Senior Phase Grade 8 • Study Guide 1/2
Mathematics CAPS
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Mathematics
exercises to refresh prior knowledge. • Revision explanations of concepts and techniques. • Detailed examples help learners to better understand new concepts. • Worked Varied exercises to entrench theory and practise mathematical skills. • Test papers and memorandums for exam preparation. • Formula sheets and accepted geometrical reasons for • quick reference. of mathematical terms. • Index facilitator’s guide contains step-by-step calculations • The and answers. • Use in school or at home.
Study Guide 1/2
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© Optimi
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Reg. No.: 2006/002404/07 ISBN: 9781779841827
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Mathematics
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Study Guide 1/2 − Grade 8
2608-E-MAM-SG01
CAPS-aligned
Prof. C Vermeulen, Lead author H Otto M Sherman
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TABLE OF CONTENTS
Solving problems involving rate ...............................................................11 Exercise 4: Solving problems involving rate ...............................................14
PREFACE...................................................................................................... viiii THEME 1: WHOLE NUMBERS........................................................................... 1 Introduction.............................................................................................................1 1.1
The commutative property .............................................................................1
1.2
The associative property ................................................................................1
1.3
The distributive property of multiplication over addition and subtraction.....2
2.
Identities and inverses..................................................................................3
2.1
Identities.........................................................................................................3
2.2
Division by zero..............................................................................................3
2.3
Inverses...........................................................................................................3
6.1
Ratio..............................................................................................................14
6.2
Writing ratios in their simplest form............................................................14
6.3
Finding an unknown quantity in a ratio........................................................15 Exercise 5: Solving problems involving ratio...............................................15
6.4
Divide an amount in a given ratio.................................................................16
6.5
Finding the full amount when part of a ratio is given..................................16
6.6
Increasing or decreasing a number in a given ratio......................................17
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Properties of whole numbers.......................................................................1
Solving problems involving ratio...............................................................14
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Exercise 6: Solving problems involving ratio...............................................18
Exercise 1: Properties of whole numbers, identities and inverses .................4 3.
Calculation techniques.................................................................................5
3.1
BODMAS rule................................................................................................5
3.2
Rounding off...................................................................................................6 Exercise 2: Calculation techniques.................................................................7
4.
Prime numbers, multiples and factors ......................................................8
4.1
Multiples and factors......................................................................................8
4.2
Divisibility rules.............................................................................................8
4.3
Prime factorisation..........................................................................................9
4.4
Highest common factor...................................................................................9
4.5
Lowest common multiple.............................................................................10
Finance: profit, loss, discount and vat......................................................19
7.1
Profit and loss ..............................................................................................19
7.2
Discount .......................................................................................................20
7.3
VAT...............................................................................................................20 Exercise 7: Profit, loss, discount and VAT...................................................21
8.
finance: exchange rate, interest and hire purchase.................................21
8.1
Exchange rate................................................................................................21
8.2
Interest..........................................................................................................22
8.3
Hire purchase................................................................................................23 Exercise 8: Exchange rate, interest and hire purchase..................................23
Self-evaluation.......................................................................................................24 Summary of theme................................................................................................25 End of theme exercise...........................................................................................26 THEME 2: INTEGERS......................................................................................29 Introduction...........................................................................................................29
Exercise 3: Prime numbers, multiples and factors........................................11 ©Optimi
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Integers........................................................................................................29 G08– Mathematics
G08 – Mathematics – Study Guide 1/2 1.1
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Counting, ordering and comparing integers.................................................29 Exercise 9: Count, order and compare integers............................................30
Exponential form........................................................................................49 Exercise 18: Write numbers in exponential and expanded form..................50
1.2
Adding integers.............................................................................................30
3.
Exponential laws.........................................................................................50
1.3
Subtracting integers......................................................................................31
3.1
Application of exponential laws...................................................................50
Exercise 10: Adding and subtracting integers..............................................32
Exercise 19: Apply the first two exponential laws........................................52
2.
Multiplication and division of integers.....................................................33
Exercise 20: Apply the third and fourth exponential laws............................53
2.1
Multiplying a positive integer by a negative integer.....................................33
2.2
Multiplying a negative integer by a negative integer....................................33
2.3
Dividing integers .........................................................................................34
3.2
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Exercise 21: Calculations using exponential laws........................................54 Exercise 22: Applying the exponential laws.................................................56
4.
Properties of integers .................................................................................35
Exercise 23: Square roots and cube roots of integers involving all four operations.........................................................................................59
Exercise 12: Properties of integers...............................................................36
5.
Exercise 13: Properties of integers ..............................................................36 Exercise 14: Calculations using all four operations......................................39 5.
Solving problems involving integers..........................................................40 Exercise 15: Problems involving integers.....................................................41
Self-evaluation.......................................................................................................43 Summary of theme................................................................................................43 End of theme exercise...........................................................................................44
THEME 3: EXPONENTS................................................................................. 46 Introduction...........................................................................................................46 1.
Mental calculations.....................................................................................46
1.1
Squares and square roots..............................................................................46
6.
Solving problems involving numbers in exponential form......................62 Exercise 25: Solving problems with numbers in exponential form..............63
7.
Scientific notation.......................................................................................64 Exercise 26: Write numbers in scientific notation and vice versa................65
Self-evaluation.......................................................................................................65 Summary of theme................................................................................................66 End of theme exercise...........................................................................................66 THEME 4: NUMERIC AND GEOMETRIC PATTERNS.....................................69 Introduction...........................................................................................................69 1.
