Grade 10 Facilitator's Guide 1/2 Mathematics

Facilitator’s Guide 1/2

10

Mathematics

CAPS

Mathematics FET Phase Grade 10 • Facilitator’s Guide 1/2 home classroom college workplace

2210-E-MAM-FG01

9 781990

948374

Mathematics Facilitator's Guide 1/2 – Grade 10

2210-E-MAM-FG01

9 781990

948374

CAPS-aligned

Prof. C Vermeulen, Lead author P de Swardt H Otto M Sherman E van Heerden L Young

1.5.5 Sum and difference of cubes....................................................................31

TABLE OF CONTENTS

Exercise 1.5.3: Sum and difference of cubes.................................................. 33 Self-evaluation............................................................................................................ 34

PREFACE..................................................................................................................... 1

1.6

THEME 1: NUMBERS AND ALGEBRAIC EXPRESSIONS........................................ 6

1.6.1 Simplifying single algebraic fractions using factors........................35

Introduction....................................................................................................................6

Exercise 1.6.1: Simplifying algebraic fractions using factors.................. 37

Algebraic expressions involving fractions .........................................35

1.1

The number system...................................................................................... 6

1.6.2 Multiplication and division of algebraic fractions...........................38

1.2

Exercise 1.1: The number system..........................................................................9

Exercise 1.6.2: Multiplication and division of algebraic fractions........ 39

1.3

Between which two integers does a surd fall?...................................11

1.6.3 Finding the lowest common multiple (LCM)......................................40

Exercise 1.2: Estimating surds............................................................................. 12

Exercise 1.6.3: Finding the LCM.......................................................................... 41

Rounding of real numbers........................................................................12

1.6.4 Adding and subtracting algebraic fractions.......................................42

Exercise 1.3: Rounding numbers........................................................................ 13

Exercise 1.6.4: Adding and subtracting algebraic fractions............................44

Self-evaluation............................................................................................................ 14

1.6.5 Simplifying expressions involving fractions within fractions......47

1.4.2 Products.........................................................................................................15

Self-evaluation............................................................................................................ 50

1.4

Multiplication of algebraic expressions..............................................14

Exercise 1.6.5: Simplifying expressions involving fractions within fractions....................................................................................... 49

1.4.1 Algebraic expressions................................................................................14

Summary of theme.................................................................................................... 51

Exercise 1.4: Multiplication of algebraic expressions................................ 17

End of theme exercise.............................................................................................. 51

Self-evaluation............................................................................................................ 21

Factorising algebraic expressions.........................................................22

THEME 2: EXPONENTS............................................................................................54

1.5.1 Common factor.............................................................................................22

Introduction................................................................................................................. 54

1.5

1.5.2 Difference of two squares.........................................................................24 2.1

1.5.3 Grouping terms (new work) ...................................................................24

Self-evaluation............................................................................................................ 61

1.5.4 Quadratic trinomial....................................................................................27

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Composite bases .........................................................................................57

Exercise 2.1: Composite bases............................................................................. 58

Exercise 1.5.1: Common factor, difference of squares and grouping...... 25 Exercise 1.5.2: Quadratic trinomials and mixed factorisation............... 29

Revision exercise........................................................................................................ 56

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G10 – Mathematics

G10 – Mathematics – Facilitator's Guide 1/2 2.2

Simplifying by factorising.........................................................................61

2.3

Exercise 2.2: Simplifying by factorising........................................................... 62

3.5

Problem-solving involving repeated patterns................................ 100 Exercise 3.5: Problem-solving involving repeated patterns................. 101

Rational exponents.....................................................................................65

Self-evaluation......................................................................................................... 101

Exercise 2.3: Rational exponents........................................................................ 65

Summary of theme................................................................................................. 102

Self-evaluation............................................................................................................ 66

End of theme exercise........................................................................................... 102

2.4 Equations.......................................................................................................66

THEME 4: EQUATIONS AND INEQUALITIES.........................................................106

2.4.1 Equations with the unknown in the base (with fractions in the exponents) .........................................................66

Introduction.............................................................................................................. 106

2.4.2 Equations with the unknown in the exponent (exponential equations)...........................................................................68 Exercise 2.4: Equations........................................................................................... 69 Self-evaluation............................................................................................................ 75 Summary of theme.................................................................................................... 75 End of theme exercise.............................................................................................. 76

