2_Gr9_LB_Book.indb   159\n2014/09/08   09:07:10 AM\n\n160\t MATHEMATICS Grade 9: Term 4\nPerson\nArm span\nHeight\nCilla\n156\n162\nMeshack\n159\n162\nTony\n161\n160\nEllen\n162\n170\nKarin\n170\n170\nSibongile\n173\n185\nGabriel\n177\n173\nAlpheus   \n178\n178\nMfiki\n188\n188\nNathi\n188\n182\nManare\n188\n192\nKhanyi\n196\n184\n2.\t What would you say about the correlation between the arm span and the height?\nMaths2_Gr9_LB_Book.indb   160\n2014/09/08   09:07:10 AM\n\nChapter 10\nInterpret, analyse and\nreport on data\n\tCHAPTER 10: INTERPRET, ANALYSE AND REPORT ON DATA\t\n161\nIn this chapter, you will develop and practise some critical data analysis skills. This means \nlooking at reported data and analysing the whole data handling cycle for this data. You \nneed to decide which way of representing data is best in a given situation. In summarising \ndata, some measures are more appropriate for different types of data. You also need to \nrecognise some ways in which bias can appear in data, including methods of collecting, \nrepresenting and summarising data.\n10.1\t Which graph is best?.............................................................................................. 163\n10.2\t The effects of summary statistics on how data is reported....................................... 167\n10.3\t Misleading graphs.................................................................................................. 168\n10.4\t Analysing extreme values and outliers..................................................................... 172\nMaths2_Gr9_LB_Book.indb   161\n2014/09/08   09:07:10 AM\n\n162\t MATHEMATICS Grade 8: Term 4\nCollect\ndata\nPose a \nquestion\nPresent \nthe data\nOrganise\nthe data\nInterpret \nand analyse \nthe data\nReport on \nthe data\nThe data\ncycle\nMaths2_Gr9_LB_Book.indb   162\n2014/09/08   09:07:10 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n163\n\tCHAPTER 10: INTERPRET, ANALYSE AND REPORT ON DATA\t\n163",
      "chapter_id": "9"
    },
    {
      "title": "Interpret, analyse and report on data",
      "content": "10\tInterpret, analyse and report on data\n10.1\t Which graph is best?\nYou have learnt that certain types of graphs are best for displaying certain kinds of \ninformation. The type of graph depends mostly on the type of data that needs to be \nrepresented. Here is a summary of the advantages of different types of graphs:\nTables show more information than graphs but  \nthe patterns are not as easy to see. They do not give a \nvisual impression of particular trends. \nPie charts show a whole divided into parts. They \nshow how the parts relate to each other and how the \nparts relate to a whole. They do not show the  \nquantities involved.\nBar graphs show the amounts or quantities  \ninvolved but do not show the relationship as  \neffectively as pie charts. They are useful for showing \nquantitative data. Bar charts allow us to compare \nthe quantities of different categories, for example, \nthe sales of different items.\nA double-bar graph is used to compare two or \nmore things for each category. For example, we could \nuse a double-bar graph to compare the differences \nbetween males and females. \nHistograms are used to represent numerical data \nthat is grouped into equal class intervals. Histograms \nare useful to show the way the data is spread out.\nBroken-line graphs show trends or changes in \nquantities over time. \nMaths2_Gr9_LB_Book.indb   163\n2014/09/08   09:07:10 AM\n\n164\t MATHEMATICS Grade 9: Term 4\nchoose the best representation\n1.\t Which kind of graph is best to represent each of the following? Explain your answers.