data/Gr7B_Mathematics_Learner_Eng.json

{
  "source_file": "Gr7B_Mathematics_Learner_Eng.txt",
  "title": "Grade 7B Mathematics",
  "table_of_contents": [
    {
      "section_id": "1",
      "title": "Numeric and geometric patterns"
    },
    {
      "section_id": "2",
      "title": "Functions and relationships 1"
    },
    {
      "section_id": "3",
      "title": "Algebraic expressions 1"
    },
    {
      "section_id": "4",
      "title": "Algebraic equations 1"
    },
    {
      "section_id": "5",
      "title": "Graphs"
    },
    {
      "section_id": "6",
      "title": "Transformation geometry"
    },
    {
      "section_id": "7",
      "title": "Geometry of 3D objects"
    },
    {
      "section_id": "8",
      "title": "Term 3: Revision and assessment"
    },
    {
      "section_id": "8",
      "title": "Integers"
    },
    {
      "section_id": "9",
      "title": "Numeric patterns"
    },
    {
      "section_id": "10",
      "title": "Functions and relationships 2"
    },
    {
      "section_id": "11",
      "title": "Algebraic expressions 2"
    },
    {
      "section_id": "12",
      "title": "Algebraic equations 2"
    },
    {
      "section_id": "13",
      "title": "Collect, organise and summarise data"
    },
    {
      "section_id": "14",
      "title": "Represent data"
    },
    {
      "section_id": "15",
      "title": "Interpret, analyse and report on data"
    },
    {
      "section_id": "16",
      "title": "Probability"
    },
    {
      "section_id": "18",
      "title": "Term 4: Revision and assessment"
    }
  ],
  "front_matter": "MATHEMATICS\nGrade 7\nBook 2\nCAPS\nLearner Book\nDeveloped and funded as an ongoing project by the Sasol Inzalo \nFoundation in partnership with the Ukuqonda Institute.\nMaths2_Gr7_LB_Book.indb   1\n2014/09/04   11:34:31 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: 978-1-920705-25-1\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:\nZhandre 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_Gr7_LB_Book.indb   2\n2014/09/04   11:34:31 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_Gr7_LB_Book.indb   3\n2014/09/04   11:34:31 AM\n\nTable of contents\nTerm 3\nChapter 1:\nNumeric and geometric patterns...........................................\t\n1\nChapter 2:\nFunctions and relationships 1.................................................\t\n17\nChapter 3:\nAlgebraic expressions 1...........................................................\t\n25\nChapter 4:\nAlgebraic equations 1..............................................................\t\n33\nChapter 5:\nGraphs .....................................................................................\t\n41\nChapter 6:\nTransformation geometry.......................................................\t\n57\nChapter 7:\nGeometry of 3D objects.........................................................\t\n79\nTerm 3: Revision and assessment...........................................\t\n97\nMaths2_Gr7_LB_Book.indb   4\n2014/09/04   11:34:31 AM\n\nTerm 4\nChapter 8:\nIntegers.....................................................................................\t 117\nChapter 9:\nNumeric patterns.....................................................................\t 133\nChapter 10: \nFunctions and relationships 2.................................................\t 141\nChapter 11:\nAlgebraic expressions 2...........................................................\t 149\nChapter 12:\nAlgebraic equations 2..............................................................\t 157\nChapter 13:\nCollect, organise and summarise data..................................\t 167\nChapter 14: \nRepresent data.........................................................................\t 189\nChapter 15:\nInterpret, analyse and report on data....................................\t 207\nChapter 16:\nProbability................................................................................\t 219\nTerm 4: Revision and assessment...........................................\t 227\nMaths2_Gr7_LB_Book.indb   5\n2014/09/04   11:34:31 AM\n\nMaths2_Gr7_LB_Book.indb   6\n2014/09/04   11:34:31 AM",
  "chapters": [
    {
      "title": "Numeric and geometric patterns",
      "content": "1\t Numeric and geometric patterns\n1.1\t\nNumber patterns in sequences\nwhat comes next?\nWhat may the next three numbers in each of these sequences be?\n\t\n4;  8;  12;  16;  20;  \n\t\n4;  8;  16;  32;  64;  \n\t\n4;  8;  14;  22;  32;  \n\t\n5;  7;  4;  8;  3;  9;  2;  \nA set of numbers in a given order is called a \nnumber sequence. In some cases each number \nin a sequence can be formed from the previous \nnumber by performing the same or a similar \naction. In such a case, we can say there is a \npattern in the sequence.\n1.\t (a)\t Write down the next three numbers in each of these sequences:\n\t\n\t\nSequence A:\t\n4;  7;  10;  13;  16;  \n\t\n\t\nSequence B:\t\n5;  10;  20;  40;  80;  \n\t\n\t\nSequence C:\t\n2;  5;  10;  17;  26;  \n\t\n(b)\t Write down how you decided what the next numbers would be in each of the \n\t\nthree sequences.\nA sequence can be formed by repeatedly adding \nor subtracting the same number. In this case the \ndifference between one term and the next is constant.\nA sequence can be formed by repeatedly multiplying \nor dividing by the same number. In this case the ratio \nbetween one term and the next is constant.\nA sequence can also be formed in such a way that  \nneither the difference nor the ratio between one term \nand the next is constant.\n\t\nThe numbers in a sequence  \nare called the terms of the \nsequence. Terms that follow \none another are said to be \nconsecutive.\nMaths2_Gr7_LB_Book.indb   3\n2014/09/04   11:34:32 AM\n\n4\t\nMATHEMATICS Grade 7: Term 3\nIn sequence A of question 1 there is a constant difference between consecutive  \nterms, as shown below.\n    Sequence A:\t\n4\t\n7\t\n10\t\n13\t\n16\n    Difference:\t\n+ 3\t\n+ 3\t\n+ 3\t\n+ 3\nIn sequence B of question 1 there is a constant ratio between consecutive terms, as \nshown below. \n    Sequence B:\t\n5\t\n10\t\n  20\t\n    40\t\n      80\n    Ratio:\t\n× 2\t\n× 2\t\n× 2\t\n× 2\nIn sequence C of question 1 there is neither a constant difference nor a constant ratio \nbetween consecutive terms. There is, however, a pattern in the differences between \nthe terms, which makes it possible to extend the sequence. Consecutive odd numbers, \nstarting with 3, are added to form the next term.\n    Sequence C:\t\n2\t\n 5\t\n  10\t\n     17\t        26\n    Difference:\t\n+ 3\t\n+ 5\t\n+ 7\t\n+ 9\n2.\t Write down the next five terms in each of the sequences below. In each case, describe \nthe relationship between consecutive terms.\n\t\n(a)\t 100;  95;  90;  85;  \n\t\n(b)\t 0,3;  0,5;  0,7;  0,9;  \n\t\n(c)\t 6;  18;  54;  162;  \n\t\n(d)\t 1;  3;  6;  10;  15;  \n\t\n(e)\t 20;  31;  42;  53;  \n\t\n(f)\t 10;   9,7;  9,4;  9,1;  \n\t\n(g)\t 18 000;  1 800;  180;  18;  \nMaths2_Gr7_LB_Book.indb   4\n2014/09/04   11:34:32 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS\t\n5\n\t\n(h)\t\n1\n48 ;  \n1\n24 ;  \n1\n12 ;  \n1\n6;  \n\t\n(i)\t 1;  4;  9;  16;  \n\t\n(j)\t 625;  125;  25;  5;\nIn all of the above cases it was possible to extend \nthe sequence by repeatedly adding or subtracting \na number to get the next term, or by repeatedly \nmultiplying or dividing by a number to get \nthe next term, or by adding different numbers \naccording to some pattern to get the next term.\nrelationships between dependent and independent variables\n1.\t (a)\t Mr Twala pays a fee to park his car in a parking lot every day. He has to pay R3 to \n\t\nenter the parking lot and then a further R2 for every hour that he leaves his car \n\t\nthere. Complete the table below to show how much his parking costs him per  \n\t\nday for various numbers of hours.\nNumber of hours\n1\n2\n3\n4\n5\n6\n7\n8\n9\nCost of parking in R \n5\n7\n9\n\t\n(b)\t How did you complete this table? Describe your method.\n\t\n(c)\t Is there another way that you could complete the table? Describe it.\n\t\n(d)\t Thembi multiplied the number of hours by 2 and then added 3 to calculate  \n\t\nthe cost for any specific number of hours. Complete the flow diagram to show \n\t\nThembi’s rule.\n\t\n\t\n\t\n\t\n\t\nThe word “recur” means “to \nhappen again”. The extension of \na number sequence by repeatedly \nperforming the same or similar \naction is called recursion. The \nrule that describes the  \nrelationship between consecutive \nterms is called a recursive rule.\n \nMaths2_Gr7_LB_Book.indb   5\n2014/09/04   11:34:33 AM\n\n6\t\nMATHEMATICS Grade 7: Term 3\nThe rule multiply by 2 and then add 3 describes the  \nrelationship between the two variables in this  \nsituation. The number of hours is the  \nindependent variable. The cost of Mr Twala’s  \nparking is the dependent variable because the amount he has to pay depends on the \nnumber of hours that he parks.\nThis rule describes how you can calculate the value of the dependent variable if the  \ncorresponding value of the independent variable is known. It differs from a recursive rule, \nwhich describes how you can calculate the value of the dependent variable that follows \non a given value of the dependent variable.\nIn the case of a number sequence, the position (number) of the term can be taken  \nas the independent variable, as shown for the sequence  15;  19;  23;  27;  31; . . .  in  \nthis table:\nTerm number\n1\n2\n3\n4\n5\n6\n7\n8\n50\nTerm\n15\n19\n23\n27\n31\n2.\t (a)\t Complete the above table.\n\t\n(b)\t How did you calculate term number 50?\n\t\n(c)\t Lungile reasoned like this:  \n\t\n\t\nI added 4 each time to complete the table. I  \n\t\ncounted backwards to see what comes before  \n\t\nterm 1. I got 11 and then I knew I had to add  \n\t\none 4 to 11 to get the first term. \n\t\n\t\nComplete the pattern below to show Lungile’s thinking:\n\t\n\t\nTerm 1:\t\n11 + 1 × 4 = 11 + 4 = 15 \n\t\n\t\nTerm 2: \t\n11 + 2 × 4 = 11 + 8 = 19\n\t\n\t\nTerm 3: \t\n\t\n\t\nTerm 4:\t\n\t\n\t\nTerm 5: \t\n\t\n\t\nTerm 6: \t\n\t\n\t\nTerm 10:\t\n\t\n\t\nTerm 50:\t\nThe R3 that is added is a constant \nin this situation. The number of hours \nand the cost are variables.\nLungile remembered that \nmultiplication is done before \naddition, unless otherwise \nindicated by brackets.\nMaths2_Gr7_LB_Book.indb   6\n2014/09/04   11:34:33 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS\t\n7\n\t\n(d)\t Describe in your own words how term number 50 can be calculated.\n\t\n(e)\t Tilly reasoned like this: The constant difference between the terms is 4. I must add  \n\t\nfour 49 times to the first term to get the 50th term. So, 15 + 49 × 4 = 15 + 196 = 211.\n\t\n\t\nComplete the pattern below to demonstrate Tilly’s thinking:\n\t\n\t\nTerm 1:\t\n15 \n\t\n\t\nTerm 2:\t\n15 + 1 × 4 = 15 + 4 = 19\n\t\n\t\nTerm 3: \t\n15 + 2 × 4 = 15 + 8 = 23\n\t\n\t\nTerm 4: \t\n\t\n\t\nTerm 5:\t\n\t\n\t\nTerm 6: \t\n\t\n\t\nTerm 10:\t\n\t\n\t\nTerm 50:\t\n\t\n(f)\t Write the rule to calculate term number 50 in your own words.\nIn the example in question 2, the term number is the independent variable and the term \nitself is the dependent variable. So, if we know the rule that links the dependent variable \nand the independent variable, we can use it to determine any term for which we know \nthe term number.\n3.\t Write a rule to calculate the term for any term number in the sequence \n15;  19;  23;  27;  31; . . .  by using \n\t\n(a)\t Lungile’s thinking.\n\t\n(b)\t Tilly’s thinking.\nWe can use n as a symbol for “any term number”. \nThe rule to calculate the term for any term number \nwhen using Lungile’s thinking will then be:\nTerm = n × 4 + 11\n\t\n(c)\t Write down the rule to calculate the term for any term number in terms of n by \n\t\nusing Tilly’s thinking.\nMaths2_Gr7_LB_Book.indb   7\n2014/09/04   11:34:33 AM\n\n8\t\nMATHEMATICS Grade 7: Term 3\n1.2\t Geometric patterns\nconstant quantities and variable quantities\nSmall yellow, blue and red tiles are combined to form larger square tiles as shown below:\n\t\nTile no. 1\t\nTile no. 2\t\nTile no. 3\t\nTile no. 4\t\nTile no. 5\n1.\t Draw tile no. 5 on the grid provided. (Shade the blue and red tiles in different ways. \nYou don’t have to use colours.)\n2.\t Complete the table.\nTile  \nno. 1\nTile  \nno. 2\nTile  \nno. 3\nTile  \nno. 4\nTile  \nno. 5\nTile  \nno. 10\nNumber of \nyellow tiles\nNumber of \nred tiles\nNumber of \nblue tiles\n\t\n3.\t How many red tiles are there in each bigger tile? \n4.\t How many yellow tiles are there in each bigger tile?\n5.\t Some of the quantities in this situation are variables and some are constants.  \n\t\nWhich are variables and which are constants?\n6.\t Was it possible to predict the pattern on tile no. 2 by looking only at tile no. 1? \nMaths2_Gr7_LB_Book.indb   8\n2014/09/04   11:34:34 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS\t\n9\nThe number of red tiles is constant and the number \nof blue tiles is constant. It is clear that the design is \nsuch that there is always a red tile in the top right \ncorner, and also in the bottom left corner, and that \nthe red tiles are always “bordered” by two blue tiles \neach. So the number of red and blue tiles is constant \nin this situation.\nThe number of yellow tiles in the arrangements \nvaries. The number of yellow tiles is a variable in \nthis situation.\npatterns with matches\n1.\t A pattern with matches is shown below.\n\t\nFigure 1\t\nFigure 2\t\nFigure 3\n\t\n\t\n(a)\t Explain how the pattern is formed.\n\t\n(b)\t Complete the table.\nFigure number\n1\n2\n3\n4\n5\n6\n7\n8\nNumber of matches\n3\n5\n7\n\t\n(c)\t What rule did you use to complete the table?\n\t\n(d)\t How many matches are needed to form figure no. 9?\t\n\t\n(e)\t How many matches are needed to form figure no. 17? Explain.\n\t\n(f)\t If you used the recursive rule to complete the table, it would have taken a long  \n\t\ntime to answer question (e) because you had to add the same number many \n\t\ntimes. Try to find an easier way to answer question (e). Describe your method.\nMaths2_Gr7_LB_Book.indb   9\n2014/09/04   11:34:34 AM\n\n10\t\nMATHEMATICS Grade 7: Term 3\n\t\n(g)\t Complete the pattern below.\n\t\n\t\nHint: It may help to think of figure no. 1 or term 1 like this:  \n\t\nThere is 1 match at the beginning and two more are added  \n\t\nevery time. It helps to “see” the two matches that are added  \n\t\neach time.\n\t\n\t\nTerm 1:\t\n1 +\t 1 × 2\t =\t 3\n\t\n\t\nTerm 2:\t\n1 +\t 2 × 2\t =\t 5\n\t\n\t\nTerm 3:\t\n1 +\t 3 × 2\t =\t 7\n\t\n\t\nTerm 4:\t\n\t\n\t\nTerm 5:\t\n\t\n\t\nTerm 10:\t\n\t\n\t\nTerm 17:\t\n\t\n(h)\t What stays the same in the pattern in (g) and what varies?\n\t\n(i)\t Use the flow diagram below to write down the rule that you can use to  \n\t\ncalculate the number of matches needed for any figure in the pattern.\n\t\nFigure number\t\n \n\t\nNumber of matches\n\t\n(j)\t Can you link the number of matches added each time to the number that you \n\t\nmultiply by in the flow diagram? Explain.\n2.\t Another pattern with matches is shown below.\n\t\nFigure 1\t\nFigure 2\t\nFigure 3\n\t\n(a)\t Explain how the pattern is formed.\nMaths2_Gr7_LB_Book.indb   10\n2014/09/04   11:34:34 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS\t\n11\n\t\n(b)\t Complete the table.\nFigure \nnumber\n1\n2\n3\n4\n5\n6\n7\n8\nNumber of \nmatches\n4\n\t\n(c)\t What rule did you use to complete the table?\n\t\n(d)\t How many matches are needed for figure 9 (or term 9)? \n\t\n(e)\t How many matches are needed for figure 20 (or term 20)? \n\t\n(f)\t What rule did you use to calculate the number of matches in question (e)?\n\t\n(g)\t Complete the pattern:\n\t\n\t\nTerm 1:\t\n1 +\t\n1 × 3\t =\t 4\n\t\n\t\nTerm 2:\t\n1 +\t\n2 × 3\t =\t 7\n\t\n\t\nTerm 3:\t\n1 +\t\n3 × 3\t =\t\n\t\n\t\nTerm 4:\t\n\t\n\t\nTerm 5:\t\n\t\n\t\nTerm 10:\t\n\t\n\t\nTerm 17:\t\n\t\n(h)\t What stays the same in the pattern in (g) and what varies?\n\t\n(i)\t Use the flow diagram below to write down the rule that you can use to  \n\t\ncalculate the number of matches needed for any figure in the pattern.\n\t\nFigure number\t\n \n\t Number of matches\n3.\t Compare the way in which the number of matches increases in question 1 to the  \nway in which it increases in question 2. What is the same and what is different?\nMaths2_Gr7_LB_Book.indb   11\n2014/09/04   11:34:34 AM\n\n12\t\nMATHEMATICS Grade 7: Term 3\nalphabetic patterns\nConsider the figures below formed with red dots.\n\t Figure 1\t\nFigure 2\t\nFigure 3\t\nFigure 4\t\nFigure 5\n1.\t How many dots are used to form figure 5? \n2.\t Draw figure 5. \n3.\t Complete the table.\nFigure \nnumber\n1\n2\n3\n4\n5\n6\n7\n8\nNumber of \ndots\n7\n12\n17\n4.\t Complete the flow diagram.\n\t\n\t\n \n\t\n5.\t What rule did you use to complete the table? Describe your rule.\n6.\t Can you think of another rule to complete the table? Describe your rule.\n7.\t Name the dependent variable and the independent variable in this situation.\nMaths2_Gr7_LB_Book.indb   12\n2014/09/04   11:34:35 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS\t\n13\nsquares and cubes\n1.\t Squares are arranged to form figures as shown below, according to a rule.\n\t\nFigure 1\t\nFigure 2\t\nFigure 3\t\nFigure 4\n\t\n(a)\t Complete the table. Then determine the differences between consecutive terms. \nFigure number\n1\n2\n3\n4\n5\n6\n7\n8\nNumber of squares\n2\n5\n\t\n+ 3\t\n\t\n(b)\t Describe the recursive rule that you can use to extend the pattern in words.\n\t\n(c)\t Nombuso played around with the differences between consecutive terms. She \n\t\nnoticed that the pattern (+ 3; + 5; + 7; …) was similar to the one that you get \n\t\nwhen you calculate the differences between square numbers. This made her \n\t\nthink that she should investigate square numbers to help her find a rule that \n\t\ncould link the figure number and the number of squares.\n\t\n\t\nComplete the following pattern along the lines of Nombuso’s thinking:\n\t\n\t\nFigure 1:\t\n1 × 1 + 1 = 1 + 1 = 2\n\t\n\t\nFigure 2: \t\n2 × 2 + 1 = 4 + 1 = 5\n\t\n\t\nFigure 3: \t\n\t\n\t\nFigure 4: \t\n\t\n\t\nFigure 5:\t\n\t\n\t\nFigure 6:\t\n\t\n\t\nFigure 7:\t\n\t\n\t\nFigure 8:\t\n\t\n\t\nFigure 50:\t\n\t\n(d)\t Write a rule to calculate the number of squares for any figure number.\nMaths2_Gr7_LB_Book.indb   13\n2014/09/04   11:34:35 AM\n\n14\t\nMATHEMATICS Grade 7: Term 3\n\t\n(e)\t Write your rule in (d) in terms of n where n is the symbol for any figure number.\n\t\n(f)\t Compare the sequence in this activity to the sequence in the previous activity  \n\t\nwhere dots were arranged to form the letter H. Describe the way in which the  \n\t\ndependent variable (the output number) changed in each of the sequences.\n2.\t Identical cubes are arranged to form stacks of cubes in the following way:\n\t\nStack 1\t\nStack 2\t\nStack 3\t\nStack 4\n\t\n(a)\t Complete the table. Then find the differences between consecutive terms. Do it  \n\t\na second and a third time. Write the differences below the arrows.\nStack number \n1\n2\n3\n4\n5\n6\n7\n8\nNumber of cubes \n2\n9\n28\n\t\n7\t\n19\t\n\t\n12\t\n\t\n\t\n(b)\t Describe the way in which you completed the table.\nMaths2_Gr7_LB_Book.indb   14\n2014/09/04   11:34:37 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS\t\n15\n\t\n(c)\t David looked carefully at the structure of the stacks and did the following to link \n\t\nthe stack number with the number of cubes in a stack. Complete the pattern.\n\t\n\t\nStack 1: \t\n1 × 1 × 1 + 1 = 1 + 1 = 2\n\t\n\t\nStack 2: \t\n2 × 2 × 2 + 1 = 8 + 1 = 9\n\t\n\t\nStack 3: \t\n3 × 3 × 3 + 1 = 27 + 1 = 28\n\t\n\t\nStack 4: \t\n4 × 4 × 4 + 1 = 64 + 1 = 65\n\t\n\t\nStack 5: \t\n\t\n\t\nStack 6: \t\n\t\n\t\nStack 7: \t\n\t\n\t\nStack 8: \t\n\t\n\t\nStack 9: \t\n\t\n\t\nStack 10: \t\n\t\n(d)\t How many cubes will there be in stack 50?\n\t\n(e)\t Write the rule that you used to calculate the number of cubes in stack 50 in words.\n\t\n(f)\t Write your rule in (e) in terms of n where n is the symbol for any stack number.\n3.\t In questions 1(a) and 2(a) you calculated the differences between the consecutive \nterms. \n\t\n(a)\t What did you find when you kept on finding the differences, as suggested in  \n\t\nquestion 2(a)?\n\t\n(b)\t Go back to question 1(a). What do you find when you keep on finding the  \n\t\ndifferences between consecutive terms, like you did in question 2(a)?\nMaths2_Gr7_LB_Book.indb   15\n2014/09/04   11:34:37 AM\n\n16\t\nMATHEMATICS Grade 7: Term 3\nmy own patterns\nUse the grid, the tables and the tile template to create and describe your own geometric \npatterns. \nPattern A\nPattern B\nMaths2_Gr7_LB_Book.indb   16\n2014/09/04   11:34:37 AM\n\nChapter 2\nFunctions and\nrelationships 1\n\t\nCHAPTER 2: FUNCTIONS AND RELATIONSHIPS 1\t\n17\nIn this chapter, you will work with formulae. A formula is a description of how one may \ncalculate the value of one variable quantity if the value or values of certain other variable \nquantities are given.\nYou will acquire the symbolic language to write formulae with two variables.\n2.1\t From counting to calculating.................................................................................... 19\n2.2\t What to calculate and how....................................................................................... 21\n2.3\t Input and output numbers....................................................................................... 23\nMaths2_Gr7_LB_Book.indb   17\n2014/09/04   11:34:37 AM\n\n18\t\nMATHEMATICS Grade 7: Term 3\nMaths2_Gr7_LB_Book.indb   18\n2014/09/04   11:34:38 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n19\n\t\nCHAPTER 2: FUNCTIONS AND RELATIONSHIPS 1\t\n19",
      "chapter_id": "1"
    },
    {
      "title": "Functions and relationships 1",
      "content": "2\t Functions and relationships 1\n2.1\t From counting to calculating\n1.\t (a)\t How many red squares and how many black squares are there in each of the  \n\t\narrangements 1, 2, 3 and 4 below? Write your answers in the table.\n1\n2\n3\n4\nArrangement number\n1\n2\n3\n4\nNumber of red squares\nNumber of black squares\n\t\n(b)\t Imagine that arrangements 5, 6 and 7 are made according to the same pattern. \n\t\nHow many red and how many black squares do you think there will be in each  \n\t\nof these arrangements? Write your answers in the above table.\n\t\n(c)\t Draw arrangements 5 and 6 on the above grid, if you have not done so already.\n\t\n(d)\t Try to figure out how many red and how many black squares there will be in  \n\t\narrangements 20, 21 and 22.\n2.\t It will be useful to have formulae to calculate the numbers of red and black squares in \ndifferent arrangements like the above.\n\t\n(a)\t Which of the formulae below can be used to calculate the numbers of red squares \n\t\nin the above arrangements? There is more than one formula that works.\n\t\n\t\ny = 2 × x + 4      y = 2 × (2 × x + 1)      y = x2 + 2      y = 4 × x + 2\nMaths2_Gr7_LB_Book.indb   19\n2014/09/04   11:34:38 AM\n\n20\t\nMATHEMATICS Grade 7: Term 3\n\t\n(b)\t Natasha decided to use the formula y = 4 × x + 2 to calculate the number of red \n\t\nsquares in an arrangement. What do the symbols x and y mean in this case?\n\t\n(c)\t Use the formula y = 4 × x + 2 to calculate the numbers of red squares in  \n\t\narrangements 20, 21 and 22. \n\t\n(d)\t If your answers differ from the answers you gave in question 1(d), you have  \n\t\nmade mistakes somewhere. Find your mistakes and correct them.\n\t\n(e)\t Complete the table.\nx\n1\n2\n3\n4\n5\n6\n7\n8\n9\n2 × (2 × x + 1)\n2 × x + 4\n4 × x + 2\n3.\t (a)\t Which of the formulae below can be used to calculate the numbers of black  \n\t\nsquares in the arrangements in question 1?\n\t\n\t\nz = (x + 2)2    z = x2 + 2    p = n2 + 2\n\t\n(b)\t Complete the table.\nx\n1\n2\n3\n4\n5\n6\n7\n8\n9\nx2 + 2\n(x + 2)2\n4.\t Hilary uses x to represent the number of squares in each side of the arrangements.\n\t\n(a)\t Which of these formulae can Hilary use to calculate the numbers of black  \n\t\nsquares in the arrangements in question 1? \n\t\n\t\ny = x2 − 4 × x + 6        y = (x − 2)2 + 2 \n\t\n(b)\t Which of these formulae can Hilary use to calculate the numbers of red squares? \n\t\n\t\ny = 3 × x − 3    y = 4 × x − 6    4 × (x − 2) + 2 \n\t\n(c)\t Complete this table to check your answers.\nx\n3\n4\n5\n6\n7\n8\n9\nx2 − 4 × x + 6\n(x − 2)2 + 2\n3 × x − 3\n4 × x − 6\n4 × (x − 2) + 2\nMaths2_Gr7_LB_Book.indb   20\n2014/09/04   11:34:38 AM\n\n\t\nCHAPTER 2: FUNCTIONS AND RELATIONSHIPS 1\t\n21\n2.2\t What to calculate and how\nrepresenting situations mathematically\n1.\t (a)\t How many minutes are there in an hour? \n\t\n(b)\t How many minutes are there in 2 hours? \n\t\n(c)\t How many minutes are there in 3 hours? \n\t\n(d)\t Explain how you determined the answers for questions 1(b) and (c).\nThe formula m = 60 × h can be used to calculate the number of minutes when the \nnumber of hours is known. The symbol h represents the number of hours and m the \nnumber of minutes.\n\t\n(e)\t Express the formula m = 60 × h in words.\n\t\n(f)\t Complete the table.\nNumber of hours\n1\n2\n3\n15\n24\nNumber of minutes\n60\n120\nHow to calculate\n60 × 1\n2.\t Three bus companies placed the following advertisements in a newspaper:\n\t\n(a)\t Which of the formulae at the top of the  \n\t\nnext page can be used to calculate the fare  \n\t\nfor a journey with Hamba Kahle Tours?\n\t\n(b)\t Which of the formulae can be used to  \n\t\ncalculate the fare for a journey with  \n\t\nSaamgaan Tours?  \nSaamgaan Tours\nWe criss-cross every province and \nstop in every town and dorpie. Pay \nonly R450 per trip plus 60c per km.\nHamba Kahle Tours\nLong distance travel is our business: \nR500 per trip plus 50c per km!\nComfort Tours\nExperience what it means to  \ntravel in style. Only R480 per trip \nplus 55c per km.\nMaths2_Gr7_LB_Book.indb   21\n2014/09/04   11:34:38 AM\n\n22\t\nMATHEMATICS Grade 7: Term 3\n\t\nSome formulae to calculate fares:\n\t\nA.\t Fare = 0,50 × distance + 500\t\n\t\n\t\nB.\t Fare = 50 × distance + 500\n\t\nC.\t Fare = 0,60 × distance + 450\t\n\t\n\t\nD.\t Fare = 60 × distance + 450\n\t\nE.\t\nFare = 55 × distance + 480\t \t\n\t\n\t\nF.\t Fare = 0,55 × distance + 480\n\t\n(c)\t Which of the above formulae can be used to \n\t\ncalculate the fare for a journey with Comfort  \n\t\nTours?\n\t\n(d)\t Complete the table by making use of the formulae below. You may use a  \n\t\ncalculator for this question.\n\t\n\t\nFare for Hamba Kahle Tours = 0,50 × distance + 500\n\t\n\t\nFare for Saamgaan Tours = 0,60 × distance + 450\n\t\n\t\nFare for Comfort Tours = 0,55 × distance + 480\nDistance in km\n150\n200\n250\n300\nHamba Kahle Tours\nSaamgaan Tours\nComfort Tours\n\t\n(e)\t Which bus company is the cheapest? Explain.\n\t\n(f)\t Complete a flow diagram for the bus company that you named in question (e):\n\t\n\t\n \n\t\n\t\n(g)\t Wandile wrote the formulae for calculating the fares for the different bus  \n\t\ncompanies using the letter symbols x and y. Say what each letter symbol stands \n\t\nfor in each of the following:\n\t\n\t\n(i)\t y = 0,50 × x + 500      \t \t\n\t\n\t\n\t\n(ii)\t y = 0,60 × x + 450     \t\n\t\n\t\n\t\n\t\n(iii)\t y = 0,55 × x + 480       \t \t\n\t\n\t\n(h)\t Which of the three bus companies would be the cheapest to use for a journey of  \n\t\n1 000 km? \nWe write 50c as R0,50 or 0,50 \nwhen we do calculations. \nMaths2_Gr7_LB_Book.indb   22\n2014/09/04   11:34:38 AM\n\n\t\nCHAPTER 2: FUNCTIONS AND RELATIONSHIPS 1\t\n23\n2.3\t Input and output numbers\nfrom formulae to tables\n1.\t For each of the tables, determine which of these formulae could have been used to \ncomplete it: \n\t\nA.\t y = 5 × x + 3\t\nB.\t\ny = 3 × x\t\nC.\t y = 3 × x + 2\n\t\nD.\t y = 4 × x\t\nE.\t\ny = 3 × x + 1\t\nF.\t\ny = 2 × x\n\t\nG.\t y = 3 × x + 10\t\nH.\t y = 2 × x − 1\t\nI.\t\ny = 5 × x\t\n(a)\nx\n1\n2\n3\n4\n5\n(b)\nx\n1\n2\n3\n4\n5\ny\n13\n16\n19\n22\n25\ny\n8\n13\n18\n23\n28\n\t\n\t\nFormula used: \n\t\n\t\n\t\nFormula used: \n(c)\nx\n1\n2\n3\n4\n5\n(d)\nx\n1\n2\n3\n4\n5\ny\n4\n8\n12\n16\n20\ny\n5\n8\n11\n14\n17\n\t\n\t\nFormula used: \n\t\n\t\n\t\nFormula used: \n(e)\nx\n1\n2\n3\n4\n5\n(f)\nx\n1\n2\n3\n4\n5\ny\n5\n10\n15\n20\n25\ny\n1\n3\n5\n7\n9\n\t\n\t\nFormula used: \n\t\n\t\n\t\nFormula used: \nWe can complete a table of values if we are given a formula. For example, for the  \nformula y = 7 × x − 3 we can complete the table below, as shown:\nFor x = 1,  y = 7 × 1 − 3\t\nFor x = 2,  y = 7 × 2 − 3\t\nFor x = 3,  y = 7 × 3 − 3 \n\t\n= 7 − 3\t\n= 14 − 3\t\n= 21 − 3\n\t\n= 4\t\n= 11\t\n= 18\nx\n1\n2\n3\n4\n5\n6\n7\ny\n4\n11\n18\n25\n32\n39\n46\n2.\t Use the given formulae to complete the tables.\n\t\n(a)\t y = 6 × x − 51\n2\nx\n1\n2\n3\n4\n5\n6\n7\ny\nMaths2_Gr7_LB_Book.indb   23\n2014/09/04   11:34:39 AM\n\n24\t\nMATHEMATICS Grade 7: Term 3\n\t\n(b)\t y = 30 × x + 1\nx\n0,1\n0,2\n0,3\n0,4\n0,5\n0,6\n0,7\ny\nfrom patterns to formulae\n1.\t Some arrangements with black and red squares and some formulae are given below.\n\t\nFormula A:   z = 2 × n + 1\t\n\t\n\t\nFormula B:   z = 2 × x − 3\n\t\nFormula C:   y = (n + 1)2 + 2\t\t\n\t\nFormula D:   y = x2 − (2 × x − 3)\n\t\n\t\n(a)\t How many black squares will there be in the next two similar arrangements?\n\t\n(b)\t Susan uses formulae B and D to calculate the numbers of red and black squares.  \n\t\nWhat do the letter symbols z, x and y mean in Susan’s work?\n\t\n(c)\t Zain uses formulae A and C to calculate the numbers of red and black squares. \n\t\nWhat do the letter symbols z, n and y mean in Zain’s work?\n2.\t Write formulae that can be used to calculate the numbers of red and black squares \nin arrangements like those below. Use letter symbols of your own choice and state \nclearly what each of your symbols represents.   \nMaths2_Gr7_LB_Book.indb   24\n2014/09/04   11:34:39 AM\n\nChapter 3\nAlgebraic expressions 1\n\t\nCHAPTER 3: ALGEBRAIC EXPRESSIONS 1\t\n25\nIn this chapter, you will learn about algebraic expressions. An algebraic expression is a \ncomputational procedure. Put differently, an algebraic expression tells you how to  \ncalculate a value. But an algebraic expression is also a value.\nYou will learn more about variable and constant quantities in this chapter and you will  \nbe required to identify these in formulae and number sentences.\n3.1\t Describing and doing computations......................................................................... 27\n3.2\t Relationships represented in formulae....................................................................... 31\nMaths2_Gr7_LB_Book.indb   25\n2014/09/04   11:34:39 AM\n\n26\t\nMATHEMATICS Grade 7: Term 3\nMaths2_Gr7_LB_Book.indb   26\n2014/09/04   11:34:40 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n27\n\t\nCHAPTER 3: ALGEBRAIC EXPRESSIONS 1\t\n27",
      "chapter_id": "2"
    },
    {
      "title": "Algebraic expressions 1",