Exercise 16: Determine squares and square roots........................................47 1.2
Squares, cubes, square roots and cube roots of rational numbers.........59 Exercise 24: Squares, cubes, square roots and cube roots of common fractions and decimal fractions......................................................61
Calculations using all four operations .....................................................37
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Calculations involving square roots and cube roots of integers.............57
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Exercise 11: Multiplying and dividing integers............................................35 3.
Calculations involving exponential laws......................................................54
Investigate and extend number patterns..................................................69 Exercise 27: Investigate and extend number patterns...................................70
Cubes and cube roots....................................................................................48 Exercise 17: Cubes and cube roots...............................................................48 iv
2. 3.
Investigate and extend geometric patterns...............................................71
Self-evaluation.......................................................................................................95
Exercise 28: Investigate and extend geometric patterns...............................71
Summary of theme................................................................................................95
Find the rule that relates the value of a term to its position in the number pattern...........................................................................................73
End of theme exercise...........................................................................................96 THEME 7: ALGEBRAIC EQUATIONS..............................................................97
Exercise 29: Find the rule that relates the value of a term to its position in the number pattern.................................................................76
Introduction...........................................................................................................97
Self-evaluation.......................................................................................................77 End of theme exercise...........................................................................................78
2.
Solving equations through inspection.......................................................98 Exercise 34: Solve equations through inspection.........................................99
3.
Expressing a situation in mathematical language..................................100
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THEME 5: FUNCTIONS AND RELATIONSHIPS............................................. 80
Exercise 35: Expressing a situation in mathematical language..................101
Introduction...........................................................................................................80 1.
What is an equation?..................................................................................97
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Summary of theme................................................................................................78
1.
Self-evaluation.....................................................................................................101
Input and output values.............................................................................80
Summary of theme..............................................................................................102
Exercise 30: Input and output values............................................................83
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Self-evaluation.......................................................................................................84
End of theme exercise.........................................................................................102
Summary of theme................................................................................................84
THEME 8 :ALGEBRAIC EXPRESSIONS........................................................104
End of theme exercise...........................................................................................84
Introduction.........................................................................................................104
THEME 6: ALGEBRAIC EXPRESSIONS......................................................... 86 Introduction...........................................................................................................86 1.
Algebraic language.....................................................................................86
1.
Exercise 36: Algebraic language................................................................105 2.
Exercise 31: Algebraic language..................................................................87 2.
Algebraic language.....................................................................................87
3.
Equivalent algebraic expressions..............................................................89
4.
Simplify algebraic expressions...................................................................91
Simplify algebraic expressions by adding and subtracting like terms...................................................................................................105 Exercise 37: Add and subtract like terms...................................................106
3.
Exercise 32: Algebraic expressions..............................................................88
Algebraic language...................................................................................104
Add and subtract algebraic expressions.................................................106 Exercise 38: Add and subtract algebraic expressions.................................107
4.
Exponential laws.......................................................................................108 Exercise 39: Exponential laws....................................................................111
Exercise 33: Simplify algebraic expressions................................................93
5.
Multiply algebraic terms..........................................................................112 Exercise 40: Multiply algebraic terms........................................................112
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G08 – Mathematics – Study Guide 1/2 6. 7.
Multiply algebraic expressions................................................................113
Self-evaluation.....................................................................................................136
Exercise 41: Multiply algebraic expressions..............................................114
Summary of theme..............................................................................................137
Divide algebraic terms..............................................................................115
End of theme exercise.........................................................................................137
Exercise 42: Divide algebraic terms...........................................................115 8.
THEME 10: CONSTRUCTIONS..................................................................... 138
Divide algebraic expressions....................................................................116
Introduction.........................................................................................................138
Exercise 43: Divide algebraic expressions.................................................117 9.
1.
Simplify algebraic expressions using multiple operations....................118
Exercise 51: Measure, construct and classify angles..................................140
Exercise 44: Simplify algebraic expressions using multiple operations.....119 Squares, cubes, square roots and cube roots..........................................120
2.
Construction of perpendicular lines.......................................................141
3.
Construction of angles without using a protractor................................143
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Revision of Grade 7 geometry..................................................................138
10.1 Squares and cubes.......................................................................................120
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Exercise 52: Construction of perpendicular lines and angles without using a protractor...........................................................................146
Exercise 45: Squares and cubes..................................................................121 10.2 Square roots and cube roots........................................................................121 Exercise 46: Square roots and cube roots...................................................122
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Self-evaluation.....................................................................................................124 Summary of theme..............................................................................................125 End of theme exercise.........................................................................................125
4.
Triangles ...................................................................................................147
4.1
Construction of triangles .........................................................................147
4.2
Properties of triangles deduced by constructing triangles..........................151 Exercise 53: Construct triangles and deduce their properties ...................151
5.
THEME 9: ALGEBRAIC EQUATIONS 2........................................................ 127 Introduction.........................................................................................................127 1.
Solve equations through inspection.........................................................127 Exercise 47: Solve equations through inspection.......................................128
2. 3. 4.
