,© 2020 University of South Africa
All rights reserved
Printed and published by the
University of South Africa
Muckleneuk, Pretoria
MIP1502/1/2021–2023
1000240
InDesign
MNB_Style
,CONTENTS
Page
LEARNING UNIT 1: ORIENTATION TO ALGEBRA, ITS NATURE, PROCEDURES AND PROCESS 1
1.1 ALGEBRA, PATTERNS AND FUNCTIONS 1
1.2 HISTORICAL DEVELOPMENT OF ALGEBRA 2
1.3 THE ENTITIES USED IN ALGEBRA 6
1.3.1 Sets of numbers and interval notation 6
1.3.2 Operations on real numbers 7
1.3.3 Expressions and equations 12
1.4 THE MAJOR PROCESSES INVOLVED IN ALGEBRA 17
1.4.1 Simplifying expressions 17
1.4.2 Solving equations 18
1.4.3 Factorising to solve quadratic equations 20
1.4.4 Graphing functions 25
TEST YOUR KNOWLEDGE 31
CONCLUSION 31
ADDITIONAL LEARNING EXPERIENCE 31
LEARNING UNIT 2: VARIOUS KEY ASPECTS OF ALGEBRA 32
2.1 DEVELOPING ALGEBRAIC THINKING THROUGH NUMERICAL STATEMENTS 32
2.1.1 Writing situations as number sentences (numerical statements) 33
2.1.2 Bringing many steps together in a single flowing numerical statement 34
2.2 ALGEBRAIC PROBLEMS IN A FAMILIAR CONTEXT 37
2.2.1 Situational contexts 37
2.2.2 Mathematical contexts 39
2.3 APPLYING MATHEMATICAL CONCEPTS AND PROCEDURES IN ALGEBRA 42
2.4 NUMBER SENTENCES IN THE CAPS CURRICULUM 47
2.4.1 Algebraic expressions 48
2.4.2 Algebraic equations 48
2.4.3 Solving algebraic equations by inspection 48
2.4.4 Solving algebraic equations by trial and improvement 49
2.4.5 Checking solutions by substitution 49
2.4.6 Problem types in the Intermediate Phase 49
2.4.7 Setting up a number sentence from word problems or number problems 52
TEST YOUR KNOWLEDGE 55
CONCLUSION 55
ADDITIONAL LEARNING EXPERIENCE 55
(iii) MIP502/1/2021–2023
, LEARNING UNIT 3: NUMERIC AND GEOMETRICAL PATTERN 56
3.1 GEOMETRICAL AND NUMERIC PATTERNS AS PREPARATION FOR FUNCTIONS 56
3.1.1 Terminology used in patterns 64
3.1.2 The relationship between symbolic (geometrical) and numerical patterns 66
3.2 DESCRIBING AND REPRESENTING PATTERNS IN A VARIETY OF FORMS 73
3.2.1 Using number lines to represent patterns 73
3.2.2 Using grids to represent number patterns 75
TEST YOUR KNOWLEDGE 84
CONCLUSION 84
ADDITIONAL LEARNING EXPERIENCE 84
LEARNING UNIT 4: FUNCTIONAL RELATIONSHIPS 85
4.1 UNDERSTANDING FUNCTIONAL RELATIONSHIPS THROUGH MODELLING 85
4.2 THE NATURE OF FUNCTIONAL RELATIONSHIPS IN PATTERNS 91
4.2.1 Arithmetic sequences 92
4.2.2 Geometric sequences 96
4.2.3 Quadratic functions 100
4.3 MODELLING AND REPRESENTING ALGEBRA USING TECHNOLOGY 101
4.3.1 On The Danger Of Indiscriminate Use Of Algebra Software 104
4.3.2 On The Advantages Of Incorporating Technology In Algebra 111
CONCLUSION 119
ADDITIONAL LEARNING EXPERIENCE 119
REFERENCES120
(iv)
All rights reserved
Printed and published by the
University of South Africa
Muckleneuk, Pretoria
MIP1502/1/2021–2023
1000240
InDesign
MNB_Style
,CONTENTS
Page
LEARNING UNIT 1: ORIENTATION TO ALGEBRA, ITS NATURE, PROCEDURES AND PROCESS 1
1.1 ALGEBRA, PATTERNS AND FUNCTIONS 1
1.2 HISTORICAL DEVELOPMENT OF ALGEBRA 2
1.3 THE ENTITIES USED IN ALGEBRA 6
1.3.1 Sets of numbers and interval notation 6
1.3.2 Operations on real numbers 7
1.3.3 Expressions and equations 12
1.4 THE MAJOR PROCESSES INVOLVED IN ALGEBRA 17
1.4.1 Simplifying expressions 17
1.4.2 Solving equations 18
1.4.3 Factorising to solve quadratic equations 20
1.4.4 Graphing functions 25
TEST YOUR KNOWLEDGE 31
CONCLUSION 31
ADDITIONAL LEARNING EXPERIENCE 31
LEARNING UNIT 2: VARIOUS KEY ASPECTS OF ALGEBRA 32
2.1 DEVELOPING ALGEBRAIC THINKING THROUGH NUMERICAL STATEMENTS 32
2.1.1 Writing situations as number sentences (numerical statements) 33
2.1.2 Bringing many steps together in a single flowing numerical statement 34
2.2 ALGEBRAIC PROBLEMS IN A FAMILIAR CONTEXT 37
2.2.1 Situational contexts 37
2.2.2 Mathematical contexts 39
2.3 APPLYING MATHEMATICAL CONCEPTS AND PROCEDURES IN ALGEBRA 42
2.4 NUMBER SENTENCES IN THE CAPS CURRICULUM 47
2.4.1 Algebraic expressions 48
2.4.2 Algebraic equations 48
2.4.3 Solving algebraic equations by inspection 48
2.4.4 Solving algebraic equations by trial and improvement 49
2.4.5 Checking solutions by substitution 49
2.4.6 Problem types in the Intermediate Phase 49
2.4.7 Setting up a number sentence from word problems or number problems 52
TEST YOUR KNOWLEDGE 55
CONCLUSION 55
ADDITIONAL LEARNING EXPERIENCE 55
(iii) MIP502/1/2021–2023
, LEARNING UNIT 3: NUMERIC AND GEOMETRICAL PATTERN 56
3.1 GEOMETRICAL AND NUMERIC PATTERNS AS PREPARATION FOR FUNCTIONS 56
3.1.1 Terminology used in patterns 64
3.1.2 The relationship between symbolic (geometrical) and numerical patterns 66
3.2 DESCRIBING AND REPRESENTING PATTERNS IN A VARIETY OF FORMS 73
3.2.1 Using number lines to represent patterns 73
3.2.2 Using grids to represent number patterns 75
TEST YOUR KNOWLEDGE 84
CONCLUSION 84
ADDITIONAL LEARNING EXPERIENCE 84
LEARNING UNIT 4: FUNCTIONAL RELATIONSHIPS 85
4.1 UNDERSTANDING FUNCTIONAL RELATIONSHIPS THROUGH MODELLING 85
4.2 THE NATURE OF FUNCTIONAL RELATIONSHIPS IN PATTERNS 91
4.2.1 Arithmetic sequences 92
4.2.2 Geometric sequences 96
4.2.3 Quadratic functions 100
4.3 MODELLING AND REPRESENTING ALGEBRA USING TECHNOLOGY 101
4.3.1 On The Danger Of Indiscriminate Use Of Algebra Software 104
4.3.2 On The Advantages Of Incorporating Technology In Algebra 111
CONCLUSION 119
ADDITIONAL LEARNING EXPERIENCE 119
REFERENCES120
(iv)