\n\t\n(a)\t Showing the value of the rand against the US dollar over several years\n\t\n(b)\t Comparing the monthly sales of six different makes of car in 2014 and 2015\n\t\n(c)\t The proportion of people of different age groups in a town\n\t\n(d)\t The quantities of different crops produced on a farm\n\t\n(e)\t The percentages of different goods sold to make up the total sales for a shop\n\t\n(f)\t The change in HIV infection rates over time\n2.\t This graph was published by Statistics South Africa to show the assets owned by \nSouth Africans. The blue bar shows the Census 2011 results and the yellow bar shows \nthe General Household Survey 2012 results.\n\t\nPercent\nVehicle\nComputer\nTV\nRefrigerator\nCensus 2011\nGeneral\nhousehold\nsurvey 2012\n0\n10\n20\n30\n40\n50\n60\n70\nPercentage of household assets owned\nWashing\nmachine\nElectric\nstove\n90\n80\nMaths2_Gr9_LB_Book.indb   164\n2014/09/08   09:07:10 AM\n\n\tCHAPTER 10: INTERPRET, ANALYSE AND REPORT ON DATA\t\n165\n\t\nGive reasons for your answers to the questions below.\n\t\n(a)\t Is it useful to show the differences in the results of Census 2011 and the General \t\n\t\nHousehold Survey 2012?\n\t\n(b)\t Is it useful to collect data on assets that people own? \n\t\n(c)\t Is it useful to show that lower percentages of people own certain assets?\n\t\n(d)\t The different coloured bars represent the two different surveys. Draw up a table \t\n\t\nto show the data in table form. (Read the percentages as accurately as you can  \n\t\nfrom the graph and round off the data to the nearest whole number for the table.)\n\t\n(e)\t Does the table show the data as effectively as the double bar chart? Give your \t \t\n\t\nown opinion.\nMaths2_Gr9_LB_Book.indb   165\n2014/09/08   09:07:10 AM\n\n166\t MATHEMATICS Grade 9: Term 4\n3.\t The table below shows the employment status of people ages 15–64 years in  \nSouth Africa. Discuss some ways of representing the data (e.g. graphs). Justify your \nanswers.\nJul–Sept 2012\nApr–June 2013\nJul–Sep 2013\nNumber of people (thousands)\nPopulation 15–64 years old\n33 017\n33 352\n33 464\nLabour force\n18 313\n18 444\n18 638\nEmployed\n13 645\n13 720\n14 028\n  Formal sector  \n  (non-agricultural)\n9 663\n9 694\n10 008\n  Informal sector  \n  (non-agricultural)\n2 197\n2 221\n2 182\n  Agriculture\n661\n712\n706\n  Private households\n1 124\n1 093\n1 132\nUnemployed\n4 668\n4 723\n4 609\nNot economically active\n14 705\n14 908\n14 826\n  Discouraged work-seekers\n2 170\n2 365\n2 240\n  Other (not economically \n  active)\n12 535\n12 543\n12 586\nUnemployment rate (%) \n25,5\n25,6\n24,7\n\t\n(a)\t The percentages of the employed, unemployed, and not economically active \t\n\t\n\t\npeople in July–September 2013.\n\t\n(b)\t The change in the employment rates over three time periods\n\t\n(c)\t The proportions of employed people who work in the formal sector, informal  \n\t\nsector, agriculture and private households.\n\t\n(d)\t The numbers of the employed and unemployed over the three time periods.\nMaths2_Gr9_LB_Book.indb   166\n2014/09/08   09:07:10 AM\n\n\tCHAPTER 10: INTERPRET, ANALYSE AND REPORT ON DATA\t\n167\n10.2\tThe effects of summary statistics on how data is\n\t\nreported\nInformation articles often use averages to report information. The articles might not use \nthe exact terms for average that you have learnt about: the mean, median and mode. \nInstead, they may use terms such as ‘most’. However, it is important to be sure which \nkind of average a report refers to, because they give us different information.\n• Remember that the mean is useful for describing a set of measurement values, but \ncan also be used for other numerical data sets. The word ‘average’ usually refers to \nthe ‘mean’ if it is not explained further. The mean is not reliable if a data set is too \nspread out. \n• The median is the value in the middle of a data set when it is arranged in order. \nHalf the values in the data set are lower than the median and half of them are \nhigher than the median. The median is often the average used when data values \nare not uniformly distributed, because the mean is affected by extreme values in \nthe data set, while the median is not. For example, house prices vary widely, so \nthe median would be a better description of the data than the mean. When the \nmedian is given in a report, the writer should state that they are using the median \nor middle value.  \n• The mode is the number that occurs most often in a set of data. For example, if we \ncollect data about people’s favourite colours, the data set would be a list of colours, \nand the mode would be the colour that comes up most often. The mode can also \nbe used for numbers. Not all data sets have a mode, because sometimes none of the \nnumbers occurs more than once. \nExample\nThe standard way of reporting house prices in South Africa and internationally is the \nmedian house price, which is used by economists in financial reports. The median is \nregarded as more useful than the mean house price because the sale of a few expensive \nhouses would increase the mean, but would not affect the median.\nIf a bank gives bonds for eight houses to the value of R100 000, and for two houses  \nto the value of R1 million, then the mean would be R280 000. This does not seem to  \nbe an accurate reflection of the value of the houses, because it is distorted by the higher \nvalues. The median house price would be R100 000, which is an accurate reflection of  \nthe prices. \nRemember that the median is the middle point, and half of the values fall below the \nmedian, and half above. If the median is lower than the mean, this shows us that there \nare high values that are distorting the mean.\nMaths2_Gr9_LB_Book.indb   167\n2014/09/08   09:07:10 AM\n\n168\t MATHEMATICS Grade 9: Term 4\nusing different summary statistics\n1.\t What kind of average is used in each of these statements?\n\t\n(a)\t The average family has 2,6 children. \n\t\n(b)\t Most families have 3 children. \n    \n\t\n(c)\t Most people prefer red cars. \n   \n\t\n(d)\t The average height for women is 1,62 m.  \n   \n\t\n(e)\t More people shop after work than at any other time during the day. \n  \n2.\t The mean monthly salary of all the staff at company ABC is R8 000 per month, but \nthe median salary is R5 000. \n\t\n(a)\t Explain why the two summary statistics are so different. \n\t\n(b)\t Which summary statistic gives a better idea of the salaries at the company? Give \t\n\t\nreasons for your answer.\n10.3\tMisleading graphs\nThe media (newspapers, magazines, television), regularly use graphs to show \ninformation. Unfortunately, the information is often manipulated to emphasise \na particular result. This may be because the writer simply wants to make his or her \nargument more obvious to the reader. \nChanging the scale of the axis \nIf you change the scale of the vertical axis on bar graphs and line graphs, you will change \nthe way the graphs look. For a bar graph, the larger the spaces between the numbers \non the vertical axis, the bigger the difference between the bars. The smaller the spaces \nbetween the numbers on the axis, the smaller the difference in the height of the bars. \nThe same is true for a line graph which will either have sharp points or be much flatter \ndepending on how you have changed the scale.