Solve equations using additive and multiplicative inverses...................128
Construction of quadrilaterals and the deduction of certain properties of quadrilaterals........................................................................................153 Exercise 54: Construct quadrilaterals and deduce their properties............157
Self-evaluation.....................................................................................................158 Summary of theme..............................................................................................159 End of theme exercise.........................................................................................159
Exercise 48: Solve equations using additive and multiplicative inverses...131
THEME 11: GEOMETRY OF 2D SHAPES...................................................... 162
Solve equations using the exponential laws............................................132
Introduction.........................................................................................................162
Exercise 49: Solve equations using exponential laws.................................133
1.
Triangles ...................................................................................................162
Use equations to solve problems..............................................................133
1.1
Classification of triangles according to their sides.....................................162
Exercise 50: Use equations to solve problems............................................136
1.2
Classification of triangles according to their angles...................................163 Exercise 55: Classify triangles according to their sides and angles...........164
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2.
Properties of triangles..............................................................................164
2.3
Perpendicular lines.....................................................................................198
Exercise 56: Use the properties of exterior and interior angles of a triangle......................................................................................165
2.4
Vertically opposite angles...........................................................................198 Exercise 64: Perpendicular lines, vertically opposite angles and adjacent angles on a straight line.........................................................199
Exercise 57: Find unknown angles using the properties of triangles.........168 3.
Quadrilaterals...........................................................................................170
3.
Parallel lines..............................................................................................201
3.1
Definitions of quadrilaterals.......................................................................170
3.1
Corresponding angles.................................................................................202
3.2
Solve problems by using the properties of quadrilaterals...........................172 3.2
Solve problems by using the properties of quadrilaterals and triangles.....175
3.3
Exercise 66: Alternate angles.....................................................................208
Exercise 59: Use the properties of triangles and quadrilaterals to find unknown values...............................................................................175 4.
Co-interior angles.......................................................................................211 Exercise 67: Co-interior angles..................................................................212 Exercise 68: Mixed exercise.......................................................................215
Congruency...............................................................................................177
Self-evaluation.....................................................................................................217
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Exercise 60: Congruency............................................................................180 5.
Alternate angles..........................................................................................206
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Exercise 58: Solve problems using the properties of various quadrilaterals..............................................................................................173
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Exercise 65: Corresponding angles............................................................204
Similarity...................................................................................................181 Exercise 61: Similarity in polygons............................................................184
Summary of theme..............................................................................................218 End of theme exercise.........................................................................................219
Self-evaluation.....................................................................................................186
JUNE: PAPER 1 (A).......................................................................................222
Summary of theme..............................................................................................187
JUNE: PAPER 1 (B).......................................................................................224
End of theme exercise.........................................................................................187
THEME 12: GEOMETRY OF STRAIGHT LINES..............................................191 Introduction.........................................................................................................191 1.
Angles around a point .............................................................................191
2.
Basic angle relationships..........................................................................192
2.1
Complementary and supplementary angles................................................192
JUNE: PAPER 2 (A)......................................................................................226 JUNE: PAPER 2 (B)...................................................................................... 231 FORMULAE..................................................................................................235 EUCLIDEAN GEOMETRY: ACCEPTED REASONS...................................... 237 INDEX...........................................................................................................239
Exercise 62: Complementary and supplementary angles...........................193 2.2
Angles around a point and on straight lines...............................................194 Exercise 63: Angles around a point and angles on a straight line..............195
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G08– Mathematics
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G08 – Mathematics – Study Guide 1/2
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The study guides and facilitator’s guides are divided into 24 themes. The themes correspond both in terms of content and time allocation with the CAPS guidelines, and represent the year plan. Study guide 1/2 and facilitator’s guide 1/2 cover themes 1 to 12 (terms 1 and 2) and study guide 2/2 and facilitator’s guide 2/2 cover themes 13 to 24 (terms 3 and 4).
PREFACE Grade 8 is the second year of the General Education and Training (GET) phase, in which Mathematics is still a compulsory subject. However, it is the first year of secondary school, and many of you would have found the transition from primary school to secondary school challenging. Rather than viewing Grade 8 work as completely new because it is the beginning of secondary school, it is important to remember that Grade 8 work continues to build upon the work done in Grade 7 (and earlier grades). Furthermore, keep in mind that you must choose between Mathematics and Mathematical Literacy from Grade 10. You sometimes choose Mathematical Literacy for the wrong reasons. This includes that you found Mathematics up to Grade 9 difficult, and that Mathematical Literacy is less challenging than Mathematics in grades 10 to 12. However, before you exercise this choice at the end of Grade 9, you should carefully consider your plans for the future, especially if you want to obtain tertiary qualifications (at a university or college). For many tertiary courses Mathematics (and not Mathematical Literacy) at Grade 12 level is a requirement for admission. Therefore it is important that you should work hard in Mathematics in Grade 8 in order to lay a good foundation for Grade 9 Mathematics, so that you can successfully proceed with Mathematics in grades 10 to 12 to keep later options open.