      "content": "3\t Algebraic expressions 1\n3.1\t Describing and doing computations\ndifferent ways of describing a computation\n1.\t The diagrams below represent arrangements of small circles. In every arrangement \nthere are two rows of circles.\n\t\nDiagram 1\t\nDiagram 2\t\nDiagram 3\t\nDiagram 4\t\nDiagram 5\n\t\n(a)\t The table below relates to the diagrams. Complete it.\nDiagram number\n1\n2\n3\n4\n5\nNumber of circles per row\nNumber of rows\nHow to calculate the total number of \ncircles per diagram (rule)\n\t\nIn every diagram, we can identify:\n• the number of rows  \n• the number of circles per row\n• the total number of circles per arrangement.\n\t\n(b)\t What remains the same in the diagrams? \t\n\t\n(c)\t What changes in the diagrams? In other words, what are the variable quantities \n\t\nin the situation?\n\t\n(d)\t Complete the flow diagram.\n21\n41\n× 2\n11\n62\n102\nMaths2_Gr7_LB_Book.indb   27\n2014/09/04   11:34:40 AM\n\n28\t\nMATHEMATICS Grade 7: Term 3\n\t\n(e)\t How many circles will diagram 11 have if the pattern is extended? Explain.\n\t\n(f)\t What does the number 2 in the rule 2 × n \n\t\nrepresent?\n\t\n(g)\t What does the letter symbol n represent in  \n\t\nthe rule 2 × n?\nConsider the sequence 1;  3;  5;  7;  9;  ...\t\nThe first odd number can be written as 2 × 1 − 1.  \nThe second odd number can be written as 2 × 2 − 1. \nThe third odd number can be written as 2 × 3 − 1.\n2.\t (a)\t What is the tenth odd number?\n\t\n(b)\t What is the thirtieth odd number?\n\t\n(c)\t What is the hundredth odd number? \n\t\n(d)\t What is the nth odd number? \n3.\t The rule 2 × n − 1 can be used to determine any odd number in the sequence \n1;  3;  5;  7;  9;  ...\n\t\nWhat does the letter symbol n represent in the rule 2 × n − 1?\nIn the questions above we have used the letter \nsymbol n to represent:\n1.\t a changing number in the rule 2 × n  \n\t\n(n represents the number of circles in a row) \n2.\t the position of the odd number in a  \n\t\nsequence in the rule 2 × n − 1.\nThe rule 2 × n can be used to \ndetermine the total number of \ncircles in a diagram. The number \n2 in the rule 2 × n remains the \nsame all the time. We say it is \na constant. n represents the \nnumber of circles per row and \nthat is a variable, because it \nchanges.\nThe numbers 2 and −1 remain \nthe same all the time; we call \nthem constants. The numbers \nin blue change according to \nthe position of the odd number \nin the sequence. We call them \nvariables.\nThe rule 2 × n can be used to \ncalculate the total number of  \ncircles in a diagram if the number  \nof circles per row is known. \nThe rule 2 × n − 1 can be used \nto determine any odd number in \nthe sequence of odd numbers if \nits position is known.\nMaths2_Gr7_LB_Book.indb   28\n2014/09/04   11:34:40 AM\n\n\t\nCHAPTER 3: ALGEBRAIC EXPRESSIONS 1\t\n29\n4.\t (a)\t Complete the flow diagram.\n+ 4\n× 5\n11\n31\n41\n125\n275\n\t\n(b)\t Which of the following instructions did you follow to calculate the output \n\t\n\t\nvalues of\t  \n+ 4\n× 5\n\t in question 4(a)? \n\t\n\t\nMake a tick mark (✓) next to the correct answer.  \n\t\n\t\nA.\t Multiply the input number by 5 and then add 4.\n\t\n\t\nB.\t Add 45 to the input number.\n\t\n\t\nC.\t Add 4 to the input number and then multiply by 5.  \n5.\t Use 10 as input number and calculate the output number for each of the word \nformulae in question 4(b).\nWe may write (x + 4) × 5 as an abbreviation for  \nadd 4 to the input number, then multiply by 5. \n(x + 4) × 5 can be called a computational instruction  \nor an algebraic expression.\nIn the expression (x + 4) × 5, the letter symbol x can be replaced by many different \ninput numbers. The symbol x represents a variable quantity or a variable. If,  \nhowever, the expression (x + 4) × 5 is equal to 35, as in the number sentence  \n(x + 4) × 5 = 35, the symbol x represents only one value, and that is 3. \nIn the expression (x + 4) × 5, the numbers 4 and 5 are constants. In the number  \nsentence (x + 4) × 5 = 35, x is an unknown value.\n6.\t Write the abbreviations for the following computational instructions by using x for \n“the input number”:\n\t\n(a)\t Half the input number plus 2 \n\t\n(b)\t Multiply the input number by 6 and subtract 2. \n\t\n(c)\t Multiply the sum of the input number and 3 by 10. \n\t\n(d)\t Subtract 4 from the input number and multiply the answer by 7. \nWe call the numbers on the left \nin the flow diagram the input \nnumbers.\nThe numbers on the right in \nthe flow diagram, and whose \nvalues depend on the input \nnumbers, are called the output \nnumbers.\nThe letter symbol x, or any \nother symbol, can be used as \nan abbreviation for “the input \nnumber”.\nMaths2_Gr7_LB_Book.indb   29\n2014/09/04   11:34:41 AM\n\n30\t\nMATHEMATICS Grade 7: Term 3\n7.\t Cardo’s teacher writes on the board: “Add 2 and then multiply the answer by 3.”  \nThe class must use 5 as an input number and apply the computational instruction.\n\t\n(a)\t Cardo uses 5 as the input number and writes: (5 + 2) × 3.\n\t\n\t\nPaul says (5 + 2) × 3 is 7 × 3 which is 21. Is Paul right?  \n\t\n(b)\t Explain your answer in (a).\n\t\n(c)\t Represent this flow diagram as an algebraic expression:\n\t\n\t\n \n+ 2\n× 3\nx\n8.\t Express each computational instruction as a flow diagram and then write the \nabbreviation (algebraic expression) with x as input number:\n\t\n(a)\t Multiply by 4 and then subtract 8.\n \n\t\n(b)\t Subtract 8 and then multiply by 4.\t\n \n\t\n(c)\t Add 15 and then divide by 5.\n \n\t\n(d)\t Divide by 5 and then add 15.\n \n9.\t Describe each computational instruction in words:\n\t\n(a)\t  \n× 4\n+ 7\n\t\n(b)\t  \n+ 7\n× 4\n\t\n(c)\t  \n× 9\n– 5\n\t\n(d)\t  \n– 5\n× 9\n10.\tTwo algebraic expressions are given in the table. Use the given input values  \n(x values) to determine the corresponding output values.\nx\n1\n2\n3\n4\n5\n6\n6 × x + 8\n14\n20\n26\n2 × x × (3 + 4)\nMaths2_Gr7_LB_Book.indb   30\n2014/09/04   11:34:42 AM\n\n\t\nCHAPTER 3: ALGEBRAIC EXPRESSIONS 1\t\n31\n3.2\t Relationships represented in formulae\nmaking sense of variables and constants in formulae\n1.\t (a)\t Chris uses the formula P = 2 × l + 2 × b to calculate the perimeters of rectangles  \n\t\nof differing lengths and breadths as indicated in the table. He also calculates the \n\t\narea of each rectangle using the formula A = l × b.\n\t\n\t\nComplete the table.\nRectangle\n1\n2\n3\n4\nLength (l)\n24\n6\n8\n12\nBreadth (b)\n1\n4\n3\n2\nPerimeter  \nP = 2 × l + 2 × b\nArea   \nA = l × b\n\t\n(b)\t Rita calculates the perimeter of a rectangle in a different way. She adds the value \n\t\nof the length of the rectangle to the value of the breadth of the rectangle and \n\t\nthen multiplies the answer by 2. \n\t\n\t\nWrite down the formula that Rita uses to calculate the perimeter of each  \n\t\nrectangle. Test whether or not Rita’s formula produces the same results as Chris’s. \n\t\nQuestions 1(c) to (e) refer to the formula P = 2 × (l + b).\n\t\n\t\n(c)\t What does the number 2 represent in the formula?\n\t\n(d)\t What is the number 2 called? \n\t\n(e)\t Which letter symbols represent variables in the formula P = 2 × (l + b)? Explain.\n\t\n(f)\t What can you say about the area of all of these rectangles?\n2.\t Sindi calculates her father’s age by using the formula F = x + 37, where x is Sindi’s age. \nHer father passed away when Sindi was 43 years old. How old was he then?\nMaths2_Gr7_LB_Book.indb   31\n2014/09/04   11:34:42 AM\n\n32\t\nMATHEMATICS Grade 7: Term 3\n3.\t Jacob wants to buy the cheapest cell phone on the market. He has already saved  \nR45 and decides to save R5 per week until he has enough money to buy the phone. \nThe formula y = 45 + 5 × w gives the amount of money (in rands) that Jacob has  \nsaved to buy the cell phone after w weeks.\n\t\n(a)\t Complete the table. The first row has been done as an example.\nNumber of \nweeks (w)\nHow to calculate \n45 + 5 × w\nAmount saved (y)\n0\n45 + 5 × 0 = 45 + 0\n45\n1\n2\n4\n5\n\t\n\t\n(b)\t The cell phone that Jacob wants to buy costs R90. Will Jacob have saved  \n\t\nenough money to be able to buy the cell phone by the eighth week? Explain.\n\t\n(c)\t Complete the table.\nFormula: \ny = 45 + 5 × w\nExplanation\nWhich are constants in the \nformula?\nWhich letter symbols \nrepresent variable quantities \nin the formula? \n4.\t In each of the following formulae, identify the symbols that represent variables and \nconstants.\nSymbols for variable(s)\nConstant(s)\n(a)\ny = 5 × x + 7\n(b)\ny = 100 + x\n(c)\ny = x ÷ 5\n(d)\ny = 5 × x\n(e)\ny = 0,7 × x + 2,3\nMaths2_Gr7_LB_Book.indb   32\n2014/09/04   11:34:42 AM\n\nChapter 4\nAlgebraic equations 1\n\t\nCHAPTER 4: ALGEBRAIC EQUATIONS 1\t\n33\nIn this chapter you will learn about solving open number sentences (or equations) by \ninspection and by the trial and improvement method. You will also represent problem \nsituations by means of number sentences as well as analyse and interpret some number \nsentences.\n4.1\t Solving by inspection................................................................................................ 35\n4.2\t Solving by the trial and improvement method.......................................................... 36\n4.3\t Describing problem situations with equations........................................................... 39\nMaths2_Gr7_LB_Book.indb   33\n2014/09/04   11:34:42 AM\n\n34\t\nMATHEMATICS Grade 7: Term 3\nMaths2_Gr7_LB_Book.indb   34\n2014/09/04   11:34:42 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n35\n\t\nCHAPTER 4: ALGEBRAIC EQUATIONS 1\t\n35",
      "chapter_id": "3"
    },
    {
      "title": "Algebraic equations 1",
      "content": "4\t Algebraic equations 1\n4.1\t Solving by inspection\nnumber puzzles\nSolve these number puzzles.\n1.\t I am thinking of a certain number. If I add 3 to that number, the answer is 13.  \nWhat is the number?\n2.\t I am thinking of a certain number. If I multiply that number by 5, the answer is 30. \nWhat is the number? \n3.\t I am thinking of a certain number. If I multiply that number by 3 and then add 4 to \nthe result, the answer is 19.\n\t\n(a)\t Is the number 3? Give a reason for your answer.\n\t\n(b)\t Is the number 4? Give a reason for your answer.\n\t\n(c)\t Is the number 5? Give a reason for your answer.\n\t\n(d)\t Is the number 6? Give a reason for your answer.\nNumber puzzles like those above can be shortened \nby using letter symbols as place holders for unknown \nnumbers. In the case of question 1 we can write the \nfollowing number sentence: x + 3 = 13.\nIn the case of a number sentence such as x + 3 = 13 \nwe cannot say whether it is true or false until we have \ndetermined the value of the unknown. The value of the \nunknown that makes the number sentence (an equation) \ntrue is called the solution of the number sentence.\nFor the number sentence x + 3 = 13, the solution is \nx = 10 because it makes the number sentence true.\nA mathematical statement \nsuch as x + 3 = 13 that could \nbe true or false depending \non the value of x, is called an \nopen number sentence or \nan equation. \nTo make a number  \nsentence true means to  \nfind its solution.\nMaths2_Gr7_LB_Book.indb   35\n2014/09/04   11:34:42 AM\n\n36\t\nMATHEMATICS Grade 7: Term 3\nthe solution is there to see\nThe solution to the number sentence x + 4 = 20 can be seen at once. The value of x is \n16 simply because 16 + 4 = 20. In this case, we say we solve the number sentence by \ninspection.\nSolve these number sentences (equations) by inspection.\n1.\t (a)\t x − 8 = 8\t\n(b)\t x + 7 = 20\n\t\n(c)\t 16\nx  = 8\t\n(d)\t\nx\n16 = 2\n\t\n(e) \t 5 × x = 40\t\n(f)\t 8 × x = 40\n2.\t (a)\t 84 ÷ x = 7\t\n(b)\t 36 ÷ x = 4\n\t\n(c)\t x + 56 = 100\t\n(d)\t 100 − x = 56\n4.2\t Solving by the trial and improvement method\nSometimes you cannot see the solution of a number sentence (an equation) at once. \nLook at the following number puzzle or equation, for example: \nI am thinking of a number. 6 × the number − 11 = 43. What is the number?\nIn this case, you will have to try many different possible solutions until you identify \nthe correct one. Here we can use a method known as trial and improvement to \ndetermine the solution. It is shown in the table below.\nPossible solution\nTest\nConclusion\nTry 5\n6 × 5 − 11 = 30 − 11 = 19\n5 is too small\nTry 10\n6 × 10 − 11 = 60 − 11 = 49\n10 is too big\nTry 8\n6 × 8 − 11 = 48 − 11 = 37\n8 is too small\nTry 9\n6 × 9 − 11 = 54 − 11 = 43\n9 is the solution\nMaths2_Gr7_LB_Book.indb   36\n2014/09/04   11:34:43 AM\n\n\t\nCHAPTER 4: ALGEBRAIC EQUATIONS 1\t\n37\nSolve the following equations by means of the trial and improvement method. In each \ncase, the solution is a number between 1 and 20.\n1.\t 2 × x + 13 = 37\t\nThe solution is x = \nPossible solution\nTest\nConclusion\n2.\t 14 × x − 21 = 77\t\nThe solution is x = \nPossible solution\nTest\nConclusion\n3.\t 7 × x + 8 = 71\t\nThe solution is x = \nPossible solution\nTest\nConclusion\n4.\t 4 × x + 7 = 31\t\nThe solution is x = \nPossible solution\nTest\nConclusion\nMaths2_Gr7_LB_Book.indb   37\n2014/09/04   11:34:43 AM\n\n38\t\nMATHEMATICS Grade 7: Term 3\n5.\t 10 × x + 11 = 141\t\nThe solution is x = \nPossible solution\nTest\nConclusion\nsolving by inspection or trial and improvement\nSolve the following equations by inspection or by the trial and improvement method:\n1.\t (a)\t x + 5 = 2 × x\t\n(b)\t k × 5 = 20 + k\n\t\n(c)\t 2 × q = 18 − q\t\n(d)\t 3 × t = t + 22\n2.\t (a)\t y + 6 = 4 × y\t\n(b)\t 5 × p = 18 + 2 × p\n\t\n(c)\t 4 × z = 18 + z\t\n(d)\t x × 5 = 20\n\t\n(e)\t 42 ÷ m = 35 − 29\t\n(f)\t 3 × x − 2 = x + 6\nMaths2_Gr7_LB_Book.indb   38\n2014/09/04   11:34:43 AM\n\n\t\nCHAPTER 4: ALGEBRAIC EQUATIONS 1\t\n39\n4.3\t Describing problem situations with equations\nfrom words to equations\nWrite an equation using a letter symbol as a placeholder for the unknown number to \ndescribe the problem in each of the situations below.\n1.\t There are 30 learners in a class. x learners are absent and 19 are present.\n2.\t There are 70 passengers on a bus. At a bus stop m passengers get off. There are now  \n23 passengers on the bus. \n3.\t A boy buys a bicycle for R1 260 on lay-by. How many payments of R90 each must he \nmake to pay for the bicycle? Let x be the number of payments to be made.\n4.\t Five people share a total cost of R240 equally amongst themselves. Let c be the cost \nper person.\n5.\t A school charges R100 a day for the use of its training facilities for athletes plus  \nR30 per athlete per day for food and use of equipment. A team of athletes paid  \nR400 for a day’s practice. Let x be the number of athletes attending the training.\n6.\t Bennie has R54 with which to buy chocolate for his friends. Each chocolate costs R6. \nHow many chocolates can he buy for that amount? Let x be the number of chocolates \nthat Bennie can buy.\n7.\t Write an equation to calculate the area of a rectangle with length 2,5 cm and breadth \n2 cm. Let A represent the area of the rectangle.\n8.\t There are 38 girls in Grade 7. This is 6 more than double the number of boys.\n9.\t Janine is 12 years old. Her father’s age is 7 years plus three times Janine’s age.\nMaths2_Gr7_LB_Book.indb   39\n2014/09/04   11:34:43 AM\n\n40\t\nMATHEMATICS Grade 7: Term 3\nmaking sense of equations\n1.\t Rajbansi Taxi Service charges R10 per kilometre travelled and a standard charge of  \nR30 per trip. Consider the equation below about a taxi trip: \n10 × t + 30 = 80\n\t\n(a)\t Explain what each number and letter symbol stands for in the equation.\n\t\n(b)\t Why is t multiplied by 10 in the equation?\n2.\t The cost of an adult’s ticket for a music concert is four times the cost of a child’s \nticket. An adult’s ticket costs R240. The equation below represents this problem: \n4 × x = 240\n\t\n(a)\t What does x represent?\t\n\t\n(b)\t Why is x multiplied by 4?\n\t\n(c)\t Solve the equation by inspection.\n\t\n(d)\t How much does a child’s ticket cost? \t\n3.\t There are 12 eggs in a carton. Consider the equation below: \n\t\n12 × c = 72\n\t\n(a)\t What does the letter symbol c represent in the equation?\n\t\n(b)\t What value of c makes the equation true?\t\n\t\n(c)\t What does the number 72 represent?\nMaths2_Gr7_LB_Book.indb   40\n2014/09/04   11:34:43 AM\n\nChapter 5\nGraphs\n\t\nCHAPTER 5: GRAPHS\t\n41\nIn this chapter, you will get familiar with a new kind of graph: the line graph. A line  \ngraph shows how the change in one variable affects another variable. You will specifically \ndeal with global graphs. These graphs show visually how variables vary, focusing on trends \nrather than detailed readings.\n5.1\t A graph can tell a story............................................................................................. 43\n5.2\t Investigating rate of change in situations.................................................................. 44\n5.3\t Interpreting graphs................................................................................................... 47\n5.4\t Drawing graphs........................................................................................................ 55\nMaths2_Gr7_LB_Book.indb   41\n2014/09/04   11:34:43 AM\n\n42\t\nMATHEMATICS Grade 7: Term 3\nMaths2_Gr7_LB_Book.indb   42\n2014/09/04   11:34:44 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n43\n\t\nCHAPTER 5: GRAPHS\t\n43",
      "chapter_id": "4"
    },
    {
      "title": "Graphs",
      "content": "5\t Graphs\n5.1\t A graph can tell a story\n1.\t Jena drew this graph to show how her feelings of hunger changed during the day. \nDescribe in a short paragraph how her day went as far as need for food is concerned.\nVery hungry\nNot hungry\nHunger\nTime of day\n2.\t Think about a specific day and things that happened to you on that day. Draw a \ngraph to show how your feelings changed during that day.\nHappy\nUnhappy\nFeelings\nTime of day\nMaths2_Gr7_LB_Book.indb   43\n2014/09/04   11:34:44 AM\n\n44\t\nMATHEMATICS Grade 7: Term 3\n5.2\t Investigating rate of change in situations\ncompare situations and represent them in a different way\n1.\t Consider the situations in (a) and (b) below and complete the tables to represent the \nrelationships.\n\t\n(a)\t Sally is saving money to buy a CD that she badly wants. She saves R4 per week.\nNumber of weeks\n1\n2\n3\n4\n5\n6\n7\n8\nMoney saved in rands\n4\n8\n\t\n(b)\t Nathi has a box of 24 chocolates. He is thinking about sharing the chocolates \n\t\nequally between different numbers of friends and is working out how many  \n\t\nchocolates each friend would get.\nNumber of friends\n1\n2\n3\n4\n5\n6\n7\n8\nChocolates per friend\n24\n12\n\t\n(c)\t On the grids below, draw bar graphs to represent the relationships in the  \n\t\nsituations described in (a) and (b). The length of each bar should represent  \n\t\nan output number.\nNumber of weeks\n1\n2\n3\n4\n5\n6\n7\n8\n10\n20\n30\nMoney saved (R)\nNumber of friends sharing\n1\n2\n3\n4\n5\n6\n7\n8\n10\n20\n30\nNumber of chocolates per friend\nMaths2_Gr7_LB_Book.indb   44\n2014/09/04   11:34:45 AM\n\n\t\nCHAPTER 5: GRAPHS\t\n45\n2.\t Consider the situations in (a) and (b) below and complete the tables to represent the \nrelationships.\n\t\n(a)\t Vusi collects acorns for a pig farmer who pays him per bag. Vusi thinks: “I wish  \n\t\nMr Bengu would agree to pay me R1 for the first bag of acorns, R2 for the second \n\t\nbag, R4 for the third bag and then keep on doubling the money for every bag.” \nNumber of bags\n1\n2\n3\n4\n5\n6\n7\nPayment (R)\n1\n2\n4\n8\n\t\n(b)\t Judy works out the areas of squares with different side lengths.\nSide length of square (cm)\n1\n2\n3\n4\n5\n6\n7\nArea of square (cm2)\n1\n4\n\t\n(c)\t On the grids below, draw bar graphs to represent the relationships in the situations \n\t\ndescribed in (a) and (b). The length of each bar should represent an output number.\nNumber of bags\n1\n2\n3\n4\n5\n6\n7\n20\n40\n60\nPayment (R)\nArea of square (cm2)\n1\n2\n3\n4\n5\n6\n7\nSide length of square (cm)\n10\n30\n50\n20\n40\n60\n10\n30\n50\nMaths2_Gr7_LB_Book.indb   45\n2014/09/04   11:34:45 AM\n\n46\t\nMATHEMATICS Grade 7: Term 3\n3.\t The input numbers for the different relationships in questions 1 and 2 are the same, \nbut the output numbers differ. Describe how the output numbers change in each of \nthe four situations.\n4.\t Describe briefly how the shape of the bar graphs differ.\n5.\t Turn back to the tables of values that you made for the four relationships in  \nquestions 1 and 2. Find out how the output values changed by calculating the \ndifferences between consecutive output values:\n\t\n(a)\t Sally’s savings:  4        8       12       16       \n\t\n\t\n4\t\n4\t\n\t\n\t\nSally’s savings grow by \nevery week.\n\t\n(b)\t Chocolates per friend:  24       12       8       \n\t\n\t\n12\t\n\t\n\t\nThe number of chocolates per friend \n\t\n(c)\t Vusi’s payment per bag:    1        2         \t\n\t\n\t\n1\t\n\t\n\t\nThe amount grows slowly at first and then \n\t\n(d)\t Areas of Judy’s squares:       1        4         9        \n\t\n\t\n3\t\n\t\n\t\nThe area of the squares \nMaths2_Gr7_LB_Book.indb   46\n2014/09/04   11:34:47 AM\n\n\t\nCHAPTER 5: GRAPHS\t\n47\nThe amount of money that Sally saves per week \nstays constant. The relationship therefore has a \nconstant rate of change. \nWith every additional friend, the number of \nchocolates per friend changes. The number of \nchocolates changes (decreases) rapidly at first \nand then it changes slower. The rate of \nchange is not constant. \nThe amount that Vusi would like to earn per bag \nof acorns grows faster and faster with each bag \nthat he collects. The rate of change increases.\nThe area of a square also increases faster and \nfaster for every cm added to the side length.\n6.\t Refer to the bar graphs in questions 1 and 2 and link the shape of the graphs to the \nrate of change of the relationship.\n5.3\t Interpreting graphs\nreading graphs\n1.\t Look carefully at the graph. \n\t\n(a)\t What does this graph tell you?\n\t\n(b)\t Explain your answer in (a).\nThe rate of change means \nhow fast or slow change  \nhappens per unit of time.\nThe shape of a bar graph shows \nthe rate of change of the  \nrelationship. If the rate of \nchange is constant, the shape  \nis a straight line. If it is  \nchanging, the shape is a curve. \nNumber of vehicles\nTime of day\nTraffic on the R44 (weekdays)\n12:00\n14:00\n16:00\n18:00\n20:00\n04:00\n06:00\n08:00\n10:00\n22:00\nMaths2_Gr7_LB_Book.indb   47\n2014/09/04   11:34:47 AM\n\n48\t\nMATHEMATICS Grade 7: Term 3\n2.\t Mr Thatcher bought three plants in containers. The salesman at the nursery told him \nthat one of the plants, Glamiolus, grows at a constant rate. The second plant, Bouncy \nBess, grows slowly at first but then grows faster and faster. The salesman was not sure \nabout the rate at which the third plant, Samara, grows.\n\t\n(a)\t What does “grows at a constant rate” mean?\n\t\nMr Thatcher measured the three plants every week and recorded the heights in a \ntable, given below. \n\t\n(b)\t Calculate the differences in height from week to week, to find the rate at which  \n\t\neach plant grows per week.\nWeek\nHeight of A (cm)\nHeight of B (cm)\nHeight of C (cm)\n1\n6\n8,3\n10,1\n2\n6,3\n10,2\n10,6\n3\n6,4\n12,2\n11,2\n4\n7,2\n14,1\n11,9\n5\n7,3\n16,2\n12,8\n6\n7,4\n18,3\n13,9\n7\n9,1\n20,2\n15,8\n\t\n(c)\t Identify the plants. Which plant is plant A, which plant is plant B and which  \n\t\nplant is plant C? Explain how you got your answers.\nMaths2_Gr7_LB_Book.indb   48\n2014/09/04   11:34:47 AM\n\n\t\nCHAPTER 5: GRAPHS\t\n49\n\t\n(d)\t The three graphs below show the growth of the three plants. Which graph  \n\t\nbelongs to which plant? Explain.\nA\nB\nC\nrevisiting change and rate of change\nIn section 5.2 you compared the way in which relationships changed. \n1.\t Consider the following situations: \n\t\nBen saves R5 per week. Sally saves R7 per week.  \n\t\nCharlie saves R5 in the first week, R6 in the second week and R7 in the third week. \n\t\nEvery week he increases the amount that he saves by R1.\n\t\n(a)\t The graph shows Ben’s savings. Draw Sally’s savings and Charlie’s savings.  \n\t\nDo it on the same sketch.\nBen\n\t\n(b)\t Describe and explain the shape of the graphs showing Sally, Ben and Charlie’s  \n\t\nsavings.\nThe rate of change in a relationship influences the \nsteepness of the graph. The higher the rate of \nchange (i.e. the faster the output numbers change), \nthe steeper the graph.\nMaths2_Gr7_LB_Book.indb   49\n2014/09/04   11:34:47 AM\n\n50\t\nMATHEMATICS Grade 7: Term 3\n2.\t Examine the following relationships. Complete the tables and calculate the \ndifferences between the output numbers indicated with arrows.\n\t\n(a)\t Christine wants to buy a book for her favourite teacher. The book costs R240.  \n\t\nThis is a lot of money for Christine to spend. She realises that if she asks her  \n\t\nfriend Beatrice to share the cost, she will have to spend only R120. She could  \n\t\neven ask more classmates to join in and share the cost. Christine investigates  \n\t\nthe situation and calculates what amount everyone must pay if they share the \n\t\ncost equally. \n\t\n\t\nNumber of learners \nsharing the cost\n1\n2\n3\n4\n6\n8\n10\n12\nAmount each learner  \nwill pay\n240\n120\n\t\n120\t\n           \n\t\n(b)\t Investigate the relationship between the length of a side of a square and the  \n\t\nperimeter of the square. \n\t\n\t\nLength of a side of  \nthe square\n1\n2\n3\n4\n6\n7\n8\n9\nPerimeter of the \nsquare\n4\n8\n12\n\t\n\t\n(c)\t Investigate the relationship between the length of a side of a square and the  \n\t\narea of the square. \nLength of a side of  \nthe square\n1\n2\n3\n4\n6\n7\n8\n9\nArea of the \nsquare\n1\n4\n9\n\t\n\t\n(d)\t A tall candle was lit and its length was measured and recorded every hour  \n\t\nwhile it was burning. \nNumber of hours the \ncandle is burning\n1\n2\n3\n4\n6\n7\n8\n9\nLength of candle  \nin cm\n33\n31\n29\n\t\nMaths2_Gr7_LB_Book.indb   50\n2014/09/04   11:34:49 AM\n\n\t\nCHAPTER 5: GRAPHS\t\n51\n3.\t Match each of the graphs below to one of the situations in question 2.\nA\nB\nC\nD\n\t\nWrite the letter of the graph next to the description of the situation:\n\t\n(a)\t Buying a book for the teacher \n\t\n(b)\t Length of a side of a square and the perimeter of the square \n\t\n(c)\t Length of a side of a square and the area of the square \n\t\n(d)\t Length of the candle and the number of hours it is burning \nWhen we investigate the growth (or change) in \na relationship, we look at the way the output \nnumbers change.\nThe change can be:\n• an increase or a decrease\n• a constant increase, for example the perimeter of a square as the side length  \nincreases\n• a constant decrease, for example the length of a burning candle\n• an increase that is not constant but happens faster and faster, for example the area  \nof a square as the side length increases\n• a decrease that is not constant but happens faster and faster, for example the  \namount of money each friend has to pay as more and more friends share the cost.\nIn the case of an increase, the graph slopes like this:                 or              or\nIn the case of a decrease, the graph slopes like this:                   or               or\nWhen the increase or decrease is constant, the \ngraph is a straight line and it is called a linear \ngraph. \nIf the increase or decrease is not constant, the \ngraph is curved and is called a non-linear graph.\nIf there is no change in the output variable, the \ngraph is a straight horizontal line.\nMaths2_Gr7_LB_Book.indb   51\n2014/09/04   11:34:49 AM\n\n52\t\nMATHEMATICS Grade 7: Term 3\n4.\t Consider the graphs in question 3 on the previous page. \n\t\n(a)\t Which graphs indicate a linear increase or decrease?\n\t\n(b)\t Which graphs indicate a decrease or increase which is not constant?\n5.\t Peter’s father drives him to school in the mornings. Below is a graph of their journey \nto school. Describe the story that the graph tells. What do you know about the route \nthat they are taking?\nGoing to school: speed changes\nCar‘s speed (in km/h)\nDistance from Peter‘s home (in km)\nA\nB C D\nE\nH\nG\nF\n10\n20\n30\n40\n50\n60\n6\n5\n4\n3\n2\n1\n0\n6.\t Consider the graph in question 5 above. Identify the parts of the graph that are \nincreasing, decreasing and constant.\n\t\n0 to A:\t\n\t\nA to B:\t\n\t\nB to C:\t\n\t\nC to D:\t\n\t\nD to E:\t\n\t\nE to F:\t\n\t\nF to G:\t\n\t\nG to H:\t\n\t\nH to 6:\t\nMaths2_Gr7_LB_Book.indb   52\n2014/09/04   11:34:49 AM\n\n\t\nCHAPTER 5: GRAPHS\t\n53\nexploring more graphs \n1.\t Janet takes a bath. The graph below shows the height of the water level in the bathtub \nas time passes. The water runs into the bath at a constant rate. Study the graph and \ndescribe what happens.\nWater level\nTime in minutes\nA\nBC\nD\nE\n2.\t The axes of the graph below are not labelled. \n\t\n(a)\t Which of the following sets of labels could fit the graph?\n\t\n\t\nA:\t vertical axis: time passed; horizontal axis: distance from home\n\t\n\t\nB:\t vertical axis: distance from home; horizontal axis: time passed\n\t\n\t\nC:\t vertical axis: rainfall; horizontal axis: temperature\n\t\n(b)\t Describe the story told by the graph, with the axes that you chose.\nThe vertical axis is the one \nthat goes from bottom to top. \nThe horizontal axis is the \none that goes from left to right. \n(Axes is the plural of axis.)\nMaths2_Gr7_LB_Book.indb   53\n2014/09/04   11:34:49 AM\n\n54\t\nMATHEMATICS Grade 7: Term 3\n3.\t The graph below shows the distance that three athletes, A, B and C, covered in a \nhurdles race in a certain time. \n\t\n(a)\t Describe what happened during the race.\nA hurdles race\nDistance (metres)\nTime (seconds)\n60\n400\nA\nB\nC\n\t\n(b)\t How far was the race? \n\t\n(c)\t Which of the athletes, A, B or C, won the race? \n\t\n(d)\t Did the best athlete of the three win? Explain your answer.\n4.\t Identify the graphs (or parts of a graph) in questions 1, 2 and 3 above that are linear \nand those that are non-linear.\nMaths2_Gr7_LB_Book.indb   54\n2014/09/04   11:34:49 AM\n\n\t\nCHAPTER 5: GRAPHS\t\n55\n5.4\t Drawing graphs\n1.\t Water is dripping at a constant rate into three containers, A, B and C, shown  \nbelow. Draw graphs to show how the height of the water in each container will  \nvary with time.\nA\nB\nC\nTime\nHeight of water\nTime\nHeight of water\nHeight of water\nTime\n2.\t Draw a graph showing the height of the water level in the swimming pool below  \nif the pool is filled with a constant stream of water.\nCross-section of swimming pool\n \nWater level\nTime\nMaths2_Gr7_LB_Book.indb   55\n2014/09/04   11:34:50 AM\n\n56\t\nMATHEMATICS Grade 7: Term 3\n3.\t Draw a graph of the speed of a racing car as it travels once around the track shown \nbelow. S is the starting point.\nSpeed\nDistance\nS\n4.\t The Western Cape gets rain during the winter months, but in summer it is usually \ndry. Draw a global graph of the average rainfall in the Western Cape during one year. \nRainfall in mm\nMonths\n5.\t Draw a graph of the following story:\n\t\nDuring a rainstorm, Lydia put a measuring cup outside to measure the rainfall.  \nAfter 10 minutes of hard rain the water level was 10 mm. It started to rain softer,  \nand after 20 more minutes the water level was 15 mm. \n\t\nWhen Lydia went back 10 minutes later, the level was 30 mm. An hour after the \nstorm started, the water level was still 30 mm. \nTime in minutes\nWater level (mm)\n10\n20\n30\n40\n50\n60\n5\n10\n15\n20\n25\n30\n35\n40\n0\nMaths2_Gr7_LB_Book.indb   56\n2014/09/04   11:34:50 AM\n\nChapter 6\nTransformation geometry\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n57\nIn this chapter, you will revise the property of symmetry and practise identifying lines of \nsymmetry in geometric figures. You will then investigate how figures can be reflected, \nrotated or translated, while the size and shape of the original figure remains the same. You \nwill also investigate how we can change the size of a figure, but still keep the angles of the \nfigure the same, to produce enlarged or reduced similar figures. In such figures, you will \nwork out the factor by which the original figure was resized.\n6.1\t Lines of symmetry..................................................................................................... 59\n6.2\t Original figures and their images.............................................................................. 62\n6.3\t Translating figures..................................................................................................... 62\n6.4\t Reflecting figures...................................................................................................... 66\n6.5\t Rotating figures........................................................................................................ 71\n6.6\t Enlarging and reducing figures................................................................................. 75\nMaths2_Gr7_LB_Book.indb   57\n2014/09/04   11:34:50 AM\n\n58\t\nMATHEMATICS Grade 7: Term 3\nMaths2_Gr7_LB_Book.indb   58\n2014/09/04   11:34:51 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n59\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n59",
      "chapter_id": "5"
    },
    {
      "title": "Transformation geometry",