\nMaths2_Gr9_LB_Book.indb   168\n2014/09/08   09:07:10 AM\n\n\tCHAPTER 10: INTERPRET, ANALYSE AND REPORT ON DATA\t\n169\nExample \nThe two broken-line graphs below show the same sales data for a business over a period \nof six months. Which graph gives the more accurate impression? \nQuantity\nSales (July to December)\n0\n100\n200\n300\n400\n500\n600\n700\n800\nA\nJ\nO\nS\nD\nN\nMonth\nGraph A\nQuantity\nSales (July to December)\n0\n350\n400\n450\n500\n550\n600\n650\n700\nA\nJ\nO\nS\nD\nN\nMonth\nGraph B\nGraph B has a different scale on the vertical axis. The vertical axis does not start at 0 and \nso two blocks on the vertical axis represent 100 items instead of only one block, as in \nGraph A. This makes it look as if the sales increased rapidly over the six months. \nNote that it is not necessarily wrong to change the scale on the axes or not to start  \nat 0. For example, graphs showing stock exchange fluctuations rarely show the origin on \nthe graph and stockbrokers are taught to interpret the graphs in that form. Sometimes \nsmall changes in data values have important effects and in these cases, it may be valid to \nchange the scale to show these. \nMaths2_Gr9_LB_Book.indb   169\n2014/09/08   09:07:11 AM\n\n170\t MATHEMATICS Grade 9: Term 4\nanalysing graphs \n1.\t This graph from Statistics South Africa shows the increase in the percentage of \nhouseholds that had access to piped water over a ten-year period. \nPercentage\nHousehold access to piped water\n82,5\n85\n87,5\n90\n92,5\nYear\n2002\n2004\n2006\n2008\n2010\n2012\n\t\n(a)\t Comment on the scale used on the vertical axis. Is this a misleading graph?\n\t\n(b)\t How could you redraw the graph so that the differences on the graph are more \t\t\n\t\nnoticeable?\n\t\n(c)\t How could you draw the graph so that the differences are less noticeable?\nMaths2_Gr9_LB_Book.indb   170\n2014/09/08   09:07:11 AM\n\n\tCHAPTER 10: INTERPRET, ANALYSE AND REPORT ON DATA\t\n171\n2.\t In this graph the height of the houses represents the number of sales. \nHome Sales\n2010\t                       2014\n\t\nDo you think that this graph is misleading? Give reason(s) for your answer.\n3.\t Look at the two graphs below: \n\t\nGraph A\t\nGraph B\n\t\nWhich graph do you think is drawn correctly? Explain your answer.\nMaths2_Gr9_LB_Book.indb   171\n2014/09/08   09:07:11 AM\n\n172\t MATHEMATICS Grade 9: Term 4\n10.4\tAnalysing extreme values and outliers\nA data item that is very different from all (or most) of \nthe other items in a data set is called an outlier.\nIt is sometimes difficult to notice outliers in numerical data. However, outliers often \nbecome clearly noticeable when data is displayed with graphs.\nMathematics marks\nHistory marks\nRaphael\nRolene\nThuni\nTebogo\nBussi\nRallai\nJoamiah\nSara\nDikgang\nBen\nSipho\nMary\nMichel\nBongilel\n \n1.\t The above scatter plot shows the performance of a group of learners in Mathematics \nand History. Which of the points on the scatter plot can be regarded as outliers? Give \nreasons for your answer.\nMaths2_Gr9_LB_Book.indb   172\n2014/09/08   09:07:11 AM\n\n\tCHAPTER 10: INTERPRET, ANALYSE AND REPORT ON DATA\t\n173\nOutliers in data sets can be very important. We need to decide whether there is a \nparticular reason for the value being so different to the others. Sometimes it gives us \nimportant information. In some cases, the data collected for that point could be wrong.\nThe scatter plot below is for data collected by a transport company. \nLoad weight (kg)\nFuel consumption (l/100 km)\n1 000\n2 000\n3 000\n4 000\n5 000\n6 000\n7 000\n40\n45\n50\n55\n60\n65\n70\nThe company uses just one type of truck. Before each transport job, the company has to \nspecify the price for the job. In order to specify a price before a job, the company needs to \nestimate how much their costs will be for doing the job. One of the main costs is the cost \nof fuel, and the main factor influencing the amount of fuel used is the distance. The load \nweight also plays a role: the greater the load weight, the higher the fuel consumption \n(litres/100 km). \nThe table on the next page gives information that was recorded for previous transport \njobs. The jobs are numbered from 1 to 16 and for each job the values of the four variables \ndistance, load weight, amount of fuel used and fuel consumption rate are given. \n2.\t (a)\t Which of the four variables are represented on the scatter plot given above?