No of hours
No of weeks
6
1,5
Whole numbers
9
2
Integers
9
2
Exponents
4,5
1
Numeric and geometric patterns
3
0,5
Functions and relationships
4,5
1
Algebraic expressions
3
0,5
Algebraic equations
9
2
Algebraic expressions
3
0,5
Algebraic equations
8
2
Constructions
8
2
Geometry of 2D shapes
9
2
Geometry of straight lines
7
1,5
Common fractions
6
1,5
Decimal fractions
5
1
Theorem of Pythagoras
5
1
Perimeter and area
5
1
Surface area and volume
10,5
2,5
Data handling
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This product consists of two study guides and two facilitator’s guides based on the concepts of Optimi’s GuidED Learning™ learning model to help you and your facilitators to be successful in the study of Mathematics. It covers all the work for Grade 8 Mathematics and complies with the requirements of the CAPS guidelines as compiled by the Department of Basic Education. The study guides and facilitator’s guides are supported by additional lesson structures on the Optimi Learning Platform (OLP). It offers continuous guidance to support and enrich your learning process. This guidance is based on the latest insights in education, cognitive psychology and neuroscience. Note that the study guides and facilitator’s guides can be used independent of the OLP.
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Below we explain how the study guides and facilitator’s guides were compiled, and how you and your facilitators can use them to achieve success in Mathematics. ©Optimi
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Theme
Assessment
Test Assignment
Test Investigation Mid-year examination
Test Assignment Project
G08– Mathematics
G08 – Mathematics – Study Guide 1/2
4
No of hours
No of weeks
6
1,5
Functions and relationships
3
0,5
Algebraic equations
9
2
Graphs
6
1,5
Transformation geometry
7
1,5
Geometry of 3D objects
4,5
1
Probability
Theme
Structure of themes
Assessment
Learning is a complex process. Millions of brain cells and neural pathways in our brains work together to store new information in the long-term memory so that we will be able to remember it later on. Long-term memory is not our only type of memory and when we learn, our working memory is just as important. Working memory is different from long-term memory and has a limited capacity. This means that one’s working memory can only handle a small amount of new information at a time.
Assignment Investigation End-of-year examination
When one learns mathematics, there is a lot of new information your brain needs to process, which can easily exhaust your working memory. This relates to the cognitive load theory. The study guides have been written and compiled in such a way that they do not overtax the working memory and therefore simplify the process of learning mathematics. Your cognitive capacity is considered at all times.
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Term
Time allocation
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* You will find the latest and most comprehensive information on assessment in the portfolio book and assessment plan.
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According to the CAPS requirements, at least 4,5 hours should be spent on teaching mathematics per week. For example, 6 hours (roughly 1,5 weeks × 4,5 hours per week) will be spent teaching Theme 1 (Whole numbers). Themes have not been sub-divided into lessons; you and your facilitator are at liberty to complete as much content per session and per week as your progress allows. If you work at a slower pace, the necessary adaptations should be done so that you will still be able to master all the work in time. Note that the teaching time referred to above does not include the time during which you should apply and practise the knowledge and concepts you have learned. For this purpose, various exercises are provided throughout each theme. These exercises involve different ways of applying and practising new knowledge and cover various degrees of difficulty. You should try to do all of these exercises. Complete solutions are provided in the facilitator’s guide.
This means that various strategies are used to ensure that you have the best possible chance to master each part of the work. In the end one can say that learning occurred when you have stored new information in your long-term memory and are able to recall and use it. The structure of the study guides supports this process and helps you to master mathematics. Each theme has the same structure in order to make it easier for you to navigate through them. Each theme has the following structure:
Introduction This briefly tells you what the theme is about without providing details or using “difficult” or unknown concepts. A comprehensive list of the learning outcomes you need to master in a specific theme is given as a summary at the end of the theme.
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Use the following scale to determine how comfortable you are with each topic in the table below:
SUB-THEMES
1. Help! I don’t feel comfortable with the topic at all. I need help. 2. Alarm! I don’t feel comfortable, but I just need more time to work through the topic again. 3. OK! I feel moderately comfortable with the topic, but I still struggle sometimes. 4. Sharp! I feel comfortable with the topic. 5. Whoo-hoo, it’s party time! I feel totally comfortable with the topic and can even answer more complicated questions about it.
New concepts and procedures are explained. Relevant previous knowledge is also dealt with here if necessary.
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Worked examples Worked examples show you how the new concepts and procedures are applied and help you understand and apply the newly taught concepts and procedures.
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Exercises
Complete the table:
Topic
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The exercises give you the opportunity to practise the concepts and procedures taught. It is important to try and complete all exercises. Complete solutions are provided in the facilitator’s guides. Questions usually progress from easy (in order to master and practise basic concepts and procedures) to difficult (more complex operations).
Mixed exercises are also provided, where you get the opportunity to practise different concepts and procedures and integrate these with previous themes.
Self-evaluation
2
3
4
5
Tip: Complete each self-evaluation as honestly as possible. If there are aspects which you have not mastered, revisit these and make sure that you do master them. Ask the facilitator for help. It is important not to move on to a next theme or sub-theme before you have mastered the topic involved, even if this means that you spend more time on a specific theme than recommended by the CAPS.
Summary of theme Here you will find a summary of what you should have mastered in the theme. This is expressed in more formal mathematical language in order to be in keeping with the CAPS (the curriculum statement).
In each theme, there is an activity where you need to reflect critically about the extent to which you have mastered certain concepts and procedures. This activity has the following format:
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G08– Mathematics
G08 – Mathematics – Study Guide 1/2 Complete ten formal assessment tasks for school-based assessment:
End of theme exercise
Forms of assessment
This is a mixed exercise involving all concepts and procedures dealt with in the theme, where this work can also be integrated with previous work. The degree of difficulty of this exercise varies. It is important that you try to complete all the exercises. Complete solutions can be found in the facilitator’s guides.