      "content": "6\t Transformation geometry\n6.1\t Lines of symmetry\nwhat is the line of symmetry?\nIn the diagrams below, the red dotted lines divide the arrows into two parts. In which \ndiagram does the red dotted line divide the arrow into two parts that are exactly the \nsame?\nArrow A\nArrow B\nIf you were to cut out arrow A and fold it along the \nred dotted line, the two parts would fit perfectly on \ntop of one another (all edges would match). The \nfold line is called a line of symmetry or an axis \nof symmetry.\nA line or axis of symmetry is a line that divides a \nfigure into two parts that have an equal number of \nsides, and all the corresponding sides and angles \nare equal. The two parts on either side of the line of \nsymmetry are mirror images of each other. We also \nsay the parts are congruent.\nA geometric figure can have no line of  \nsymmetry, one line of symmetry, or more than one \nline of symmetry.\nCongruent figures are \nfigures that are the same size \nand shape. All the sides and \nangles of the figures match.\nMaths2_Gr7_LB_Book.indb   59\n2014/09/04   11:34:51 AM\n\n60\t\nMATHEMATICS Grade 7: Term 3\nidentifying lines of symmetry\n1.\t (a)\t Make a tick next to each figure in which the red line is a line of symmetry.\n\t\n(b)\t In the figures where the red line is not a line of symmetry, draw in a line of  \n\t\nsymmetry if this is possible. If there is more than one line of symmetry, \tdraw it  \n\t\nin too. If a figure doesn’t have any lines of symmetry, write this above the figure.\n\t\nA\t\nB \nC \nD\t\nE \nF \nMaths2_Gr7_LB_Book.indb   60\n2014/09/04   11:34:51 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n61\n2.\t Draw lines of symmetry in the following geometric figures. Also write down how \nmany lines of symmetry there are in each figure.\nA\nB\nC\nD\nE\nF\nMaths2_Gr7_LB_Book.indb   61\n2014/09/04   11:34:52 AM\n\n62\t\nMATHEMATICS Grade 7: Term 3\n3.\t In each diagram, the dotted line is the axis of symmetry. Complete each figure.\n\t\nA                   \t\nB\n6.2\t Original figures and their images\nFigures can be moved around in different ways – they can be shifted, swung around  \nand turned over. When the movement is done, the figure in its new position is called  \nthe image of the original figure. \nFigures can be moved in three ways: through \ntranslation, reflection and rotation. These \ntransformations are often referred to as “sliding” \n(shifting), “flipping” (turning over) and “turning” \n(swinging) respectively.\n6.3\t Translating figures\nHere are two original figures and their images after the figures were translated:\nOriginal\nImage\nOriginal\nImage\nL\nK\nD\nD'\nJ'\nK'\nJ\nE\nE'\nL'\nG\nG'\nF\nF'\nMaths2_Gr7_LB_Book.indb   62\n2014/09/04   11:34:52 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n63\nWhen we name the image, we use the same letters for the points that correspond to \nthose of the original figure, but we add the prime symbol ( ' ) after each letter. The image \nof ∆JKL is ∆J'K'L'. The image of parallelogram DEFG is parallelogram D'E'F'G'.\ninvestigating the properties of translation\nIn a translation, all the points on the figure move in the same direction by the same \ndistance. For example, look at ∆JKL on the previous page. All of its points have moved  \n6 units to the right. Also look at parallelogram DEFG on the previous page. All of its \npoints have moved 3 units to the right and 5 units down.\n1.\t Look at ∆ABC below. \n\t\n(a)\t Translate each of the points A, B and C 5 units to the right and 2 units down.  \n\t\nThen join the translated points to form the image ∆A'B'C'.\nA\nC\nB\n\t\nLook at the completed translation. \n\t\n(b)\t Are the side lengths of the original triangle and those of its image the same? \n\t\n(c)\t Is the area of the original triangle the same as the area of its image? \t\nMaths2_Gr7_LB_Book.indb   63\n2014/09/04   11:34:52 AM\n\n64\t\nMATHEMATICS Grade 7: Term 3\n2.\t Look at ∆PQR below.\n\t\n(a)\t Translate each of the points P, Q and R 4 units to the right and 2 units up. \n\t\nThen join the translated points to form the image ∆P'Q'R'.\nP\nQ\nR\nOriginal\n\t\n(b)\t Join point P and its image, point Q and its image, and point R and its image.\n\t\n(c)\t Are the line segments that join the original points to their image points equal  \n\t\nin length?\n\t\n(d)\t Are the line segments that join the original points to their image points  \n\t\nparallel? \nProperties of translation\nUse the diagram on the right to check if the \nfollowing is true:\n• The line segments that connect the vertices \nof the original figure to those of the image \nare all equal in length: \nPP' = RR' = QQ'\n• The line segments that connect the vertices \nof the original figure to those of the image \nare all parallel to one another: \nPP' || RR' || QQ'\n• When a figure is translated, its shape and  \nsize do not change. The original and its  \nimage are therefore congruent.\nP\nQ\nR\nOriginal\nP'\nQ'\nR'\nImage\nMaths2_Gr7_LB_Book.indb   64\n2014/09/04   11:34:53 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n65\npractise translating figures\n1.\t Translate the following figure 8 units to the left and 2 units down.\nM\nL\nN\nJ\nI\nK\n2.\t Translate the following figure 6 units to the right and 1 unit down.\nA\nB\nC\nD\nE\nF\nG\n3.\t Describe the translation in each of the following diagrams:\n\t\n(a)\t\nOriginal\nImage\nP\nQ\nR\nS\nP'\nQ'\nR'\nS'\nMaths2_Gr7_LB_Book.indb   65\n2014/09/04   11:34:53 AM\n\n66\t\nMATHEMATICS Grade 7: Term 3\n\t\n(b)\t\n\t\n(c)\t\nOriginal\nImage\nQ\nS\nP\nR\nS'\nR'\nP'\nQ'\nOriginal\nImage\nB\nD\nA\nC\nB'\nD'\nA'\nC'\n6.4\t Reflecting figures\nWhen a figure is reflected, it is flipped or turned over. The image that is produced is the \nmirror image of the original figure. The line of reflection is like a mirror in which the \noriginal figure is reflected.\nThe image is produced on the opposite side of the line of reflection. Each point on \nthe original figure and its corresponding point on the image are the same distance away \nfrom the line of reflection.\ninvestigating the properties of reflection\nThe diagrams below and on the next page show examples of figures that have been \ncorrectly and incorrectly reflected in the lines of reflection.\nC\nB\nA\nA'\nA'\nC'\nA\nB\nB'\nB'\nC\nC'\nLine of reflection\nLine of reflection\nCorrect reflection\nIncorrect reflection\nMaths2_Gr7_LB_Book.indb   66\n2014/09/04   11:34:53 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n67\nF\nE\nD'\nD'\nE'\nD\nF'\nF'\nE'\nLine of reflection\nLine of reflection\nCorrect reflection\nIncorrect reflection\nF\nE\nD\nH\nK\nG'\nH'\nG'\nG\nH'\nK'\nK'\nCorrect reflection\nIncorrect reflection\nLine of reflection\nLine of reflection\nH\nK\nG\n1.\t Write down the distance from each of the following points to the line of reflection.\n \nOriginal figure\nCorrect reflection\nIncorrect reflection\nA: 2 units\nA': \nA': \nB: \nB': \nB': \nC: \nC': \nC': \nD: \nD': \nD': \nE: \nE': \nE': \nF: \nF': \nF': \nG: \nG': \nG': \nH: \nH': \nH': \nK: \nK': \nK': \nMaths2_Gr7_LB_Book.indb   67\n2014/09/04   11:34:54 AM\n\n68\t\nMATHEMATICS Grade 7: Term 3\n2.\t Look at each set of correct reflections.\n\t\n(a)\t Are the side lengths of the image the same as those of the original figure? \n\t\n(b)\t Are the size and shape of the image the same as the size and shape of the  \n\t\noriginal figure? \n3.\t (a)\t In each diagram showing the correct reflection, draw a dotted line to join each \n\t\npoint on the original figure to its corresponding reflected point (A to A', B to B',  \n\t\nC to C' and so on).\n\t\n(b)\t Is the line that joins the original point to its correct reflection perpendicular  \n\t\nto the line of reflection? \n4.\t (a)\t In each diagram showing the incorrect reflection, draw a dotted line to join  \n\t\neach point on the original figure to its corresponding reflected point.\n\t\n(b)\t Is the line that joins the original point to its incorrect reflection perpendicular  \n\t\nto the line of reflection?\nProperties of reflection\nThe diagram on the right shows ∆FHG and  \nits reflection ∆F'H'G'. Notice the following  \nproperties of reflection:\n• The image of ∆FHG lies on the opposite  \nside of the line of reflection.\n• The distance from the original point to  \nthe line of reflection is the same as the  \ndistance from the reflected point to the  \nline of reflection: GE = G'E; FC = F'C and  \nHD = H'D\n• The line that connects an original point to its \nimage is always perpendicular (⊥) to the line of \nreflection: HH' ⊥ line of reflection; FF' ⊥ line of \nreflection, and GG' ⊥ line of reflection.\n• When a figure is reflected, its shape and size do  \nnot change. The original and its image are \ntherefore congruent.\nOriginal\nLine of reflection\nF\nH\nG\nF'\nH'\nG'\nD\nC\nE\nImage\nMaths2_Gr7_LB_Book.indb   68\n2014/09/04   11:34:54 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n69\npractise reflecting figures\n1.\t Reflect the following figures in the given line of reflection. (Hint: First reflect the \npoints; then join the reflected points.)\n\t\n(a)\nG\nE\nLine of reflection\nC\nF\nD\nH\n\t\n(b)\nG\nE\nJ\nLine of reflection\nC\nK\nF\nD\nH\nI\nMaths2_Gr7_LB_Book.indb   69\n2014/09/04   11:34:54 AM\n\n70\t\nMATHEMATICS Grade 7: Term 3\n\t\n(c) \nN\nM\nO\nLine of reflection\nL\nK\n2.\t Draw the line of reflection.\n\t\n(a)\nF\nE\nD\nF'\nC'\nD'\nImage\nOriginal\nC\nG\nG'\nE'\n(b)\nI\nG\nH\nI'\nG'\nH'\nImage\nOriginal\n\t\n(c) \nImage\nOriginal\nA\nB\nD\nA'\nB'\nD'\nC\nC'\nMaths2_Gr7_LB_Book.indb   70\n2014/09/04   11:34:54 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n71\n6.5\t Rotating figures\nWhen a figure is rotated it is turned in a clockwise  \ndirection or in an anticlockwise direction around a  \nparticular point. This point is called the centre of  \nrotation and could be inside the figure or outside  \nof the figure. \nThe following diagrams show ∆ABC rotated \n90° clockwise and 90° anticlockwise about different  \ncentres of rotation.\nCentre of rotation is at C\nCentre of rotation is at B\nanticlockwise\n90° \nclockwise\n90° \nclockwise\n90° \nanticlockwise\n90° \nB\nA\nC\nB'\nA'\nC'\nB''\nA''\nC''\nB\nA\nC\nB'\nA'\nC'\nB''\nA''\nC''\nCentre of rotation is at A\nCentre of rotation is at P\nanticlockwise\n90° \nclockwise\n90° \nclockwise\n90° \nanticlockwise\n90° \nB\nA\nC\nB'\nA'\nC'\nB''\nA''\nC''\nB\nA\nC\nB'\nA'\nC'\nB''\nA''\nC''\nP\nClockwise\nAnticlockwise\nIn this case, about means \n“around”.\nMaths2_Gr7_LB_Book.indb   71\n2014/09/04   11:34:55 AM\n\n72\t\nMATHEMATICS Grade 7: Term 3\ninvestigating the properties of rotation\nIn the following diagrams, the centre of rotation is point A. ∆PRS has been rotated \nanticlockwise through 90° about point A. \n1.\t Lines have been drawn to join A to  \n\t\npoint S, and A to point S'.\n\t\n(a)\t Measure the distance from  \n\t\nA to S.\n\t\n(b)\t Measure the distance from  \n\t\nA to S'.\n\t\n(c)\t What do you notice about the  \n\t\ndistances in (a) and (b) above?\n\t\n(d)\t Measure the size of the angle  \n\t\nSAS'. What do you notice?\n2.\t Lines have been drawn to join  \n\t\nA to P, and A to P'.\n\t\n(a)\t Measure the distance from  \n\t\nA to P.\n\t\n(b)\t Measure the distance from  \n\t\nA to P'.\n\t\n(c)\t What do you notice about the  \n\t\ndistances in (a) and (b) above?\n\t\n(d)\t Measure the size of the angle  \n\t\nPAP'. What do you notice?\nP\nR\nS\nOriginal\nP'\nR'\nS'\nA\nCentre of \nrotation\nImage\nP\nR\nS\nOriginal\nP'\nR'\nS'\nA\nCentre of \nrotation\nImage\nMaths2_Gr7_LB_Book.indb   72\n2014/09/04   11:34:55 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n73\n3.\t Lines have been drawn to join A to R,  \n\t\nand A to R'.\n\t\n(a)\t Measure the distance from A to R.\n\t\n(b)\t Measure the distance from A to R'.\n\t\n(c)\t What do you notice about the  \n\t\ndistances in (a) and (b) above?\n\t\n(d)\t Measure the size of the angle RAR'.  \n\t\nWhat do you notice?\n4.\t In any of the diagrams in questions 1 to 3 above, measure the sides of the original \ntriangle and the corresponding sides of the image. What do you notice?\nProperties of rotation\n• The distance from the centre of rotation to  \nany point on the original is equal to the \ndistance from the centre of rotation to the \ncorresponding point on the image. In the \ndiagram on the right: PA = PA', PB = PB'  \nand PC = PC'.\n• The angle formed by the connecting lines \nbetween any point on the original figure, the \ncentre of rotation and the corresponding point \non the image is equal to the angle of rotation. \nFor example, if the image is rotated through  \n90°, this angle will be equal to 90°. If the  \nimage is rotated through 45°, the angle will  \nbe 45°.\n• When a figure is rotated, its shape and size do \nnot change.\nP\nR\nS\nOriginal\nP'\nR'\nS'\nA\nCentre of \nrotation\nImage\n(Centre of \nrotation)\nB\nA\nC\nC'\nA'\nB'\nP\nMaths2_Gr7_LB_Book.indb   73\n2014/09/04   11:34:55 AM\n\n74\t\nMATHEMATICS Grade 7: Term 3\npractise rotating figures\n1.\t Rotate triangle ∆ABC 90° clockwise  \n\t\nabout point P as follows:\n\t\n(a)\t Plot the image of each vertex on  \n\t\nthe grid. Remember:\n• The image point must be the  \nsame distance from P as the  \noriginal point.\n• The angle that is formed between  \nthe line connecting an original  \npoint to point P and the line  \nconnecting its image point to  \npoint P must be the same as the angle  \nof rotation. In this case, it must be 90°.\n\t\n(b)\t Join the image points to create ∆A'B'C'.\n2.\t Rotate KLMN 180° about point T.\nT\nK\nM\nN\nCentre of \nrotation\nL\n3.\t Rotate ∆XYZ 90° anticlockwise about point F.\nF\nY\nX\nZ\nCentre of rotation\nP\nA\nB\nCentre of rotation\nC\nMaths2_Gr7_LB_Book.indb   74\n2014/09/04   11:34:56 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n75\n6.6\t Enlarging and reducing figures\nEnlarging a figure means that we make it bigger in a specific way. Reducing a figure \nmeans that we make it smaller in a specific way. Enlarging or reducing figures is also \ncalled resizing.\ninvestigate the properties of enlargements and reductions\n1.\t Look at the following rectangles and answer the questions below.\nA\nB\nC\nE\nF\nG\nL\nK\nJ\nD\nH\nM\n\t\n(a)\t Rectangle EFGH:\n\t\n\t\nHow many times is FG longer than BC?\t\n\t\n\t\nHow many times is EF longer than AB?\t\n\t\n(b)\t Rectangle JKLM:\n\t\n\t\nHow many times is KL longer than BC?\t\n\t\n\t\nHow many times is JK longer than AB?\t\nWhen the lengths of all the sides of a figure are \nmultiplied by the same number to produce a \nsecond figure, the second figure is an enlargement \nor reduction of the first figure.\nThe number by which the sides are multiplied to \nproduce an enlargement or reduction is called the \nscale factor. The scale factor in question 1(b)  \nabove is 2. We say that figure ABCD has been  \nenlarged (or resized) by a scale factor of 2 to  \nproduce figure JKLM.\nFigure EFGH is not an enlargement of figure ABCD \nbecause not all its sides have been increased by the \nsame scale factor.\nMaths2_Gr7_LB_Book.indb   75\n2014/09/04   11:34:56 AM\n\n76\t\nMATHEMATICS Grade 7: Term 3\nThe scale factor\n• When the scale factor is 1, the image is the same \nsize as the original.\n• When the scale factor is <1, the image is a \nreduction. For example, if the scale factor is 1\n2 or \n0,5, each side of the image is half the length of its \ncorresponding side in the original figure.\n• When the scale factor is >1, the image is an \nenlargement. For example, if the scale factor is 2, \neach side of the image is double the length of its \ncorresponding side in the original figure.\n2.\t Look at the following triangles and answer the questions that follow.\nA\nB\nC\nE\nF\nG\nI\nK\nJ\n \n(a)\t How many times is:\n• FG longer than BC?\t\n  \n•  JK shorter than BC? \n• EF longer than AB?\t\n  \n•  IJ shorter than AB? \n• EG longer than AC?\t\n  \n•  IK shorter than AC? \n\t\n(b)\t Is ∆EFG an enlargement of ∆ABC? Explain your answer.\n\t\n(c)\t Is ∆IJK a reduction of ∆ABC? Explain your answer.\nSimilar figures\nWhen figures are enlarged or reduced, the \nenlarged or reduced image is similar to the \noriginal figure. ∆ABC, ∆EFG and ∆IJK above are \nall similar. We also say that the lengths of their \ncorresponding sides are in proportion.\nIf two or more figures are similar:\n• their corresponding angles are \nequal, and\n• their corresponding sides are \nlonger or shorter by the same \nscale factor.\nMaths2_Gr7_LB_Book.indb   76\n2014/09/04   11:34:56 AM\n\n\t\nCHAPTER 6: TRANSFORMATION GEOMETRY\t\n77\npractise resizing figures\n1.\t State whether the following scale factors will produce a larger or smaller image:\n\t\n(a)\t 5\t\n\t\n(b)\t 0,25\t\n\t\n(c)\t 1,2\t\n\t\n(d)\t 3\n8\t\n2.\t Enlarge the triangle below with a scale factor of 2.\n3.\t Resize the following figure. Use a scale factor of 0,5.\n4.\t Resize the figure below. Use a scale factor of 1\n3.\nMaths2_Gr7_LB_Book.indb   77\n2014/09/04   11:34:57 AM\n\n78\t\nMATHEMATICS Grade 7: Term 3\n5.\t (a)\t Which image below is similar to the original? \n\t\n(b)\t State the scale factor by which it has been resized. \nOriginal\nImage 1\nImage 2\n6.\t What scale factors were used to produce image 1 and image 2 from the original?\n\t\nImage 1: \n\t\nImage 2: \nOriginal\nImage 1\nImage 2\nMaths2_Gr7_LB_Book.indb   78\n2014/09/04   11:34:57 AM\n\nChapter 7\nGeometry of 3D objects\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n79\nIn this chapter, you will revise the work you have done on 3D objects in Grade 6. This \nincludes describing, sorting and comparing 3D objects by focusing on the number and \nshapes of their faces, their number of vertices and their number of edges. The 3D objects \nyou will work with in this chapter are cubes, rectangular prisms, triangular prisms, pyramids \nand cylinders.\nAfter classifying various 3D objects, you will build cardboard or paper models of  \ndifferent cubes and prisms. In order to do this, you will need to know how to draw nets  \n(or flat patterns) for these 3D objects. As you do the drawings of the nets of the 3D objects \nand as you construct the objects, you will find that you have to think carefully about which \nsides of the shapes in the net have to match up. This will give you clues about how long \neach of these sides has to be.\n7.1\t Classifying 3D objects............................................................................................... 81\n7.2\t Prisms and pyramids................................................................................................. 83\n7.3\t Describing, sorting and comparing 3D objects......................................................... 86\n7.4\t Nets of 3D objects.................................................................................................... 88\n7.5\t Using nets to construct cubes and prisms................................................................. 93\nMaths2_Gr7_LB_Book.indb   79\n2014/09/04   11:34:57 AM\n\n80\t\nMATHEMATICS Grade 7: Term 3\nMaths2_Gr7_LB_Book.indb   80\n2014/09/04   11:34:58 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n81\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n81",
      "chapter_id": "6"
    },
    {
      "title": "Geometry of 3D objects",
      "content": "7\t Geometry of 3D objects\n7.1\t\nClassifying 3D objects\nThere are two main groups of objects with three dimensions (length, width and \nheight), namely those with curved surfaces and those with flat surfaces. Spheres \n(balls), cylinders and cones are examples of objects with curved surfaces. Objects with \nonly flat surfaces are called polyhedra.\nExamples of objects with curved surfaces\t\nExamples of objects with flat surfaces only\nwhat is a polyhedron?\nA polyhedron is a three-dimensional object (or 3D object) made of flat surfaces only. It \nhas no curved surfaces. It consists of faces, edges and vertices.\nA face is the flat surface of a 3D object.\nAn edge is the segment where two faces of a \npolyhedron intersect. \nA vertex is the point where the edges meet.\n\t\nFaces\t\nEdges\t\nVertices\nWe say: one polyhedron; \ntwo or more polyhedra.\nWe say: one vertex; two or \nmore vertices.\nMaths2_Gr7_LB_Book.indb   81\n2014/09/04   11:34:58 AM\n\n82\t\nMATHEMATICS Grade 7: Term 3\nA cube has 6 faces, 12 edges and  \n8 vertices.\nThis pyramid has 5 faces, 8 edges and  \n5 vertices.\nidentifying and describing 3D OBJECTS\n1.\t Identify parts (a) to (d) on the figure correctly. \n\t\n(a)\t\n\t\n(b) \n\t\n(c)\t\n\t\n(d)\t\n2.\t Which of the following objects are polyhedra?\n\t\n(a)\t \t\n(b)\t \t\n(c)\n\t\n(d)\t \t\n(e)\t \t\n(f)\t \t\n3.\t How many faces, edges and vertices does each of the following polyhedra have?\n\t\n(a)\t \t\n(b)\t \t\n(c)\t\n\t\n\t\nFaces: \n\t\n\t\nFaces: \n\t\n\t\nFaces: \n\t\n\t\nEdges: \n\t\n\t\nEdges: \n\t\n\t\nEdges: \n\t\n\t\nVertices: \n\t\n\t\nVertices: \n\t\n\t\nVertices: \n4.\t Six learners each used play dough to make a 3D object. Use the descriptions on the \nnext page to match each 3D object to the learner who made it.\n\t\n(a)\t \t\n(b)\t \t\n(c)\n(a)\n(b)\n(c)\n(d)\nMaths2_Gr7_LB_Book.indb   82\n2014/09/04   11:34:59 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n83\n\t\n(d)\t \t\n(e)\t \t\n(f)\t \t\n• Tumi’s object has 6 vertices and 5 faces.\n• Debbie’s object has 8 vertices and 12 edges.\n• Brad made an object that has 7 faces and 10 vertices.\n• Xola made an object with no vertices.\n• Mpuka’s object has 8 edges and 5 faces.\n• Maggie made an object with 2 circles and no vertices.\n7.2\t\nPrisms and pyramids\ndifference between prisms and pyramids\nPrisms and pyramids are two special groups of polyhedra.\nPrisms\nA prism is a polyhedron with two faces that are  \ncongruent and parallel polygons. These faces are  \ncalled bases and they are connected by lateral  \nfaces that are parallelograms. \nIn the case of right prisms the bases are connected by rectangles which are  \nperpendicular to the base and the top. This means the lateral faces of a right prism  \nmake a 90° angle with the bases.\nA prism is named according to the shape of its base. So a prism whose base is a  \ntriangle is called a triangular prism; a prism whose base is a rectangle or square is called  \na rectangular prism; and a prism with a pentagonal base is called a pentagonal prism.\nAny pair of faces in a prism that are congruent and parallel can be the bases of that \nprism. A cube is a special type of prism. It has 6 congruent faces; therefore any of its faces \ncan be a base.\n\t\nTriangular prism\t\nRectangular prism\t\nPentagonal prism\t\nCube\nCongruent means exactly \nthe same shape and size.\nLateral faces are faces that \naren’t bases.\nMaths2_Gr7_LB_Book.indb   83\n2014/09/04   11:35:00 AM\n\n84\t\nMATHEMATICS Grade 7: Term 3\nThese are two oblique prisms – the one on the left is a triangular prism and the one on \nthe right is an oblique pentagonal prism.\nPyramids\nA pyramid has only one base. The lateral faces of a pyramid  \nare always triangles. These triangles meet at the same vertex  \nat the top. This vertex is called the apex of the pyramid. In a  \nright pyramid, the lateral faces are isosceles triangles.\nThere are many types of pyramids. The type of pyramid  \nis determined by the shape of its base. For example, a  \ntriangular-based pyramid has a triangle as its base, a  \nsquare-based pyramid has a square as its base and a  \nhexagonal-based pyramid has a hexagon as its base.\nSquare-based pyramid \nIt has 5 faces: \n1 square,  \n   4 triangles\nHexagonal-based pyramid \nIt has 7 faces: \n1 hexagon,  \n6 triangles\nIf a pyramid is not a right pyramid, it is called an oblique pyramid, like the two  \nshown below. The lateral faces of an oblique pyramid are not necessarily isosceles \ntriangles.\napex\nA triangular-based pyramid \nis also called a triangular \npyramid; a square-based \npyramid is also called \na square pyramid; a \nhexagonal-based pyramid \nis also called a hexagonal \npyramid, etc.\nMaths2_Gr7_LB_Book.indb   84\n2014/09/04   11:35:00 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n85\nidentifying prisms and pyramids\n1.\t Shade the base of each of the following figures and write down whether it is a prism \nor a pyramid. In some cases there is more than one possibility for the base.\nA\nB\nC\nD\nE\nF\nG\nH\nA:\t\nB:\t\nC:\t\nD:\t\nE:\t\nF:\t\nG:\t\nH:\t\n2.\t Join each 3D object with its correct name.\n\tHexagonal-based\t\nTriangular\t\nSquare-based\t\nCube\t\nPentagonal prism\n\t\npyramid\t\nprism\t\npyramid\nMaths2_Gr7_LB_Book.indb   85\n2014/09/04   11:35:01 AM\n\n86\t\nMATHEMATICS Grade 7: Term 3\n7.3\t\nDescribing, sorting and comparing 3D objects\npractise describing and classifying 3D objects\n1.\t For each of the following 3D objects:\n\t\n(a)\t name the object, and\n\t\n(b)\t describe the number of faces it has and the shapes of these faces.\nMaths2_Gr7_LB_Book.indb   86\n2014/09/04   11:35:01 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n87\n2.\t (a)\t Sort the 3D objects below into prisms, pyramids and cylinders by writing down  \n\t\nthe correct letters.\n\t\n\t\nPrisms: \n\t\n\t\nPyramids: \n\t\n\t\nCylinders: \n\t\n(b)\t Further divide the prisms into three groups and name each group by writing  \n\t\ndown the letters.\n\t\n\t\nCubes: \n\t\n\t\nRectangular prisms: \n\t\n\t\nTriangular prisms: \nA\nB\nC\nD\nE\nF\nG\nH\nI\nJ\nK\nL\nM\nN\nMaths2_Gr7_LB_Book.indb   87\n2014/09/04   11:35:01 AM\n\n88\t\nMATHEMATICS Grade 7: Term 3\n7.4\t\nNets of 3D objects\nwhat is a net?\nIn mathematics, a net is a flat pattern that can be folded to form a 3D object. Different \n3D objects have different nets. Sometimes the same 3D object can have different nets. \nHere are examples of different 3D objects and their nets.\n\t\nTriangular pyramid\t\nRectangular prism\t\nCube\nIn this section, you are going to focus on the net of a cube. In order for a net to form a \ncube, it must consist of 6 equal squares. But not all net patterns that consist of 6 squares \nwill fold into a cube. \nOnly one of the nets below will fold into a cube. Write down which one it is. \nA\nB\nC\nD\nMaths2_Gr7_LB_Book.indb   88\n2014/09/04   11:35:01 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n89\na cube or not a cube\nIn order to decide whether or not a net will fold into a cube, you have to imagine what \nwill happen when you fold the net. Read through the following steps.\n1.\t We can label the faces of a cube: bottom face (X), right face (R), back face (B), left  \nface (L), top face (T) and front face (F). We will use these terms in the rest of the steps.\nR\nB\nL\nT\nF\n2.\t Start by choosing one square of the net as the bottom face (marked with an X).\n3.\t Look at the square to the right of the X. (It is coloured blue.) If you fold the net on the \nred line, the blue square will be the right face of the cube.\nfold\n4.\t Look at the next square to the left of the right face. If you fold this square on the red \nline, it will become the front face of the cube. \nR\nR\nF\nfold\nMaths2_Gr7_LB_Book.indb   89\n2014/09/04   11:35:02 AM\n\n90\t\nMATHEMATICS Grade 7: Term 3\n5.\t Look at the square to the left of the  \nfront face. It will fold to become the  \nleft face of the cube.\n6.\t The square to the left of the left face  \nof the cube will become the back  \nface of the cube.\n7.\t The last square will form the top. \n8.\t Therefore you can label the squares on the net as follows:\nT\nF\nR\nL\nB\nT\nF\nR\nL\nB\n9.\t Since each square on the net corresponds with a face of the cube, this net can be \nfolded into a cube.\nidentifying the nets of a cube\nFor each of the following nets, determine whether it will fold into a cube or not by \nlabelling the squares to match the faces of a cube.\n1.\t\n\t\n\t\n\t\n\t\n2.\t\n3.\t\n\t\n\t\n\t\n\t\n4.\t\nF\nR\nfold\nL\nF\nR\nL\nF\nR\nB\nT\nMaths2_Gr7_LB_Book.indb   90\n2014/09/04   11:35:02 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n91\nnets of other 3D objects\n1.\t In each of the following cases, label the faces on the net according to the labels on the \n3D object. \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)\t\nL\nR\nF\nB\nT\nx\nL\nR\nF\nB\n\t\n(c)\nL\nR\nF\nB\n2.\t Decide whether the following nets will form 3D objects.\n\t\n(a)\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(b)\t\n\t\n(c)\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(d)\t\nMaths2_Gr7_LB_Book.indb   91\n2014/09/04   11:35:03 AM\n\n92\t\nMATHEMATICS Grade 7: Term 3\n3.\t Identify the following nets:\n\t\n(a)\t \t\n(b)\t \t\n(c)\t\n\t\n(d)\t \t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(e)\t\n4.\t (a)\t Identify the shapes that the net on the right consists of.\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\n\t\n\t\nH:\t\n\t\n(b)\t How many rectangular faces does the net have? \n\t\n(c)\t How many other shapes does the net have? What are they? \n\t\n(d)\t Will this form a pyramid or a prism? \n\t\n(e)\t How do you know? \n\t\n(f)\t Name the 3D object that the net will form. \nA\nB\nC\nD\nE\nF\nH\nG\nMaths2_Gr7_LB_Book.indb   92\n2014/09/04   11:35:03 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n93\n5.\t Answer the following questions about this net.\n\t\n(a)\t Which shapes does this net consist of?\n\t\n(b)\t How many triangular faces does the net have?\t\n\t\n(c)\t How many other shapes does the net have?\t\n\t\n(d)\t Will this form a pyramid or a prism?\t\n\t\n(e)\t How do you know?\n\t\n(f)\t Name the 3D object that the net will form.\n7.5\t\nUsing nets to construct cubes and prisms\nhow to draw a net of a prism\nLook at the triangular prism alongside. Its faces are  \n2 right-angled triangles and 3 rectangles.  \n(Since the base has 3 edges, there will be 3 rectangles.)\n1.\t Draw a rectangle, which will be the bottom of the  \nprism. Then add a right-angled triangle at the top  \nand a mirror image of the triangle at the bottom of the rectangle.\n11 cm\n4 cm\n5 cm\n3 cm\nMaths2_Gr7_LB_Book.indb   93\n2014/09/04   11:35:04 AM\n\n94\t\nMATHEMATICS Grade 7: Term 3\n2.\t Draw the other two rectangles next to the centre one. The rectangle on the right \nmust fit onto the side opposite the right angle in the triangle when it is folded, so \nthat rectangle will be the biggest. The rectangle on the left is the smallest of the three \nrectangles.\npractise drawing nets and constructing 3d models\n1.\t Complete the following nets:\n\t\n(a)\t\n\t\n(b)\nMaths2_Gr7_LB_Book.indb   94\n2014/09/04   11:35:04 AM\n\n\t\nCHAPTER 7: GEOMETRY OF 3D OBJECTS\t\n95\n2.\t Draw nets for the following objects:\n\t\n(a)\n\t\n(b)\t\n\t\n(c)\nMaths2_Gr7_LB_Book.indb   95\n2014/09/04   11:35:04 AM\n\n96\t\nMATHEMATICS Grade 7: Term 3\n3.\t (a)\t Copy each of the nets that you drew in questions 1 and 2 above onto  \n\t\ncardboard or paper.\n\t\n(b)\t Cut out the nets, and fold and paste them to make each 3D object.\n\t\n(c)\t Write down what you found difficult in making your 3D models and how you \n\t\novercame this difficulty.\nMaths2_Gr7_LB_Book.indb   96\n2014/09/04   11:35:04 AM\n\nTerm 3\nRevision and assessment",
      "chapter_id": "7"
    },
    {
      "title": "Term 3: Revision and assessment",