\n\t\n(b)\t What are the values of these two variables for the point indicated by the blue  \n\t\narrow on the scatter plot?\nMaths2_Gr9_LB_Book.indb   173\n2014/09/08   09:07:11 AM\n\n174\t MATHEMATICS Grade 9: Term 4\nJob number\nDistance (km)\nLoad weight  \n(kg)\nFuel used \n(litres)\nFuel \nconsumption \n(litres/100 km)\n1\n1 304\n5 445\n879\n67.4\n2\n1 320\n2 954\n639\n48.4\n3\n1 151\n4 705\n698\n60.6\n4\n1 371\n4 378\n787\n57.4\n5\n325\n3 673\n176\n54.2\n6\n1 630\n5 995\n1 113\n68.3\n7\n1 023\n5 357\n600\n58.7\n8\n620\n4 988\n382\n61.6\n9\n73\n1 992\n35\n47.9\n10\n1 071\n5 529\n680\n63.5\n11\n370\n4 140\n218\n58.9\n12\n1 423\n4 062\n843\n59.2\n13\n394\n4 068\n221\n56.1\n14\n1 536\n1 678\n682\n44.4\n15\n1 633\n3 736\n887\n54.3\n16\n435\n3 644\n241\n55.4\n3.\t (a)\t Consider the scatter plot and the data set. What is the effect of load weight on \t \t\n\t\nfuel consumption?\n\t\n(b)\t Is job 7 an exception in this respect? Explain your answer.\n4.\t Further investigations revealed that the driver for jobs 2 and 7 was the same person, \nand that he was not the driver for any other jobs. What may this indicate?\nMaths2_Gr9_LB_Book.indb   174\n2014/09/08   09:07:11 AM\n\n\tCHAPTER 10: INTERPRET, ANALYSE AND REPORT ON DATA\t\n175\nfind outliers\nResearchers collected data on the population of some African countries plus the \nSeychelles, the income per person, and the percentage of the income spent on health.\nCountry\nTotal population \n(in 1 000s)\nTotal annual national \nincome per person \n(US$)\nPercentage of \nincome spent on \nhealth\nAngola\n18 498\n4 830\n4,6\nBotswana\n1 950\n13 310\n10,3\nDRC \n66 020\n280\n2,0\nLesotho\n2 067\n1 970\n8,2\nMalawi\n15 263\n810\n6,2\nMauritius\n1 288\n12 580\n5,7\nMozambique\n22 894\n770\n5,7\nNamibia\n2 171\n6 250\n5,9\nSeychelles\n84\n19 650\n4,0\nSouth Africa\n50 110\n9 790\n8,5\nSwaziland\n1 185\n5 000\n6,3\nTanzania\n43 739\n1 260\n5,1\nZambia\n12 935\n1 230\n4,8\n1.\t What are the three variables in this table?\n2.\t Why do you think it is important to look at income per person in this case, rather \nthan the total income?\nMaths2_Gr9_LB_Book.indb   175\n2014/09/08   09:07:12 AM\n\n176\t MATHEMATICS Grade 9: Term 4\n3.\t Plot the points for the national income per person and the percentage spent on \nhealth care for each country. \n4.\t Write a short report on the data in the table and what the scatter plot shows you \nabout the data. Comment on the general trend and any outliers. \nMaths2_Gr9_LB_Book.indb   176\n2014/09/08   09:07:12 AM\n\nChapter 11\nProbability\n\t\nCHAPTER 11: PROBABILITY\t\n177\nIn this chapter you will learn about the idea of probability, and what information  \nprobabilities provide about what may happen in future. You will also learn about  \ncompound events.\n11.1\t Simple events......................................................................................................... 179\n11.2\t Compound events.................................................................................................. 184\nMaths2_Gr9_LB_Book.indb   177\n2014/09/08   09:07:12 AM\n\n178\t MATHEMATICS Grade 9: Term 4\nMaths2_Gr9_LB_Book.indb   178\n2014/09/08   09:07:13 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n179\n\t\nCHAPTER 11: PROBABILITY\t\n179",
      "chapter_id": "10"
    },
    {
      "title": "Probability",