Test
Mixed exercises such as these in this guide form a very important component of mastering mathematics. There is a big difference between the ability to recognise the work and the ability to recall it. When you are able to recognise your work, you will often say “Oh, of course!” but you struggle to remember this when you are writing an examination. When you are able to recall your work, this means that you have captured that knowledge in your long-term memory and are able to remember and use it. Mixed exercises enable you to not only recognise the work, but also recall it from your long-term memory.
Term 1
Term 2
Term 3
Tasks Weight Term per year 4
1
1
1
3
Examination Assignment
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School-based assessment
1
1
1
1
Investigation
1
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Project
Total
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When you practise the same type of problem over and over, you often get lazy and do not reflect upon the exercise anymore. You are convinced that you know exactly what type of problem you need to solve. But in a test or exam, all these problems are mixed up and then it might be difficult to know what to do. When mixed exercises form part of your learning process, you learn to identify and complete a problem correctly. This means that you are truly prepared for tests or exams, because you can recall your work instead of merely recognising it.
Assessment requirements
Minimum requirements per term
Visit Impaq’s online platform for the assessment plan and comprehensive information about the compilation and mark allocation of tests, assignments and examinations. The number of assignments, marks and relative weight is subject to change.
3
1
2
1 2
3
3
40%
1 2
End of the year examination
10* 1
60%
*To be completed before the end of the year examination
The two papers in the middle of and at the end of the year are compiled as follows (in accordance with CAPS): Midyear exam: Paper 1
Topics
Marks
% Mark
Solving 7±2 problems
15 ± 2
Concepts
Whole numbers
Properties of numbers
Calculations using whole numbers
Multiples and factors
Integers
Counting, ordering and comparing integers
Calculations with integers
Solving problems
Tip: Be aware of the CAPS requirements and plan the year’s assessment accordingly.
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1
10 ± 2 23 ± 2
Paper 1
Comparing and representing numbers in exponential form
Calculations using numbers Solving problems in exponential form
Paper 1
Topics
11 ± 2 23 ± 2
Concepts Calculations using whole numbers
Multiples and factors
Solving problems
7±2
11 ± 2
Integers
Properties of integers
Calculations with integers
Solving problems
4±2
7±2
Common fractions
Calculations Solving using problems fractions
Equivalent 5±2 forms
9±2
Decimal fractions
Calculations with Solving problems decimal fractions
Equivalent 4±2 forms
7±2
Comparing and Calculations using representing numbers in exponential numbers in form exponential form
Solving problems
7±2
11 ± 2
3±2
6±2
10 ± 2
16 ± 2
7±2
11 ± 2
7±2
11 ± 2
75
100
5±2
11 ± 2
Functions and relationships
4±2
Algebraic language
Algebraic equations
Solving equations
Expand and simplify algebraic expression
7±2
Total Paper 2
Topics Geometry of straight line Geometry of 2 D shapes Total
16 ± 2 12 ± 2
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Algebraic expressions
Concepts
8±2
Angle relationships
Solving problems
Classifying 2D shapes
Similar and congruent 2D shapes Solving problems
8±2
60
100
Marks
% Mark
Investigate and extend patterns
53 ± 2
Numeric and geometric patterns
47 ± 2
Algebraic expressions
Algebraic language
100
Algebraic equations
Solving equations
Graphs
Interpreting graphs
28 ± 2 60
Total ©Optimi
Percentages
Exponents
32 ± 2
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% Mark
Properties of numbers
Numeric and geometric Investigate and extend patterns patterns Input and output values
Marks
Whole numbers
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Exponents
Marks
Concepts
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Topics
End of year exam:
% Mark
Expand and simplify algebraic expression
Drawing graphs
G08– Mathematics
G08 – Mathematics – Study Guide 1/2
Marks
% Mark
Solving problems
9±2
15 ± 2
Similar and Solving congruent 2D problems shapes
8±2
13 ± 2
Geometry of straight line
Angle relationships
Geometry of 2D shapes
Classifying 2D shapes
Geometry of 3D Objects
Classifying 3D objects
Transformation Transformations Geometry
Enlargements and reductions
• • •
Maths 4 Africa, available at www.maths4africa.co.za The Siyavula textbook, available online for free at www.siyavula.com Pythagoras, available at www.fisichem.co.za.
Calculator 7±2
12 ± 2
6±2
10 ± 2
The Theorem of Pythagoras
Use the Theorem of Pythagoras
Data Handling
Organise and Represent Interpret Analyse Report 10 ± 2 summarise data data data data data
Probability
Probability
5±2
We recommend the CASIO fx-82ES (Plus) or CASIO fx-82ZA. However, any scientific, non-programmable and non-graphing calculator is suitable. Tip: Ensure that you have a suitable calculator.
8±2
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Total
Any other books can be used along with this textbook for extra exercises and explanations, including:
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Concepts
Supplementary books
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Paper 2
Topics
18 ± 2
5±2
8±2
60
100
Tip: Make sure that you know which themes are covered in which paper and what the relevant weighting of each is. Ensure that papers comply with this weight allocation. Note: The composition of the exams are subject to change. Always refer to the portfolio book and assessment plan for updated information about the composition of the e𝑥amination papers.