      "content": "TERM 3: REVISION AND ASSESSMENT\t\n97\nRevision............................................................................................................................. 98\n• Numeric and geometric patterns.................................................................................. 98\n• Functions and relationships 1..................................................................................... 100\n• Algebraic expressions 1.............................................................................................. 101\n• Algebraic equations 1................................................................................................. 102\n• Graphs....................................................................................................................... 103\n• Transformation geometry........................................................................................... 105\n• Geometry of 3D objects............................................................................................. 109\nAssessment...................................................................................................................... 111\nMaths2_Gr7_LB_Book.indb   97\n2014/09/04   11:35:04 AM\n\n98\t\nMATHEMATICS Grade 7: Term 3\nRevision\nShow all your steps of working. \nnumeric and geometric patterns \n1.\t For each of the following sequences, (i) describe in words the relationship between \nthe terms in the sequence, and (ii) use the relationship to find the next three terms  \nin the sequence.\n\t\n(a)\t 23;  19;  15;  …\n\t\n(b)\t 0,1;   0,2;   0,4;   0,8;   …\n\t\n(c)\t 1\n2;  3\n2;  21\n2; …\n\t\n(d)\t 1;  4;  9;  16;  …\n\t\n(e)\t 2;  4;  7;  11;  … \n\t\n(f)\t 2;  9;  28;  65; …\n\t\n(g)\t 21 200;  2 120;  212;  …\nMaths2_Gr7_LB_Book.indb   98\n2014/09/04   11:35:05 AM\n\n\t\nTERM 3: REVISION AND ASSESSMENT\t\n99\n2.\t Write down the first three terms of a sequence that fits the description given:\n\t\n(a)\t Each term is 2,3 bigger than the previous term.\n\t\n(b)\t Each term is 1\n3 smaller than the next term.\n\t\n(c)\t Each term is half the previous term.\n3.\t (a)\t Write down the values of a to d:\nTerm number\n1\n2\n3\n4\n5\n10\nValue of the term\n7,2\n7,7\n8,2\na\n9,2\nb\nTerm number\n1\n2\n3\nc\nd\n7\nValue of the term\n1\n3\n9\n81\n243\n729\n\t\n(b)\t Explain how you obtained the values of b and d.\n4.\t Below are the first three arrangements in a pattern created with matches.\n1\n2\n3\n\t\n(a)\t Complete the table below:\nNumber of the arrangement\n1\n2\n3\n4\n15\nNumber of matches needed\n3\n5\n121\n\t\n(b)\t Write in words the rule that describes the number of matches needed for each \n\t\nnew arrangement.\nMaths2_Gr7_LB_Book.indb   99\n2014/09/04   11:35:05 AM\n\n100\t MATHEMATICS Grade 7: Term 3\nfunctions and relationships 1\n1.\t Use the formula for the area of a rectangle (A = l × b) to calculate the following:\n\t\n(a)\t The area, if the length is 0,4 m and the breadth is 0,3 m\n\t\n(b)\t The length, if the area is 12,4 cm2 and the breadth is 4 cm\n\t\n(c)\t The breadth, if the area is 14,4 m2 and the length is 12 m\n2.\t The formula for the total surface area of a  \ncylinder of base radius r and height h is:  \nA = 2 × π × r2 + 2 × π × r × h\n\t\nUse this formula to calculate the surface area if  \n\t\nπ = 22\n7 , the base radius is 7 cm and the height is 3 cm.\n3.\t The formula for finding the temperature in degrees Celsius (°C) is C = 5\n9 × (F − 32), \nwhere F is the temperature in degrees Fahrenheit (°F). What is the temperature in °C  \nif it is 59 °F?\nπ or pi (pronounced “pie”) is a\nnumber approximately equal to 22\n7 .\nYou will learn about pi in Grade 8.\nr\nh\nMaths2_Gr7_LB_Book.indb   100\n2014/09/04   11:35:06 AM\n\n\t\nTERM 3: REVISION AND ASSESSMENT\t\n101\nalgebraic expressions 1\n1.\t If Skumbuzo is x years old, write down in terms of x the ages of the following  \n\t\npeople:\n\t\n(a)\t Suzie, who is 2 years older than Skumbuzo\t\n\t\n(b)\t Mohau, who is two years older than Suzie\t\n\t\n(c)\t Lintle, who is twice as old as Skumbuzo\t \t\n2.\t The relationship between a girl’s age (y years old) and that of her mother is y + 27. \nHow old is the girl’s mother when the girl is 13 years old?\n3.\t Write numbers in the boxes that make the statements true. All the numbers that you \nfill in should be decimal numbers.\n\t\n(a)\t If x = 4,5, then x + 7,7 = \n \n\t\n(b)\t If x = 2,6, then 2 × x = \n \n\t\n(c)\t If x = \n , then x\n3 = \n   \n4.\t Match each computational instruction to the correct expression. The first one has  \n\t\nbeen done for you.\n5 − y\nAdd y to 5\n5\ny\nSubtract 5 from y\n5 + y\nMultiply y by 5\ny − 5\nDivide 5 by y\ny\n5\nSubtract 5 from y and multiply  \nthe answer by 5\ny3\nCube y\n5 × y\ny2\n5 × (y − 5) \nMaths2_Gr7_LB_Book.indb   101\n2014/09/04   11:35:07 AM\n\n102\t MATHEMATICS Grade 7: Term 3\nalgebraic equations 1\n1.\t Write an equation (an open number sentence) that fits the given description:\n\t\n(a)\t Christian is x years old, his best friend Refilwe is y years old, and the sum of their \n\t\nages is 27.\n\t\n(b)\t Car a is R5 000 cheaper than car b.\n2.\t Here is an equation: 3 + c = d\n\t\n(a)\t Write down a pair of numbers that makes the equation true.\n\t\n(b)\t Write down a different pair of numbers that makes the equation true. The  \n\t\nnumbers should be common fractions.\n3.\t Solve for x:\n\t\n(a)\t x − 6 = 15\t\n(b)\t 3 × x = 45\t\n(c)\t 80\nx  = 4\n4.\t You are given that 3 × x + 5 = 11. Write down the value of:\n\t\n(a)\t 3 × x + 4\t\n(b)\t (3 × x + 5)2\n5.\t If d = c3 + 12, calculate the value of d when c has a value of:\n\t\n(a)\t 3\t\n(b)\t 6\nMaths2_Gr7_LB_Book.indb   102\n2014/09/04   11:35:07 AM\n\n\t\nTERM 3: REVISION AND ASSESSMENT\t\n103\ngraphs\n1.\t Study the following graph, showing the distance travelled by a car on the N1, and \nthen answer the questions that follow:\nDistance\n(km)\nTime (hours)\n0\n0\n1\n2\n3\n4\n5\n6\n100\n200\n300\n400\n500\n600\n\t\n(a)\t Ahmed says, “This is a linear graph.” Is he correct? Explain your answer.\n\t\n(b)\t Sindi says the graph is increasing. Is she correct? Explain your answer.\n\t\n(c)\t How far has the car travelled after 1,5 hours?\n\t\n(d)\t Complete the following table of values by reading off from the graph:\nTime (hours)\n1\n2\n3\n4\n5\n6\nDistance (km)\n\t\n(e)\t If the car continued in the same way as shown on the graph, how far will it have \n\t\ntravelled after 10 hours?\nMaths2_Gr7_LB_Book.indb   103\n2014/09/04   11:35:07 AM\n\n104\t MATHEMATICS Grade 7: Term 3\n2.\t Study the following graph, showing the average maximum temperatures in Cape \nTown, and then answer the questions that follow:\n\t\nTemperature (˚C)\nMonths\nJan\nFeb\nMar\nApr\nMay\nJun\nJul\nAug\nSep\nOct\nNov\nDec\n0\n5\n10\n15\n20\n25\n30\n\t\n(a)\t Describe the trend in the maximum temperatures from February to June. \n\t\n(b)\t Describe the trend in the maximum temperatures from June to August.\n\t\n(c)\t Describe the trend in the maximum temperatures from August to October.\n\t\n(d)\t Which is the hottest month in Cape Town? \n\t\n\t\nWhat is the average maximum temperature in this month?\n\t\n(e)\t Write down the names of the coldest months in Cape Town.\n\t\n\t\nWhat is the average maximum temperature during these months?\nMaths2_Gr7_LB_Book.indb   104\n2014/09/04   11:35:07 AM\n\n\t\nTERM 3: REVISION AND ASSESSMENT\t\n105\n3.\t Draw graphs that fit each of the following descriptions. Label the axes. You don’t  \nhave to show any values on the axes. \n(a)\t A speed-time graph for a vehicle \t  \ntravelling at a constant speed\n\t\n(b)\t A temperature-time graph for one day \n(midnight to midnight) in Durban\n\t\n\t\ntransformation geometry\n1.\t (a)\t Make tick marks in the relevant boxes in this table, to show which  \n\t\ntransformations will produce congruent figures:\nTranslation\nReflection\nRotation\nEnlargement\nCongruent figures\n\t\n(b)\t Which of the transformations will produce similar figures that are not congruent?\n2.\t Perform the described transformation on shape ABC on each of the following grids, \nand label the image vertices A', B' and C': \n\t\n(a)\t Translation of 2 units to the left and 1 unit down \nA\nB\nC\nMaths2_Gr7_LB_Book.indb   105\n2014/09/04   11:35:08 AM\n\n106\t MATHEMATICS Grade 7: Term 3\n\t\n(b)\t Reflection in the dotted line\nA\nB\nC\n \n(c)\t Rotation of 90° clockwise around vertex A \nA\nB\nC\nMaths2_Gr7_LB_Book.indb   106\n2014/09/04   11:35:08 AM\n\n\t\nTERM 3: REVISION AND ASSESSMENT\t\n107\n\t\n(d)\t Enlargement of factor 2, with vertex B as the centre of enlargement\nA\nB\nC\n3.\t Describe in as much detail as possible the single transformation which maps  \ntriangle A onto:\n\t\n(a)\t triangle B\n\t\n(b)\t triangle C.\nA\nB\nC\nMaths2_Gr7_LB_Book.indb   107\n2014/09/04   11:35:08 AM\n\n108\t MATHEMATICS Grade 7: Term 3\n4.\t In this image, quadrilateral ABCD has been enlarged to quadrilateral AEFG:\nA\nB\nC\nF\nD\nE\nG\n\t\n(a)\t Write down the factor of enlargement. \n\t\n(b)\t What kind of quadrilateral is ABCD? \n\t\n(c)\t What kind of quadrilateral is AEFG? \n\t\n(d)\t Are the two quadrilaterals (ABCD and AEFG) congruent, or are they similar, or  \n\t\nboth?\n5.\t Consider the square grid below, with some blocks shaded in.\n\t\n(a)\t How many lines of symmetry does the shape have? \n\t\n\t\nDraw them in on the diagram, using dashed lines. \t\nMaths2_Gr7_LB_Book.indb   108\n2014/09/04   11:35:09 AM\n\n\t\nTERM 3: REVISION AND ASSESSMENT\t\n109\n\t\n(b)\t Shade in some more blocks on the grid below so that the shape has exactly two \n\t\nlines of symmetry.  \n \n\t\n(c)\t Shade in some blocks on the grid below so that the shape has exactly one line  \n\t\nof symmetry.\n \ngeometry of 3D objects\n1.\t What is the name of the solid (3D object) that:\n\t\n(a)\t has only square faces? \n\t\n(b)\t has a combination of square faces and triangular faces? \n\t\n(c)\t has 3 faces and no vertices? \n\t\n(d)\t has 4 triangular faces only? \n2.\t (a)\t Complete the table.\nNumber of  \nfaces\nNumber of \nvertices\nNumber of  \nedges\nRectangular prism\nTriangular prism\nMaths2_Gr7_LB_Book.indb   109\n2014/09/04   11:35:09 AM\n\n110\t MATHEMATICS Grade 7: Term 3\n\t\n(b)\t Euler was a Swiss mathematician who lived in the 18th century. He noticed that  \n\t\nthe number of faces (F) plus the number of vertices (V) minus the number of  \n\t\nedges (E) is almost always equal to 2 for solids. Thus: F + V − E = 2.\n\t\n\t\nCheck whether Euler’s formula works for the two prisms in question (a). Show \n\t\nall your working.\n3.\t Study the image below, showing the net of a dodecahedron:\n\t\nWrite down the number of faces and vertices a dodecahedron has.\n4.\t On the grid below, draw a possible net for a rectangular prism that is 4 blocks long,  \n3 blocks wide and 2 blocks high.\n\t\nMaths2_Gr7_LB_Book.indb   110\n2014/09/04   11:35:09 AM\n\n\t\nTERM 3: REVISION AND ASSESSMENT\t\n111\nAssessment\nIn this section, the numbers in brackets at the end of a question indicate the number of \nmarks that the question is worth. Use this information to help you determine how much \nworking is needed. \nThe total number of marks allocated to the assessment is 50.\n1.\t For each sequence, (i) describe in words the relationship between the terms in the \nsequence, and (ii) use the relationship to find the next three terms in the sequence:\n\t\n(a)\t 28,3;  31,1;  33,9;  …\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(2)\n\t\n(b)\t 2\n5; 6\n5; 18\n5 \t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(2)\n2.\t (a)\t Complete the table.\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(2)\nTerm number\n1\n2\n3\n4\n7\nValue of the term\n0\n3\n8\n15\n143\n\t\n(b)\t Describe in words the rule by which you could find any term in the sequence  \n\t\nshown in the table.\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(1)\n3.\t (a)\t  If a = b3 − (2 × c2 + 8), what is the value of a if b = 5 and c = 4?\b\n(3)\n\t\n(b)\t Use the formula p = 4\n2\n×\n−\ns\na\nt\n to calculate the value of t if p = 36, s = 45 \n\t\nand a = 6.\b\n(2)\nMaths2_Gr7_LB_Book.indb   111\n2014/09/04   11:35:10 AM\n\n112\t MATHEMATICS Grade 7: Term 3\n4.\t A soccer ball costs x rand. Write down in terms of x the costs of the following items:\n\t\n(a)\t five soccer balls \n\t\n(b)\t a rugby ball that costs twice as much as the soccer ball \n\t\n(c)\t a dress that costs R150 more than the soccer ball \n\t\n(d)\t three soccer balls and a dress \n   (4)\n5.\t Solve for x:\n\t\n(a)\t x + 18 = 52\t\n(b)\t 5 × x = 60\t\n(c)\t x\n3  = 12\t\n\t\n        (3)\n6.\t You are given that 2 × x + 8 = 15. Write down the value of 2 × x + 10.\t\n(1)\n7.\t Study the following graph, showing the average minimum temperatures in Cape \nTown, and then answer the questions that follow:\n\t\nTemperature (˚C)\nMonths\nJan Feb Mar Apr May Jun\nJul\nAug Sep Oct Nov Dec\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n\t\n(a)\t Describe the trend in the minimum temperatures from July to December.\t\n\t\n(1)\n\t\n(b)\t Which month has the lowest minimum temperature, and what is the average  \n\t\nminimum temperature in this month?\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(2)\n\t\n(c)\t Write down the names of the months with the highest minimum  \n\t\ntemperatures, and their average minimum temperatures.\t\n\t\n\t\n\t\n\t\n\t\n(2)\nMaths2_Gr7_LB_Book.indb   112\n2014/09/04   11:35:10 AM\n\n\t\nTERM 3: REVISION AND ASSESSMENT\t\n113\n8.\t Draw graphs that fit each of the following descriptions. Remember to label the axes. \nYou don’t have to show any values on the axes. \n(a)\t A graph for the total cost of different \t \nmasses of stewing beef                       (2)\n\t\n(b)\t A speed-time graph for a vehicle \nthat is accelerating                              (2)\n\t\n\t\n9.\t Perform the described transformation on shape ABC and label the image vertices A', \nB' and C':\n\t\n(a)\t Translation by 3 units to the right and 2 units up\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(2)\nA\nB\nC\nMaths2_Gr7_LB_Book.indb   113\n2014/09/04   11:35:11 AM\n\n114\t MATHEMATICS Grade 7: Term 3\n\t\n(b)\t Reflection in the dotted line\t \t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(2)\nA\nB\nC\n\t\n(c)\t Rotation of 90° clockwise around vertex C\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(2)\nA\nB\nC\nMaths2_Gr7_LB_Book.indb   114\n2014/09/04   11:35:11 AM\n\n\t\nTERM 3: REVISION AND ASSESSMENT\t\n115\n10.\t(a)\t Draw in all the lines of symmetry (use dashed lines) on the following shape.\t\n(2)\n \n(b)\t Add one line from one side of the shape to the other to each of the following  \n\t\ndiagrams, so that the shape has:\n\t\n\t\n(i)\t exactly one line of symmetry\t \t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(1)\n\t\n\t\n\t\n \n\t\n(ii)\t no lines of symmetry \t \t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(1)\n\t\n\t\n\t\nMaths2_Gr7_LB_Book.indb   115\n2014/09/04   11:35:11 AM\n\n116\t MATHEMATICS Grade 7: Term 3\n11.\tComplete the table.\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(4)\nNumber of faces\nNumber of vertices\nNumber of edges\nCube\nSquare-based \npyramid\n12.\t(a)\t On the grid below, draw a net for a cube which has a side length of 2 units.\t \t\n(3)\n\t\n(b)\t On the grid below, draw a completely different net for the same cube. It should \n\t\nnot simply be, for example, a rotated or reflected version of the one you drew in \n\t\nquestion (a).\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(3)\nMaths2_Gr7_LB_Book.indb   116\n2014/09/04   11:35:12 AM\n\nChapter 8\nIntegers\n\t\nCHAPTER 8: INTEGERS\t\n117\nIn this chapter you will work with numbers smaller than 0. These numbers are called  \nnegative numbers. Mathematicians have agreed that negative numbers should have  \ncertain properties that will make them useful for various purposes. You will learn about \nthese properties and how they make it possible to do calculations with negative numbers.\n8.1\t The need for numbers called integers..................................................................... 119\n8.2\t Finding numbers that make statements true........................................................... 125\n8.3\t Adding and subtracting integers............................................................................. 128\nMaths2_Gr7_LB_Book.indb   117\n2014/09/04   11:35:12 AM\n\n118\t MATHEMATICS Grade 7: Term 4\nDo what you can.\n  5 − 0 = ?\n5 − 7 = ?\n5 + 5 = ?\n  5 − 1 = ?\n5 − 6 = ?\n5 + 4 = ?\n  5 − 2 = ?\n5 − 5 = ?\n5 + 3 = ?\n  5 − 3 = ?\n5 − 4 = ?\n5 + 2 = ?\n  5 − 4 = ?\n5 − 3 = ?\n5 + 1 = ?\n  5 − 5 = ?\n5 − 2 = ?\n5 + 0 = ?\n  5 − 6 = ?\n5 − 1 = ?\n5 + ? = ?\n  5 − 7 = ?\n5 − 0 = ?\n5 + ? = ?\n  5 − 8 = ?\n5 − ? = ?\n5 + ? = ?\n  5 − 9 = ?\n5 − ? = ?\n5 + ? = ?\n5 − 10 = ?\n5 − ? = ?\n5 + ? = ?\n5 − 11 = ?\n5 − ? = ?\n5 + ? = ?\n5 − 12 = ?\n5 − ? = ?\n5 + ? = ?\n5 − 13 = ?\n5 − ? = ?\n5 + ? = ?\nMaths2_Gr7_LB_Book.indb   118\n2014/09/04   11:35:12 AM\n\n\t\nCHAPTER 8: INTEGERS\t\n119\n\t\nCHAPTER 8: INTEGERS\t\n119",
      "chapter_id": "8"
    },
    {
      "title": "Integers",
      "content": "8\t Integers\n8.1\t The need for numbers called integers\nNumbers are used for many different purposes. We use numbers to say how many \nobjects there are in a collection, for example the number of desks in a classroom. For \nthis purpose we use the counting numbers 1,  2,  3,  4  . . .  Numbers are also used to \ndescribe size, for example the lengths of objects. For this purpose we need more than the \ncounting numbers, we also need fractions. Another purpose of numbers is to indicate \nposition, for example the position of the right end of the red line on the pictures below. \nNumbers also occur as the solutions to equations, and the natural numbers and  \nfractions do not provide solutions for all equations. For example, there is no natural \nnumber or fraction that is the solution to the equation 10 − x = 20. The number that  \nprovides the solution to this equation must have the property that when you subtract it, \nit has the same effect as when you add 10!\nWith a view to have numbers that can serve more purposes than counting and  \nmeasuring, mathematicians have decided to also think of another kind of numbers \nwhich are called integers. The integers include the natural numbers, but for each  \nnatural number, for example 24, there is also another number called the additive  \ninverse. For example, −24 is the additive inverse of 24. When you add a number to its \nadditive inverse, the answer is 0. For example, 24 + (−24) = 0.\nsaying how cold it is\nOne of the uses of integers is for the measurement of temperature. If we say that the \ntemperature is 0 when water freezes to become ice, we need numbers smaller than 0 to \ndescribe the temperature when it gets even colder than when water freezes. When water \nstarts boiling, its temperature is 100 degrees on the scale called the Celsius scale.\nLiquids expand when heated, and shrink when cooled down. So when it is warm, the \nliquid in a thin tube may almost fill the tube:\nWhen it is cold, the column of liquid will be quite short.\nThis property of liquid is used to measure temperature, and an instrument like the above \nis called a thermometer.\nMaths2_Gr7_LB_Book.indb   119\n2014/09/04   11:35:12 AM\n\n120\t MATHEMATICS Grade 7: Term 4\nThis is what a thermometer will show when it is put in water that is boiling. It shows a \ntemperature of 100 degrees Celsius, which is written as 100 °C.\n–80 –60 –40 –20\n0\n20\n40\n60\n80 100 120 140 160 180 200 220\nOn the diagram below, you can see what a thermometer will show if it is in water that is \nstarting to freeze. It shows a temperature of 0 °C.\n–80 –60 –40 –20\n0\n20\n40\n60\n80 100 120 140 160 180 200 220\nOn the next diagram you can see what a thermometer will show when the temperature  \nis −40 °C, which is colder than any winter night you may have experienced.\n–80 –60 –40 –20\n0\n20\n40\n60\n80 100 120 140 160 180 200 220\n1.\t Write down the temperature that is shown on each of the thermometers below.\n–80 –60 –40 –20\n0\n20\n40\n60\n80 100 120 140 160 180 200 220\n\t\n(a)\t\n–80 –60 –40 –20\n0\n20\n40\n60\n80 100 120 140 160 180 200 220\n\t\n(b)\t\n–80 –60 –40 –20\n0\n20\n40\n60\n80 100 120 140 160 180 200 220\n\t\n(c)\t\n–80 –60 –40 –20\n0\n20\n40\n60\n80 100 120 140 160 180 200 220\n\t\n(d)\t\n–80 –60 –40 –20\n0\n20\n40\n60\n80 100 120 140 160 180 200 220\n\t\n(e)\t\n–80 –60 –40 –20\n0\n20\n40\n60\n80 100 120 140 160 180 200 220\n\t\n(f)\t\nMaths2_Gr7_LB_Book.indb   120\n2014/09/04   11:35:13 AM\n\n\t\nCHAPTER 8: INTEGERS\t\n121\n2.\t (a)\t The temperature of water in a pot is 20 °C. It is heated so that it gets 30 °C \n\t\n\t\nwarmer. What is the temperature of the water now? \n\t\n(b)\t The temperature of water in a bottle is 80 °C. During the night it cools down to \n\t\n\t\n30 °C. By how much has it cooled down? \n\t\n(c)\t In the middle of a very cold winter night the temperature outside is −20 °C.  \n\t\nAt nine o’clock in the morning it has become 30 degrees warmer. What is the \n\t\n\t\ntemperature at nine o’clock? \n3.\t (a)\t The temperature is 8 °C. What will the temperature be if it gets 10 degrees \n\t\n\t\ncolder? \n\t\n(b)\t The temperature is 8 °C. What will the temperature be if it gets 20 degrees \n\t\n\t\ncolder? \n\t\n(c)\t The temperature is −8 °C. What will the temperature be if it gets 10 degrees \n\t\n\t\nwarmer? \n\t\n(d)\t The temperature is −24 °C. What will the temperature be if it gets 10 degrees \n\t\n\t\nwarmer? \n4.\t Some numbers are shown on the number lines below. Fill in the missing numbers.\n–9\n–5\n–4\n–2\n–1\n0\n1\n2\n3\n4\n9\n0\n1\n2\n3\n–10 –5\n0\n5\n10\n15\nsaying how much money it is\nSimon is in Grade 5. He saved money in a tin. When he turned 10, his grandmother  \ngave him R100. He also opened his savings tin on his tenth birthday and there was  \nR260 in the tin. Simon was very happy. He said to himself: “I am very rich!”\nSimon decides to buy some things that he has always wanted. This is what he decides \nto buy:\n• a soccer ball at R160\n• a pair of sunglasses at R180\n• a book about animals at R90 \t\n1.\t How much money did Simon have in total on the day that he thought he was  \nrich? \nMaths2_Gr7_LB_Book.indb   121\n2014/09/04   11:35:13 AM\n\n122\t MATHEMATICS Grade 7: Term 4\n2.\t What is the total cost of the three items he wants to buy?\n3.\t Simon decides to first buy the soccer ball only. How much money will he have after \npaying for the soccer ball?\n4.\t How much money will Simon have if he buys the soccer ball and the sunglasses?\n5.\t How much money will Simon have if he buys the soccer ball and the sunglasses and \nthe book about animals?\nSimon did these calculations while he was thinking about buying the various items:\n\t\n\t\n\t\nR360 − R160 = R200\n\t\n\t\n\t\nR200 − R180 = R20\n\t\n\t\n\t\nR20 − R90 = (−) R70 ?\n6.\t Fatima owns a small shop. One afternoon when she closed the shop, she had R120 \ncash, clients owed her R90, and she owed her suppliers R310. In Fatima’s view her \nfinancial position was as follows: R120 + R90 − R310 = −R100.\n\t\n(a)\t On another day, Fatima ended the business day with R210 cash, clients owed  \n\t\nher R180 and she owed her suppliers R160. What was her financial position?\n\t\n(b)\t On another day, Fatima ended the business day with R150 cash, clients owed  \n\t\nher R130 and she owed her suppliers R460. What was her financial position?\nAbout 500 years ago, some mathematicians proposed  \nthat a “negative number” may be used to describe the  \nresult in a situation like the above, where a number is  \nsubtracted from a number smaller than it.\nFor example, we may say 10 − 20 = (−10)\nThis proposal was soon accepted by other mathematicians, and it is now used all over  \nthe world. \nMathematicians are people who \ndo mathematics for a living. \nMathematics is their profession, \nlike health care is the profession \nof nurses and medical doctors.\nMaths2_Gr7_LB_Book.indb   122\n2014/09/04   11:35:13 AM\n\n\t\nCHAPTER 8: INTEGERS\t\n123\n7.\t Continue the lists of numbers below to complete the table.\n(a)\n(b)\n(c)\n(d)\n(e)\n(f)\n(g)\n10\n100\n3\n−3\n−20\n150\n0\n9\n90\n6\n−6\n−18\n125\n−5\n8\n80\n9\n−9\n−16\n100\n−10\n7\n70\n12\n−12\n−14\n75\n−15\n6\n60\n15\n−15\n50\n−20\n5\n50\n−25\n4\n40\n3\n30\n2\n20\n1\n10\n0\n0\n−1\n8.\t Calculate each of the following:\n\t\n(a)\t 16 − 20 = \n\t\n(b)\t 16 − 30 = \n\t\n(c)\t 16 − 40 = \n\t\n(d)\t 16 − 60 = \n\t\n(e)\t 16 − 200 = \n\t\n(f)\t 5 − 1 000 = \n9.\t Jeminah has R200 in a savings account and R40 in her purse. Her brother owes her \nR50. How rich is she? In other words, how much money does she have?\n10. Oops! Jeminah forgot that she borrowed R60 from her mother, and that she still has \nto pay R150 for a dress she bought last month. So how rich (or poor) is she really? In \nother words, how much money does she actually have?\n11.\tIn fact, Jeminah’s financial situation is even worse. She has received an outstanding \nbill from her doctor, for R250. So how much money does she really have?\nMaths2_Gr7_LB_Book.indb   123\n2014/09/04   11:35:14 AM\n\n124\t MATHEMATICS Grade 7: Term 4\nordering and comparing integers\n1.\t On a certain day the following minimum temperatures were provided by the weather \nbureau:\n\t\nBethlehem\t\t\n−4 °C \t \t\n\t\n\t\n\t\nBloemfontein\t \t\n−6 °C\n\t\nCape Town\t\t\n7 °C\t\n\t\n\t\n\t\n\t\nDordrecht\t \t\n\t\n−9 ºC\n\t\nDurban\t\t\n\t\n12 °C \t \t\n\t\n\t\n\t\nJohannesburg\t \t\n  0 °C\n\t\nPretoria\t\n\t\n4 °C\t\n\t\n\t\n\t\n\t\nQueenstown\t\n\t\n−1 °C\n\t\nArrange the temperatures from the coldest to the warmest.\n2.\t Place the following numbers on the number line as accurately as you can:\n\t\n50;  −2;  −23;  5;  −36\n3.\t In each case, place the numbers in the boxes provided:\n\t\n(a)\t 125 000;   −178 000;   −100 900;   180 500\n–200 000\n200 000\n\t\n(b)\t −1 055 500;   −1 010 100;   −1 100 100;   −1 032 800;  −1 077 500\n–1 111 000\n–1 000 000\n4.\t Insert one of the symbols > or < to indicate which number is the smaller of the two. \n\t\n(a)\t 978 543 \n  978 534\t \t\n\t\n\t\n\t\n(b)  −1 043 724 \n  −1 034 724\n\t\n\t\n(c)\t −864 026 \n  −864 169\t\n\t\n\t\n\t\n(d)  −103 232 \n  −104 326\n\t\n\t\n(e)\t −710 742 \n  710 741\t \t\n\t\n\t\n\t\n(f)  −904 700 \n  −904 704\nMaths2_Gr7_LB_Book.indb   124\n2014/09/04   11:35:14 AM\n\n\t\nCHAPTER 8: INTEGERS\t\n125\n8.2\t Finding numbers that make statements true\nThe numbers 1, 2, 3, 4 and so on that we use for counting are called the natural \nnumbers. Natural numbers are whole numbers – they do not contain fraction parts.\n1.\t Is there a natural number that can be put in the brackets below to make the  \nstatement true?\n\t\n12 + (\n) = 17\n2.\t In each case below, insert a natural number in the space between the brackets that \nwill make the statement true. \n\t\n(a)\t 15 + (\n) = 21\t\n\t\n(b)\t 15 − (\n) = 10\n\t\n(c)\t (\n) + 10 = 34\t\n\t\n(d)\t (\n) − 10 = 34\n\t\n(e)\t 3 × (\n) = 18\n3.\t (a)\t Can you think of a natural number that will make this statement true?\n\t\n\t\n2 × ( …. ) = 5 \n\t\n(b)\t Can you think of any other number that will make the statement true?\n4.\t (a)\t Can you think of a natural number that will \n\t\nmake this statement true?\n\t\n\t\n8 + ( …. ) = 5 \n\t\n(b)\t Can you think of any other number that  \n\t\nwill make the statement true?\nWe are looking for a number that will make the following statement true:\n8 + ( .… ) = 5\nConsider this plan: \nLet us agree that we will call this number negative 3 and write it as (−3).  \nIf we agree to this, we can say 8 + (−3) = 5.\nThis may seem a bit strange to you. You do not have to agree now. But even if you do \nnot agree, let us explore how this plan may work for other numbers. What answers will a \nperson who agrees to the plan give to the following question?\nHere is a different way to ask the \nsame questions:\n(a)\tWhat is x if 15 + x = 21?\n(b)\tWhat is x if 15 − x = 10?\n(c)\tWhat is x if x + 10 = 34?\n(d)\tWhat is x if x − 10 = 34? \n(e)\tWhat is x if 3 × x = 18?\nWe normally think of adding \nas making something bigger. \nQuestion 4(a) requires us to \nchange our mind about this.  \nWe have to consider the  \npossibility that adding a number \nmay make something smaller.\nMaths2_Gr7_LB_Book.indb   125\n2014/09/04   11:35:14 AM\n\n126\t MATHEMATICS Grade 7: Term 4\n5.\t Calculate each of the following:\n\t\n(a)\t 10 + (−3)\t\n\t\n(b)\t 12 + (−3)\t\n\t\n(c)\t 12 + (−5)\t\n\t\n(d)\t 10 + (−9)\t\n\t\n(e)\t 8 + (−8)\t\n\t\n(f)\t 1 + (−1)\t\nYou possibly agree that \n\t\n5 + (−5) = 0    10 + (−10) = 0    and    20 + (−20) = 0\nWe may say that for each “positive” number there is a corresponding or opposite \nnegative number. Two positive and negative numbers that correspond, for example  \n3 and (−3), are called additive inverses. They wipe each other out when you add them.\nWhen you add any number to its additive inverse, \nthe answer is 0. For example, 120 + (−120) = 0\nSo, the set of integers consists of all the natural \nnumbers and their additive inverses and zero.\n6.\t Write the additive inverse of each of the following numbers:\n\t\n(a)\t 24\t \t\n\t\n\t\n\t\n\t\n\t\n(b)\t −24\t\n\t\n(c)\t −103\t\n\t\n\t\n\t\n\t\n\t\n(d)\t 2 348\t\nThe idea of additive inverses may be used to explain why 8 + (−5) is equal to 3:\n8 + (−5) = 3 + 5 + (−5) = 3 + 0 = 3\n7.\t Use the idea of additive inverses to explain why each of these statements is true:\n\t\n(a)\t 43 + (−30) = 13\t\t\n\t\n\t\n\t\n\t\n\t\n(b)   150 + (−80) = 70\n8.\t Calculate each of the following:\n\t\n(a)\t 10 + 4 + (−4) \n\t\n(b)\t 10 + (−4) + 4 \n\t\n(c)\t 3 + 8 + (−8)   \n\t\n(d)\t 3 + (−8) + 8   \n9.\t Calculate each of the following:\n\t\n(a)\t 18 + 12 = \n\t\n(b)\t 12 + 18 = \n\t\n\t\n(c)\t 2 + 4 + 6 = \n\t\n(d)\t 6 + 4 + 2 = \n\t\n(e)\t 2 + 6 + 4 = \n\t\n(f)\t 4 + 2 + 6 = \n\t\n(g)\t 4 + 6 + 2 = \n\t\n(h)\t 6 + 2 + 4 = \n\t\n(i)\t 6 + (−2) + 4 = \n\t\n(j)\t 4 + 6 + (−2) = \n\t\n(k)\t 4 + (−2) + 6 = \n\t\n(l)\t (−2) + 4 + 6 = \n\t\n(m)\t6 + 4 + (−2) = \n\t\n(n)\t (−2) + 6 + 4 = \nWhat may each of the following \nbe equal to?\n5 + (−8) \n(−5) + (−8) \nThe number zero is regarded as \nan integer.\nNatural numbers can be  \narranged in any order to add \nand subtract them. It would \nmake things easy if we agree \nthat this should also be the case \nfor negative numbers.\nMaths2_Gr7_LB_Book.indb   126\n2014/09/04   11:35:14 AM\n\n\t\nCHAPTER 8: INTEGERS\t\n127\n10. Calculate each of the following:\n\t\n(a)\t (−5) + 10 \n\t\n(b)\t 10 + (−5) \n\t\n(c)\t (−8) + 20 \n\t\n(d)\t 20 − 8 \n\t\n(e)\t 30 + (−10) \n\t\n(f)\t 30 + (−20) \n\t\n(g)\t 30 + (−30) \n\t\n(h)\t 30 + (−40) \n\t\n(i)\t 10 + (−5) + (−3) \n\t\n(j)\t (−5) + 7 + (−3) + 5 \n\t\n(k)\t (−5) + 2 + (−7) + 4 \n11.\tIn each case find the number that makes the statement true. Give your answer by \nwriting a closed number sentence.\n\t\n(a)\t 20 + (an unknown number) = 50\n\t\n(b)\t 50 + (an unknown number) = 20\n\t\n(c)\t 20 + (an unknown number) = 10\n\t\n(d)\t (an unknown number) + (−25) = 50\n\t\n(e)\t (an unknown number) + (−25) = (−50)\n12. Use the idea of additive inverses to explain why each of the following statements  \nis true:\n\t\n(a)\t 43 + (−50) = −7\t\n(b)\t 60  + (−85) = −25\nStatements that are true for many different numbers\nFor how many different pairs of numbers can the following statement be true, if only natural \n(positive) numbers are allowed?   \n\t\n\t\n\t\n\t\n a number + another number = 10\n \nFor how many different pairs of numbers can the statement be true if negative numbers are  \nalso allowed?   \nStatements like these are also \ncalled number sentences.\nAn incomplete number  \nsentence, where some numbers \nare not known at first, is  \nsometimes called an open \nnumber sentence:\n8 − (a number) = 10\nA closed number sentence \nis where all the numbers are \nknown:\n8 + 2 = 10\nMaths2_Gr7_LB_Book.indb   127\n2014/09/04   11:35:14 AM\n\n128\t MATHEMATICS Grade 7: Term 4\n8.3\t Adding and subtracting integers\nproperties of integers\n1.\t Calculate:\n\t\n(a)\t 80 + (−60) = \n\t\n(b)\t 500 + (−200) + (−200) = \n2.\t (a)\t Do you agree that 20 + (−5) = 15? \n\t\n(b)\t What do you think 20 − (−5) should be?\n3.\t (a)\t Is 100 + (−20) + (−20) = 60, or does it equal \n\t\n\t\nsomething else? \n\t\n(b)\t What do you think (−20) + (−20) should be equal to? \n4.\t Complete the following as far as you can:\n(a)\n(b)\n(c)\n5 − 9 = \n5 + 9 = \n9 − 3 = \n5 − 8 = \n5 + 8 = \n8 − 3 = \n5 − 7 = \n5 + 7 = \n7 − 3 = \n5 − 6 = \n5 + 6 = \n6 − 3 = \n5 − 5 = \n5 + 5 = \n5 − 3 = \n5 − 4 = \n5 + 4 = \n4 − 3 = \n5 − 3 = \n5 + 3 = \n3 − 3 = \n5 − 2 = \n5 + 2 = \n2 − 3 = \n5 − 1 = \n5 + 1 = \n1 − 3 = \n5 − 0 = \n5 + 0 = \n0 − 3 = \n5 − (−1) = \n5 + (−1) = \n(−1) − 3 = \n5 − (−2) = \n5 + (−2) = \n(−2) − 3 = \n5 − (−3) = \n5 + (−3) = \n(−3) − 3 = \n5 − (−4) = \n5 + (−4) = \n(−4) − 3 = \n5 − (−5) = \n5 + (−5) = \n(−5) − 3 = \nWe normally think of addition \nand subtraction as actions that \nhave opposite effects: what the \none does is the opposite or  \ninverse of what the other does. \nMaths2_Gr7_LB_Book.indb   128\n2014/09/04   11:35:14 AM\n\n\t\nCHAPTER 8: INTEGERS\t\n129\n5.\t Calculate each of the following:\n\t\n(a)\t 20 − 20 = \n\t\n(b)\t 50 − 20 = \n\t\n(c)\t (−20) − (−20) = \n\t\n(d)\t (−50) − (−20) = \n6.\t In each case suggest a number that may make the statement true. Also give an \nargument to support your proposal.\n\t\n(a)\t 20 + (a number) = 8 \n\t\n(b)\t 20 + (a number) = 28\n\t\n(c)\t 20 − (a number) = 28\n\t\n(d)\t 20 − (a number) = 12\nSome history\nThe following statement is true if the number is 5: \n\t\n15 − (a certain number) = 10 \nA few centuries ago, some mathematicians decided they wanted to have numbers that will also \nmake sentences like the following true:\n\t\n15 + (a certain number) = 10 \nBut to go from 15 to 10 you have to subtract 5.\nThe number we need to make the sentence 15 + (a certain number) = 10 true must have the  \nfollowing strange property:\n\t\nIf you add this number, it should have the same effect as to subtract 5.\nNow the mathematicians of a few centuries ago really wanted to have numbers for which such \nstrange sentences would be true. So they thought:\n\t\n“Let us just decide, and agree amongst ourselves, that the number we call  \n\t\nnegative 5 will have the property that if you add it to another number,  \n\t\nthe effect will be the same as when you subtract the natural number 5.”\nThis means that the mathematicians agreed that 15 + (−5) is equal to 15 − 5.\nStated differently, instead of adding negative 5 to a number, you may subtract 5.\nMaths2_Gr7_LB_Book.indb   129\n2014/09/04   11:35:14 AM\n\n130\t MATHEMATICS Grade 7: Term 4\nWe may agree that subtracting a negative number  \nhas the same effect as adding the additive inverse of \nthe negative number. If we stick to this agreement, \nthe following two calculations should have the  \nsame answer:\n\t \t\n\t\n10 − (−7)  and  10 + 7\n7.