Introduction................................................................................................................. 81 Revision exercise........................................................................................................ 82

3.1

The general term of a number pattern................................................83

3.2

Exercise 3.1: The general term of a number pattern.................................. 86

Using the general term..............................................................................87

3.3

Exercise 3.2: Using the general term................................................................. 88

3.4

Linear equations.......................................................................................109

4.2

Exercise 4.1: Linear equations.......................................................................... 110

Quadratic equations................................................................................113

Exercise 4.1: Quadratic equations................................................................... 115

Self-evaluation......................................................................................................... 120

Linear equation systems........................................................................121

Exercise 4.3: Linear equation systems.......................................................... 124 Self-evaluation......................................................................................................... 130

4.4

Literal equatons (changing the subject of a formula).................. 131

4.5

Exercise 4.4: Literal equations.......................................................................... 132

Linear inequalities...................................................................................133

Exercise 4.5: Linear inequalities...................................................................... 135

Self-evaluation......................................................................................................... 138

The formula for the general term of a linear sequence..................90

Summary of theme................................................................................................. 138

Exercise 3.3: The formula for the general term of a linear sequence.. 92

End of theme exercise........................................................................................... 139

Self-evaluation............................................................................................................ 94

Geometric patterns.....................................................................................95

Exercise 3.4: Geometric patterns........................................................................ 97

4.1

4.3

THEME 3: NUMBER PATTERNS.............................................................................. 81

Revision exercise..................................................................................................... 106

ii

THEME 5: TRIGONOMETRY...................................................................................142 5.1

5.2

5.3

5.4

Introduction.............................................................................................................. 142

The definitions of the trigonometric ratios sin θ, cos θ and tan θ in right-angled triangles.....................................................143

Exercise 5.1: Basic trigonometric ratios in right-angled triangles .......................................................................................... 144

5.7

Exercise 5.6: Finding the values of trigonometric ratios using a calculator.................................................................................................... 176

Solving trigonometric equations (calculating the sizes of angles) ..........................................................177

Exercise 5.7: Solving trigonometric equations for acute angles......... 178

Solving right-angled triangles..............................................................180

Exercise 5.8: Solving right-angled triangles................................................ 182

Self-evaluation......................................................................................................... 185

Exercise 5.2: Inverse trigonometric ratios in right-angled triangles.148

Summary of theme................................................................................................. 185

Self-evaluation......................................................................................................... 150

End of theme exercise .......................................................................................... 185

The definitions of the trigonometric ratios sin θ, cos θ and tan θ and their inverses in the Cartesian plane............................. 151

THEME 6: FUNCTIONS...........................................................................................192

Exercise 5.3 Applying the trigonometric ratios in the Cartesian plane.......154

Using the theorem of pythagoras and trigonometric ratioS...... 158

6.1

Exercise 5.4: Using the theorem of Pythagoras and trigonometric ratios.............................................................................................. 162

6.2

Introduction.............................................................................................................. 192 Functions and relations.........................................................................193

Exercise 6.1: Functions and relations............................................................ 195

Linear functions........................................................................................197

6.2.1 Investigating the properties of linear functions............................ 197

Self-evaluation......................................................................................................... 170

Development exercise........................................................................................... 197

Finding the trigonometric ratios of the special angles without using a calculator ....................................................................170

Direction and steepness of the graph............................................................. 201 Intercepts with the axes is missing................................................................. 202

The trigonometric ratios of 45°........................................................................ 170

Domain and rage..................................................................................................... 202

The trigonometric ratios of 60° and 30°....................................................... 171 The trigonometric ratios of 0° and 90°......................................................... 171

6.2.2 Function notation.....................................................................................204

Self-evaluation......................................................................................................... 174

Exercise 6.2.1: Drawing straight line graphs.............................................. 207

6.2.3 Parallel and perpendicular lines........................................................204

Exercise 5.5: Working with the special angles without using a calculator.................................................................................................... 173

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Finding the values of trigonometrical ratios using a calculator ...175

5.8

Definitions of the inverse trigonometric ratios, cosec θ, sec θ and cot θ, in right-angled triangles....................................................146

Revision exercise..................................................................................................... 159