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Numbers, operations and relationships (1)
REMEMBER
THEME 1
The set of whole numbers ={0; 1; 2; …}
WHOLE NUMBERS
1.1
The commutative property REMEMBER
Introduction
• • •
•
•
1.
properties of whole numbers: ◦
commutative, associative and distributive properties
◦
the numbers 0 and 1 as identities
calculations with whole numbers involving all four operations different calculation techniques and strategies using prime factorisation of numbers to find: ◦
the lowest common multiple (LCM)
◦
the highest common factor (HCF)
solving problems involving: ◦
ratio
◦
rate
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•
•
The commutative property for addition. If we add two whole numbers, we can change the order of the numbers. Examples: 7 + 3 = 10and 3 + 7 = 10
•
The commutative property for multiplication If we multiply two whole numbers, we can change the order of the numbers. Examples: 4 × 3 = 12and 3 × 4 = 12
•
The commutative property does not work for subtraction or division. Examples:
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In this theme, we revise and extend work covered in Grade 7 regarding:
E
To commute means to go backwards and forwards. The numbers can move backwards and forwards.
10 − 3 = 7, but 3 − 10 =a negative number. The answers are not the same. 24 ÷ 4 = 6, but 4 ÷ 24 =a fraction. The answers are not the same.
1.2 The associative property
solving problems involving money.
REMEMBER To associate with someone means to get together. The brackets hold the numbers together.
PROPERTIES OF WHOLE NUMBERS •
In this sub-theme, we briefly review the properties of whole numbers using addition and multiplication. ©Optimi
1
The associative property for addition. If we add three whole numbers, we can group the numbers differently and still get the same answer. G08 – Mathematics
Theme 1: Whole numbers
G08 – Mathematics – Study Guide 1/2
•
Example: ( 5 + 2)+ 3 = 7 + 3 = 10and 5 + ( 2 + 3) = 5 + 5 = 10
The distributive law also works from behind the bracket, e.g.
The associative property for multiplication. If we multiply three whole numbers, we can group the numbers differently and still get the same answer.
(4 + 6) × 2 = 4 × 2 + 6 × 2 = 8 + 12 = 20 Why is this? It is because of the commutative property. We can swop 2 and the bracket around.
Example: ( 5 × 2)× 3 = 10 × 3 = 30and 5 × ( 2 × 3) = 5 × 6 = 30 •
The associative property does not work for subtraction or division. Examples:
If we multiply the difference of two numbers by another number, the distributive property also works, e.g. 3 × ( 7 − 5) = 3 × 7 − 3 × 5.
( 10− 3) − 2 = 7 − 2 = 5, but 10 − (3 − 2) = 10 − 1 = 9. The answers are not
NOTE
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the same.
(24 ÷ 4) ÷ 2 = 6 ÷ 2 = 3, but 24 ÷ ( 4 ÷ 2) = 24 ÷ 2 = 12. The answers are not the same.
PL
REMEMBER
SA M
1.3 The distributive property of multiplication over addition and subtraction
We can use these properties to make calculations simpler, especially when we do algebra. The commutative and associative properties work for addition and multiplication, but not for subtraction or division. The distributive property works for multiplication over addition or subtraction.
To distribute means to give out. The number in front is distributed to each number in the brackets. If we multiply the sum of two numbers by another number: •
we can first add the two numbers and then multiply the answer by the other number, e.g. 2 × (4 + 6) = 2 × 10 = 20
•
or we can multiply each number in the brackets by the other number using the distributive property, e.g.
Worked example 1: Properties of whole numbers
Complete the statements and indicate which property you used. a)
173 + 14 = 14 + ___
b)
(3 × 6) × ___ = 3 × (6 × 11)
c)
17 × (32 + 8) = 17 × 32 + ___
d)
86 × 12 = ___ × 86
e)
17 + (6 + 12) = (17 + ___) + 12
f)
(7 − 3) × 8 = 7 × 8 − ___
Solutions
2 × (4 + 6) = 2 × 4 + 2 × 6 = 8 + 12 = 20
2
a)
173 + 14 = 14 + 173
b)
(3 × 6) × 11 = 3 × (6 × 11)
Use the commutative property for addition Use the associative property for multiplication
c)
17 × (32 + 8) = 17 × 32 + 17 × 8
d)
86 × 12 = 12 × 86
e)
17 + (6 + 12) = (17 + 6) + 12
f)
( 7 − 3) × 8 = 7 × 8 − 3 × 8
2.
IDENTITIES AND INVERSES
Use the distributive property
Why can’t we divide by 0?
Use the commutative property for multiplication
Look at this pattern. We will divide 120 by smaller and smaller numbers, and see what happens to the answers.
Use the associative property for addition Use the distributive property
120 ÷ 120 = 1; 120 ÷ 60 = 2; 120 ÷ 10 = 12; 120 ÷ 6 = 20; 1 20 ÷ 4 = 30; 120 ÷ 2 = 60; 120 ÷ 1 = 120
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The blue numbers (divisors) are decreasing, while the red numbers (answers) are increasing. If the number you divide by becomes extremely small, the answer becomes extremely large. We can never reach a number large enough to provide an answer to 120 ÷ 0.
PL
In this sub-theme, we work with identities and inverses. We also introduce a new concept: division by zero.
2.3 Inverses
2.1 Identities
If we do an operation on a number and it stays the same (identical), we have used an identity. Addition If we add 0 to any number, the number stays the same. Example: 7 + 0 = 7
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•
The additive identity (the identity for addition) is 0.