\t Calculate.\n\t\n(a)\t 20 − (−10) = \n\t\n(b)\t 100 − (−100) = \n\t\n(c)\t 20 + (−10) = \n\t\n(d)\t 100 + (−100) = \n\t\n(e)\t (−20) − (−10) = \n\t\n(f)\t (−100) − (−100) = \n\t\n(g)\t (−20) + (−10) = \n\t\n(h)\t (−100) + (−100) = \n8.\t Complete the following as far as you can:\n(a)\n(b)\n(c)\n5 − (−9) = \n(−5) + 9 = \n9 − (−3) = \n5 − (−8) = \n(−5) + 8 = \n8 − (−3) = \n5 − (−7) = \n(−5) + 7 = \n7 − (−3) = \n5 − (−6) = \n(−5) + 6 = \n6 − (−3) = \n5 − (−5) = \n(−5) + 5 = \n5 − (−3) = \n5 − (−4) = \n(−5) + 4 = \n4 − (−3) = \n5 − (−3) = \n(−5) + 3 = \n3 − (−3) = \n5 − (−2) = \n(−5) + 2 = \n2 − (−3) = \n5 − (−1) = \n(−5) + 1 = \n1 − (−3) = \n5 − 0 = \n(−5) + 0 = \n0 − (−3) = \n5 − 1 = \n(−5) + (−1) = \n(−1) − (−3) = \n5 − 2 = \n(−5) + (−2) = \n(−2) − (−3) = \n5 − 3 = \n(−5) + (−3) = \n(−3) − (−3) = \n5 − 4 = \n(−5) + (−4) = \n(−4) − (−3) = \n5 − 5 = \n(−5) + (−5) = \n(−5) − (−3) = \nMaths2_Gr7_LB_Book.indb   130\n2014/09/04   11:35:14 AM\n\n\t\nCHAPTER 8: INTEGERS\t\n131\n9.\t In each case, state whether the statement is true or false and give a numerical \nexample to demonstrate your answer.\n\t\n(a)\t Subtracting a positive number from a negative number has the same effect as  \n\t\nadding the additive inverse of the positive number.\n\t\n(b)\t Adding a negative number to a positive number has the same effect as adding  \n\t\nthe additive inverse of the negative number.\n\t\n(c)\t Subtracting a negative number from a positive number has the same effect as  \n\t\nsubtracting the additive inverse of the negative number.\n\t\n(d)\t Adding a negative number to a positive number has the same effect as subtracting \n\t\nthe additive inverse of the negative number.\n\t\n(e)\t Adding a positive number to a negative number has the same effect as adding  \n\t\nthe additive inverse of the positive number.\n\t\n(f)\t Adding a positive number to a negative number has the same effect as subtracting \n\t\nthe additive inverse of the positive number.\n\t\n(g)\t Subtracting a positive number from a negative number has the same effect as  \n\t\nsubtracting the additive inverse of the positive number.\n\t\n(h)\t Subtracting a negative number from a positive number has the same effect as \n\t\nadding the additive inverse of the negative number.\nproperties of operations\n1.\t Calculate the following:\n\t\n(a)\t (−3) + (−5) = \n\t\n(b)\t (−5) + (−3) = \n\t\n(c)\t 5 + (−7) = \n\t\n(d)\t (−7) + 5 = \n\t\n(e)\t (−13) + 17 = \n\t\n(f)\t 17 + (−13) = \n\t\n(g)\t 15 + 19 = \n\t\n(h)\t 19 + 15 = \n\t\n(i)\t (−21) + (−15) = \n\t\n(j)\t (−15) + (−21) = \nMaths2_Gr7_LB_Book.indb   131\n2014/09/04   11:35:14 AM\n\n132\t MATHEMATICS Grade 7: Term 4\nIn chapter 1 of Book 1 (which was about whole numbers) we said:\nAddition is commutative: the numbers can be swopped around. \nOr, in symbols: a + b = b + a, where a and b are whole numbers.\n2.\t (a)\t Would you say addition is also commutative when the numbers are integers?\n\t\n(b)\t Explain your answer.\n3.\t Calculate the following:\n\t\n(a)\t 9 − 5 = \n\t\n\t\n(b)\t 5 − 9 = \n\t\n(c)\t (−7) − 3 = \n\t\n\t\n(d)\t 3 − (−7) = \n\t\n(e)\t 15 − (−12) = \n\t\n\t\n(f)\t\n(−12) − 15 = \n\t\n(g)\t (−40) − (−23) = \n\t\n\t\n(h)\t (−23) − (−40) = \n4.\t (a)\t Do you think subtraction is commutative? \n\t\n(b)\t Explain your answer.\nIn Book 1, chapter 1 we also said:\nWhen three or more whole numbers are added, the order in which you perform the \ncalculations makes no difference. We say: Addition is associative.\n5.\t Do you think addition is also associative when we work with integers? Investigate.\nMaths2_Gr7_LB_Book.indb   132\n2014/09/04   11:35:14 AM\n\nChapter 9\nNumeric patterns\n\t\nCHAPTER 9: NUMERIC PATTERNS\t\n133\nIn this chapter you will analyse, extend and form number patterns with integers, including \nnegative numbers.\n9.1\t Investigating and extending numeric patterns........................................................ 135\n9.2\t Making patterns from rules..................................................................................... 137\n9.3\t Making patterns from expressions.......................................................................... 138\nMaths2_Gr7_LB_Book.indb   133\n2014/09/04   11:35:15 AM\n\n134\t MATHEMATICS Grade 7: Term 4\n−100\n−91\n−82\n−73\n−64\n−55\n−46\n−37\n−100\n−92\n−84\n−76\n−68\n−60\n−52\n−44\n−100\n−93\n−86\n−79\n−72\n−65\n−58\n−51\n−100\n−94\n−88\n−82\n−76\n−70\n−64\n−58\n−100\n−95\n−90\n−85\n−80\n−75\n−70\n−65\n−100\n−96\n−92\n−88\n−84\n−80\n−76\n−72\n−100\n−97\n−94\n−91\n−88\n−85\n−82\n−79\n−100\n−98\n−96\n−94\n−92\n−90\n−88\n−86\n−100\n−99\n−98\n−97\n−96\n−95\n−94\n−93\n−100\n−100\n−100\n−100\n−100\n−100\n−100\n−100\n−100\n−101\n−102\n−103\n−104\n−105\n−106\n−107\n−100\n−102\n−104\n−106\n−108\n−110\n−112\n−114\n−100\n−103\n−106\n−109\n−112\n−115\n−118\n−121\n−100\n−104\n−108\n−112\n−116\n−120\n−124\n−128\n−100\n−105\n−110\n−115\n−120\n−125\n−130\n−135\n−100\n−106\n−112\n−118\n−124\n−130\n−136\n−142\n−100\n−107\n−114\n−121\n−128\n−135\n−142\n−149\n−100\n−108\n−116\n−124\n−132\n−140\n−148\n−156\nMaths2_Gr7_LB_Book.indb   134\n2014/09/04   11:35:15 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n135\n\t\nCHAPTER 9: NUMERIC PATTERNS\t\n135",
      "chapter_id": "8"
    },
    {
      "title": "Numeric patterns",
      "content": "9\t Numeric patterns\n9.1\t Investigating and extending numeric patterns\npatterns in two directions\n1.\t The numbers in each row of the table form a sequence, but not all the numbers  \nare given. \nA\n4\n6\n8\n10\nB\n10\n8\n6\n4\nC\n5\n8\n11\n14\nD\n20\n17\n13\n8\n\t\n(a)\t Fill in the missing numbers.\n\t\n(b)\t What is the constant difference in sequence A? \n\t\n(c)\t What is the constant difference in sequence C? \n2.\t The first term of a certain sequence is 100 and the constant difference is 20. \n\t\n(a)\t What is the second term, and the third term, and the fourth term? \n\t\n(b)\t What is the 10th term in this sequence? \nA constant-difference sequence is formed by adding \nthe constant difference each time to form the next \nterm.\n3.\t The first term of a certain sequence is 100 and the constant difference is −20. \n\t\n(a)\t What is the second term, and the third term, and the fourth term? \n\t\n(b)\t What is the 10th term in this sequence? \n4.\t (a)\t What is the constant difference in sequence B in question1? \n\t\n(b)\t What is the constant difference in sequence D in question1? \nMaths2_Gr7_LB_Book.indb   135\n2014/09/04   11:35:15 AM\n\n136\t MATHEMATICS Grade 7: Term 4\n5.\t The sixth terms of sequences E, F and G are given in the table. Fill in the other terms.\nTerm number\n1\n2\n3\n4\n5\n6\n7\n8\n9\nE with constant difference 10\n30\nF with constant difference −5\n30\nG with constant difference −10\n30\n6.\t Investigate each of the patterns below. Find the pattern and write the next four  \nterms in the sequence.\n\t\n(a)\t \t\n1\t\n\t\n4\t\n\t\n9\t\n\t\n16\t \t\n25\t \t\n\t\n \n\t\n(b)\t \t\n3\t\n\t\n6\t\n\t\n11\t \t\n18\t \t\n27\t \t\n\t\n \n\t\n(c)\t \t\n20\t \t\n19\t \t\n17\t \t\n14\t \t\n10\t \t\n\t\n \n\t\n(d)\t \t\n20\t \t\n25\t \t\n29\t \t\n32\t \t\n34\t \t\n\t\n \n7.\t Make some numeric patterns of your own.\n\t\n(a)\t\n\t\n(b)\n\t\n(c)\n\t\n(d)\t\n\t\n(e)\n\t\n(f)\n\t\n(g)\n\t\n(h)\nMaths2_Gr7_LB_Book.indb   136\n2014/09/04   11:35:15 AM\n\n\t\nCHAPTER 9: NUMERIC PATTERNS\t\n137\n9.2\t Making patterns from rules\n1.\t (a)\t Start at 30. Add −5 and write the answer. Add −5 again and write the answer.  \n\t\nContinue until you have a number sequence with 10 terms.\n\t\n(b)\t Start at −30. Add −5 and write the answer. Add −5 again and write the answer.  \n\t\nContinue until you have a number sequence with 10 terms.\n\t\n(c)\t Start at −30. Add 5 and write the answer. Add 5 again and write the answer.  \n\t\nContinue until you have a number sequence with 10 terms.\n2.\t (a)\t The first term of a sequence is −10 and there is a constant difference of 5  \n\t\nbetween the terms. Write down the first ten terms of the sequence.\n\t\n(b)\t The first term of a sequence is −10 and there is a constant difference of −5  \n\t\nbetween the terms. Write down the first ten terms of the sequence.\n3.\t Choose a number to be your first term and another number to be a constant \ndifference. Write the first ten terms of your sequence.\n4.\t Choose a number smaller than −10 to be your first term and another number to be a \nconstant difference. Write the first ten terms of your sequence.\n5.\t Choose a number to be your first term and a negative number to be a constant \ndifference. Write the first ten terms of your sequence.\n6.\t Choose a negative number to be your first term and another negative number to be a \nconstant difference. Write the first ten terms of your sequence.\n7.\t Choose a number to be your tenth term and another number to be a constant \ndifference. Write the first ten terms of your sequence.\n8.\t Choose a negative number to be your tenth term and another negative number to be \na constant difference. Write the first ten terms of your sequence.\nMaths2_Gr7_LB_Book.indb   137\n2014/09/04   11:35:15 AM\n\n138\t MATHEMATICS Grade 7: Term 4\n9.3\t Making patterns from expressions\n1.\t (a)\t Complete the table.\nx\n0\n1\n2\n3\n4\n5\n6\n7\n8\n2 × x − 10\n\t\n(b)\t Do the output values of 2 × x − 10 in the above table form a pattern with a  \n\t\nconstant difference? If they do, what is the constant difference?\n\t\n(c)\t Complete the table.\nx\n0\n1\n2\n3\n4\n5\n6\n7\n8\n3 × x − 20\n\t\n(d)\t What is the constant difference in (c)?\n\t\n(e)\t Complete the table.\nx\n0\n1\n2\n3\n4\n5\n6\n7\n8\n2 − 3 × x \n\t\n(f)\t What is the constant difference in (e)?\n\t\n(g)\t Complete the table.\nx\n0\n1\n2\n3\n4\n5\n6\n7\n8\n1 − 2 × x\n\t\n(h)\t What is the constant difference in (g)?\n2.\t Look at the pattern:  −15;  −19;  −23;  −27;  −31;  . . .\t\n\t\nIn this pattern, −19 is followed by −23 and −23 is followed by −27.\n\t\n(a)\t What number in the pattern is followed by −19? \n\t\n(b)\t What number in the pattern is followed by −31? \n  \n\t\n(c)\t In the pattern, −19 follows on −15 and −23 follows on −19.\n\t\n\t\nWhat number follows on −31? \nMaths2_Gr7_LB_Book.indb   138\n2014/09/04   11:35:15 AM\n\n\t\nCHAPTER 9: NUMERIC PATTERNS\t\n139\n3.\t A certain pattern is formed by a common difference of 6.\n\t\n(a)\t What number follows on 23 in this pattern? \n\t\n(b)\t What number is followed by 23 in this pattern? \n\t\n(c)\t What number follows on 47 in this pattern? \n\t\n(d)\t What number is followed by 47 in this pattern? \nConsider the sequence:  10    6    2    −2    −6    . . .    . . .    . . .\t  \nIn this sequence, 2 follows on 6. They are called consecutive terms.\nWhen one number follows another in a sequence \nthey are called consecutive terms.\n4.\t Write down any two consecutive terms in the pattern formed by 2 × x + 3, when the \ninput numbers are consecutive whole numbers.\n5.\t Each of the patterns below was formed by using one of the following expressions. \nEstablish which pattern belongs to each expression.\n\t\n(a)\t 2 × x + 5\t\n\t\n(b)\t 3 × x + 2\t\n\t\n(c)\t 4 × x + 1\t\n\t\n(d)\t 5 × x + 6\t\n\t\n(e)\t 6 × x − 5\t\n\t\n(f)\t 7 × x − 2\t\n\t\n(g)\t 1 − 4 × x\t\n\t\n(h)\t 5 − 5 × x\t\n\t\n(i)\t −5 − 6 × x\t\n\t\nA.\t 6\t\n\t\n11\t \t\n16\t \t\n21\t \t\n26\t \t\n\t\n\t\nExpression \n\t\nB.\t\n13\t \t\n17\t \t\n21\t \t\n25\t \t\n29\t \t\n\t\n\t\nExpression \n\t\nC.\t 20\t \t\n23\t \t\n26\t \t\n29\t \t\n32\t \t\n\t\n\t\nExpression \n\t\nD.\t 1\t\n\t\n−3\t \t\n−7\t       −11        −15\t \t\n\t\n\t\nExpression \n\t\nE.\t\n31\t \t\n33\t \t\n35\t \t\n37\t \t\n39\t \t\n\t\n\t\nExpression \n\t\nF.   −20\t       −25\t      −30\t      −35\t      −40\t \t\n\t\n\t\nExpression \n\t\nG.\t 25\t \t\n31\t \t\n37\t \t\n43\t \t\n49\t \t\n\t\n\t\nExpression \n\t\nH.\t 26\t \t\n33\t \t\n40\t \t\n47\t \t\n54\t \t\n\t\n\t\nExpression \n\t\nI.   −11\t       −17\t      −23\t      −29\t      −35\t \t\n\t\n\t\nExpression \nMaths2_Gr7_LB_Book.indb   139\n2014/09/04   11:35:15 AM\n\n140\t MATHEMATICS Grade 7: Term 4\nSequence I in question 5 is a decreasing sequence; the numbers become smaller as the \nsequence progresses:\n\t\n–11\t\n–17\t\n–23\t\n–29\t\n−35\t\nSequence H is an increasing sequence; each term is bigger than the previous term:\n\t\n 26\t\n  33\t\n  40\t\n  47\t\n  54\t\n\t\n6.\t (a)\t Which sequences in question 5 are increasing sequences? \n\t\n(b)\t Which sequences in question 5 are decreasing sequences? \n7.\t (a)\t By how much does sequence A increase from one term to the next? \n\t\n(b)\t By how much does sequence B increase from one term to the next? \n\t\n(c)\t Which of the sequences in question 5 increases by the biggest amount from  \n\t\none term to the next, and by how much does it increase? \nSequence G increases by 6 from term to term, and sequence E increases only by 2. \nWe may say that sequence G increases faster than sequence E.\n8.\t (a)\t Which of the sequences in question 5 decreases fastest? \n\t\n(b)\t Which of the sequences in question 5 decreases slowest? \n9.\t (a)\t Write 5 consecutive terms of a sequence which decreases faster than  \n\t\nsequence D in question 5. \n\t\n(b)\t Write 5 consecutive terms of a sequence which increases slower than  \n\t\nsequence B in question 5. \n10.\t(a)\t Each of the expressions below can be used to produce a sequence. Which of the \n\t\nexpressions will produce the sequence that increases fastest? \n\t\n\t\n3 × x + 5      2 × x + 10      6 × x – 1      20 + 3 × x      4 × x − 9\n\t\n(b)\t Think of a way in which you can test your answer, and do it.\n11.\tIn each case state whether the sequence will be decreasing or increasing.\n\t\n10 + 3 × x      10 − 3 × x      10 × x + 3      3 × x − 10\nMaths2_Gr7_LB_Book.indb   140\n2014/09/04   11:35:15 AM\n\nChapter 10\nFunctions and \nrelationships 2\n\t\nCHAPTER 10: FUNCTIONS AND RELATIONSHIPS 2\t\n141\nThe way in which an output number can be calculated is called the rule for the  \nrelationship. The rule can be described in words or with a formula, and in some cases  \nwith a flow diagram. \nThe work in this chapter builds on the work that you did last term, in chapter 2.\n10.1\t Relationships between variables.............................................................................. 143\n10.2\t Integers in the rules for relationships....................................................................... 146\nMaths2_Gr7_LB_Book.indb   141\n2014/09/04   11:35:15 AM\n\n142\t MATHEMATICS Grade 7: Term 4\nMaths2_Gr7_LB_Book.indb   142\n2014/09/04   11:35:16 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n143\n\t\nCHAPTER 10: FUNCTIONS AND RELATIONSHIPS 2\t\n143",
      "chapter_id": "9"
    },
    {
      "title": "Functions and relationships 2",
      "content": "10\tFunctions and relationships 2\n10.1\t Relationships between variables\ndifferent ways to represent the rule for a relationship\nA relationship between two variables consists of two sets of numbers as shown in the two \nrows of the table below. The first row contains the input numbers and the second row \ncontains the output numbers. \nx\n1\n2\n3\n4\n5\n6\n7\n8\n9\ny\n32\n39\n46\n53\n60\n67\n74\n81\n88\nFor the relationship shown in the table, any output number can be calculated by \nmultiplying the input number by 7 and adding 25 to the answer.\nThe way in which an output number can be \ncalculated is called the rule for the relationship. \nThe rule can be described in words or with \na formula, and in some cases with a flow \ndiagram.\nThe rule “multiply by 7 and add 25” can be represented with this flow diagram:\n \n× 7\n+ 25\nThe same rule can also be represented with the formula below:\ny = 7 × x + 25\n1.\t Calculate the value of 7 × x + 25 for each of the following values of x:\n\t\n(a)\t x = 10\t\n(b)\t x = 20\n\t\n(c)\t x = 5\t\n(d)\t x = 15\n2.\t (a)\t What is the value of 3 × x − 5 if x = 10?\t\n\t\n(b)\t What is the value of 3 × x − 5 if x = 20?\t\n\t\n(c)\t What is the value of 3 × x − 5 if x = 25?\t\n\t\n(d)\t What is the value of 3 × x − 5 if x = 100?\t\nThe input numbers may also be \ncalled the values of the input \nvariable, and the output \nnumbers may also be called the \nvalues of the output variable.\nMaths2_Gr7_LB_Book.indb   143\n2014/09/04   11:35:16 AM\n\n144\t MATHEMATICS Grade 7: Term 4\n3.\t Complete the table for the values of x and 3 × x − 5 given in the table.\nx\n0\n1\n2\n5\n15\n50\n200\n3 × x − 5\n61\n595\n994\n4.\t When you worked out the input number that corresponds to the output number  \n994 in question 3, you solved the equation 3 × x − 5 = 994.\n\t\nWrite the equation that you solved when you worked out the input number that \ncorresponds to the output number 61.\n5.\t (a)\t Express each of the rules below in words.\n\t\n\t\nA\t\n\t\n\t\n\t\n\t\nB\t\n\t\n\t\n\t\n\t\nC\t\n\t\n\t\n(b)\t Which of the above flow diagrams represent the same calculations as the  \n\t\nexpression 3 × x − 5?    \nInstead of 3 × x − 5 we may write 3x − 5.  \n3x means 3 × x.\nThe multiplication sign can be left out.  \nInstead of 3 × (x − 5) we may write 3(x − 5).\n6.\t (a)\t Which of the formulae below provide the same information as flow diagram B  \n\t\nin question 5? \n\t\n\t\ny = 5x − 3\t \t\n\t\n\t\n\t\ny = 3 + 5x\t \t\n\t\n \t\n\t\ny = 5(x − 3)\n\t\n\t\ny = 3x − 5\t \t\n\t\n\t\n\t\ny = 5x + 3\t \t\n\t\n\t\n\t\ny = 3(x − 5)\n\t\n(b)\t Which of the above formulae provide the same information as flow diagram A  \n\t\nin question 5?\n \n× 5\n+ 3\n \n– 5\n× 3\n \n× 3\n– 5\nMaths2_Gr7_LB_Book.indb   144\n2014/09/04   11:35:17 AM\n\n\t\nCHAPTER 10: FUNCTIONS AND RELATIONSHIPS 2\t\n145\nformulae for tables\n1.\t The table below shows the values of y that correspond to some of the given values  \nof x. In this case, the output numbers form a pattern with a constant difference if  \nthe input numbers are the natural numbers.\nx\n1\n2\n3\n4\n5\n6\n7\n8\ny\n13\n21\n37\n45\n53\n\t\n(a)\t Find the output numbers that correspond to the input numbers 3, 7 and 8.\n\t\n(b)\t Find the output numbers that correspond to the input numbers 20, 21 and 22.\n\t\n(c)\t Which of the formulae below is the rule for the relationship between x and y in \n\t\nthe above table?\n\t\n\t\ny = 10x + 3      y = 8x + 5      y = 6x + 7      y = 4x + 9      y = 2x + 11\n2.\t Complete the tables below for the formulae in question 1(c).\nx\n1\n2\n3\n4\n5\n6\n7\n8\n10x + 3\nx\n1\n2\n3\n4\n5\n6\n7\n8\n8x + 5\nx\n1\n2\n3\n4\n5\n6\n7\n8\n6x + 7\nx\n1\n2\n3\n4\n5\n6\n7\n8\n4x + 9\nx\n1\n2\n3\n4\n5\n6\n7\n8\n2x + 11\nMaths2_Gr7_LB_Book.indb   145\n2014/09/04   11:35:17 AM\n\n146\t MATHEMATICS Grade 7: Term 4\n3.\t In each table in question 2, the output numbers form a number pattern with a \nconstant difference between consecutive terms. What is the constant difference in \nthe pattern generated by each of the following expressions, when the input numbers \nare consecutive natural numbers? Write your answers in the table below. Also fill in \nthe values of the expressions for x = 0 in the last column.\nExpression\nConstant difference between \noutput numbers\nValue of the expression  \nfor x = 0\n2x + 11\n4x + 9\n6x + 7\n8x + 5\n10x + 3\n4.\t What do you think the constant differences between consecutive output numbers, \nand the values of the expressions for x = 0 may be in each of the following cases, \nwhen the input numbers are consecutive natural numbers?\nExpression\nConstant difference between \noutput numbers\nValue of the expression  \nfor x = 0\n5x + 7\n3x + 10\n12x + 5\n5x − 5\n(−10x) + 3\n10.2\tIntegers in the rules for relationships\nrules that may look strange at first\n1.\t Complete the flow diagrams.\n\t\nA.\t 5   \n× 3\n– 5\n \n    \t\n\t\nB.\t 5 \n× 3\n+ (–5)\n \n\t\nC.\t 5   \n– 3\n× 5\n \n\t \t\n\t\nD.\t 5 \n× 5\n+ (–3)\n \n\t\nE.\t\n5 \n× 3\n– (–5)\n \n\t\t\n\t\nF.\t 5   \n× 3\n+ 5\n \n\t\nMaths2_Gr7_LB_Book.indb   146\n2014/09/04   11:35:17 AM\n\n\t\nCHAPTER 10: FUNCTIONS AND RELATIONSHIPS 2\t\n147\n2.\t Describe each rule in question 1 in words, for example \n“multiply by 6 and then add −3”.\n\t\nA.\t\n\t\nB.\t\n\t\nC.\t\n\t\nD.\t\n\t\nE.\t\n\t\nF.\t\nThe rule multiply by 6 and subtract the answer from 100 can be expressed with the formula \ny = 100 − 6x. This formula can also be written as y = 100 + (−6x) or as y = (−6x) + 100. \nThe brackets around the −6x can be left out, so the last formula above can also be  \nwritten as y = −6x + 100.\n3.\t Calculate y if y = −10x + 3, for each of the following values of x:\n\t\n(a)\t x = 5\t\n(b)\t x = 10\n\t\n(c)\t x = 20\t\n(d)\t x = 1\n4.\t Describe each of the rules in question 1 with a formula, for example y = 5x + 8.  \n\t\nA.\t\n\t\nB.\t\n\t\nC.\t\n\t\nD.\t\n\t\nE.\t\n\t\nF.\t\n5.\t In each case below, predict which of the different expressions will produce the  \nsame results. You will test your predictions later, and can then mark your own \nanswers for this question.\n\t\n(a)\t 20 − 5x\t\t\n\t\n5x − 20\t\t\n\t\n(−5x) + 20\t \t\n\t\n20 + (−5x)\t \t\n\t\n\t\n\t\n(b)\t 20 + 5x\t\t\n\t\n5x + 20\t\t\n\t\n20x + 5\t\t\n\t\n\t\n20 − (−5x)\t \t\n\t\n5(x + 4)\n\t\n(c)\t 5x − 20\t\t\n\t\n20x − 5\t\t\n\t\n(−20) − (−5x)\t\n\t\n−((−5x) + 20)\t\n\t\nMaths2_Gr7_LB_Book.indb   147\n2014/09/04   11:35:17 AM\n\n148\t MATHEMATICS Grade 7: Term 4\n6.\t Complete the table below and then use the results to carefully check your answers  \nto question 5.\nx\n0\n1\n5\n10\n100\n20 − 5x\n5x − 20\n(−5x) + 20\n20 + (−5x)\n20 + 5x\n5x + 20\n20x + 5\n20 − (−5x)\n5(x + 4)\n5x − 20\n20x − 5\n(−20) − (−5x)\n−((−5x) + 20)\n7.\t In each case below, use your results in the above table or other methods to establish \nfor which values of x the two expressions have the same value(s).\n\t\n(a)\t 20 − 5x   and   20 + 5x\n\t\n(b)\t 20 − 5x   and   (−5x) + 20\n\t\n(c)\t 5x − 20   and   (−20) − (−5x)\n\t\n(d)\t 5(x + 4)   and   5x − 20\n\t\n(e)\t 20 + 5x   and   20 − (−5x)\nMaths2_Gr7_LB_Book.indb   148\n2014/09/04   11:35:18 AM\n\nChapter 11\nAlgebraic expressions 2\n\t\nCHAPTER 11: ALGEBRAIC EXPRESSIONS 2\t\n149\nYou already know that an algebraic expression is a computational procedure. It tells you \nwhat calculations you must do with the value of one variable, to produce the value of \nanother variable. In this chapter, we extend the work you have already done to include \nalgebraic expressions with integer constants, including negative numbers.\n11.1\t Interpret rules to calculate values of a variable........................................................ 151\n11.2\t Slightly different kinds of rules................................................................................ 154\nMaths2_Gr7_LB_Book.indb   149\n2014/09/04   11:35:18 AM\n\n150\t MATHEMATICS Grade 7: Term 4\n45\n−228\n41\n−208\n37\n−188\n33\n−168\n29\n−148\n25\n−128\n(−3) −5 × x\n=\n21\n−108\n17\n−88\n13\n−68\n9\n−48\n5\n−28\n1\n−8\nMaths2_Gr7_LB_Book.indb   150\n2014/09/04   11:35:18 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n151\n\t\nCHAPTER 11: ALGEBRAIC EXPRESSIONS 2\t\n151",
      "chapter_id": "10"
    },
    {
      "title": "Algebraic expressions 2",
      "content": "11\tAlgebraic expressions 2\n11.1\t Interpret rules to calculate values of a variable\nrules in verbal and symbolic form\n1.\t Do this to each of the numbers in the top row of the table, and write your answers  \nin the bottom row: multiply the input number by 20 and add 50 to the answer.\nx\n1\n2\n3\n4\n5\n6\n7\n8\n9\ny\nThe sentence multiply the input number by 20 and add 50 to the answer is the rule that \ndescribes how the output number that corresponds to each input number in the above \nrelationship between the variables x and y can be calculated.\nThe same rule can be described with the algebraic  \nexpression 20x + 50. In this expression, the symbol  \nx represents the input variable (the values of x). The  \nnumbers 20 and 50 are constant; they remain the  \nsame for all the different values of x.\nThe rule add 50 to the input number and multiply the  \nanswer by 20 can be described with the expression  \n20(x + 50).\n2.\t Describe each of the following rules in words.\n\t\n(a)\t 15x + 30\t\n\t\n(b)\t 30 + 15x\t\n\t\n(c)\t 15(x + 30)\t\n\t\n(d)\t 15(x + 2)\t\n\t\n(e)\t 15x − 30\t\n\t\n(f)\t 15(x − 30)\t\n\t\n(g)\t 15(x − 2)\t\n3.\t What is the difference between 3(x + 5) and 3x + 5?\nIf there are no brackets in an \nexpression, multiplication is \ndone first, even if it appears \nlater in the expression like in  \n30 + 5x.\nIf there are brackets in an \nalgebraic expression, the \noperations in brackets are to \nbe done first.\nMaths2_Gr7_LB_Book.indb   151\n2014/09/04   11:35:18 AM\n\n152\t MATHEMATICS Grade 7: Term 4\n4.\t Complete the table.\nx\n1\n2\n3\n4\n5\n6\n7\n8\n9\n15x + 30\n30 + 15x\n15(x + 30)\n15(x + 2)\n5.\t Complete the table.\nx\n30\n40\n50\n60\n70\n80\n90\n15x − 30\n15(x − 30)\n15(x − 2)\n6.\t (a)\t Investigate which of the following rules will produce the same output numbers.  \n\t\nYou need to check for several different input numbers.\n\t\n\t\nA:\t Multiply the input number by 10 and then add 20.\n\t\n\t\nB:\t Add 20 to the input number and then multiply by 10.\n\t\n\t\nC:\t Add 2 to the input number and then multiply by 10.\n\t\n\t\nD:\t Multiply the input number by 3, add 15, add 7 times the input number,  \n\t\n\t\nand then add 5.\nx\nA\nB\nC\nD\n\t\n(b)\t Describe each of the above rules with an algebraic expression.\n\t\n\t\nA:\t\n\t\n\t\nB:\t\n \n\t\n\t\nC:\t\n \n\t\n\t\nD:\t\nMaths2_Gr7_LB_Book.indb   152\n2014/09/04   11:35:18 AM\n\n\t\nCHAPTER 11: ALGEBRAIC EXPRESSIONS 2\t\n153\n7.\t (a)\t Which of these rules do you think will produce the same output numbers?\n\t\n\t\nA: 5x + 20\t \t\n\t\n\t\nB: 4x + 19\t \t\n\t\n\t\nC: 5(x + 20)\t\n\t\n\t\nD: 20 + 5x\t \t\n\t\n\t\nE: 5(x + 4)\t \t\n\t\n\t\nF: 3x + 7 + 2x + 13\n\t\n(b)\t Express each of the above rules in words.\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(c)\t Complete this table for the rules given in (a).\nx\n0\n5\n10\n15\n5x + 20\n4x + 19\n5(x + 20)\n20 + 5x\n5(x + 4)\n3x + 7 + 2x + 13\n\t\n(d)\t Use your completed table to check your answer in question (a).\n8.\t (a)\t Which of these rules do you think will produce the same output numbers?\n\t\n\t\nA: 5x − 20\t \t\n\t\n\t\nB: 20 − 5x\t \t\n\t\n\t\nC: 5(x − 20)\t\n\t\n\t\nD: 3x − 18\t \t\n\t\n\t\nE: 5(x − 4)\t \t\n\t\n\t\nF: 9x + 10 − 4x − 30\n\t\n(b)\t Express each of the above rules in words.\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\nMaths2_Gr7_LB_Book.indb   153\n2014/09/04   11:35:18 AM\n\n154\t MATHEMATICS Grade 7: Term 4\n\t\n(c)\t Complete this table for the rules given in (a).\nx\n20\n30\n40\n50\n60\n70\n80\n90\n5x − 20\n20 − 5x\n5(x − 20)\n3x − 18\n5(x − 4)\n9x + 10 − 4x − 30\n\t\n(d)\t Use your completed table to check your answer to question (a).\n11.2\t Slightly different kinds of rules\nsubtract positive and negative quantities\n1.\t Complete the table.\nx\n1\n10\n5\n20\n25\n10x\n50 − 10x\n20 − 10x\n0 − 10x\n2.\t (a)\t Complete the table. \nx\n0\n5\n10\n15\n20\n25\n30\n10x − 5\n5x − 10\n100 − 5x\n−100 + 5x\n5x − 100\n5 − 10x\n\t\n(b)\t The values of 10x − 5 increase as the values of x increase from 0 to 30.  \n\t\nFor which expressions in (a) do the values decrease when x is increased?\n\t\n(c)\t Do the values of −100 + 5x increase or decrease when x is increased from 0 to 30?\nMaths2_Gr7_LB_Book.indb   154\n2014/09/04   11:35:18 AM\n\n\t\nCHAPTER 11: ALGEBRAIC EXPRESSIONS 2\t\n155\n3.\t (a)\t The values of the expression 5x − 10 increase when x is increased from 0 to 30. \n\t\n\t\nDo you think the values will increase further when x is increased beyond 30, \n\t\n\t\nor will they start to decrease at some stage? \n\t\n(b)\t Do you think the values of the expression 100 − 3x will increase when x is  \n\t\nincreased from 0 to 30? Explain why you think they will or will not.\nThe additive inverse of a number may be indicated by writing a negative sign before the \nnumber. For example, the additive inverse of I can be written as −8.\n4.\t Write the additive inverse of each of the  \nfollowing numbers: \n\t\n20\t\n30\t\n−25\t\n−20\t\n40\n5.\t Different values for x are given in the first row of the table below. Write the additive \ninverses of the x values in the second row, and then complete the table.\nx\n5\n10\n15\n20\n25\n30\nthe additive inverse of x\n20 + (the additive inverse of x)\n20 − (the additive inverse of x)\n20 + x\n20 − x\n6.\t Complete the table.\nx\n−5\n−10\n−15\n−20\n−25\n−30\nthe additive inverse of x\n20 + (the additive inverse of x)\n20 − (the additive inverse of x)\n20 + x\n20 − x\n7.\t Complete the table.\nx\n3\n2\n1\n0\n−1\n−2\n−3\n−x\n5 + (−x)\n5 − (−x)\n5 − x\n5 + x\nWhen a number is added to \nthe number called its additive \ninverse, the answer is 0. For \nexample, 45 + (−45) = 0 and \n(−12) + 12 = 0.\nMaths2_Gr7_LB_Book.indb   155\n2014/09/04   11:35:18 AM\n\n156\t MATHEMATICS Grade 7: Term 4\nexpressions with additive inverses\n1.\t Complete the table. \nx\n1\n5\n10\n20\n25\n5x\nthe additive inverse of 5x\n20 + (the additive inverse of 5x)\n20 − (the additive inverse of 5x)\n3x\n−3x\n10 + (−3x)\n10 − 3x\n10 − (−3x)\n2.\t Complete the table below.  \nNote that (−10x) indicates the additive inverse of 10x.\nx\n1\n2\n3\n4\n−4\n−3\n−2\n10x − 1 000\n1 000 − (−10x)\n1 000 − 10x\n(−10x) + 1 000\n10x + 1 000\n10x + (−1 000)\n(−10x) − 1 000\n1 000 + (−10x)\n1 000 + 10x\n10x − (+1 000)\nInstead of (−10x) − 1 000 we may write −10x − 1 000, in other words the brackets around \nthe additive inverse may be left out. \nSimilarly, (−10x) + 1 000 may be written as −10x + 1 000.\n3.\t Complete the table. \nx\n1\n5\n10\n20\n25\n30\n−5x + 20\n−5x + (−20)\nMaths2_Gr7_LB_Book.indb   156\n2014/09/04   11:35:18 AM\n\nChapter 12\nAlgebraic equations 2\n\t\nCHAPTER 12: ALGEBRAIC EQUATIONS 2\t\n157\nYou have already done some work on equations in Term 3. In this term, we extend the \nwork you have already done to include negative numbers.\n12.1\t Describing problem situations................................................................................. 159\n12.2\t Analysing and interpreting equations...................................................................... 160\n12.3\t Solving and completing equations.......................................................................... 161\n12.4\t Identifying variables and constants......................................................................... 165\n12.5\t Numerical values of expressions.............................................................................. 166\nMaths2_Gr7_LB_Book.indb   157\n2014/09/04   11:35:18 AM\n\n158\t MATHEMATICS Grade 7: Term 4\n5\n×\n21\n− 3\n=\n102\n5\n×\n?\n− 3\n=\n97\n5\n×\n?\n− 3\n=\n92\n5\n×\n?\n− 3\n=\n87\n5\n×\n?\n− 3\n=\n82\n5\n×\nx\n− 3\n=\n77\n5\n×\n?\n− 3\n=\n72\n5\n×\n?\n− 3\n=\n67\n5\n×\n?\n− 3\n=\n62\n5\n×\n?\n− 3\n=\n57\n5\n×\n?\n− 3\n=\n52\n5\n×\n?\n− 3\n=\n47\nMaths2_Gr7_LB_Book.indb   158\n2014/09/04   11:35:19 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n159\n\t\nCHAPTER 12: ALGEBRAIC EQUATIONS 2\t\n159",
      "chapter_id": "11"
    },
    {
      "title": "Algebraic equations 2",