      "content": "11\tProbability\n11.1\t Simple events\nrevision\nyellow\ngreen\npink\nblue\nred\nbrown\ngrey\nblack\n1.\t (a)\t Suppose the 8 coloured buttons above are in a bag and you draw one button from \t\n\t\nthe bag without looking. Can you tell what colour you will draw? \n\t\n(b)\t Suppose you repeatedly draw a button from the bag, note its colour, then put it \t\n\t\nback. Can you tell in approximately what fraction of all the trials the button will \t\n\t\nbe yellow?  \nArchie has a theory. Because the 8 possible outcomes  \nare equally likely, he believes that if you perform 8  \ntrials in a situation like the above you will draw each  \ncolour once. \n2.\t If Archie’s theory is correct, how many times will  \neach colour be drawn if 40 trials are performed?\n3.\t If Archie’s theory is correct, in what fraction of the  \ntotal number of trials will each colour be drawn?\nEach time you draw a button \nfrom the bag without looking \nyou perform a trial. If you do \nthis and put the button back, \nand repeat the same actions 8 \ntimes, you have performed 8 \ntrials.\nThe number of times an event \noccurs during a set of trials is \ncalled the frequency of the \nevent.\nWhen the frequency of an \nevent is expressed as a fraction \nof the total number of trials, it is \ncalled the relative frequency.\nMaths2_Gr9_LB_Book.indb   179\n2014/09/08   09:07:13 AM\n\n180\t MATHEMATICS Grade 9: Term 4\n4.\t If Archie’s theory is correct, how many times will each of the colours be drawn if \na total of 40 trials is performed? Write your answers in the second row of the  \ntable below. Write the predicted relative frequencies in row 3 as fortieths, and \nin row 4 as twohundredths. \ncolour\nyellow\ngreen\npink\nblue\nred\nbrown\ngrey\nblack\nfrequencies predicted \nby Archie\nrelative frequencies \npredicted by Archie \nexpressed in \n40ths\nrelative frequencies \npredicted by Archie \nexpressed in 200ths\nThe relative frequency for each colour that Archie predicted is called the probability  \nof drawing that colour. If all the outcomes are equally likely, then\nprobability of an outcome = \n1\nthe total number of equally-likely outcomes\nYou will now investigate whether Archie’s theory is correct.\n5.\t (a)\t Make 8 small cards and write the name of one of the above colours on each card, \t\n\t\nso that you have cards with the eight colour names. Perform 8 trials to check \n\t\nwhether Archie’s theory is correct. Record your results (your tally marks 1 and  \n\t\nyour frequencies 1) in the relevant row of the table below.\n\t\n(b)\t Find out what any four of your classmates found when they did the experiment. \t\n\t\nEnter their results in your table too (Friend 1, 2, 3, 4 frequencies).\nTable for the results of the experiments\ncolour\nyellow\ngreen\npink\nblue\nred\nbrown\ngrey\nblack\nyour tally marks (1)\nyour frequencies (1)\nFriend 1 frequencies\nFriend 2 frequencies\nFriend 3 frequencies\nFriend 4 frequencies\nTotal frequencies for \n5 experiments \nMaths2_Gr9_LB_Book.indb   180\n2014/09/08   09:07:13 AM\n\n\t\nCHAPTER 11: PROBABILITY\t\n181\n6.\t (a)\t What was the total number of trials in the five experiments you recorded in the \t\n\t\nabove table? \n\t\n(b)\t What is the total of the frequencies for the different colours, in the last row of \t \t\n\t\nyour table? \n \n7.\t Is Archie’s theory correct? \n \nBettina has a different theory to Archie’s. She believes that if one does many trials with \nthe eight buttons in a bag, each colour will be drawn in approximately one-eighth of \nthe cases. In other words Bettina believes that the relative frequency of each outcome \nwill be close to the probability of that outcome, but may not be equal to it. \n8.\t (a)\t You and your four classmates performed 40 trials in total. Enter the results in the \t\n\t\nsecond row of the table below. Also express each frequency as a fraction of 40, in \t\n\t\nfortieths and in twohundredths.\ncolour\nyellow\ngreen\npink\nblue\nred\nbrown\ngrey\nblack\nactual frequencies \nobtained in your \nexperiments  \n(40 trials) \nrelative frequencies \nas 40ths\nrelative frequencies \nas 200ths\nprobability as \n200ths\n\t\n(b)\t Do your experiments show that Bettina’s theory is correct or not?\nJayden believes that when more trials are performed, the relative frequencies \nwill get closer to the probabilities. \nYou will now do an investigation to investigate whether Jayden’s theory is true.