5.5

5.6

6.2.4 Drawing sketch graphs of linear functions...................................... 205

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G10 – Mathematics

G10 – Mathematics – Facilitator's Guide 1/2 6.2.5 Finding the equation of a straight line graph................................. 215

6.4.5 Domain and range....................................................................................244

Exercise 6.2.2: Finding the equation of a straight line graph.............. 216

6.4.6 Drawing sketch graphs of hyperbolas............................................... 244

Self-evaluation......................................................................................................... 223

Exercise 6.4.1: Hyperbolas.................................................................................. 248

6.2.6 Intersecting lines......................................................................................218

6.4.7 Finding the equation of a hyperbola.................................................. 246

Exercise 6.2.3: Intersecting lines.................................................................... 220

6.4.8 Restrictions on hyperbolas...................................................................247

6.3

Quadratic functions.................................................................................223

6.3.1 Investigating the properties of quadratic functions..................... 224

6.5

Exponential functions.............................................................................251

6.5.1 Investigating the properties of exponential functions................ 252

Development exercise........................................................................................... 224

6.3.2 The influence of ​a​on the graph of a parabola ............................... 227

Development exercise........................................................................................... 252

6.3.3 The turning point of a parabola..........................................................227

6.5.2 The influence of ​a​in ​y = ​a​​ 𝑥​​ on the graph of an exponential function...............................................................................255

6.3.4 The influence of ​q​ on the graph of a parabola................................ 227

6.5.3 The influence of ​b​in ​y = b . ​a​ 𝑥​on the graph of an exponential function...............................................................................256

6.3.5 Domain and range....................................................................................228 6.3.6 Axis of symmetry......................................................................................228

6.5.4 The influence of ​q ​in ​y = ​a​ 𝑥​ + q​on the graph of an exponential function...............................................................................256

6.3.7 Intercepts with the axes.........................................................................228 6.3.8 Drawing sketch graphs of parabolas.................................................. 229

6.5.5 Domain and range....................................................................................256

Exercise 6.3.1: Drawing sketch graphs of parabolas............................... 231

6.5.6 Reflections..................................................................................................257

6.3.9 Finding the equation of a parabola ................................................... 234

6.4

Self-evaluation......................................................................................................... 251

Exercise 6.5.1: Properties of exponential graphs...................................... 257

Exercise 6.3.2: Finding the equation of a parabola.................................. 236

6.5.7 Drawing sketch graphs of exponential functions.......................... 260

Self-evaluation......................................................................................................... 239

Exercise 6.5.2: Drawing sketch graphs of exponential functions....... 260

Hyperbolic functions...............................................................................239

6.5.8 Finding the equation of an exponential graph............................... 263

6.4.1 Investigating the properties of hyperbolic functions................... 240

Exercise 6.5.3: Finding the equation of an exponential function....... 264

Development exercise........................................................................................... 240

6.4.2 The influence of ​a​ on the graph of a hyperbola.............................. 243

6.6

6.4.3 The influence of ​q​ on the graph of a hyperbola.............................. 244

Self-evaluation......................................................................................................... 267

Summary: the effect of ​a​and q on each type of graph................. 268

Summary of theme................................................................................................. 270

6.4.4 Asymptotes.................................................................................................244 iv

End of theme exercise........................................................................................... 270

THEME 7: TRIGONOMETRIC FUNCTIONS............................................................281

7.1

Revision exercise: Basic geometry: lines and angles............................... 323

Introduction.............................................................................................................. 281

Property of triangles............................................................................................. 324

Revision exercise..................................................................................................... 282

Triangles: Similarity.............................................................................................. 324

Drawing accurate graphs of the sin, cos and tan function.......... 284

Triangles: Congruence.......................................................................................... 325

Exercise 7.1.1: Drawing accurate graphs of the sine function y = sin 𝑥....................................................................................................................... 285

7.2

7.3

Exercise 7.1.2: Drawing accurate graphs of the cosine function y = cos 𝑥....................................................................................................................... 288

8.2

Self evaluation.......................................................................................................... 295

8.3

Exercise 7.1.3: Drawing accurate graphs of the tangent function y = tan 𝑥....................................................................................................................... 291 Drawing and interpreting sketch graphs of trigonometric functions.........................................................................296

Exercise 8.1: Basic geometry: lines, angles and triangles..................... 326 Self evaluation.......................................................................................................... 334