      "content": "12\tAlgebraic equations 2\n12.1\t Describing problem situations\nA closed number sentence is a true statement about numbers, for example  \n21 + 5 = 26. All the numbers are given. \nIn an open number sentence, for example  \n15 + x = 21, one or more of the numbers are  \nunknown.\n1.\t Jan is 3 years older than his sister Amanda. Amanda is 14 years old. Write a closed \nnumber sentence to show Jan’s age.\n2.\t Numbers are said to be consecutive if they follow one another. The numbers  \n−1,  0,  1 are consecutive. The sum of −1,  0  and 1 is 0.\n\t\n(a)\t Write a closed number sentence that shows two consecutive numbers that  \n\t\nadd up to −33.\n\t\n(b)\t Write a closed number sentence that shows two consecutive numbers whose  \n\t\nproduct is 6.\n3.\t A cell phone costs R500 after a discount of R150 is given. Write a closed number \nsentence to show the original price of the cell phone.\n4.\t When the bus leaves the terminal, it is carrying 55 people. At the first bus stop 12 \npeople get off the bus and 9 people get in. At the second bus stop, 12 people get in \nand 9 people get off the bus. Write a closed number sentence to show the number  \nof people that are now in the bus.\n5.\t A rectangle is shown on the right.  \nWrite a closed number sentence to calculate  \nthe following:\n\t\n(a)\t the area of the rectangle \t \t\n\t\n \n\t\n(b)\t the perimeter of the rectangle\t\t\nAn open number sentence is \nalso called an equation.\n6 cm\n2 cm\nMaths2_Gr7_LB_Book.indb   159\n2014/09/04   11:35:19 AM\n\n160\t MATHEMATICS Grade 7: Term 4\n12.2\t Analysing and interpreting equations\n1.\t The cost of a school uniform in rands is represented by x. An alteration fee of R20 is \nalso charged. Mr Malan paid R520 for both the school uniform and the alterations \t\ndone on it. \n\t\n(a)\t Which equation describes this situation?\n\t\n\t\nA.\t 20 × x = 520\t\nB.\t x − 20 = 520\t\nC.\t x + 20 = 520\t\nD.\t 20 + 20 = x\n\t\n(b)\t What is the price of the uniform? \n2.\t Five learners should each receive the same number of sweets. There are 60 sweets in \ntotal that they have to share.\n\t\n(a)\t Which equation describes this situation?\n\t\n\t\nA.\t 5 + s = 60\t \t\nB.\t 5s = 60\t\t\n\t\nC.\t s − 5 = 60\t \t\nD.\t s\n5 = 60 \n\t\n(b)\t How many sweets does each learner get?\n\t\n(c)\t What does the letter s represent in the equation you have chosen?\n3.\t A taxi picks up n passengers at the airport and drives to the nearest hotel. When it \nleaves the hotel, the number of passengers in the taxi has decreased by 6. There are \nnow 7 passengers in the taxi.\n\t\n(a)\t Which equation describes this situation?\n\t\n\t\nA.\t n − 6 = 7\t\n\t\nB.\t 7 − n = 6\t\n\t\nC.\t n + 6 = 7\t\n\t\nD.\t n − 7 = 6\n\t\n(b)\t How many passengers were in the taxi when it left the airport?\n4.\t Write a closed number sentence for calculating the  \nperimeter of an equilateral triangle whose sides are  \n5 cm long.\n5.\t Write a closed number sentence to calculate  \nthe perimeter of the triangle shown on the right.\nRemember: An equilateral \ntriangle is a triangle in which \nall three sides are equal.\n3 cm\n4 cm\n5 cm\nMaths2_Gr7_LB_Book.indb   160\n2014/09/04   11:35:19 AM\n\n\t\nCHAPTER 12: ALGEBRAIC EQUATIONS 2\t\n161\n12.3\t Solving and completing equations\t\nsolve by inspection\n1.\t The number sentences given below are not true. Make the number sentences true by \nchanging the numbers in blue.\n\t\n(a)\t 13 + 7 = 22\t\n(b)\t 50 + (−50) = −100\t\n(c)\t 7 × 8 = 54\n\t\n(d)\t 9 − (−3) = 6\t\n(e)\t −5 + 12 = −7\t\n(f)\t 4 × 6 = 28\n\t\n(g)\t 6 − 9 = 3\t\n(h)\t 9 − 6 = −3\t\n(i)\t 5 + (−12) = 7\n\t\n(j)\t 10 + (−2) = 12\t\n(k)\t (−1) − (−1) = −2\t\n(l)\t 0 + (−2) = 0\n2.\t Consider the equations given below. Check whether the value given in brackets is  \nthe solution. Simply write yes or no with an explanation.\n\t\n(a)\t x + 3 = 0  (x = −3)\n\t\n(b)\t 3 − x = 4  (x = 1)\n\t\n(c)\t −5 + x + x = −11  (x = −2)\n\t\n(d)\t 3 − x = 4  (x = −1)\n3.\t Find the value of the unknown that makes the equation true in each case:\n\t\n(a)\t x  +  6 = 8\t\n(b)\t x  +  6 =  4\t\n(c)\t x  +  6 = 0\n\t\n(d)\t 6 − x = 8\t\n(e)\t 6 − x = 4\t\n(f)\t 6 − x = 0\n\t\n(g)\t x\n4 = 2 \t\n(h)\t x = 4 × 2\t\n(i)\t\nx\n2 = 1\n4\nTo check whether a given \nvalue is the solution or not \nwe have to answer the \nfollowing question in our \nminds: Does the given \nvalue make the equation \ntrue? If it does, we say such \na value is the solution.\nMaths2_Gr7_LB_Book.indb   161\n2014/09/04   11:35:20 AM\n\n162\t MATHEMATICS Grade 7: Term 4\n4.\t Three possible solutions are given in brackets below each equation, but only one is \ncorrect. Find the correct solution in each case.\n\t\n(a)\t x + 27 = 27\t\n(b)\t 12 = 4 − x\t\n(c)\t x + 3 = 0\n\t\n\t\n{−27; 0; 1}\t\n\t\n{8; 16; −8}\t\n\t\n{−3; 0; 3}\n\t\n(d)\t 5 − x = 10\t\n(e)\t 5 + x = 10\t\n(f)\t −5 + x = 10\n\t\n\t\n{−5; 0; 5}\t\n\t\n{−5; 0 ; 5}\t\n\t\n{−5; −15; 15}\n\t\n(g)\t −5 − x = 10 \t\n(h)\t −5 − x = 0 \t\n(i)\t 5 − x = −10 \n\t\n\t\n{−5; −15; 15}\t\n\t\n{−5; −15; 15}\t\n\t\n{−5; −15; 15}\n\t\n(j)\t x = 10\n10 \t\n(k)\t 10x = 0\t\n(l)\t\nx\n10 = 0 \n\t\n\t\n{0; 1; 100}\t\n\t\n{0; 1; 1\n10}\t\n\t\n{0; 1; 10}\n5.\t What value for x would make each equation below true?\n\t\n(a)\t Let x = .... then x + 3 = 10\t\n(b)\t Let x = …. then x + 3 = −4\n\t\n(c)\t x + x + x = −6 is true for x = ….\t\n(d)\t x + x + x + x = −8 is true for x = ….\n6.\t In each case, fill in the table until you can see for what value of x the equation  \ngiven above the table is true. You may add more x values of your own choice. To  \nsave time and work, you may skip columns that you think will not help you to find \nthe solution.\n\t\n(a)\t 37 − 4x = 5\nx\n1\n10\n5\n6\n7\n37 − 4x\n\t\n(b)\t 50 − 7x = 22\nx\n1\n10\n5\n6\n50 − 7x\n\t\n(c)\t 100 − 3x = 49\nx\n10\n20\n25\n15\n16\n100 − 3x\n\t\nMaths2_Gr7_LB_Book.indb   162\n2014/09/04   11:35:20 AM\n\n\t\nCHAPTER 12: ALGEBRAIC EQUATIONS 2\t\n163\nsolve by trial and improvement\nWe can think of an equation as a question asking \nfor a value that we can assign to the unknown  \nto make the equation true.\nConsider the equation 82 + m = 23. We need to assign values to m until we find a value \nthat makes the equation true, as shown in the table below.\nEquation\nTrue/False\nLet m = −50\n82 + (−50) = 82 − 50 = 32\nFalse\nLet m = −30\n82 + (−30) = 82 − 30 = 52\nFalse\nLet m = −60\n82 + (−60) = 82 − 60 = 22\nFalse\nLet m = −59\n82 + (−59) = 82 − 59 = 23\nTrue\nSo m = −59 because 82 + (−59) = 82 − 59 = 23\n1.\t Determine the value of t that makes the equation 28 − t = 82 true by making use of \nthe trial and improvement method.\nEquation\nTrue/False\n\t\nSolution: \n2.\t Consider the equation w + 32 = −68. Use the trial and improvement method to find \nthe solution of the equation.\nEquation\nTrue/False\n\t\nSolution: \nMaths2_Gr7_LB_Book.indb   163\n2014/09/04   11:35:20 AM\n\n164\t MATHEMATICS Grade 7: Term 4\n3.\t The equation 200 − 5t = 110 is given. What value of t makes the equation true?  \nUse the table below to determine the solution.\nEquation\nTrue/False\n\t\nSolution: \n4.\t What value of p makes the equation 18p = 90 true?\nEquation\nTrue/False\n\t\nSolution: \n5.\t What value of x makes the equation 88 − 6x = 46 true?\nEquation\nTrue/False\n\t\nSolution: \nMaths2_Gr7_LB_Book.indb   164\n2014/09/04   11:35:20 AM\n\n\t\nCHAPTER 12: ALGEBRAIC EQUATIONS 2\t\n165\n12.4\t Identifying variables and constants\n1.\t The mass of an empty truck is 2 680 kg. The truck is used to transport cement.  \nEach pocket of cement has a mass of 90 kg. \n\t\nThe combined mass of the truck and the cement can be calculated by means of  \nthe formula: y = 90 × x + 2 680.\n\t\nUse the terms variable or constant to describe the meaning of each symbol  \nused in the formula. Explain your answer. \n\t\n(a)\t y\t\n\t\n\t\n\t\n(b)\t 90\t\t\n\t\n\t\n(c)\t x\t\n\t\n\t\n\t\n(d)\t 2 680\n2.\t A steel spring is suspended from a stand. Mass pieces of equal  \nmass are hooked onto the bottom end of the spring. The length  \nof the spring is measured with 1 mass piece hooked, 2 mass  \npieces hooked, 3 mass pieces hooked and so on. The results are  \nshown in the table below.\nNumber of mass pieces\n1\n2\n3\n4\n5\n7\n10\nLength of spring in cm\n48\n56\n64\n72\n80\n96\n120\n\t\nThe formula y = 8x + 40 is used to predict the length of the spring for the various \nnumber of mass pieces hooked.\n\t\nUse the terms variable or constant to describe each symbol used in the formula. \nExplain your answer.\n\t\n(a)\t y\t\n\t\n\t\n\t\n(b)\t 8\t \t\n\t\n\t\n(c)\t x\t\n\t\n\t\n\t\n(d)\t 40\n \n0\n1\n2\n3\n4\n5\n6\nMaths2_Gr7_LB_Book.indb   165\n2014/09/04   11:35:21 AM\n\n166\t MATHEMATICS Grade 7: Term 4\n12.5\t Numerical values of expressions\nsubstituting numbers into expressions\n1.\t (a)\t Calculate the values of each expression for the given values of x, and write your \n\t\nanswers in the table. \nx\n0\n2\n5\n10\n20\n50\n100\n100 − 9x\n100 − 8x\n100 − 7x\n100 − 6x\n100 − 5x\n100 − 4x\n100 − 3x\n\t\n(b)\t Which sequence in the above table decreases fastest, and which sequence  \n\t\ndecreases slowest?\n2.\t (a)\t Complete the table.\nx\n1\n2\n3\n4\n5\n6\n7\n2x + 3\n3x − 3\n3x − 2\n3x − 1\n\t\n(b)\t For which value of x is 2x + 3 equal to 3x − 1? \n\t\n(c)\t For which values of x is 2x + 3 smaller than 3x − 1?   \n\t\n(d)\t Do you think 2x + 3 is smaller than 3x − 1 for all values of x greater than 4?  \n\t\nYou may try a few numbers to help you think about this.\n\t\n(e)\t Which sequence increases fastest, the sequence generated by 2x + 3 or the  \n\t\nsequence generated by 3x − 3?\nMaths2_Gr7_LB_Book.indb   166\n2014/09/04   11:35:21 AM\n\nChapter 13\nCollect, organise and\nsummarise data\n\tCHAPTER 13: COLLECT, ORGANISE AND SUMMARISE DATA\t\n167\nData handling is the part of Mathematics that deals with numbers and facts that we  \ncollect about the world around us. Data can be many different things, for example  \npeople’s opinions on politics or the success rates of treating people with a certain kind  \nof medicine. We use data to help us make decisions and solve problems about the  \nworld around us.\nIn this chapter you will focus on: collecting data using questionnaires; organising  \ndata, which includes using stem-and-leaf displays and grouping data into intervals;  \nand then summarising data by determining the mode, median, mean and range of sets  \nof numerical data.\n13.1\t Collecting data....................................................................................................... 169\n13.2\t Organising data...................................................................................................... 174\n13.3\t Summarising data................................................................................................... 184\nMaths2_Gr7_LB_Book.indb   167\n2014/09/04   11:35:21 AM\n\n168\t MATHEMATICS Grade 7: Term 4\nCollect\ndata\nPose a\nquestion\nRepresent \nthe data\nOrganise \nthe data\nInterpret \nand analyse \nthe data\nReport on \nthe data\nThe data\ncycle\nMaths2_Gr7_LB_Book.indb   168\n2014/09/04   11:35:21 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n169\n\tCHAPTER 13: COLLECT, ORGANISE AND SUMMARISE DATA\t\n169\n13\tCollect, organise and summarise\n\t\n\t data\n13.1\t Collecting data\nThink of something that you really want to know about your own community or  \nabout children your age in other schools. For example, “How many Grade 7 learners \nin South Africa have access to a computer?” What would you find interesting to know \nabout?\nWhen you start the cycle of data handling, you start with at least one question. But \nthere can of course be many more questions. \nOnce you have a research idea in mind, you can start planning how you will collect the \ndata. When you collect data, you need to consider:\n• what question you are asking\n• where you will find the data to answer the question (for example from people such \nas your peers, family or the wider community; or from published sources such as \nnewspapers, books or magazines)\n• how you will collect the data (for example by using questionnaires or conducting \ninterviews)\n• who you will collect the data from (the entire population or a sample).\npopulations and samples: from whom to collect data\nIn data handling, population refers to the whole \ngroup you are asking the question about.  \nSample refers to a small number of the group that \nyou think will represent the whole group.\nHere is an example: Thandeka wants to know about the home languages of all Grade 7s  \nacross the whole of South Africa. All Grade 7s in all of South Africa would be the \npopulation of that data. But it is not possible to reach every single Grade 7 learner in \nSouth Africa, so Thandeka could choose a sample of Grade 7 learners. For example,  \nshe could choose to collect data from her own Grade 7 class and from two other Grade 7 \nclasses from two other schools. \nBut if Thandeka chose her own Grade 7 class and only two other classes from other \nschools, her sample would not really give information about learners across the whole  \nof South Africa, because the learners in all three of the schools could be from the same \nlanguage group. \nMaths2_Gr7_LB_Book.indb   169\n2014/09/04   11:35:21 AM\n\n170\t MATHEMATICS Grade 7: Term 4\nSo, how can you try to make sure that a sample gives information about the whole \npopulation? In other words, how can you make sure that your sample is representative \nof the population?\n1.\t Choose a big enough sample. Generally, the bigger the sample is the more likely it is to \nrepresent the characteristics of the population.\n2.\t Ensure that you do not take a sample from only one of the groups within the population.  \nFor example, if you want to find out if people like watching soccer, you cannot survey \npeople at a Chiefs versus Pirates match. The majority of these people will almost \ncertainly be there because they love watching soccer!\nExample\nGanief wishes to find out if learners at his school like the style and colour of their school \nuniform and surveys 10 learners in Grade 7. There are 2 000 learners at the school.\nGive two reasons to explain why the sample chosen is not likely to be representative  \nof the population.\nAnswer\n1.\t The sample is too small.\n2.\t He is only getting the views of Grade 7s, not of the learners in any of the other grades \n(who might have very different views).\nthinking about populations and samples\n1.\t Here are some research questions. Use a P to show which statement describes  \nthe population and an S to show which statement describes a sample of the \npopulation.\n\t\n(a)\t What percentage of plants in the vegetable patch is affected by disease?\n\t\n\t\n  All the plants in the vegetable patch\t\n\t\n\t\n  Every fourth or fifth plant in the vegetable patch\n\t\n(b)\t How often do teenagers recycle plastic?\n\t\n\t\n  Every teenager in South Africa\t\n\t\n\t\n  About 40 teenagers in the community\t\n\t\n(c)\t How many hours of sleep do 10-year-olds in my community get per night?\n\t\n\t\n  All 10-year-olds in the community\t\n\t\n\t\n  About ten 10-year-olds in the community\nMaths2_Gr7_LB_Book.indb   170\n2014/09/04   11:35:21 AM\n\n\tCHAPTER 13: COLLECT, ORGANISE AND SUMMARISE DATA\t\n171\n2.\t You want to know the most popular colour of the learners in your school.\n\t\n(a)\t Write down the population of your data collection.\n\t\n(b)\t Write down what sample you would use.\n3.\t Census@School took place in 2001 and 2009. These were surveys that Statistics \nSouth Africa did to show learners how information about people is collected and \nanalysed. The Census@School wanted to know personal, community and household \ninformation about learners from Grades 3 to 12. This is how they chose their sample:\n• A sample of 2 500 schools was selected from the Department of Basic Education’s \ndatabase of approximately 26 000 registered schools.\n• The schools were divided into groups depending on their province, school type (primary: \nGrades 3 to 7 only; intermediate: Grades 5 to 9 only; secondary: Grades 8 to 12 only; \ncombined: Grades 3 to 12), and education district.\n• A sample of schools was selected from each of these groups.\n• Approximately 790 000 learners participated in the Census@School 2009. \n\t\nThis information was included in their final report. \n\t\n(a)\t What percentage was the sample of all the schools in the country?\n\t\n(b)\t Why do you think they separated the schools into groups first?\n\t\n(c)\t Do you think the information that they obtained from this survey would be  \n\t\ninteresting to you? Explain.\n4.\t Unathi goes to River View Girls’ Primary School. She wants to find out whether \n13-year-olds in her town prefer rugby or netball. She surveys 10 learners from \neach of the three Grade 7 classes at her school. Is the sample chosen likely to be \nrepresentative of the population (13-year-olds in her town)? Explain your answer. \nMaths2_Gr7_LB_Book.indb   171\n2014/09/04   11:35:21 AM\n\n172\t MATHEMATICS Grade 7: Term 4\nconstructing questionnaires: how to collect data\nA questionnaire is a sheet with questions used  \nto collect data from people. Each respondent in  \nthe sample completes a questionnaire. The \nquestions on the sheet can be structured  \ndifferently, for example:\n• The questions may require “yes” or “no” answers.\n• A selection of answers (multiple-choice answers) \nmay be provided for respondents to choose from.\n• The respondents may enter their own views or \ninformation on the questionnaire.\nThe type of responses you need (for example a simple “yes” or “no” or more detailed \ninformation) depends on the data you intend to collect. \nLook at the examples below. Notice how each question is worded to be as clear as  \npossible and to allow the data to be collected easily. (The questions in examples 4 and 5 \nwere used by Census@School in their 2009 questionnaire.)\nExample 1\t\nExample 2\nDo you help with chores at home?\n   Yes\t\t\n   No\nWhich of these chores do you help with?\n   cleaning dishes\t\n\t\n   washing clothes\n   sweeping/vacuuming\t\n   making beds\nExample 3\nHow old are you?\n   5–8 years\t \t\n   9–11 years\n   12–15 years\t\n   16–19 years\nExample 4\n14.  Tick the box if you have:\n1 \n   Running water inside home\t\n\t\n6 \n   A cell phone\n2 \n   Electricity inside home\t \t\n\t\n7 \n   Access to a computer\n3 \n   A radio at home\t\n\t\n\t\n8 \n   Access to the internet\n4 \n   A TV at home\t \t\n\t\n\t\n9 \n   Access to a library\n5 \n   A telephone at home\nA respondent is a person \nwho fills in a questionnaire or \nfrom whom you collect data.\nNote in example 3 how all the \nages from 5 to 19 are covered, \nbut without any overlaps.\nMaths2_Gr7_LB_Book.indb   172\n2014/09/04   11:35:21 AM\n\n\tCHAPTER 13: COLLECT, ORGANISE AND SUMMARISE DATA\t\n173\nExample 5\n6.\t How tall are you without your shoes on? Answer to the nearest cm.\n\t\n \n \n centimetres\n7.\t What is the length of your right foot, without a shoe? Answer to the  \n\t\nnearest cm.\n\t\n \n \n centimetres\n8.\t What is your arm span? (Open arms wide, measure the distance across  \n\t\nyour back from the tip of your right hand middle finger to the tip of  \n\t\nyour left hand middle finger.) Answer to the nearest cm.\n\t\n \n \n centimetres\nmaking questionnaires\n1.\t (a)\t Refue wants to find out how much pocket money learners in her class receive  \n\t\neach month. She draws up the following multiple-choice question:\nHow much pocket money do you get?\n  0–10           \n 10–20           \n 20–30           \n 30–40\n\t\n\t\nExplain why this question is not clear. Give at least three reasons.\n\t\n(b)\t Draw up the multiple-choice question so that it will allow Refue to collect the  \n\t\ndata that she needs.  \t \t\n\t\nMaths2_Gr7_LB_Book.indb   173\n2014/09/04   11:35:21 AM\n\n174\t MATHEMATICS Grade 7: Term 4\n2.\t You want to find out which sports learners at your school play.\n\t\n(a)\t Describe the population of your data.\n\t\n(b)\t Describe the sample you will use.\n3.\t Make a question with yes/no or multiple-choice responses to help you collect the  \n\t\ndata you need:\n\t\n\t\n4.\t Collect your data from your population or the sample you chose. Keep your data for \nthe next chapter.\n13.2\t Organising data\nTo organise data that we have collected, we can use tally marks and tables, dot plots, and \nstem-and-leaf displays. We can also group the data when there are many data values. The \nways that we organise the data depends on the type of data we collected.\ndifferent types of data\nLook at the five examples of questions for questionnaires on pages 172 and 173.\n1.\t Which of the examples will give you data that looks like this?\n\t\nYes\n1 235 learners\nNo\n1 265 learners\nMaths2_Gr7_LB_Book.indb   174\n2014/09/04   11:35:21 AM\n\n\tCHAPTER 13: COLLECT, ORGANISE AND SUMMARISE DATA\t\n175\n2.\t Which of the examples might give you data that looks like this?\n\t\n132 cm; 141 cm; 160 cm; 132 cm; 154 cm; 145 cm; 147 cm; 129 cm; 121 cm;  \n143 cm; 135 cm; 154 cm; 156 cm; 133 cm; 156 cm; 123 cm; 137 cm etc.\n3.\t What could the data for example 4 look like? Fill in this table to give a possible \nexample for 30 learners. Use numbers that you have made up. \nNumber of learners\n4.\t Which of the examples might give you a data set that looks like this?\t\n5–8 years\n15\n9–11 years\n45\n12–15 years\n32\n16–19 years\n28\nThe type of data in questions 1 and 3 is called \ncategorical data. This is often described by  \nwords. The categories don’t have to be given in  \norder. \nThe type of data in questions 2 and 4 is called  \nnumerical data. Numerical data can be whole \nnumbers only, or it can include fractions.\nFor both of these kinds of data, your results give you a list of responses. You will soon \nlearn how to organise these responses.\nMaths2_Gr7_LB_Book.indb   175\n2014/09/04   11:35:21 AM\n\n176\t MATHEMATICS Grade 7: Term 4\n5.\t Classify the following data sets as categorical or numerical.\n\t\n(a)\t the number of pages in books\t\t\n\t\n\t\n\t\n\t\n\t\n(b)\t the length of learners’ arm spans\t \t\n\t\n\t\n\t\n\t\n(c)\t learners’ favourite soccer teams\t\n\t\n\t\n\t\n\t\n\t\n(d)\t the time it takes 13-year-olds to run 1,5 km\t\n\t\n\t\n(e)\t the cost of different types of cell phone\t\n\t\n\t\n\t\n(f)\t colours of new cars manufactured\t\n\t\n\t\n\t\norganising categorical data\nThandeka asked the following question: “Which of South Africa’s official languages are \nthe home languages of the learners in my class?”\nThandeka drew up a table with each learner’s name. She then asked each learner what \nhis or her home language was, and wrote it down as follows:\nName\nLanguage\nName\nLanguage\nName\nLanguage\nNonkhanyiso isiXhosa\nMarike\nAfrikaans\nHerbert\nSepedi\nAnna\nAfrikaans\nJennifer\nSepedi\nThabo\nisiXhosa\nMpho\nNdebele\nNomonde\nisiXhosa\nNomi\nisiXhosa\nNontobeko\nisiZulu\nThandeka\nSepedi\nManare\nSepedi\nJonathan\nEnglish\nSiza\nisiZulu\nUnathi\nSesotho\nSibongile\nisiZulu\nPrince\nSesotho\nGabriel\nNdebele\nDumisani\nisiZulu\nDuma\nisiZulu\nMarlene\nAfrikaans\nMatshediso\nSesotho\nThandile\nSepedi\nSimon\nSesotho\nChokocha\nSepedi\nNicholas\nSesotho\nMiriam\nSetswana\nKhanyisile\nisiXhosa\nJabulani\nisiZulu\nSibusiso\nisiZulu\nRamphamba\nTshivenda\nNomhle\nisiXhosa\nMishack\nisiZulu\nPortia\nisiZulu\nFrederik\nAfrikaans\nPeter\nSetswana\nErik\nAfrikaans\nLola\nAfrikaans\nMaya\nAfrikaans\nJan\nAfrikaans\nZinzi\nisiXhosa\nThobile\nSesotho\nPalesa\nisiZulu\nJacob\nSetswana\nWe don’t need the learners’ names in the data. This data could be written as a list of the \nlanguages, like this:\nisiXhosa, Afrikaans, Sepedi, Afrikaans, Sepedi, isiXhosa, Ndebele, isiXhosa, isiXhosa, isiZulu,  \nSepedi, Sepedi, English, isiZulu, Sesotho, isiZulu, Sesotho, Ndebele, isiZulu, isiZulu, Afrikaans, \nSesotho, Sepedi, Sesotho, Sepedi, Sesotho, Setswana, isiXhosa, isiZulu, isiZulu, Tshivenda,  \nisiXhosa, isiZulu, isiZulu, Afrikaans, Setswana, Afrikaans, Afrikaans, Afrikaans, Afrikaans, isiXhosa, \nSesotho, isiZulu, Setswana\nMaths2_Gr7_LB_Book.indb   176\n2014/09/04   11:35:22 AM\n\n\tCHAPTER 13: COLLECT, ORGANISE AND SUMMARISE DATA\t\n177\nNow work with this data set to see what story it is telling you. What do you notice about \nthe data? \n1.\t What do you need to find out from this list of languages?\n2.\t Does it matter what order you write the languages in? Why or why not?\n3.\t (a)\t Use Thandeka’s table. In the space below, draw a dot above each language to  \n\t\nshow every learner who speaks that language. The languages are in alphabetical \n\t\norder. Try to space out the dots evenly. The dots for Afrikaans have been drawn \n\t\nfor you. A graph like this is called a dot plot.\nAfrikaans English\nisiXhosa\nisiZulu Ndebele Sepedi Sesotho Setswana Siswati Tshivenda Xitsonga\nLanguages\n\t\n(b)\t Which languages have the same numbers of learners?\n\t\n(c)\t List the languages in order from the language spoken by the most learners to  \n\t\nthe language spoken by the fewest learners.\nMaths2_Gr7_LB_Book.indb   177\n2014/09/04   11:35:22 AM\n\n178\t MATHEMATICS Grade 7: Term 4\nYou can also record results in a tally table. To do \nthis, you draw a single line ( | ) for each item you \ncount. This line is called a tally mark.\nYou group tally marks in groups of five. The fifth \ntally mark is always drawn horizontally to show that \nthe group of five is complete. Then you start a new \ngroup. This makes it easy to quickly count how many \ntally marks there are in a particular category.\n4.\t (a)\t Complete the table.\nHome language of learners in the Grade 7 class\nLanguage\nNumber of speakers of each home language\nTotal\nAfrikaans\n||||  |||\n8\nEnglish\n|\n1\nisiXhosa\nisiZulu\nNdebele\nSepedi\nSesotho\nSetswana\nSiswati\nTshivenda\nXitsonga\nTotal (whole class)\n\t\n(b)\t How many learners altogether were asked about their home language? \n\t\n(c)\t Which home language occurs most often in this class? \n\t\n(d)\t Which languages are not spoken as a home language by any of the learners in  \n\t\nthis class? \n\t\n(e)\t Write a short paragraph to describe the home languages in Thandeka’s class.\nExamples of tally marks:\nA count of three = |||\nA count of four = ||||\nA count of five = ||||\nA count of seven = ||||  ||\nMaths2_Gr7_LB_Book.indb   178\n2014/09/04   11:35:22 AM\n\n\tCHAPTER 13: COLLECT, ORGANISE AND SUMMARISE DATA\t\n179\nDot plots and tally tables are used for numerical data too. You can write data values \non prepared tally tables or dot plots as you record them. This sorts the data at the same \ntime as it is recorded. \nintroducing stem-and-leaf displays\nA stem-and-leaf display (also called a stem-\nand-leaf plot) is a way of listing numerical data \nusing two columns divided by a vertical line. Each \nnumber is split across the columns. \nFor example, if the numbers in a set of data \nconsist of digits for tens and units (such as 23, 25, \n34), the column on the right (the leaf column) \nshows the units digits of the numbers, and the \ncolumn on the left (the stem column) shows the \ntens digits of the numbers.\nExample 1\nShow the following data set as a stem-and-leaf  \ndisplay:\n13, 56, 20, 35, 47, 53, 12, 51, 53, 49, 34, 53\nFirst, we order the values in the data set from  \nsmallest to biggest:\n12, 13, 20, 34, 35, 47, 49, 51, 53, 53, 53, 56\nThe stem-and-leaf display of the above data set looks like this:\n\t\t\n\t\n\t\n           Key: 1 | 2 means 12\n1 2, 3\n2 0\n3 4, 5\n4 7, 9\n5 1, 3, 3, 3, 6\nNumerical data is data that \nconsists of numbers.\nIn this example, the tens \ndigits range from 1 to 5, \nso we list these in the stem \ncolumn. Then we fill in \nthe units digits in the leaf \ncolumn.\nValues with the same stem are written \nin the same row. Numbers with the \nsame tens value are separated by \na space or a comma. The first row \nshows the numbers 12 and 13, the \nsecond row shows 20, and the last \nrow shows 51, 53, 53, 53 and 56.\nIn this example, the \nstem column shows the \ntens digit of each value.\nHere the leaf column \nshows the units digit \nof each value.\nStems\nLeaves\nMaths2_Gr7_LB_Book.indb   179\n2014/09/04   11:35:22 AM\n\n180\t MATHEMATICS Grade 7: Term 4\nExample 2\nThis stem-and-leaf display shows the units digits as the leaves, and both the hundreds \nand tens digits as the stems:\n\t\n10 2, 5\n11 0, 6\n   Key: 10 | 2 means 102\n12 1, 4, 4\n13\n14 7, 9\n15\n16 1, 3, 8\nThe values shown are: 102, 105, 110, 116, 121, 124, 124, 147, 149, 161, 163, 168.\nNote that if there is a 0 in the leaf column it means the unit digit is a 0, as in 110 above. \nWhen there is nothing written in the leaf column next to a stem, it means that there \naren’t any numbers with that particular stem. In the case of stem 13 above, for example, \nit means there are no values between 129 and 140.\nWhen you draw stem-and-leaf displays, it is important that the numbers line up  \nvertically so that you can compare the leaves. Draw lines to help you. (Or use grid paper, \nif you have some.)\ndot plots and stem-and-leaf displays\n1.\t Look at the following stem-and-leaf display and answer the questions below.\n13 1, 9\nKey: 13 | 1 means 131\n14 0\n15\n16 2, 3, 5, 5, 5\n17 6, 8, 8\n18\n19 4, 6, 7\n\t\n(a)\t Write down the values in the data set shown by the stem-and-leaf display.\n\t\n(b)\t Do most of the values fall in the 160s or 170s? \n\t\n(c)\t Which value occurs the most times? \n\t\n(d)\t Add the following values to the stem-and-leaf display: 143, 167 and 199.\n\t\n(e)\t There are no values in the 150s. Can we add the following to the stem-and-leaf  \n\t\ndisplay to show that there are no values in the 150s? Explain your answer.\n\t\n\t\n\t\n15 | 0\nMaths2_Gr7_LB_Book.indb   180\n2014/09/04   11:35:22 AM\n\n\tCHAPTER 13: COLLECT, ORGANISE AND SUMMARISE DATA\t\n181\n2.\t (a)\t Arrange the values in the following data set in order from smallest to largest:\n\t\n\t\n378, 360, 390, 378, 378, 400, 379, 382, 354, 394, 399, 395, 378, 361, 375\n\t\n(b)\t Organise the data set as a stem-and-leaf display.\n\t\n\t\n\t\nKey: \n\t\n(c)\t Which value occurs most often? \n3.\t (a)\t The data sets below show the sales of two new makes of cars (Jupiter and  \n\t\nMercury) over 24 months. Draw a dot plot for each set on the number lines. \n\t\n\t\nMercury: 23, 27, 30, 27, 32, 31, 32, 32, 35, 33, 28, 39, 32, 29, 35, 36, 33, 25, 35, \n\t\n\t\n37, 26, 28, 36, 30\n\t\n\t\nJupiter: 31, 44, 30, 36, 37, 34, 43, 38, 37, 35, 36, 34, 31, 32, 40, 36, 31, 44, 26, \n\t\n\t\n30, 37, 43, 42, 33\nMercury\nJupiter\nNumber of cars sold\nNumber of cars sold\n20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45\n20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45\n\t\n(b)\t If you look at the dots for Mercury and the dots for Jupiter, what can you see  \n\t\nabout the sales of the two cars? What does this mean?\nMaths2_Gr7_LB_Book.indb   181\n2014/09/04   11:35:22 AM\n\n182\t MATHEMATICS Grade 7: Term 4\nsomething to think about\nWhat kind of graph does the stem-and-leaf display look like if you turn it by 90°?\n1\n2, 5\n2\n0, 6\n3\n1, 4, 4, 5, 5, 9\n4\n5\n2, 7, 8, 9\n6\n1, 3, 6, 7, 7\n7\n1, 3, 8\ngrouping data into intervals\nWhen a data set contains many data items, we sometimes group the data items to help \nus organise the data. For example, the following data set shows the number of milk \nbottles collected by 24 learners for recycling:\n9, 10, 13, 23, 24, 26, 26, 27, 30, 31, 34, 40, 42, 49, 50, 53, 61, 64, 67, 67, 68, 69, 91, 94\nWe can group the data into categories called class \nintervals, such as 0–9, 10–19, 20–29, and so on. \nWe can then count how many times a value occurs \nin each interval. The number of times a value \noccurs in an interval is called its frequency.\nThis table shows the grouped data and the frequency of the values in each interval.\nInterval\n0–9\n10–19 20–29 30–39 40–49 50–59 60–69 70–79 80–89 90–99\nFrequency\n1\n2\n5\n3\n3\n2\n6\n0\n0\n2\nThe table shows that 1 learner collected 0–9 bottles, 2 learners collected 10–19 bottles, \n5 learners collected 20–29 bottles, and so on. We can clearly see that most learners (6) \ncollected 60–69 bottles.\n9\n5\n7\n5\n9\n7\n4\n8\n6\n8\n5\n6\n4\n7\n3\n3\n2\n0\n1\n2\n1\n1\n1\n2\n3\n4\n5\n6\n7\nMaths2_Gr7_LB_Book.indb   182\n2014/09/04   11:35:22 AM\n\n\tCHAPTER 13: COLLECT, ORGANISE AND SUMMARISE DATA\t\n183\nworking with grouped data\n1.\t Anita collected data from a sample of Grade 7 learners about how far they live from \nthe nearest grocery store. Below are the results. The values are in kilometres, correct  \nto one decimal figure.\n0,1\n0,1\n0,2\n0,2\n0,2\n0,2\n0,3\n0,3\n0,3\n0,4\n0,4\n0,5\n0,5\n0,5\n0,6\n0,6\n0,7\n0,7\n0,7\n0,8\n0,8\n0,8\n0,9\n0,9\n0,9\n1\n1\n1\n1,5\n1,5\n2\n2\n2\n2\n2,5\n2,5\n3\n3\n3\n3,5\n3,5\n4\n4\n4,\n4,5\n5\n5\n6\n6\n7\n7\n8\n8\n9\n10\n10\n15\n20\n23\n30\n\t\n(a)\t Complete the table alongside to indicate  \n\t\nhow many of the values appear in each of  \n\t\nthe given intervals.\n\t\n(b)\t How far do most of the learners live from  \n\t\nthe nearest grocery store? \n2.\t Here are the heights of 50 Grade 7 boys at a school (in centimetres):\n165\n148\n150\n160\n165\n150\n156\n155\n164\n162\n160\n158\n138\n158\n140\n146\n160\n148\n152\n139\n165\n148\n152\n139\n165\n148\n160\n163\n178\n138\n142\n179\n156\n160\n160\n171\n140\n160\n164\n135\n159\n143\n167\n138\n163\n164\n155\n160\n167\n165\n\t\n(a)\t Draw a stem-and-leaf display to show this set of data.\n\t\n\t\nKey:  \n\t\n(b)\t Write a short paragraph to describe the data set.\nInterval\nFrequency\nless than 1,0 km\n1,0–5,9 km\n6,0–9,9 km\n10 km or further\nMaths2_Gr7_LB_Book.indb   183\n2014/09/04   11:35:22 AM\n\n184\t MATHEMATICS Grade 7: Term 4\n\t\n(c)\t Complete the frequency table below for the grouped data from your stem-and- \n\t\nleaf display in question (a).  \nClass interval (cm)\nFrequency\n130–139\n140–149\n150–159\n160–169\n170–179\nTotal\n13.3\t Summarising data\nWhen you have collected data, you often need to tell someone what you have found out. \nPeople want to know what your conclusions are, without looking at all of the data you \nhave collected.