We can also apply this identity in reverse by subtracting: 7 − 0 = 7. •
•
Addition and subtraction are the inverses of each other. If you add 3 and then subtract 3, you are back where you began. Example: □ + 3 − 3 = □ + 0 = □
•
Multiplication and division are the inverses of each other. If you multiply by 3 and then divide by 3, you are back where you began. Example: □ × 3 ÷ 3 = □ × 1 = □
•
We can use inverse operations to check calculations. Addition is the inverse operation of subtraction: Example: 130 − 80 = 50 Check: 50 + 80 = 130
Multiplication If we multiply any non-zero number by 1, the number stays the same. Example: 7 × 1 = 7
Multiplication is the inverse operation of division. Example: 240 ÷ 8 = 30
Check: 30 × 8 = 240
The multiplicative identity (the identity for multiplication) is 1.
Worked example 2: Identities and inverses
We can also apply this identity in reverse by dividing: 7 ÷ 1 = 7.
2.2 Division by zero
Indicate whether each statement is true or false. Explain your answer and give the correct answer whenever a statement is false.
Division by zero is undefined. Try this on your calculator: 4 ÷ 0. Did you get a message like ‘Cannot be done’ or ‘Math error’? ©Optimi
a) 3
578 × 0 = 578 G08 – Mathematics
Theme 1: Whole numbers
G08 – Mathematics – Study Guide 1/2 b)
731 ÷ 0 = 0
2.2
c)
1 × 439 = 439
d)
1 × 0 = 1
2.3 9 × 6
e)
346 + 119 − 119 = 346
f)
952 × 621 ÷ 621 = 952
2.5 280 ÷ 40
g)
679 − 125 = 554, so 554 − 679 = 125
2.6 320 ÷ 8
h)
328 × 2 = 656, so 656 ÷ 328 = 2
2.7
2.4
c)
True
d)
False
e)
True
f)
True
g)
False
h)
True
E
False
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b)
3 × 12
2.9 360 ÷ 12
The multiplicative identity is 1, not 0(578 × 0 = 0) Division by 0 is undefined
2.10 12 × 7
1 is the multiplicative identity
2.11 63 ÷ 7
1 is the multiplicative identity of non-zero numbers only
Subtraction is the inverse of addition
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False
144 ÷ 12
2.8 360 ÷ 6
Solutions a)
72 ÷ 9
2.12 60 × 4
Division is the inverse of multiplication
A
B
C
D
E
F
G
H
I
J
K
L
Subtraction is the inverse of addition, so 679 − 554 = 125
7
8
9
12
30
36
40
54
56
60
84
240
Division is the inverse of multiplication
Exercise 1: Properties of whole numbers, identities and inverses
3.
Complete the statements. Indicate which property you have used in each case.
3.1 79 + 21 = 21 + ___ 3.2
( 5 × 2) × ___ = 5 × (2 × 7)
All questions must be answered without using a calculator.
3.3 11 × ( 12 + 9) = 11 × 12 + ___
1.
3.4 31 × 15 = ___ × 31
2.
List all whole numbers in the given set. {350; 2 __12 ; –7; 50; 0,75; 291}
3.5 20 + ( 3 + 13) = ( ___ + 3) + 13
Mental speed test. Match the calculation with the letter of the answer in the table as quickly as you can.
3.6
2.1 8 × 7 4
( 9 − 4) × 6 = 9 × 6 − ___
4.
Check your understanding of identities. Write down the answers to the following:
4.1
173 × 1
3.1 BODMAS rule You probably remember the BODMAS rule, which tells us the correct order of operations.
4.2 468 × 0 4.3 329 ÷ 0 4.4
1 × 720
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4.5 0 + 1
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4.6 0 × 1
Here, the word ‘operation’ doesn’t refer to surgery, but to actions involving numbers, such as adding, subtracting, multiplying, squaring, etc.
4.7 528 + 0 528 + 1
3.
CALCULATION TECHNIQUES
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4.8
In this sub-theme, we review the order of operations and calculation techniques.
REMEMBER Order of calculations using BODMAS B Brackets
O Other operations like squares, square roots, cubes, etc.
Worked example 3: Applying the BODMAS rule Simplify the expressions by doing only one calculation per step. a)
2 + 3 × 10 − 6 + 5(3 − 2)
b)
48 ÷ (4 + 2) − 2 × 2 + 7
Solutions a)
2 + 3 × 10 − 6 + 5(3 − 2)
D and M
Division and multiplication from left to right
= 2 + 3 × 10 − 6 + 5(1)
A and S
Addition and subtraction from left to right
= 2 + 3 × 10 − 6 + 5 = 2 + 30 − 6 + 5 = 32 − 6 + 5
Work out brackets
Multiply Add and subtract from left to right
= 26 + 5 = 31 ©Optimi
5
G08 – Mathematics
Theme 1: Whole numbers
G08 – Mathematics – Study Guide 1/2 b)
48 ÷ (4 + 2) − 2 × 2 + 7
b)
= 48 ÷ 6 − 2 × 2 + 7
Work out brackets
= 8 − 2 × 2 + 7
Divide and multiply from left to right
= 72 − 70 = 2
= 8 − 4 + 7 = 4 + 7
REMEMBER
REMEMBER
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If the number after the key digit is 4 or less, we round down. This means the key digit stays the same.