\nMaths2_Gr9_LB_Book.indb   181\n2014/09/08   09:07:13 AM\n\n182\t MATHEMATICS Grade 9: Term 4\ninvestigate what happens when more trials are done\n1.\t Perform 40 trials by drawing one card at a time from eight small cards with the names \nof the colours written on them, and enter your results in the second and third rows of \nthe table below.\ncolour\nyellow\ngreen\npink\nblue\nred\nbrown\ngrey\nblack\ntally marks\nfrequencies\nrelative frequencies \nas 40ths\nrelative frequencies \nas 200ths\nprobabilities as \n200ths\n2.\t Make a copy of the above table, without the row for tally marks, and without the  \nrow for the relative frequencies as fourtieths and the row for the probabilities, on  \na loose sheet of paper. Exchange it with a classmate. Enter the results of your \nclassmate on table 1 and 2 on the next page. Also enter your own results for  \nquestion 1 on the tables.\n3.\t Get hold of the data reports of three other classmates, and enter these on the tables \non the next page too.\n4.\t Add the frequencies of the various colours in the five sets of data for 40 trials each, \nand calculate the relative frequencies expressed as twohundredths.\n5.\t Is the range of relative frequencies for 200 trials smaller than the ranges for the \nfive different sets of 40 trials each? What does this indicate with respect to Jayden’s \ntheory?\nWhen only a small number of trials are done, the \nactual relative frequencies for different outcomes \nmay differ a lot from the probabilities of the \noutcomes.\nWhen many trials are done, the actual relative \nfrequencies of the different outcomes are quite close \nto the probabilities of the outcomes.\nMaths2_Gr9_LB_Book.indb   182\n2014/09/08   09:07:13 AM\n\n\t\nCHAPTER 11: PROBABILITY\t\n183\nTable 1: Frequencies for 5 sets of 40 trials each\ncolour\nyellow\ngreen\npink\nblue\nred\nbrown\ngrey\nblack\nfrequencies for your \nown 40 trials in \nquestion 1\nfrequencies for 40 trials \nby classmate 1\nfrequencies for 40 trials \nby classmate 2\nfrequencies for 40 trials \nby classmate 3\nfrequencies for 40 trials \nby classmate 4\ntotal frequencies for \n200 trials\nrelative frequencies for \n200 trials as \n200ths\nTable 2: Relative frequencies for each of the 5 sets of 40 trials each\n(expressed as 200ths)\ncolour\nyellow\ngreen\npink\nblue\nred\nbrown\ngrey\nblack\nrelative frequencies for \nyour own 40 trials\nrelative frequencies for \n40 trials by classmate 1\nrelative frequencies for \n40 trials by classmate 2\nrelative frequencies for \n40 trials by classmate 3\nrelative frequencies for \n40 trials by classmate 4\n6.\t How many different three-digit numbers can be formed with the symbols 3 and 5, \nif no other symbols are used? You may use one, two or three of the symbols in each \nnumber, and you may repeat the same symbol. \nMaths2_Gr9_LB_Book.indb   183\n2014/09/08   09:07:13 AM\n\n184\t MATHEMATICS Grade 9: Term 4\n11.2\t Compound events\ntossing a coin and giving birth\n1.\t Simon threw a coin and the outcome was heads. He will now throw the coin again.\n\t\n(a)\t What are the possible outcomes? \n\t\n(b)\t What is the probability of each of the possible outcomes? \n \n\t\n(c)\t What are the possible outcomes if Simon throws the coin for a third time? \n\t\n(d)\t What is the probability of each of the possible outcomes for the third throw?\nWhat happens when a coin is thrown for a second \ntime has nothing to do with what happened when it \nwas thrown the first time.\nThe first throw and the second throws are called \nindependent events: what happened on the first \nthrow cannot influence what will happen on the \nsecond throw.\n2.