The mid-point theorem..........................................................................335

Exercise 8.2: The mid-point theorem............................................................. 337

Self evaluation.......................................................................................................... 344

Special quadrilaterals.............................................................................345

Exercise 8.3: Special quadrilaterals................................................................ 352

Self evaluation.......................................................................................................... 358

Exercise 7.2: Drawing and interpreting sketch graphs of the trigonometric functions....................................................................................... 301

Summary of theme................................................................................................. 358 End of theme exercise........................................................................................... 359

Finding the equations of trigonometric graphs and interpreting graphs.................................................................................306

PAPERS AND MEMOS...........................................................................................362

Exercise 7.3: Finding the equations of trigo­nometric graphs and interpreting graphs................................................................................................ 308

FORMULAE..............................................................................................................375 EUCLIDIAN GEOMETRY: ACCEPTED REASONS..................................................378

Self evaluation.......................................................................................................... 311

INDEX......................................................................................................................381

Summary of theme................................................................................................. 312 End of theme exercise........................................................................................... 312

THEME 8: EUCLIDEAN GEOMETRY......................................................................316 8.1

Introduction.............................................................................................................. 317 Basic geometry: Lines, angles and triangles................................... 318

Lines............................................................................................................................. 319

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Triangles..................................................................................................................... 322

v

G10 – Mathematics

PREFACE

Term

In Grade 10, mathematics is an optional subject (as an alternative to mathematical literacy) for the first time. There may be various reasons why you might choose mathematics as a subject, for example to prepare you for a field of study where Grade 12 mathematics is a prerequisite, or a career in which a background in mathematics would be advantageous.

1

(11 weeks)

In general, mathematics in Grade 10 to 12 involves more abstract concepts and more complex procedures than in Grade 1 to 9. Mastering mathematics in Grade 10 to 12 requires more time, commitment, critical thought and reflection than in Grade 1 to 9.

2

This product consists of two study guides and two facilitator’s guides, which are based on the concepts of Optimi’s GuidEd Learning™ model to help you achieve success in your study of mathematics. These books cover all work required for Grade 10 mathematics and have been compiled in accordance with the CAPS guidelines as required by the Department of Basic Education.

(11 weeks)

Number of weeks 3

1. Algebraic expressions

2

3. Number patterns

2 2

3 4

1 3 3 2

The study guides are supported by supplementary lesson structures on the Optimi Learning Platform (OLP), which is an online platform. These lesson structures offer 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 can also be used independently of the OLP.

3

(10 weeks)

The study guides and facilitator’s guides are divided into 15 themes. Study guide 1/2 and facilitator’s guide 1/2 cover themes 1 to 8 (terms 1 and 2) and study guide 2/2 and facilitator’s guide 2/2 cover themes 9 to 15 (terms 3 and 4).The themes correspond with the CAPS guidelines with regard to content and time allocation and represent the year plan.

2 2 2

1 1

4

(10 weeks)

Theme

2

4 4

2. Exponents

4. Equations and inequalities 5. Trigonometry 6. Functions

7. Trigonometric functions 8. Euclidean geometry

Assessment* Investigation or project Test

Assignment or test

Practice papers are included in the textbook.

June examination

10. Finance and growth

Test

9. Analytical geometry 11. Statistics

12. Problems in two dimensions 13. Euclidean geometry 14. Measurement 15. Probability Revision

Practice papers are included in the textbook.

Test Test End-of-year examination

* You will find the latest and most comprehensive information on assessment in the portfolio book and assessment plan. © Optimi

1

G10 – Mathematics

G10 – Mathematics – Facilitator's Guide 1/2

Time allocation

Each theme has the same structure in order to make it easier for you to navigate through them. The structure is as follows:

According to the CAPS requirements, at least 4,5 hours should be spent on teaching mathematics per week. For example, 13,5 hours (three weeks × 4,5 hours per week) will be spent on teaching Theme 1 (algebraic expressions).

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.

Introduction

What this theme is about 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.

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.

Prior knowledge

This section tells you what existing knowledge you need to master the theme involved.

The learning activities available in the OLP’s lesson structures involve different formats and levels of interaction. The resources not only support the learning process, but also offer you the opportunity to practise new knowledge.

Tip: The more e𝑥ercises you do, the greater the chance that you will achieve success in mathematics.