‘Increase by’, ‘the sum of’ and ‘more than’ all mean addition.
If the number after the key digit is more than 4, we round up. This means the key digit increases by 1.
‘The product of’ means multiplication.
Worked example 5: Rounding off
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Worked example 4: Applying the BODMAS rule to words
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‘Decrease by’, ‘the difference between’ and ‘less than’ all mean subtraction.
Rewrite each statement using mathematical operations. Then calculate the value of the expression. a)
Increase the product of 13 and 10 by 15.
b)
70 less than the product of 12 and 6
a)
The product of 13 and 10 = 13 × 10 ‘Increase by 15’ means +15 Combine the parts: 13 × 10 + 15 = 130 + 15 = 145
First multiply; then subtract
3.2 Rounding off
Add and subtract from left to right
= 11
Solutions
The product of 12 and 6 = 12 × 6 ‘70 less than’ means − 70 Combine the parts: 12 × 6 − 70
First multiply; then add
6
Round off the following numbers: a)
6 275 to the nearest ten
b)
7 249 to the nearest hundred
Solutions a)
6 275: The key digit is the tens digit, which is 7. The digit after 7 is 5, so we round up. The answer is 6 280.
b)
7 249: The key digit is the hundreds digit, which is 2. The digit after 2 is 4, so we round down. The answer is 7 200.
Worked example 6: Rounding off and compensation involving multiplication Consider the following: 48 × 19 b)
Check your answer using estimation.
c)
Check your answer using rounding off and compensation.
d)
Which answer is the least accurate?
a)
380 + 513
b)
7 930 − 5 675
Solutions
E
Calculate the answer using the column method.
a)
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= 400 + 500 − 7
4
8
1
9
4
3
2
4
8
0
9
1
2
Round off: 50 × 20 = 1 000
= 900 − 7
b)
7 930 − 5 675
c)
= (7 930 + 25) − (5 675 + 25) = 7 955 − 5 700 = 2 255
The answer (912) is quite close to 1 000, so we can accept this estimation. 48 × 19 = 48 × (20 − 1) = 48 × 20 − 48 × 1 = 960 − 48
Round off 19 and then compensate by subtracting 1
1.
Use rounding off and compensation to do the following calculations:
1.1 74 × 59
The answer to b) is the least accurate. Rounding off only gives us a rough check.
©Optimi
Add 25 to the second number and compensate by adding 25 to the first number too. We choose 25 to make the calculation easier: 7 5 + 25 = 100
Exercise 2: Calculation techniques
Use the distributive property
= 912 d)
Round off by adding 20 to the first number and subtracting 13 from the second. This is the same as adding 7. So compensate by subtracting 7.
= 893
SA M
7
×
c)
380 + 513
= (380 + 20)+ (513 − 13)− 7
a)
b)
Use rounding off and compensation to calculate the following:
To compensate means to make up for a change.
a)
Solutions
Worked example 7: Rounding off and compensation involving addition and subtraction
7
1.2
680 × 72
1.3
272 + 614
1.4
3 857 − 2 190 G08 – Mathematics
Theme 1: Whole numbers
G08 – Mathematics – Study Guide 1/2 2.
Calculate the following using the column method. Check your answers using a calculator.
4.2 Divisibility rules
2.1 74 + 103 + 96 + 250 2.2 47 × 56
Divisible by 2
If the last digit is an even number
Divisible by 3
If the sum of the individual digits is a multiple of 3
Divisible by 4
If the last two digits form a multiple of 4
Divisible by 5
If the last digit is 0 or 5
Rewrite each statement using mathematical operations. Then calculate the value of the expression.
3.1
Find the sum of 15 increased by 200 and the product of 12 and 70.
Divisible by 6
If the number is divisible by 2 and 3
3.2
Double the difference between 30 and 77.
Divisible by 8
If the last three digits form a multiple of 8
Divisible by 9
If the sum of the individual digits is a multiple of 9
4.
PRIME NUMBERS, MULTIPLES AND FACTORS
Divisible by 10
If the last digit is 0
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3.
Worked example 8: Divisibility
4.1 Multiples and factors • •
• •
SA M
In this sub-theme, we revise prime numbers, write a number as a product of prime factors, and learn to find the HCF and LCM of several numbers. A number has A multiple of a number is a number into which the Few Factors but given number can divide without leaving a remainder. Many Multiples. Multiples of 6 are {6; 12; 18; 24; 30; ...}. A factor of a number is a number that can divide into that number without leaving a remainder. We can also say that factors are numbers that are multiplied together to give a product: 3 × 4 = 12. All the factors of 12 are {1; 2; 3; 4; 6; 12}. If a number has two factors only (itself and 1), it is called a prime number. Note: 1 is not a prime number. Examples of prime numbers: {2; 3; 19; 53} Numbers with more than two factors, for example 18, are called composite numbers.
8
Indicate whether each of the following numbers can divide into 4 164. Provide reasons for your answers. a)
3
b)
4
c)
8
d)
9
Solutions a)
3: Yes
4 + 1 + 6 + 4 = 15, which is a multiple of 3
b)
4: Yes
The last two digits (64) form a multiple of 4
c)
8: No
The last three digits (164) do not form a multiple of 8
d)
9: No
The sum of the digits is 15, which is not a multiple of 9