\nIt is often useful to summarise a set of numerical data by using one value. For example, \nwhich value best summarises or describes the following data set?\n    0  1  1  5  8  8  9  9  10  10  10  11  11\nStatisticians use any of three values that show the  \nmost central values in the set, or the value around  \nwhich the other values tend to cluster. These  \nvalues are called the measures of central  \ntendency or summary statistics.\n• The mode is the value that occurs the most \nfrequently in the data set. In the example \nabove, the mode is 10 because it occurs the \nmost times (three times).\n• The median is the value exactly in the middle  \nof the data set when the data values are \narranged in order from smallest to largest. For \nthe data set above, the median is 9 because \nthere are six values to the right of the first 9 and \nsix values to the left of it.\n• The mean (average) is the total (sum) of the \nvalues divided by the number of values in the \ndata set. So: \n\t\n        Mean = \nTotal of values\nNumber of values = 93\n13  = 7,15\nStatisticians are mathematicians \nwho specialise in collecting, \norganising and analysing data.\nA data set can have more than \none mode.\nIf the data set consists of an \neven number of items, the \nmedian = sum of the two \nmiddle values divided by 2.\nIn the data set above, either 10  \n(mode), 9 (median) or 7,15 \n(mean) could be used to  \nrepresent the entire data set.\nMaths2_Gr7_LB_Book.indb   184\n2014/09/04   11:35:23 AM\n\n\tCHAPTER 13: COLLECT, ORGANISE AND SUMMARISE DATA\t\n185\nunderstanding the mean\nThis activity will help you to understand how the mean represents the whole set of data.\nMake piles of blocks of different heights: \n \nThen move blocks from the higher piles to the lower ones to make all the piles equal: \nYou have just found the mean: Each pile now has 4 blocks in it. But how do you do this if you  \nonly have the numbers 5, 6, 3, 2 and 4 to work with? You add them up and then divide the \nanswer by the total number of values (numbers):\n\t\n\t\n5 + 6 + 3 + 2 + 4 = 20              20 ÷ 5 = 4  \nWhat this means is that you are finding a single number that you can use in place of all the  \ndifferent numbers and still get the same total.\n \nIt is also useful to know how big the spread of the data is.\nThe range of a data set is the difference between the \nhighest value and the lowest value. For example, for \nthe data set on the previous page, the range is: \n\t \t\n11 − 0 = 11\nThe bigger the range, the more the data is spread out. \nThe smaller the range, the more the data is clustered \naround similar values.\nMaths2_Gr7_LB_Book.indb   185\n2014/09/04   11:35:23 AM\n\n186\t MATHEMATICS Grade 7: Term 4\ndetermining the mode, median, mean and range\n1.\t The following data set shows the shoe sizes of a sample of learners at a school:\n\t\n1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6\n\t\n(a)\t What is the mode of the data set? \n\t\n(b)\t What is the median of the data set? \n\t\n(c)\t What is the mean? (Round off to the nearest whole number.)\n\t\n(d)\t What is the range of the data set? \t\t\n2.\t The following data set shows the number of siblings (that is, brothers and sisters)  \nthat the learners in a sample of Grade 7 learners have:\n\t\n0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5\n\t\n(a)\t How many learners are in the sample? \n\t\n(b)\t What is the mode of the data set? \n\t\n(c)\t What is the median of the data set? \n\t\n(d)\t What is the mean? (Round off to the nearest whole number.)\n\t\n(e)\t What is the range of the data set? \n3.\t The following data set shows the number of hours worked in a week by a sample of \nparents at School A:\n\t\n15, 16, 20, 25, 25, 30, 40, 40, 40, 40, 40, 42, 45, 45, 48, 48\n\t\n(a)\t How many parents are in the sample? \n\t\n(b)\t What is the mode of the data set? \n\t\n(c)\t What is the median of the data set?\n\t\n(d)\t What is the mean? (Round off to one decimal place.)\n\t\n(e)\t What is the range of the data set? \nRemember, if the number of \nitems in a data set is even, the \nmedian = the sum of the two \nmiddle numbers divided by 2.\nMaths2_Gr7_LB_Book.indb   186\n2014/09/04   11:35:23 AM\n\n\tCHAPTER 13: COLLECT, ORGANISE AND SUMMARISE DATA\t\n187\n4.\t The following data set shows the number of hours worked in a week by a sample of \nparents at School B:\n\t\n25, 30, 35, 35, 35, 40, 40, 40, 40, 40, 42, 45, 45, 45, 48, 50\n\t\n(a)\t How many parents are in the sample? \n\t\n(b)\t What is the mode of the data set? \n\t\n(c)\t What is the median of the data set? \n\t\n(d)\t What is the mean? (Round off to one decimal place.)\n\t\n(e)\t What is the range of the data set? \n5.\t The following is a list of test scores of learners in a Grade 7 class: \n40, 42, 44, 13, 10, 23, 68, 31, 69, 91, 30, 49, 50, 53, 67, 94, 61, 64, 67, 34\n\t\n(a)\t Arrange the scores from the lowest to the highest.\n\t\n(b)\t How many learners are in the population? \n\t\n(c)\t What is the mode of the data set? \n\t\n(d)\t What is the median of the data set? \n\t\n(e)\t What is the mean?\n\t\n(f)\t What is the range of the data set? \t\nMaths2_Gr7_LB_Book.indb   187\n2014/09/04   11:35:23 AM\n\n188\t MATHEMATICS Grade 7: Term 4\n6.\t A hockey player recorded the number of goals she scored in her last 30 matches:\n\t\n1\t\n1\t\n3\t\n2\t\n0\t\n0\t\n4\t\n2\t\n2\t\n4\t\n3\t\n1\t\n0\t\n1\t\n0\n\t\n2\t\n1\t\n5\t\n1\t\n3\t\n7\t\n2\t\n2\t\n2\t\n4\t\n3\t\n1\t\n1\t\n0\t\n3\n\t\n(a)\t Draw a dot plot on the number line below to organise these data values.\nNumber of goals scored\n0\n1\n2\n3\n4\n5\n6\n7\n\t\nNow use the dot plot to answer these questions.\n\t\n\t\n(b)\t Which of the values are quite different to the other values? \n\t\n(c)\t Which number of goals has she scored the highest number of times?  \n\t\n(d)\t Which numbers of goals did she score in the two groups with five matches each?\n\t\n(e)\t Use the dot plot to find the mode of the data. \n\t\n(f)\t Use the dot plot to find the median.\n\t\n(g)\t What is the mean of the goals? \nMaths2_Gr7_LB_Book.indb   188\n2014/09/04   11:35:23 AM\n\nChapter 14\nRepresent data\n\t\nCHAPTER 14: REPRESENT DATA\t\n189\nWhen we have collected and organised our data, we often represent it as a graph. This \nhelps us to see the data and patterns in the data more easily. In this chapter, you will  \nrevise bar graphs, double bar graphs and pie charts, which you have learnt about in  \nprevious grades. You will learn about a new type of graph called a histogram, and how  \nthis differs from a bar graph. You will also learn how to draw your own pie charts by  \nestimating fractions of a whole circle.\n14.1\t Bar graphs and double bar graphs.......................................................................... 191\n14.2\t Histograms............................................................................................................. 194\n14.3\t Pie charts................................................................................................................ 202\nMaths2_Gr7_LB_Book.indb   189\n2014/09/04   11:35:23 AM\n\n190\t MATHEMATICS Grade 7: Term 4\nCollect\ndata\nPose a\nquestion\nRepresent \nthe data\nOrganise \nthe data\nInterpret \nand analyse \nthe data\nReport on \nthe data\nThe data\ncycle\nMaths2_Gr7_LB_Book.indb   190\n2014/09/04   11:35:23 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n191\n\t\nCHAPTER 14: REPRESENT DATA\t\n191\n14\tRepresent data\nNow that we have collected and organised a set of data, we want to show the results in a \nuseful way. \nRemember when you drew dot plots in the previous chapter, you could see which \ncategories or measurements occurred many times and which occurred only a few times. \nThere are a few different graphs that show the important things about the data in such a \nway that you can see them easily. You need to be able to draw these graphs.\n14.1\t Bar graphs and double bar graphs\ndrawing a bar graph\nA bar graph shows categories (or classes) of data along the horizontal axis, and the \nfrequency of each category along the vertical axis. (Sometimes the axes are swopped \naround.) Here is an example of a bar graph.\nFrequency of \neach category\nCategories \nof data\nTitle of graph\nHeight of bar \nindicating the \nfrequency of \neach category\n45\n50\n40\n35\n30\n25\n20\n15\n10\n5\n0\nSports played by learners at my school\nNumber of learners\nSports\nSoccer\nHockey\nNetball\nCricket\nOther\nGo back to section 13.2 of chapter 13, where you drew a dot plot and made a tally table \nof Thandeka’s data about languages spoken in her class. Use the data to draw a bar graph \non the set of axes on the next page. Draw the bars to the correct height by looking at the \nnumbers on the vertical axis.\nMaths2_Gr7_LB_Book.indb   191\n2014/09/04   11:35:23 AM\n\n192\t MATHEMATICS Grade 7: Term 4\n0\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\nHome languages of the Grade 7 class\nisiZulu\nusing double bar graphs\nA double bar graph shows two sets of data for each category (or class). For example, \nthe double bar graph below shows data collected from girls for each category, and data \ncollected from boys for each category.\nTwo bars are shown in \neach category. The blue \nbars show the data for \nboys and the red bars \nshow the data for girls.\nA key (or a legend) \nexplains the colours \nused to distinguish the \ntwo sets of data.\n0\n5\n10\n15\n20\n25\n30\nSports played by boys and girls at my school\nNumber of learners\nSports\nSoccer\nHockey\nNetball\nCricket\nOther\nBoys\nGirls\nMaths2_Gr7_LB_Book.indb   192\n2014/09/04   11:35:24 AM\n\n\t\nCHAPTER 14: REPRESENT DATA\t\n193\n1.\t Look at the data below and answer the questions that follow.\n0\n20\n40\n60\n80\n100\n120\n140\n160\n180\n200\nNumber of schools, by province, participating in a school survey\nNumber of schools\nEastern Cape\nSecondary schools\nPrimary schools\nFree State\nGauteng\nKwaZulu-Natal\nLimpopo\nMpumalanga\nNorthern Cape\nNorth West\nWestern Cape\nProvinces\n\t\n(a)\t Did more primary schools or more secondary schools participate in the survey?\n\t\n(b)\t Which province had fewer than 50 secondary schools participating in the  \n\t\nsurvey?\n\t\n(c)\t Which provinces had more than 150 of its primary schools participating in  \n\t\nthe survey?\n2.\t Draw a double bar graph to show the following data. Use the grid on the next page.\nFacilities available at schools in Province A and Province B\nFacility\nPercentage of schools  \nin Province A\nPercentage of schools \nin Province B\nElectricity\n73\n50\nRunning water\n68\n45\nComputers\n60\n20\nInternet\n30\n10\nMaths2_Gr7_LB_Book.indb   193\n2014/09/04   11:35:24 AM\n\n194\t MATHEMATICS Grade 7: Term 4\n14.2\t Histograms\na situation where data has to be organised\n1.\t Mr Makae wants to buy an orange farm. Three farms are available, each with an \norchard of orange trees, and the three farms cost about the same. There are 40 orange \ntrees on each farm. The total mass of oranges (in kg) harvested from each tree on each \nfarm over the last 3 years is given below. Which farm should he buy?\n\t\nFarm A:\n426\n628\n467\n413\n862\n585\n652\n600\n734\n611\n741\n605\n536\n643\n833\n438\n613\n704\n623\n719\n719\n701\n501\n768\n642\n444\n751\n579\n695\n726\n616\n619\n441\n703\n902\n947\n785\n952\n725\n721\n\t\nFarm B: \n822\n736\n773\n674\n884\n463\n644\n433\n688\n487\n884\n530\n448\n410\n982\n638\n492\n638\n725\n621\n743\n661\n744\n530\n560\n745\n455\n943\n760\n734\n888\n457\n621\n969\n507\n500\n542\n831\n576\n801\n\t\nFarm C:\n438\n530\n743\n947\n450\n777\n859\n748\n473\n724\n750\n852\n428\n464\n725\n554\n758\n997\n467\n743\n722\n438\n779\n690\n785\n543\n752\n898\n474\n483\n460\n772\n544\n756\n491\n576\n482\n744\n701\n803\nMaths2_Gr7_LB_Book.indb   194\n2014/09/04   11:35:24 AM\n\n\t\nCHAPTER 14: REPRESENT DATA\t\n195\n2.\t How can the data about the orange trees on the three farms be organised so that the \nfarmer has a clear picture of the difference between the orchards on the three farms? \nFor now just write down how you think the data may be organised. You will organise \nthe data later when you do the questions that follow.\n3.\t Complete these tally and frequency tables for the data about the masses of oranges \nharvested on the three orange farms.\nMasses of oranges harvested from different trees on Farm A\nMass of oranges harvested from each tree. \nThese are called class intervals.\nNumber of trees that produced \nmasses in the interval\nTotal\n400 kg or more but less than 500 kg\n||||  |\n500 kg or more but less than 600 kg\n||||\n600 kg or more but less than 700 kg\n||||  ||||  ||\n700 kg or more but less than 800 kg\n||||  ||||  |||\nMasses of orange harvested from different trees on Farm B\nClass interval\nNumber of trees that produced \nmasses in the interval\nTotal\n400 kg or more but less than 500 kg\n||||  |||\n8\n500 kg or more but less than 600 kg\n||||  ||\n7\n600 kg or more but less than 700 kg\n||||  |||\n700 kg or more but less than 800 kg\n||||  |||\n800 kg or more but less than 900 kg\n  \n900 kg or more but less than 1 000 kg\nMasses of oranges harvested from different trees on Farm C\nClass interval\nNumber of trees that produced \nmasses in the interval\nTotal\n400 kg or more but less than 500 kg\n  \n  \n700 kg or more but less than 800 kg\n  \n  \n  \nOn the next page, you will learn how to draw graphs of the data for the three farms.\nMaths2_Gr7_LB_Book.indb   195\n2014/09/04   11:35:24 AM\n\n196\t MATHEMATICS Grade 7: Term 4\nThe data for Farm A is represented on this graph.\nHistogram of the total masses of oranges from trees on Farm A\nFrequency\n0\n5\n10\n15\n20\nMass intervals\n300\n400\n500\n600\n700\n800\n900\n1 000\nThis type of graph is called a histogram.\n(The columns in a histogram are normally not coloured differently, or even coloured \nat all. In this histogram the columns are coloured only because some questions are asked \nabout them in question 4 below.)\nThe numbers 400 on the left and 500 on the right of the light yellow column indicate \nthat masses of 400 kg or more but less than 500 kg are counted in that interval.\nThe height of each column represents the number of masses (the frequency) that fall \nin that interval.\n4.\t (a)\t A total of 536 kg of oranges was harvested from one of the trees on Farm A over  \n\t\na period of the 3 years. In which column on the above histogram is this tree  \n\t\nrepresented? Explain your answer.\n\t\n(b)\t Which masses are represented in the red column?\n\t\n(c)\t Which class interval is represented by the light blue column on the above histogram?\n\t\n(d)\t How many masses are represented by the green column?    \n\t\n(e)\t Which column represents the highest frequency?\nMaths2_Gr7_LB_Book.indb   196\n2014/09/04   11:35:24 AM\n\n\t\nCHAPTER 14: REPRESENT DATA\t\n197\n5.\t Complete the histograms below.\nHistogram of the total masses of oranges from trees on Farm B\nFrequency\n0\n5\n10\n15\n20\nMass intervals\n300\n400\n500\n600\n700\n800\n900 1 000\nHistogram of the total masses of oranges from trees on Farm C\nFrequency\n0\n5\n10\n15\n20\nMass intervals\n300\n400\n500\n600\n700\n800\n900\n1 000\nThe different class intervals are consecutive and cannot have values that overlap.  \nFor example, we can group heights into class intervals of 10 cm, as shown below:\nHeight (m)\nHeights that fall in the class interval\nFrequency\n1,20–1,30\n1,20;  1,25;  1,29\n3\n1,30–1,40\n1,30;  1,31;  1,35;  1,39\n4\n1,40–1,50\n1,40;  1,46;  1,48;  1,48;  1,49\n5\n1,50–1,60\n1,53;  1,53;  1,57;  1,58;  1,59;  1,59\n6\nWe follow the convention that the top value  \n(also called the upper boundary) of each class \ninterval is not included in the interval.\n  \nSo the height of 1,20 m falls \ninto the 1,20–1,30 m interval, \nbut the height 1,30 m falls into \nthe 1,30–1,40 m interval.\nMaths2_Gr7_LB_Book.indb   197\n2014/09/04   11:35:24 AM\n\n198\t MATHEMATICS Grade 7: Term 4\ninterpreting a histogram\nStudy the histogram showing the numbers of members, in different age groups, of a \nsports club. Then answer the questions that follow.\nAges of members (years)\nFrequency\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\n24\n26\n28\n30\n32\n34\n36\n20\n30\n50\n70\n40\n60\n80\n0\n1.\t Complete a frequency table for the information.\n2.\t How many of the members are in their fifties? \n3.\t How many members does the club have?\n4.\t When you drew a bar graph, it did not matter what order the bars were in. Does the \norder of the columns on the histogram matter? Explain.\nMaths2_Gr7_LB_Book.indb   198\n2014/09/04   11:35:25 AM\n\n\t\nCHAPTER 14: REPRESENT DATA\t\n199\nNotice that you cannot see the individual data values in a histogram – they have been \n“lost”. For example, below you can see a stem-and-leaf display and a histogram of the \nsame data set:\nFishes’ masses in g\n15  7,  8\n16  2,  3,  3\n16  5,  5,  6,  7,  8,  8,  9,  9\n17  1,  2,  3\n17  6,  7,  7,  8,  8\n18  0,  3\n18  6,  7\nKey: 15|7 means 157\nMass in g\nFishes’ masses\nNumber of fishes\n0\n1\n2\n3\n4\n5\n6\n7\n8\n155\n160\n165\n170\n175\n180\n185\n190\n9\n10\nA histogram usually has many more data values than a stem-and-leaf display – too  \nmany to show in a stem-and-leaf display. It would, for example, be difficult to put the 84 \nvalues for the members of the sports club onto a stem-and-leaf display.\ndrawing more histograms\n1.\t The table shows how long it takes learners \nfrom a Grade 7 class at Western Primary to \ntravel to school each day. In question (d)  \nyou will represent the data in the table  \nwith a histogram.\n\t\n(a)\t How many learners were asked about their  \n\t\ntravelling hours?\n\t\n(b)\t Look at the grid provided in question (d). What do you have to consider in order \n\t\nto help you decide on a scale division for the vertical axis?\n\t\n(c)\t What scale will you use on the horizontal axis? Explain your answer.\nTime \n(minutes)\nFrequency\n0–10\n7\n10–20\n18\n20–30\n11\n30–40\n3\nMaths2_Gr7_LB_Book.indb   199\n2014/09/04   11:35:25 AM\n\n200\t MATHEMATICS Grade 7: Term 4\n\t\n(d)\t Draw a histogram of the data. \n2.\t The table shows how much money different  \nvendors earn selling their goods every week. \n\t\n(a)\t How many vendors were asked about  \n\t\ntheir earnings? \n\t\n(b)\t Look at the grid below. Decide on a scale for  \n\t\nthe vertical axis of a histogram and indicate  \n\t\nit on the axis. \n\t\n(c)\t Decide on a scale for the horizontal axis and indicate it on the axis.\n\t\n(d)\t Complete the histogram showing the data. \nMoney (R)\nFrequency\n0–100\n6\n100–200\n9\n200–300\n11\n300–400\n7\n400–500\n5\nMaths2_Gr7_LB_Book.indb   200\n2014/09/04   11:35:25 AM\n\n\t\nCHAPTER 14: REPRESENT DATA\t\n201\n3.\t In a Natural Sciences class, learners planted beans and measured the heights of the \nbean plants after two months. Here is the data they collected (in cm):\n\t\n34  65  72  42  37  29  78  43  79  91  43  45  28  42  79\n\t\n34  92  87  40  43  43  78  82  47  85  43  32  86  76\t\n\t\n(a)\t Complete this frequency table:\nHeight of bean plants (cm)\nTally\nFrequency\n20–30\n30–40\n40–50\n50–60\n60–70\n70–80\n80–90\n90–100\nTotal\n\t\n\t\n(b)\t Draw a histogram of this data.\nMaths2_Gr7_LB_Book.indb   201\n2014/09/04   11:35:25 AM\n\n202\t MATHEMATICS Grade 7: Term 4\n14.3\t Pie charts\nA pie chart consists of a circle divided into slices (sectors), where the slices show how \nthe different categories of data make up the whole set of data. Bigger categories of data \nhave bigger slices of the circle.\nLook at the example of a pie chart below.\nCategories \nof data\nNumber of learners playing sports\n30\n20\n50\n100\nHockey\nBasketball\nSoccer\nCricket\nThe pie chart shows the following:\n• A total of 200 learners were asked about the sports they played: \n20 + 30 + 50 + 100 = 200\n• The key shows the four categories of data:\n–\t soccer\n–\t hockey\n–\t basketball\n–\t cricket.\n• 100 of the 200 learners play hockey. This is the largest category, and gets the biggest \nslice (half of the whole).\n• 20 of the 200 learners play basketball. This is the smallest category, and gets the \nsmallest slice (one tenth of the whole).\nYou will learn how to draw accurate pie charts in later grades. In this grade, you will \nestimate the portions of a pie chart that each category of data requires.\nMaths2_Gr7_LB_Book.indb   202\n2014/09/04   11:35:25 AM\n\n\t\nCHAPTER 14: REPRESENT DATA\t\n203\nestimating sizes of slices in a pie chart\n1.\t (a)\t Write down the fraction of a whole that each slice in the following diagrams \n\t\nshows.\nHalve each slice\nHalve each slice\n1\n2\n\t\n1\n2  = \n %\t\n\t\nHalve each slice\n\t\n\t\n\t\nHalve each slice\n\t\n\t\n\t\n\t\n(b)\t Below each diagram in question 1(a), write down what percentage each  \n\t\nfraction is equal to.\nYou can use the diagrams above to estimate the sizes of slices when drawing your own  \npie charts.\nMaths2_Gr7_LB_Book.indb   203\n2014/09/04   11:35:26 AM\n\n204\t MATHEMATICS Grade 7: Term 4\n2.\t Use the data in each of the following tables to complete the pie charts. You must:\n• label the major sector\n• divide the other sector into the parts that represent the other languages\n• label each sector.\n\t\n(a)\nProvince: Western Cape\nMajor \nlanguages\nFrequency \n(in % )\nAfrikaans\n50%\nEnglish\n20%\nisiXhosa\n25%\nOther\n5%\nAfrikaans\n(50%)\n\t\n\t\n(b)\nProvince: KwaZulu-Natal\nMajor \nlanguages\nFrequency \n(in %)\nEnglish\n15%\nisiZulu\n80%\nOther\n5%\n\t\n(c)\nProvince: Limpopo\nMajor \nlanguages\nFrequency  \n(in %)\nSepedi\n50%\nTshivenda\n15%\nXitsonga\n20%\nOther\n15%\nMaths2_Gr7_LB_Book.indb   204\n2014/09/04   11:35:26 AM\n\n\t\nCHAPTER 14: REPRESENT DATA\t\n205\nrepresenting data as fractions and percentages in pie charts\nTo represent data in a pie chart, you need to know how to convert (change) the \nfrequencies of the different categories into a fraction or percentage of the total.\n1.\t The learners in Class A were asked how many languages they could speak. The table \nshows the data that was collected.\n\t\n(a)\t Complete the ‘Fraction’ column by determining what fraction of the whole  \n\t\neach category is.\n\t\n(b)\t Complete the ‘Percentage’ column by  \n\t\nconverting the fraction to a percentage.\nNumber of languages spoken by learners in Class A\nLanguages\nFrequency\nFraction\nPercentage\nOne language\n10\n10\n40  \n=\n \n1\n4\n25%\nTwo languages\n20\nThree languages\n6\nFour languages\n2\nMore than four \nlanguages\n2\nTotal \n40\n40\n40\n100%\n\t\n(c)\t Draw a pie chart of the data in your completed table. Use a circular object to  \n\t\ndraw the circle. Then estimate the sizes of the various slices of the pie chart.\nRemember, to convert a common \nfraction to a percentage you have \nto multiply by 100%.\nMaths2_Gr7_LB_Book.indb   205\n2014/09/04   11:35:27 AM\n\n206\t MATHEMATICS Grade 7: Term 4\n2.\t The learners in Class B were asked how many languages they could speak. The table \nshows the data that was collected.\n\t\n(a)\t Complete the ‘Fraction’ column by determining what fraction of the whole  \n\t\neach category is.\n\t\n(b)\t Complete the ‘Percentage’ column by converting the fraction to a percentage.\nNumber of languages spoken by learners in Class B\nLanguages\nFrequency\nFraction\nPercentage\nOne language\n12\n12\n60  \n=\n \n1\n5\n20%\nTwo languages\n30\nThree languages\n12\nFour languages\n3\nMore than four \nlanguages\n3\nTotal \n60\n60\n60\n100%\n\t\n(c)\t Draw a pie chart to represent the data in your completed table.\nMaths2_Gr7_LB_Book.indb   206\n2014/09/04   11:35:27 AM\n\nChapter 15\nInterpret, analyse and \nreport on data\n\tCHAPTER 15: INTERPRET, ANALYSE AND REPORT ON DATA\t\n207\nBy now you should be able to read and interpret data represented in words, bar graphs, \ndouble bar graphs, pie charts and histograms. The activities in this chapter will give you \nmore practice in interpreting and analysing such data. At the same time, you will be asked \nto think critically about the data, especially how the ways in which data is presented can \nmislead the reader into drawing inaccurate conclusions. You will also practise reporting on \ndata by writing short paragraphs to summarise the data presented to you.\n15.1\t Interpreting and reporting on data......................................................................... 209\n15.2\t Identifying bias and misleading data....................................................................... 212\nMaths2_Gr7_LB_Book.indb   207\n2014/09/04   11:35:27 AM\n\n208\t MATHEMATICS Grade 7: Term 4\nCollect\ndata\nPose a\nquestion\nRepresent \nthe data\nOrganise \nthe data\nInterpret \nand analyse \nthe data\nReport on \nthe data\nThe data\ncycle\nMaths2_Gr7_LB_Book.indb   208\n2014/09/04   11:35:27 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n209\n\tCHAPTER 15: INTERPRET, ANALYSE AND REPORT ON DATA\t\n209\n15\tInterpret, analyse and \n\t\n\t report on data\n15.1\t Interpreting and reporting on data \ncritically reading and reporting on data\n1.\t Read the following paragraph and answer the questions that follow.\nIn 2009, a sample of 2 500 schools from about 26 000 schools across South Africa took  \npart in a survey to provide data about learners and schools. The sample included schools \nfrom each province as follows: 415 schools from the Eastern Cape, 238 from the Free  \nState, 265 from Gauteng, 386 from KwaZulu-Natal, 326 from Limpopo, 248 from \nMpumalanga, 129 from the Northern Cape, 275 from North West and 218 from the  \nWestern Cape.\nAdapted from: Census @ School Results 2009, Statistics South Africa\n\t\n(a)\t What was the population of the survey? \n\t\n(b)\t What was the sample of the survey? \n\t\n(c)\t Which province were most of the schools from? \n\t\n(d)\t Which province were the fewest schools from? \n\t\n(e)\t Complete the first two columns of the table by listing the provinces in order \n\t\nfrom the province that had the most schools to the province that had the  \n\t\nfewest schools participating in the survey.\nProvince\nNumber of schools\nPercentage of all schools\nMaths2_Gr7_LB_Book.indb   209\n2014/09/04   11:35:27 AM\n\n210\t MATHEMATICS Grade 7: Term 4\n\t\n(f)\t Complete the last column by working out the percentage of the whole that  \n\t\nthe schools in each province make up. You may use your calculator for this  \n\t\nquestion. (Round off to one decimal place.)\n\t\n(g)\t Write three to five lines as a summary report of the data described in the  \n\t\nparagraph on the previous page. The summary should give an idea of the  \n\t\nhighest and lowest data items, as this indicates the range of the data.\n2.\t The graph below shows the percentage of male and female learners at schools in \nGrades 3 to 8 in 2009.\nPercentage of male and female learners in Grades 3 to 8\nPercentage (%)\nMale\nFemale\n0\n10\n20\n30\n40\n50\n60\nGrade\n3\n4\n5\n6\n7\n8\n51,6\n48,4\n52,1\n47,9\n51,0\n49,0\n50,3\n49,7\n50,6\n49,4\n51,1\n48,9\n(Source: Census @ School Results 2009, Statistics South Africa)\n\t\n(a)\t Which grade has the highest percentage of females? \n\t\n(b)\t Which grade has the lowest percentage of females? \n\t\n(c)\t Which grade has the highest percentage of males? \n\t\n(d)\t Which grade has the lowest percentage of males? \n\t\n(e)\t If 150 000 Grade 6 learners took part in the survey, how many girls and how  \n\t\nmany boys were there in Grade 6? You may use your calculator.\nMaths2_Gr7_LB_Book.indb   210\n2014/09/04   11:35:28 AM\n\n\tCHAPTER 15: INTERPRET, ANALYSE AND REPORT ON DATA\t\n211\n\t\n(f)\t Complete the following summary report:\nThe graph shows that the number of male learners seems to (decrease/increase)  \nthe higher the grade. For example, in Grade 3, \n % learners were male \ncompared to \n % in Grade 8. The number of female learners seems to \n(decrease/increase) the higher the grade. For example, in Grade 3, \n % \nlearners were female compared to \n % in Grade 8.\n\t\n(g)\t Based on the graph, would you expect there to be more or fewer males in \n\t\nGrade 10? Explain your answer.\n\t\n(h)\t Based on the graph, would you expect there to be more or fewer females in  \n\t\nGrade 10? Explain your answer.\n3.\t The following pie chart shows the land area of each province in 2011.\nNorthern Cape\n30,5%\nWestern Cape\n10,6%\nEastern Cape\n13,8%\nFree State\n10,6%\nMpumalanga\n6,3%\nKwaZulu-Natal\n7,7%\nLimpopo\n10,3%\nGauteng\n1,4%\nNorth West\n8,7%\n(Source: Census 2011: Census in brief, Statistics South Africa)\n\t\n(a)\t Which province has the largest land area? \n\t\n(b)\t Which province has the smallest land area? \n\t\n(c)\t Which three provinces have more or less the same land area?\nMaths2_Gr7_LB_Book.indb   211\n2014/09/04   11:35:28 AM\n\n212\t MATHEMATICS Grade 7: Term 4\n\t\n(d)\t How much bigger is the Northern Cape than Gauteng? (Use a calculator.)\n\t\n(e)\t Are we able to tell from the pie chart which province has the largest  \n\t\npopulation? Explain your answer.\n\t\n(f)\t If the total land area of South Africa is 1 200 000 km2, how many square  \n\t\nkilometres are the largest and the smallest provinces?\n\t\n(g)\t Write a short paragraph to summarise the data shown in the pie chart.\n15.2\t Identifying bias and misleading data\nSometimes the ways in which data is presented  \ncould be intentionally or unintentionally biased  \nor misleading. As you work through the following  \nactivities, think carefully about:\n• data that is not necessarily shown by the graph\n• when, how and where the data was collected\n• which scales are used on the graphs\n• which summary statistics (mean, median and mode) are used to summarise the data.\nBias means that a person  \nprefers a certain idea and  \npossibly does not give equal \nchance to a different idea.\nMaths2_Gr7_LB_Book.indb   212\n2014/09/04   11:35:28 AM\n\n\tCHAPTER 15: INTERPRET, ANALYSE AND REPORT ON DATA\t\n213\ncritically analysing data\n1.\t Look at the bar graph below and answer the following questions:\nMost popular burgers\nBurger A\nNumber of people\nBurger B\nBurger C\nBurger D\n102\n104\n106\n108\n110\n\t\n(a)\t Which burger is the clear favourite? \n\t\n(b)\t The height of the bars indicate that burger A is liked by five times as many  \n\t\npeople as burger B. Is this true? Look at the vertical scale.\n\t\n(c)\t In your exercise book, redraw the bar graph, but show the full vertical scale.\n2.\t Look at the pie chart. \n\t\n(a)\t What is the second most  \n\t\ncommon mode of transport  \n\t\nthat learners use?\n\t\n(b)\t Which mode of transport is the  \n\t\nleast common one? \n\t\n(c)\t Is the pie chart misleading in  \n\t\nany way? Explain.\nLearners’ modes of transport to school\nBus\n17%\nTrain\n19%\nTaxi\n22%\nCar\n16%\nBicycle\n8%\nWalk\n18%\nMaths2_Gr7_LB_Book.indb   213\n2014/09/04   11:35:28 AM\n\n214\t MATHEMATICS Grade 7: Term 4\n3.\t Ilse and Moletsi wanted to find out more about the number of hours people spend \nwatching TV on a particular public holiday. Ilse did her survey on the public holiday \nfrom 13:00 to 15:00. She visited a supermarket and asked adult respondents to \ncomplete her questionnaire. Moletsi did his survey on the same day from 17:00 to \n19:00. He went from door to door in his neighbourhood and asked the children to \ncomplete his questionnaire.\nNumber of hours watching TV\nPercentage (%)\n0\n1\n2\n3\n4\n5\n10\n15\n20\n25\n30\n35\n40\n45\n50\n55\n60\n65\nIlse’s data\nNumber of hours watching TV\nPercentage (%)\n0\n1\n2\n3\n4\n5\n10\n15\n20\n25\n30\n35\n40\n45\n50\n55\n60\n65\nMoletsi’s data\nMaths2_Gr7_LB_Book.indb   214\n2014/09/04   11:35:28 AM\n\n\tCHAPTER 15: INTERPRET, ANALYSE AND REPORT ON DATA\t\n215\n\t\n(a)\t According to Ilse’s data, how long did most people spend watching TV on the \n\t\npublic holiday?\t\n\t\n(b)\t According to Moletsi’s data, how long did most people spend watching TV on  \n\t\nthe public holiday? \t\n\t\n(c)\t Write a paragraph to summarise and compare Ilse’s data and Moletsi’s data.\n\t\n(d)\t How could the time when the data was collected have affected the data?\n\t\n(e)\t How could the place where the data was collected have affected the data?\n\t\n(f)\t How could the people from whom data was collected have affected the data?\nMaths2_Gr7_LB_Book.indb   215\n2014/09/04   11:35:28 AM\n\n216\t MATHEMATICS Grade 7: Term 4\n4.\t Look at the following graphs and answer the questions that follow:\n0\n100\n200\n300\n400\n500\n600\n700\n800\nMonths\nNumber of CDs sold\nNumber of CDs sold by Music Note \nin Year 1\nJan–Mar Apr–Jun Jul–Sep Oct–Dec\n0\n50\n100\n150\n200\n250\n300\n350\n400\nMonths\nNumber of CDs sold\nNumber of CDs sold by Music Note \nin Year 2\nJan–Mar Apr–Jun Jul–Sep Oct–Dec\n\t\n(a)\t What does each of the graphs show?