Structure of themes 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.

When one learns mathematics, there is a lot of new information your brain needs to process, which can easily exhaust your working memory. The study guides have been written and compiled in such a way that it does not overtax the working memory and therefore simplifies the process of learning mathematics.

2

Revision

Exercises

This may involve one of the following: 1.

2.

3.

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.

revision of the concepts, definitions and procedures required as previous knowledge, an exercise or activity with solutions so that you can test your previous knowledge yourself, or a combination of the above.

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.

Do not neglect this revision. It is important to work through this section thoroughly. Mathematical concepts often follow on one another and if basic knowledge is lacking or has not been mastered sufficiently, this will handicap the formation of new knowledge.

Summary of theme

Following the introductory part of the theme, new knowledge is dealt with in sub-themes. Each sub-theme has the following structure:

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).

SUB-THEME

Introduction

End of theme exercise

New concepts and procedures are explained. Relevant previous knowledge is also dealt with here if necessary.

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 and complete all the exercises. Complete solutions can be found in the facilitator’s guide.

Worked examples

Mixed exercises such as these in this textbook form a very important component of mastering mathematics. There is a big difference between the ability to recognise your 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

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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3

G10 – Mathematics

G10 – Mathematics – Facilitator's Guide 1/2

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.

Self-evaluation

When you practise the same type of sum or 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 sum or 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 sum or problem correctly. This means that you are truly prepared for tests or exams, because you can recall your work instead of merely recognising it.

In each theme, and usually following each sub-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:

Use the following scale to determine how comfortable you are with each topic in the table below:

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.

Complete the table:

Topic

1

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.

4

Exam papers

Supplementary books

Visit Impaq’s online platform for the assessment plan and comprehensive information about the compilation and mark allocation of tests, assignments and examinations.

Any other books can be used along with this textbook for extra exercises and explanations, including: • 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.

Tip: Make sure that you know which themes are covered in which paper. The themes covered in the e𝑥amination papers are subject to change. Always refer to the portfolio book and assessment plan for updated information about the composition of the e𝑥amination papers.

Paper 1

Tip: Use supplementary resources for further e𝑥planations, e𝑥amples, and especially e𝑥tra e𝑥ercises.

Paper 2

Calculator

Algebraic expressions, equations and Euclidean geometry and inequalities, exponents (Theme 1, 2 measurement (Theme 8, 13 and 14) and 4) Analytical geometry (Theme 9) Number patterns (Theme 3) Trigonometry (Theme 5 and 7) Functions and graphs (Theme 6) Statistics (Theme 11) Finance and growth (Theme 10)

We recommend the CASIO fx-82ES (Plus) or CASIO fx-82ZA. However, any scientific, non-programmable and non-graphing calculator is suitable.

Probability (Theme 15)

Note:

• No graphing or programmable calculators are allowed (for example to factorise or find the roots of equations). Calculators should only be used to do standard numeric calculations and to verify calculations done by hand.

• Formula sheets are not provided during tests and final exams in Grade 10.

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5

G10 – Mathematics

Theme 1: Numbers and algebraic expressions

G10 – Mathematics – Facilitator's Guide 1/2

THEME 1

Introduction

NUMBERS AND ALGEBRAIC EXPRESSIONS

In this theme learners will learn more about: • • • • • •

Learning requirements according to CAPS Learners should be able to: 1.

2.

3.

4.

5.

6.

understand that real numbers can be rational or irrational establish between which two integers a given simple surd lies round real numbers to an appropriate degree of accuracy multiply a binomial by a trinomial factorise expressions using techniques taught in Grade 9 as well as:

Prior knowledge

In order to master this theme, learners should already know: • • • •

• trinomial (more advanced trinomials) • grouping in pairs • sum and difference of two cubes

simplify algebraic fractions by factorising denominators, which may include cubes (limited to sum and difference of cubes).

Term

Duration Paper Weight

different types of numbers how to estimate the values of certain numbers how to round off numbers how to multiply algebraic expressions how to find factors of algebraic expressions how to simplify algebraic fractions. what types of numbers there are how we classify numbers how to multiply simple algebraic expressions how to simplify simple fractions.

1.1 THE NUMBER SYSTEM

1

Introduction

3 weeks

This sub-theme is a summary of work covered in Grades 8 and 9. If learners should find it difficult to complete this section, you should first revise the work covered in these grades.