\t (a)\t If an event has four different equally-likely outcomes, what is the probability of \t\n\t\neach of the four outcomes? \n   \n\t\n(b)\t Does that mean that if the event is repeated 4 times, each of the four outcomes \t\t\n\t\nwill happen once? \n  \n\t\n(c)\t Does your answer in (a) means that if the event is repeated 100 times, each of \t \t\n\t\nthe four outcomes will happen 25 times? \n  \n3.\t (a)\t What are the possible outcomes when two coins are thrown? Use the two-way \t\n\t\ntable below to answer this question. One possible outcome is already given.\nHeads\nTails\nHeads\nH T\nTails\n\t\n(b)\t Do you think these four outcomes are equally likely? \n  \n\t\n(c)\t What is the probability of each of the four outcomes? \n  \n\t\n(d)\t What is the probability of getting a head and a tail? \nMaths2_Gr9_LB_Book.indb   184\n2014/09/08   09:07:13 AM\n\n\t\nCHAPTER 11: PROBABILITY\t\n185\n4.\t Let us consider the possible outcomes if three coins are thrown. \n\t\nBelow is a tree diagram that can help you figure out what the different possible \t \t\n\t\noutcomes are. Complete the diagram by filling in the missing information.\nFirst coin\nSecond coin\nThird coin\nOutcome\nheads\ntails\nheads\ntails\nheads\ntails\nheads\nHHH\nHHT\nHTH\nHTT\ntails\n5.\t (a)\t Do you think the eight different outcomes in question 4 are equally likely?  \n\t\n(b)\t What is the probability of each of the eight outcomes? \n  \n\t\n(c)\t What is the probability of throwing two heads and one tail? \n\t\n\t\n6.\t In question 6 on page 183 you were asked to write down the various numbers that \ncan be formed by using symbols 3 and 5. Think of all the four-letter codes that you \ncan form by using only two letters, P and Q. Any letter can be used more than once \nin one code. First think about how you will go about finding all the possibilities in a \nsystematic way and then try to set up a tree diagram to help you.\n\t\n(a)\t Draw a tree diagram in your exercise book to help you to solve this problem. List \t\n\t\nall the outcomes.\nMaths2_Gr9_LB_Book.indb   185\n2014/09/08   09:07:14 AM\n\n186\t MATHEMATICS Grade 9: Term 4\n\t\n(b)\t If the codes are formed by randomly choosing the letters, what is the probability \t\n\t\nthat the code will consist of the using the same letter four times? \n\t\n(c)\t What is the probability that the code will consist of two P’s and two Q’s? \n \nWhen a woman is pregnant, the baby can be a boy or a girl. Suppose we make the \nassumption that the two possibilities are equally likely, so the probability of a boy is 1\n2  \nand the probability of a girl is 1\n2 . \n7.\t (a)\t Complete this two-way table to show the possible outcomes of the gender of the \t\n\t\ntwo children in a family \nBoy\nGirl\n\t\n(b)\t List the possible outcomes.\n\t\n(c)\t What is the probability that the two children in the family will be of the same \t \t\n\t\ngender?\n\t\n(d)\t What is the probability that the eldest child will be a boy and then they will have \t\n\t\na girl?\n8.\t A certain woman already has one child, which is a  \nboy. She now expects a second child. What is the  \nprobability of it being a boy again, if we make the  \nassumption that a baby being a boy or a girl are  \nequally likely events? \n9.\t (a)\t A woman gets married and plans to have a baby in one year and another \t\n\t\n\t\n\t\nbaby in the next year. What is the probability that both babies will be girls? \n   \n\t\n(b)\t A woman gets married and plans to have a baby in each of the first three years \t \t\n\t\nof the marriage. What is the probability that she will have a boy in the first year,\n \t\n\t\nand girls in the second and third years? \n   \nThe assumption that a boy or a \ngirl being born are equally likely \nevents may not actually be true. \nHowever, probabilities can only \nbe calculated and used to make \npredictions if it is assumed that \noutcomes are equally likely.\nMaths2_Gr9_LB_Book.indb   186\n2014/09/08   09:07:14 AM",
      "chapter_id": "11"
    }
  ]
}