\n\t\n(b)\t How many CDs were sold in July to September of Year 1? \n\t\n(c)\t How many CDs were sold in July to September of Year 2? \n\t\n(d)\t The heights of the bars indicate that Music Note sold more CDs in October to  \n\t\nDecember of Year 2 than in the same months of Year 1. Is this the case?\n\t\n(e)\t How many CDs were sold altogether in Year 1? \n\t\n(f)\t How many CDs were sold altogether in Year 2?\n\t\n(g)\t Explain why the heights of the bars seem to indicate that Music Note sold more \n\t\nor less the same number of CDs in both years, which is not true. \nMaths2_Gr7_LB_Book.indb   216\n2014/09/04   11:35:29 AM\n\n\tCHAPTER 15: INTERPRET, ANALYSE AND REPORT ON DATA\t\n217\n5.\t The following table shows the Mathematics marks of Class A and Class B. \nClass A\n94, 42, 23, 67, 67, 68, 13, 53, 44, 34, 64, 69, 50, 31, 91, 40, 10, 30,  \n49, 61\nClass B\n74, 26, 65, 45, 71, 77, 58, 35, 39, 45, 68, 45, 57, 62, 29, 55, 23, 56,  \n38, 36, 50, 64, 58, 32, 42\n\t\n(a)\t Find the range of each set of data.\n\t\n\t\nClass A: \n\t\nClass B: \n\t\n(b)\t What can you say about the two classes by looking at the range of marks?\n\t\n(c)\t Calculate the mean (average) Mathematics mark for each class. You may use  \n\t\nyour calculator.\nClass \nTotal marks\nNumber of marks\nMean\nClass A\nClass B\n\t\n(d)\t Compare the two sets of data using the means.\n\t\n(e)\t Find the median for each class.\nClass\nMarks from highest to lowest\nMiddle \nposition\nMedian\nClass A\nClass B\n\t\n(f)\t Compare the two sets of data using the medians.\nMaths2_Gr7_LB_Book.indb   217\n2014/09/04   11:35:29 AM\n\n218\t MATHEMATICS Grade 7: Term 4\n\t\n(g)\t Find the mode for each class.\nClass\nHighest frequency\nMode\nClass A\nClass B\n\t\n(h)\t Compare the two sets of data using the mode.\n\t\n(i)\t Which of the following do you think best represents each set of data: mean,  \n\t\nmedian or mode? Explain your answer.\nMaths2_Gr7_LB_Book.indb   218\n2014/09/04   11:35:29 AM\n\nChapter 16\nProbability\n\t\nCHAPTER 16: PROBABILITY\t\n219\nProbability theory deals with situations that can have many possible outcomes, only one of \nwhich actually occurs. For example, when you throw a dice, only one face shows – but any \nof the others could have shown. Or when you cross a street you usually get to the other \nside without being struck by a car – but it could have happened.\n16.1\t Possible and actual outcomes, and frequencies....................................................... 221\n16.2\t Relative frequencies................................................................................................ 222\n16.3\t More trials and relative frequencies......................................................................... 224\n\t\nMaths2_Gr7_LB_Book.indb   219\n2014/09/04   11:35:29 AM\n\n220\t MATHEMATICS Grade 7: Term 4\n1\n1\n4\n4\n3\n2\n4\n1\n3\n4\n5\n4\n1\n1\n2\n3\n1\n6\n4\n4\n3\n1\n5\n5\n5\n3\n2\n2\n5\n3\n3\n2\n4\n1\n2\n4\n4\n5\n4\n1\n2\n6\n2\n2\n5\n5\n2\n4\n6\n2\n5\n4\n5\n5\n3\n4\n1\n5\n4\n5\n3\n3\n2\n3\n1\n1\n2\n1\n4\n6\n1\n1\n2\n4\n2\n1\n5\n6\n2\n4\n4\n1\n5\n1\n4\n1\n1\n6\n5\n1\n2\n6\n2\n5\n6\n1\n3\n2\n2\n2\n4\n1\n1\n4\n5\n1\n3\n3\n4\n3\n6\n5\n5\n2\n3\n3\n4\n3\n5\n5\n3\n2\n1\n5\n4\n4\n4\n5\n2\n6\n6\n2\n4\n4\n2\n1\n3\n1\n5\n4\n5\n5\n3\n4\n3\n1\n4\n4\n1\n5\n2\n2\n1\n6\n2\n4\n1\n1\n6\n5\n5\n5\n3\n4\n4\n2\n4\n3\n3\n1\n4\n6\n1\n2\n2\n1\n3\n4\n3\n6\n1\n5\n5\n4\n1\n1\n2\n3\n2\n2\n6\n3\n3\n4\n4\n3\n4\n1\n6\n6\n6\n4\n5\n2\n4\n2\n1\n6\n4\n4\n2\n5\n3\n4\n4\n5\n1\n3\n6\n3\n2\n5\n6\n2\n4\n5\n2\n3\n1\n6\n3\n5\n5\n1\n5\n6\n2\n1\n3\n1\n2\n2\n3\n1\n3\n2\n2\n6\n1\n3\n2\n2\n4\n4\n4\n4\n3\n4\n6\n3\n3\n3\n5\n3\n6\n4\n6\n6\n3\n1\n1\n5\n6\n1\n5\n5\n1\n6\n4\n6\n4\n5\n4\n2\n2\n6\n6\n3\n3\n1\n1\n3\n5\n3\n2\n2\n1\n6\n4\n6\n2\n1\n6\n2\n6\n3\n1\n6\n6\n4\n4\n5\n3\n2\n6\n6\n4\n3\n1\n5\n1\n1\n1\n1\n3\n1\n4\n2\n5\n5\n2\n3\n4\n4\n1\n5\n2\n5\n3\n6\n4\n3\n5\n3\n2\n2\n6\n3\n4\n1\n1\n2\n1\n3\n4\n4\n4\n2\n2\n1\n2\n4\n3\n2\n2\n2\n6\n4\n2\n4\n2\n1\n3\n2\n5\n6\n3\n6\n2\n3\n2\n4\n6\n6\n3\n6\n5\n4\n2\n1\n5\n4\n3\n3\n1\n4\n6\n4\n4\n4\n1\n6\n1\n3\n3\n2\n4\n1\n5\n3\n2\n3\n4\n6\n3\n2\n5\n2\n6\n2\n5\n1\n5\n3\n4\n3\n2\n3\n1\n1\n1\n6\n1\n5\n3\n4\n3\n3\n6\n1\n5\n2\n2\n1\n3\n3\n3\n5\n5\n5\n1\n1\n3\n6\n2\n3\n1\n3\n3\n3\n5\n1\n2\n2\n3\n4\n6\n1\n5\n4\n4\n3\n3\n6\n6\n2\n2\n2\n2\n5\n6\n1\n1\n2\n3\n5\n6\n5\n4\n6\n6\n1\n5\n2\n2\n4\n4\n6\n2\n2\n4\n1\n1\n4\n1\n3\n2\n4\n5\n5\n6\n4\n2\n5\n1\n6\n3\n4\n4\n4\n2\n4\n5\n3\n3\n5\n5\n4\n2\n4\n3\n5\n1\n3\n3\n1\n2\n5\n2\n5\n2\n5\n1\n3\n6\n4\n2\n2\n2\n3\n2\n1\n3\n1\n4\n1\n1\n6\n3\n1\n6\n1\n3\n1\n3\n5\n1\n3\n1\n1\n5\n1\n6\n3\n4\n2\n1\n3\n5\n1\n6\n1\n3\n3\n5\n6\n4\n3\n1\n4\n1\n4\n1\n6\n2\n1\n5\n5\n5\n4\n1\n2\n2\n4\n5\n2\n1\n6\n5\n5\n1\n2\n2\n1\n3\n5\n3\n4\n4\n5\n1\n1\n6\n2\n3\n6\n1\n6\n3\n5\n1\n2\n6\n2\n4\n1\n4\n4\n1\n1\n1\n3\n5\n1\n3\n6\n6\n4\n1\n4\n6\n3\n2\n2\n3\n5\n1\n6\n2\n5\n5\n2\n3\n4\n2\n3\n5\n4\n4\n2\n6\n1\n2\n3\n4\n3\n1\n1\n5\n1\n6\n3\n5\n4\n1\n2\n6\n1\n6\n5\n4\n6\n6\n4\n1\n5\n1\n2\n1\n6\n1\n1\n4\n5\n4\n6\n2\n2\n2\n2\n6\n6\n5\n6\n1\n1\n1\n1\n4\n5\n4\n3\n3\n2\n1\n3\n3\n5\n3\n2\n4\n2\n1\n4\n6\n4\n2\n6\n6\n1\n5\n3\n5\n2\n2\n5\n4\n1\n5\n4\n5\n5\n4\n5\n3\n4\n2\n6\n4\n2\n4\n4\n5\n2\n2\n5\n5\n4\n5\n1\n4\n1\n4\n2\n6\n3\n1\n2\n6\n3\n3\n4\n6\n3\n2\n2\n2\n3\n6\n3\n2\n4\n5\n6\n5\n4\n6\n4\n1\n4\n6\n1\n6\n6\n1\n2\n1\n5\n1\n6\n6\n4\n4\n4\n3\n2\n4\n2\n5\n2\n1\n5\n4\n1\n3\n1\n2\n6\n5\n2\n2\n4\n1\n3\n4\n6\n3\n1\n2\n3\n6\n1\n1\n6\n1\n5\n5\n5\n1\n1\n4\n6\n1\n1\n5\n2\n1\n2\n6\n2\n3\n5\n2\n2\n5\n1\n2\n1\n3\n4\n6\n1\n1\n3\n3\n2\n2\n4\n3\n5\n2\n4\n4\n5\n1\n2\n3\n3\n2\n5\n6\n4\n4\n2\n1\n1\n4\n1\n5\n5\n4\n6\n5\n1\n6\n1\n4\n3\n3\n2\n6\n6\n4\n2\n3\n6\n2\n2\n5\n4\n2\n5\n6\n2\n1\n5\n3\n6\n3\n5\n5\n2\n2\n5\n6\n1\n3\n6\n1\n5\n6\n3\n4\n4\n4\n3\n1\n1\n2\n1\n2\n5\n5\n1\n6\n5\n1\n6\n3\n3\n6\n6\n4\n1\n2\n3\n2\n2\n4\n6\n6\n4\n6\n1\n4\n2\n3\n2\n4\n6\n4\n3\n5\n4\n1\n6\n2\n5\n1\n6\n4\n5\n2\n1\n1\n3\n5\n3\n2\n2\n5\n3\n1\n2\n4\n1\n5\n4\n4\n3\n4\n4\n2\n6\n2\n2\n5\n3\n2\n4\n4\n2\n5\n4\n5\n5\n3\n4\n2\n5\n4\n5\n2\n3\n2\n5\n4\n2\n5\n6\n2\n6\n1\n1\n2\n4\n2\n1\n5\n5\n2\n4\n4\n1\n5\n1\n4\n1\n1\n6\n5\n1\n3\n6\n2\n2\n6\n1\n3\n5\n2\n3\n6\n1\n2\n1\n3\n4\n3\n5\n4\n3\n6\n5\n5\n3\n5\n3\n4\n3\n5\n2\n3\n2\n1\n5\n4\n4\n4\n5\n2\n6\n6\n2\n4\n4\n2\n1\n5\n1\n5\n4\n5\n5\n3\n2\n3\n1\n4\n4\n1\n5\n2\n2\n1\n3\n2\n4\n1\n5\n6\n5\n5\n5\n3\n4\n4\n2\n4\n3\n3\n1\n4\n6\n1\n5\n2\n1\n4\n4\n3\n6\n1\n5\n5\n4\n5\n1\n2\n6\n2\n1\n6\n2\n3\n6\n1\n6\n3\n5\n5\n2\n6\n2\n4\n1\n6\n2\n1\n2\n1\n3\n5\n3\n3\n6\n6\n4\n4\n5\n1\n2\n1\n2\n3\n4\n5\n1\n6\n3\n3\n2\n6\n6\n4\n5\n5\n5\n5\n2\n1\n2\n2\n1\n2\nMaths2_Gr7_LB_Book.indb   220\n2014/09/04   11:35:30 AM\n\n\t\nCHAPTER 1: NUMERIC AND GEOMETRIC PATTERNS 1\t\n221\n\t\nCHAPTER 16: PROBABILITY\t\n221\n16\tProbability \n16.1\t Possible and actual outcomes, and frequencies\nwhat can you expect?\nYou will soon do an experiment. To do the experiment you need a bag like a plastic \nshopping bag or a brown paper bag. You also need three objects of the same size and \nshape, like three buttons, bottle tops or small square pieces of cardboard. The three \nobjects must look different, for example they should have different colours such as \nyellow, red and blue. If you use cardboard squares, you can write “yellow”, “red” and \n“blue” on them.\n1.\t (a)\t Put your three objects in your bag. You will later draw one object out of the bag, \n\t\nwithout looking inside. Can you say whether the object that you will draw will  \n\t\nbe the yellow one, the blue one or the red one?\n\t\n(b)\t Discuss this with two classmates.\n2.\t (a)\t Now draw an object out of the bag, write down its colour, and put it back. \n\t\n(b)\t You will soon do this 12 times. Can you say how many times you will draw each \n\t\nof the three colours? If you think you can, write your prediction below.\n\t\n(c)\t Compare your predictions with two classmates.\n\t\n(d)\t Can you think of any reason why you may draw blue more often than red or  \n\t\nyellow, when you do the experiment described in (b)?\n3.\t (a)\t Draw an object out of the bag, write down its colour, and put it back. Do this 12 \n\t\ntimes and write down the colour each time.\n\t\n(b)\t Write your results in the table below.\nOutcome\nYellow\nRed\nBlue\nNumber of times obtained\nMaths2_Gr7_LB_Book.indb   221\n2014/09/04   11:35:30 AM\n\n222\t MATHEMATICS Grade 7: Term 4\nWhat you did in question 3 is called a probability \nexperiment. Each time you drew an object out of \nthe bag, you performed a trial.\nEach time you performed a trial, three different \nthings could have happened. These are called the \npossible outcomes.\nEach time you performed a trial, one of the possible \noutcomes actually occurred. This is called the actual \noutcome.\nThe number of times that a specific outcome  \noccurred during an experiment is called the actual \nfrequency of that outcome.\n4.\t (a)\t What were the possible outcomes in the experiment that you did in question 3?\n\t\n(b)\t How many trials did you perform in the experiment? \n \n\t\n(c)\t What was the actual outcome in the third trial that you performed?\n\t\n(d)\t What was the actual frequency of drawing a blue object during the 12 trials in  \n\t\nthe experiment that you did? \n16.2\tRelative frequencies\nThomas also did the experiment in question 3 on page 221 but he performed more trials \nand his results were as follows:\nOutcome\nYellow\nRed\nBlue\nNumber of times obtained\n5\n7\n8\n1.\t (a)\t How many trials did Thomas perform in total? \n\t\n(b)\t What fraction of the trials produced yellow as an outcome? \n\t\n(c)\t What fraction of the trials produced red as an outcome? \n\t\n(d)\t What fraction of the trials produced blue as an outcome? \nMaths2_Gr7_LB_Book.indb   222\n2014/09/04   11:35:30 AM\n\n\t\nCHAPTER 16: PROBABILITY\t\n223\nThe fraction of the trials in an experiment that  \nproduce a specific outcome is called the relative  \nfrequency of that outcome.\nRelative frequency of an outcome = number of times the outcome occurred\ntotal number of trials\nA relative frequency can be expressed as a common fraction, as a decimal or as a \npercentage. The relative frequencies in the results of the experiment Thomas did \n(question 1) were one quarter for yellow, 7 twentieths for red and 2 fifths for blue. \nExpressed as percentages, the relative frequencies were 25%, 35% and 40%. The range \nof Thomas’s relative frequencies, expressed as percentages, is 15% (40% − 25%).\n2.\t (a)\t Use your calculator to calculate the relative frequencies that you obtained for  \n\t\nthe three different outcomes in the experiment you did in question 3 on page \n\t\n221. Express them both as fractions and percentages.\n\t\n(b)\t Calculate the range of the relative frequencies of the three outcomes for the  \n\t\nresults of the experiment you did in question 3.\n\t\n(c)\t You will soon repeat the experiment with 3 possible outcomes and 12 trials  \n\t\nthat you did. Do you think you will get the same results than when you first  \n\t\ndid the experiment? \n3.\t (a)\t Join with three or four classmates to work as a team, and discuss question 2(c).\n\t\n(b)\t Assign the “names” A, B, C, D and E (if you are 5) to the team members and  \n\t\ncomplete the table below for the experiment you did in question 3. Give the \n\t\nrelative frequencies as percentages. Note that to calculate the relative  \n\t\nfrequencies for the totals as percentages, you have to use your calculators.\nActual frequencies\nRelative frequencies %\nRange\nYellow\nRed\nBlue\nYellow\nRed\nBlue\nExperiment 1 by A\nExperiment 1 by B\nExperiment 1 by C\nExperiment 1 by D\nExperiment 1 by E\nTotals for experiment 1\n\t\n(c)\t Which of the ranges is the smallest? \nMaths2_Gr7_LB_Book.indb   223\n2014/09/04   11:35:30 AM\n\n224\t MATHEMATICS Grade 7: Term 4\n16.3\tMore trials and relative frequencies\nwhat happens when you conduct many trials?\n1.\t Join up with your teammates of the previous session. Each of you will soon repeat  \nthe experiment you did previously. You will put a yellow object, a red object and a \nblue object in a bag, draw one object and note the colour. You will do this 12 times. \n\t\nThis will be experiment 2. \n\t\n(a)\t Do you expect that the results will in some ways be the same as for the  \n\t\nexperiment in which you did this in the previous section? Do not talk to  \n\t\nyour teammates yet. Form your own opinion, and also consider why you  \n\t\nthink the results will be different or the same.\n\t\n(b)\t Share your ideas with your teammates.\nYou will soon repeat the experiment and write the results in the rows for “experiment \n2” on the table on the next page. You will repeat it once more and write the results \nin the rows for “experiment 3”. If you have time left, you may repeat it once more as \n“experiment 4”.\n2.\t (a)\t Look at the table on the next page. Certain rows are for the outcomes that you \n\t\nand your teammates obtain. The shaded rows are for adding different sets of  \n\t\noutcomes together. Think about what may happen and predict in what rows  \n\t\nthe ranges will be smaller than in other rows, and in what row the range will be \n\t\nthe smallest of all.\n\t\n(b)\t Share your ideas with your teammates.\n3.\t (a)\t Copy the totals for “experiment 1” into the first row of the table on the next  \n\t\npage. Do the experiment described in question 1 and enter the results in the  \n\t\nrows for “experiment 2”. Calculate the relative frequencies and the range.\n\t\n(b)\t Add in the results of your teammates, add up the totals and calculate the relative \n\t\nfrequencies and the range of the totals.\n4.\t Repeat question 3, and enter the results in the rows for “experiment 3”.\nMaths2_Gr7_LB_Book.indb   224\n2014/09/04   11:35:30 AM\n\n\t\nCHAPTER 16: PROBABILITY\t\n225\nActual frequencies\nRelative frequencies %\nRange\nYellow\nRed\nBlue\nYellow\nRed\nBlue\n1\nTotals for experiment 1\n2\nExperiment 2 by A\n3\nExperiment 2 by B\n4\nExperiment 2 by C\n5\nExperiment 2 by D\n6\nExperiment 2 by E\n7\nTotals for experiment 2\n8\nTotals for experiments  \n1 and 2 combined\n9\nExperiment 3 by A\n10\nExperiment 3 by B\n11\nExperiment 3 by C\n12\nExperiment 3 by D\n13\nExperiment 3 by E\n14\nTotals for experiment 3\n15\nTotals for experiments \n1, 2 and 3 combined\n16\nExperiment 4 by A\n17\nExperiment 4 by B\n18\nExperiment 4 by C\n19\nExperiment 4 by D\n20\nExperiment 4 by E\n21\nTotals for experiment 4\n22\nTotals for experiments \n1, 2, 3 and 4 combined\nMaths2_Gr7_LB_Book.indb   225\n2014/09/04   11:35:30 AM\n\n226\t MATHEMATICS Grade 7: Term 4\nMaths2_Gr7_LB_Book.indb   226\n2014/09/04   11:35:31 AM\n\nTerm 4\nRevision and assessment\n\t\nTERM 4: REVISION AND ASSESSMENT\t\n227\nRevision........................................................................................................................... 228\n• Integers...................................................................................................................... 228\n• Numeric patterns....................................................................................................... 229\n• Functions and relationships 2..................................................................................... 229\n• Algebraic expressions 2.............................................................................................. 230\n• Algebraic equations 2................................................................................................. 231\n• Collect, organise and summarise data........................................................................ 231\n• Represent data........................................................................................................... 233\n• Interpret, analyse and report on data......................................................................... 235\n• Probability.................................................................................................................. 237\nAssessment...................................................................................................................... 238\nMaths2_Gr7_LB_Book.indb   227\n2014/09/04   11:35:31 AM\n\n228\t MATHEMATICS Grade 7: Term 4\nRevision\nDo not use a calculator for any of the questions in this section. Show all your steps of \nworking.\nintegers\n1.\t Write down the next 3 terms of the following sequences:\n\t\n(a)\t 3; 2; 1; 0; …\t\n\t\n\t\n(b)\t −10; −12; −14; …\t\n2.\t Fill in either <, > or = to make the statement true:\n\t\n(a)\t −10 \n  −11\t\n(b)\t −8 + 2 \n  −8 − 2\n3.\t Rewrite the following numbers in order from smallest to largest: 3; −3; 0; −6; 4.\n4.\t Write down the values of all integers that are bigger than −12 and smaller than −8.\n5.\t Calculate the following:\n\t\n(a)\t −4 + 5\t\n(b)\t 7 − 3\t\n(c)\t 8 + 4\n\t\n(d)\t 5 − (−3)\t\n(e)\t −2 + (−4)\t\n(f)\t −1 − 2 – 3 − 4\n\t\n(g)\t −40 − (−24)\t\n(h)\t −24 − (−40)\t\n(i)\t −13 − 15\n6.\t If I descend from 10 m above sea level to 5 m below sea level, how many metres have  \nI descended?\n7.\t If a submarine that is 15 m below sea level rises 7 m, how far below sea level would  \nthe submarine still be?\nMaths2_Gr7_LB_Book.indb   228\n2014/09/04   11:35:31 AM\n\n\t\nTERM 4: REVISION AND ASSESSMENT\t\n229\nnumeric patterns \n1.\t Write down the first four terms of a sequence that fits the description given:\n\t\n(a)\t The sequence starts at −14, and each term is 3 bigger than the previous term.\n\t\n(b)\t The sequence starts at 5, and each term is 4 less than the previous term.\n2.\t Describe in words the relationship between the terms in the sequence. Then use the \nrelationship to find the next 3 terms in the sequence.\n\t\n(a)\t −90; −94; −98; …\n\t\n(b)\t −4; −5; −9; −14; …\n3.\t (a)\t Complete the table.\nx\n0\n1\n2\n3\n4\n5\n6\n7\n8\n2 − 4x\n2\n−6\n\t\n(b)\t Do the output values form a pattern with a constant difference? If so, what is  \n\t\nthe constant difference?\nfunctions and relationships 2\n1.\t Study the flow diagrams and fill in all the missing numbers:\n(a)\n–11\n–8\n–2\n+ 7\n–12\n–10\n1\n3\n(b)\n3\n–8\n– 6\n5\n0\n–7\n–10\n–12\nMaths2_Gr7_LB_Book.indb   229\n2014/09/04   11:35:32 AM\n\n230\t MATHEMATICS Grade 7: Term 4\n2.\t Fill in the rule.\n–22\n–30\n–32\n–39\n–18\n–9\n–17\n–19\n–26\n–5\n3.\t Study the tables and fill in all the missing numbers:\n(a)\nTerm number\n1\n2\n3\n4\n8\nValue of the term\n−9\n−12\n−15\n(b)\nTerm number\n1\n2\n3\n4\n7\nValue of the term\n5\n0\n−5\n−50\nalgebraic expressions 2\n1.\t Write numbers in the boxes to make the statements true:\n\t\n(a)\t If x = 5, then x − 12 = \n\t\n(b)\t If x = −4, then x + x = \n\t\n(c)\t If x = −2 and y = 4, then x − y = \n\t\n(d)\t If x = −8 and y = −3, then x − y = \n \t\n2.\t Estelle is visiting Paris in July, and she SMSes Sonja back in Pretoria that the \ntemperature at 6 a.m. is 13 °C. Sonja SMSes back that the temperature in Pretoria is  \n13 − x where x is 15 degrees. What is the temperature in Pretoria?\n3.\t Consider the following situation and identify the variable quantities and the constants:\n\t\nA camping site charges R50 as entrance fee for a vehicle and thereafter R60 per night. \nThe Brinks use the formula 60x + 50 to calculate the total cost if they want to camp \nfor x nights.\nMaths2_Gr7_LB_Book.indb   230\n2014/09/04   11:35:32 AM\n\n\t\nTERM 4: REVISION AND ASSESSMENT\t\n231\nalgebraic equations 2\n1.\t Solve for x:\n\t\n(a)\t x − 6 = −2\t\n(b)\t x + 4 = −2\t\n(c)\t x + x = −8\n2.\t Here is an equation: 2 + c = d\n\t\n(a)\t Write down a pair of integers that makes the equation true. One of the integers  \n\t\nshould be positive and the other negative.\n\t\n(b)\t Write down a pair of negative integers that makes the equation true. \n3.\t You are given that x − 6 = −15. Write down the value of:\n\t\n(a)\t x − 9\t\n(b)\t x + 2\n4.\t If d = c − 5, calculate the value of c when d has a value of −2.",
      "chapter_id": "12"
    },
    {
      "title": "Collect, organise and summarise data",
      "content": "collect, organise and summarise data\n1.\t Write down whether each of the described groups represents a sample or a population:\n\t\n(a)\t the learners of Star Primary\t\n\t\n(b)\t 30 drivers of Toyota Corollas\t\n2.\t Is the sample chosen for each study appropriate? Why do you say so?\n\t\n(a)\t The study is “favourite music amongst teenagers” and the sample is all  \n\t\nteenagers in Grade 7 at your school.\nMaths2_Gr7_LB_Book.indb   231\n2014/09/04   11:35:32 AM\n\n232\t MATHEMATICS Grade 7: Term 4\n\t\n(b)\t The study is “what food should the school tuck shop stock” and the sample is  \n\t\n6 learners in each of the grades at your school.\n3.\t Is the following multiple-choice question a good one? If it is not good, explain why \nand re-write it.\n\t\nHow old are you?\n\t\n(a)\t 6–10 years old\t \t\n\t\n(b)\t 10–15 years old\t\t\n\t\n(c)\t 15–20 years old\t\t\n\t\n(d)\t 20+ years old\t\n\t\n4.\t The learners of one Grade 7 class at Star Primary were asked how many pets they had, \nand this is the data that was generated:\n\t\n0; 3; 0; 1; 2; 3; 2; 1; 0; 0; 1; 3; 2; 2; 1; 1; 0; 1; 1; 2; 1; 4; 0; 1; 2; 1; 3; 0\n\t\n(a)\t Summarise this data in a tally and frequency table.\nNumber of pets\nTally\nFrequency\n\t\n(b)\t Write down the modal number of pets. \n\t\n(c)\t Write down the range of the number of pets. \n\t\n(d)\t Determine the mean number of pets, correct to 1 decimal place. \nMaths2_Gr7_LB_Book.indb   232\n2014/09/04   11:35:32 AM\n\n\t\nTERM 4: REVISION AND ASSESSMENT\t\n233\n5.\t The following stem-and-leaf display represents the amount of money (in rands) in \nsome learners’ purses/wallets:\n0\n0\n3\n4\n7\n9\n1\n0\n2\n4\n7\n9\n2\n0\n1\n2\n8\n3\n0\n2\n2\n7\n9\n4\n0\n2\n6\n7\n7\n8\n9\n5\n3\n4\nKey:   3|2 means 32\n\t\n(a)\t How many learners were sampled? \n\t\n(b)\t What is the maximum amount of money in the purses/wallets? \n\t\n(c)\t Determine the median of the amount of money in the purses/wallets.",
      "chapter_id": "13"
    },
    {
      "title": "Represent data",
      "content": "represent data\n1.\t The 120 learners in Grade 7 at a boys’ school were asked to name their favourite \nwinter sport, and the data was represented by the pie chart below.\n30%\n15%\n25%\nsoccer\nhockey\nrugby\nother\n30%\n\t\n(a)\t What percentage of the boys like hockey the most? \n\t\n(b)\t How many boys like soccer the most?\n\t\n(c)\t How many boys like hockey the most?\nMaths2_Gr7_LB_Book.indb   233\n2014/09/04   11:35:32 AM\n\n234\t MATHEMATICS Grade 7: Term 4\n2.\t Seventy-five people were asked to name their favourite South African soccer teams. \nThe bar graph shows the results, with two bars missing. The number of people that \nlike Ajax Cape Town or Bloemfontein Celtic the most are equal. Draw in the  \nmissing bars.\n18\n20\n16\n14\n12\n10\n8\n6\n4\n2\n0\nNumber of people\nFavourite team\nKaizer\nChiefs\nOrlando\nPirates\nAmazulu Ajax Cape\nTown\nBloem\nCeltic\nOther\n3.\t Draw a histogram to show the ages of a sample of people, as recorded in this \nfrequency table:\nAge group (years)\n< 20\n20–29\n30–39\n40–49\n50–59\n60–69\nFrequency\n5\n7\n6\n8\n4\n3\nMaths2_Gr7_LB_Book.indb   234\n2014/09/04   11:35:32 AM\n\n\t\nTERM 4: REVISION AND ASSESSMENT\t\n235",
      "chapter_id": "14"
    },
    {
      "title": "Interpret, analyse and report on data",
      "content": "interpret, analyse and report on data\n1.\t The information collected from boys about their favourite winter sport (see question \n1 on page 233) could be represented in a number of different ways, for example by a \npie chart or in a bar graph.\n30%\n15%\n25%\nsoccer\nhockey\nrugby\nother\n30%\n40\n30\n20\n10\n0\nPercentage of boys\nFavourite sport\nsoccer hockey\nrugby\nother\n\t\nWhat type of graph do you think is best suited to represent the data? Why do you  \nsay so?\n2.\t Jan wants to find out how important the following activities are in people’s lives: \nshopping, playing sport, watching TV, and doing hobbies. He goes to the shopping \nmall near his house and asks 20 teenagers to rank these activities in order of \nimportance. Describe some possible sources of bias in sourcing and collecting  \nthe data.\nMaths2_Gr7_LB_Book.indb   235\n2014/09/04   11:35:33 AM\n\n236\t MATHEMATICS Grade 7: Term 4\n3.\t A survey of favourite fast foods was undertaken. The data that was collected is \nrepresented in this bar graph:\n950\n955\n960\n965\n970\n975\n980\n985\n990\n995\n1 000\nNumber of votes\nFavourite food\nHot dogs\nHamburgers\nPizza\n\t\nDavid says, “The graph shows clearly that hamburgers are by far the least favourite \nfast food.” Do you agree with David? Explain your answer.\n4.\t The following data is collected for a project. It represents the number of times nine \ndifferent people have accessed their Facebook page in the past week: \n\t\n5; 8; 8; 10; 15; 17; 18; 23; 59. \n\t\nWhich measure of central tendency (the mode, mean or median) will best represent \nthe ‘average’ of the data? Explain your answer.\nMaths2_Gr7_LB_Book.indb   236\n2014/09/04   11:35:33 AM\n\n\t\nTERM 4: REVISION AND ASSESSMENT\t\n237",
      "chapter_id": "15"
    },
    {
      "title": "Probability",
      "content": "probability\n1.\t Rendani has 12 T-shirts: 3 are black; 4 are white; 2 are blue and 3 are green. If, one \nmorning, he picks a T-shirt at random from his cupboard, what is the probability \n(given as a fraction in simplest form) that:\n\t\n(a)\t he chooses a white T-shirt? \n\t\n(b)\t he chooses a T-shirt that is not blue? \n2.\t In a bag there are only black, green and yellow counters. You are going to take one \ncounter out of the bag at random. If you are told that:\n• the probability that it will be black is less than a third, and \n• the probability that it will be green is twice the probability it will be black\n\t\nwrite down one example of how many counters of each colour there might be in  \nthe bag.\n3.\t You are going to throw two dice at the same time, and the outcome will be the sum  \nof the numbers showing on each dice. For example, if a 3 and a 4 is thrown the \noutcome will be 7.\n\t\n(a)\t List all the possible outcomes of the experiment.\n\t\n(b)\t Write down the probability the outcome will be 2.\n\t\n(c)\t Write down the probability the outcome will be 10.\nMaths2_Gr7_LB_Book.indb   237\n2014/09/04   11:35:33 AM\n\n238\t MATHEMATICS Grade 7: Term 4\nAssessment\nIn this section, the numbers in brackets at the end of a question show the number of \nmarks that the question is worth. Use this information to help you determine how much \nworking is needed. \nThe total number of marks allocated to the assessment is 60.\n1.\t Write the numbers below in order from smallest to largest. \t \t\n\t\n\t\n\t\n\t\n\t\n(2)\n\t\n1; −4; 0; −2; 4\n2.\t List all the integers that are between −17 and −22.\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(2)\n3.\t Calculate the following:\t\n\t\n\t\n\t\n\t\n(4)\n\t\n(a)\t 8 − 10 \t\n(b)\t 5 + (−4)\n\t\n(c)\t −2 − 3 − 4 \t\n(d)\t −3 − (−2)\n4.\t If the temperature at sunset is 12 °C, and it drops by 16 degrees overnight, what is  \nthe temperature at dawn?\t \t\n\t\n\t\n\t\n(2)\n5.\t Fill in either <, > or = to make the statement true.\t\n\t\n\t\n(2)\n\t\n(a)\t −1 234 \n  −1 235 \t\n(b)\t 2 − (−1) \n  1 − (−2) \n6.\t Describe in words the relationship between the terms in the sequence. Then use  \n\t\nthe relationship to find the next 3 terms in the sequence.\t\t\n\t\n(6)\n\t\n(a)\t −81; −77; −73; …\n\t\n(b)\t 10; 4; 6; −2; 8; …\nMaths2_Gr7_LB_Book.indb   238\n2014/09/04   11:35:33 AM",
      "chapter_id": "16"
    },
    {
      "title": "Term 4: Revision and assessment",
      "content": "TERM 4: REVISION AND ASSESSMENT\t\n239\n7.\t Study the flow diagram and fill in all the missing numbers:\t\n\t\n        (3)\n4\n–20\n– 9\n10\n0\n–10\n–12\n–29\n–21\n8.\t Write numbers in the boxes to make the statements true.\t \t\n\t\n(4)\n\t\n(a)\t If x = 10, then x − 14 = \n\t \t\n\t\n(b)\t If x = −2, then x + x = \n \n\t\n(c)\t If x = −1 and y = 7, then x − y = \n  \n\t\n(d)\t If x = −5 and y = −1, then x − y = \n  \n9.\t Solve for x.\t\n\t\n\t\n\t\n\t\n(3)\n\t\n(a)\t x − 9 = −2\t\n(b)\t x + 7 = −5\t\n(c)\t x + x + x = −6\n10. Three possible solutions are given in brackets next to the equation, but only one  \n\t\nis correct. Which one is correct?\t\n\t\n\t\n(1)\n\t\n−5 − x = 10        {−5; −15; 15}\n11. Here is an equation: e + f = −5\n\t\n(a)\t Write down a pair of integers that makes the equation true. One of the integers  \n\t\nshould be positive and the other negative.\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(2)\n\t\n(b)\t Write down a pair of negative integers that makes the equation true.\t\n\t\n\t\n(2)\nMaths2_Gr7_LB_Book.indb   239\n2014/09/04   11:35:33 AM\n\n240\t MATHEMATICS Grade 7: Term 4\n12. An inter-schools shot-put competition was held and the best throws (in metres)  \nwere recorded as a stem-and-leaf display:\n5\n0\n1\n1\n1\n1\n3\n9\n6\n5\n6\n7\n7\n7\n9\n9\n7\n1\n4\n5\n7\n9\n8\n1\n1\n6\n9\n0\n0\n2\n7\n7\n8\nKey:   5|1 means 5,1\n\t\n(a)\t How many shot-putters were there in the competition? \n  (1)\n\t\n(b)\t Only those able to throw 7 m or more in the first two throws were allowed to  \n\t\ncontinue in the competition, to throw a further three times. \n\t\n\t\nHow many competitors were permitted to continue? \n  (1)\n\t\n(c)\t What is the modal distance of the throws? \n  (1)\n\t\n(d)\t What is the range? \n  (1)\n13. The graph shows the Mathematics results of four learners in two exams: the Term 2 \nexam and the Term 4 exam.\n0\n10\n20\n30\n40\n50\n60\n70\n80\nPercentage\nTerm 2\nSusan\nMalachai\nButi\nTumi\nTerm 4\n\t\n(a)\t What is the name of this type of graph? \n  (1)\n\t\n(b)\t Whose results remained constant? \n  (1)\n\t\n(c)\t Who performed best in Term 4, and what was his or her percentage?\t\n\t\n\t\n(2)\nMaths2_Gr7_LB_Book.indb   240\n2014/09/04   11:35:33 AM\n\n\t\nTERM 4: REVISION AND ASSESSMENT\t\n241\n14. In a survey, 200 people were asked about the quality of services they received from \ntheir local municipality. The bar graph summarises their responses.\nvery poor\ndon’t know\nsatisfactory\npoor\nvery good\n\t\n(a)\t Which response was the most common? \n  (1)\n\t\n(b)\t About how many people thought the services received were either very  \n\t\ngood or satisfactory?\t \t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(2)\n\t\n(c)\t Siyoli says that about 80 people think the services received were poor.  \n\t\nIs Siyoli right or wrong, or is it impossible to tell from the information  \n\t\nprovided?\t \t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(1)\n\t\n(d)\t If you had been the person responsible for this survey, what feedback  \n\t\nwould you give the local municipality?\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(1)\n15. Ashwell wanted to collect information on which fast food was the most popular \namongst 12- to 13-year-olds. He collected information by asking 10 of his friends \nwhich fast food they liked the most. \n\t\nDiscuss any problems with Ashwell’s process of data collection, and suggest  \n\t\nbetter alternatives.\t\n\t\n\t\n\t\n\t\n(4)\nMaths2_Gr7_LB_Book.indb   241\n2014/09/04   11:35:33 AM\n\n242\t MATHEMATICS Grade 7: Term 4\n16.\tYou are going to throw two dice at the same time, and the outcome will be the \t\nproduct of the numbers shown on the dice. For example, if a 3 and a 4 is thrown  \nthe outcome will be 12.\n\t\n(a)\t List all the possible outcomes of the experiment.\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(3)\n\t\n(b)\t Write down the probability that the outcome will be 6. \n\t\n\t\nGive your answer as a fraction in simplest form. \n  (3)\n17.\tA bag contains 20 counters of 3 different colours. I am going to take one counter  \nfrom the bag, without looking.\n\t\n(a)\t Fill in the missing information in this table:\t \t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(3)\nColour of  \ncounters\nNumber of counters\nProbability of \nchoosing this colour\nred\n5\nwhite\n1\n4\nblue\n10\n\t\n(b)\t Suppose that before taking a counter out of the bag, I add an extra white counter \n\t\nto the bag. How will this affect the probability that I will take a red counter? \n\t\n\t\nTick one of these options:\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n(1)\n• It will increase the probability.\t\t\n\t\n• It will decrease the probability.\t\n\t\n• The probability will remain the same.\t\n• It is not possible to tell.\t \t\n\t\n\t\nMaths2_Gr7_LB_Book.indb   242\n2014/09/04   11:35:34 AM",
      "chapter_id": "18"
    }
  ]
}