1

Algebraic expressions form part of algebra, of which the weighting is 30±3 of Paper 1.

6

Different types of numbers

Note

​ℕ = ​{1 ; 2 ; 3 ; 4 ; 5 ; …}​ =​natural numbers

• A rational number is any number that can be written as ​_ ba ​, where ​a​and ​b​ are integers.

• The following are rational numbers:

​ℕ​ 0​ = ​{0 ; 1 ; 2 ; 3 ; 4 ; 5 ; …}​ =​whole numbers

◦ fractions of which both the numerator and denominator are integers, e.g. 73​_ ​ ◦ integers, e.g. − ​ 5​ ◦ decimal numbers that end, e.g. ​0,125​

​ℤ = ​{… ; − 2 ; − 1 ; 0 ; 1 ; 2 ; 3 ; …}​ =​ integers

_____________ ​ℚ = ​{numbers that can be written as a non-zero integer ​ ​} ​ = rational numbers​ an integer

◦ decimal numbers that repeat, e.g. ​0, 151515…​

_____________ ​ℚ′ ​ = ​{numbers that cannot be written as a ​ non-zero integer

​}​ = irrational numbers​

an integer

• Irrational numbers are not rational. They cannot be written with an integer numerator and denominator, e.g. ​0,8672345…​ • If the ​nth ​ root of a number cannot be written as a rational value, this ​nth ​ 3 _ root is called a surd, e.g. ​√ 5 ​​.

(non-terminating and non-repetitive decimal numbers)

​ℝ = ​{rational and irrational numbers}​ =​real numbers

​ℝ′ ​ = ​{numbers that do not exist in the real number system}​ =​non-real numbers

Worked example 1

Summary of the real number system

Real numbers

Rational numbers

Integers

Rewrite ​0, ​1˙ ​2˙ ​ as a common fraction.

Irrational numbers

Solution

In order to rewrite a recurring fraction as a common fraction, you need to manipulate the recurring fraction to lose the recurring “tail”.

Fractures

Let ​𝑥 = 0,1212121212…​

• Negative integers • Zero • Positive integers (natural numbers)

​∴ 100𝑥 = 12,1212121212…​ ​

REMEMBER

_0 𝑥​ ​ = 0

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_

​√ − 3 ​ is non-real

​∴

_𝑥 ​ is undefined​ 0

​∴ 7

−𝑥 = 0,1212121212…​

​× 100​to get integer ​+​recurring “tail” Subtract

​ 99𝑥 = 12​ 12 𝑥 = ​_ 99 ​​

_4 𝑥 = 33 ​​

Simplify

G10 – Mathematics

Theme 1: Numbers and algebraic expressions

G10 – Mathematics – Facilitator's Guide 1/2

Worked example 2 3

​

​

− 10y = 25,121212…​

​∴ ​∴

√

0,​​3 ̇ ​​

​× 1 000​and × ​ 10​to get integer ​+​recurring “tail”

_

Remember that the 5 ​ ​is not recurring

990y = 2487,000000…​ 2 487 y = ​_ ​​ 990

y = ​_ 829 ​​ 330

ℝ

_ ​​ 02 ​​ __ 9 ​​ _ ​16 ​ ​​

Solution

​∴ 1 000y = 2512,121212…​

_

ℚ'

​​ √ 9 ​​

Rewrite ​2,5​1˙ ​2˙ ​ as a common fraction.

Let ​y = 2,512121212…​

_ ​​ 17 ​​

ℚ

Solution

​​√ 50 ​​

In order to determine where these numbers fit into the number system, you can use your calculator to find the decimal fraction where applicable:

Always remember to simplify completely

_ ​​ 17 ​​

Worked example 3 3

Using your knowledge of the number system, complete the following table by making a  in the appropriate block(s):

_

This number is written in the form _ ​​ ab ​​; therefore it is a rational number (ℚ).

It exists (your calculator does not give you a math error message); therefore it is a real number (ℝ).

​​ √ 9 ​​ = 0,2080083823… … (non-finite, non-recurring decimal fraction) This number cannot be written in the form _ ​​ab ​​therefore it is an irrational number (ℚ').

It exists (your calculator does not give you a math error message); therefore it is a real number (ℝ).

8

_ ​​ 02 ​​

__

= 0 (zero divided by any non-zero number = zero)

This number is written in the form _ ​​ ab ​​therefore it is a rational number (ℚ).

√

ü ü

ü

ü

ü

ü

_

​​√ 50 ​​

ü

ü

Note that real numbers (ℝ) are either rational (ℚ) or irrational (ℚ') numbers.

1

​ ​_ = 3 ​​ This number is written in the form _ ​​ ab ​​therefore it is a rational number (ℚ).

Exercise 1.1: The number system

It exists (your calculator does not give you a math error message); therefore it is a real number (ℝ).

​​√ 50 ​​ =3,684031499… (non-finite, non-recurring decimal fraction)

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ü ü

0,​​3 ̇ ​​

0,​​3 ̇ ​​ = 0,333333333… (non-finite, recurring decimal fraction)

_

ü

_ ​​ 02 ​​ __ 9 ​​ _ ​16 ​ ​​

√

It exists (your calculator does not give you a math error message); therefore it is a real number (ℝ).

ℝ

ü

_

​​ √ 9 ​​

_ ​​ ab ​​therefore it is a rational

ℚ'

_ ​​ 17 ​​

3

It exists (your calculator does not give you a math error message); therefore it is a real number (ℝ).

9 ​​ _ ​16 ​ ​​ = ​ ​_ 34 ​​ This number is written in the form number (ℚ).

ℚ

This number cannot be written in the form _ ​​ab ​​therefore it is an irrational number (ℚ').

It exists (your calculator does not give you a math error message); therefore it is a real number (ℝ).

1.

Is the number zero a positive or negative number?

2.

What type of number is ​√ 8 ​?

Zero is neither positive nor negative. _

_

_

_

​​√ 8 ​ = ​√ 4 × 2 ​ = 2 ​√ 2 ​​

​∴​a real, irrational number

9

G10 – Mathematics

Theme 1: Numbers and algebraic expressions

G10 – Mathematics – Facilitator's Guide 1/2 3.

4.

5.

6.

_

What type of number is ​√ −8 ​?

7.

The square root of a negative number does not exist in the real number system. ∴ a non-real number

3

7.1

_

_

×10 to get integer + recurring “tail”

9𝑥 = 6 6 𝑥 = ​​ _ ​​ 9

_

7.2

What type of number is ​√ −8 ​?

3

Let 𝑥 = 0,​​6​ ̇ =0,666666…

– 𝑥 = 0,666666…

​​√ 8 ​ = 2​ ∴ a real number, a rational number, an integer, a whole number and a natural number (natural number implicates: real number, rational number, integer, whole number) 3

0,​​6​̇

∴10𝑥 = 6,666666…

_

What type of number is ​√ 8 ​ ?

3

Rewrite the following as common fractions:

​√ −8 ​ = ​−​2

2 𝑥 = ​​ _ ​​ 3

3,1​​5 ̇ ​​​​6 ​​̇

Let 𝑦 = 3,1​​5̇ ​​​​6 ​​̇​̇ = 3,1565656…

∴1 000𝑦 = 3 156,565656…

∴ a real number, a rational number and an integer

– 10𝑦 = 31,565656…

Without using a calculator, determine all the number types that 2 ___ ​​ 10 ​​ 27 belongs to.

×1 000 to get integer + recurring “tail” ×10 to get recurring “tail”

990𝑦 = 3 125 3 125 ∴ 𝑦 = ​​ _ ​ 990

2 ___ ​​ 10 ​​ = ___ ​​ 64 ​​ 27 27

625 ∴𝑦 = ​​ _ ​ 198

∴ a real, rational number (64÷27 gives a non-finite, recurring decimal number as an answer)

REMEMBER Always remember to simplify completely.

10

8.

For which value(s) of 𝑥 will f(𝑥) be non-real if: For the expression to be non-real, the value in the root will be a negative number. Substitute the given elements to determine which value will give a negative answer:

But to estimate the values of other surds, such as ​​√ 18 ​​, without using a calculator, learners should first understand the following:

_

√

_

_

​ 9 ​ –(–5) ​​ = ​​ ___ ​ 9 ​ +5 ​​ = ​​ ___ ​64 ​ ​​ > 0 f(–5) = ​​ ___ 11 11 11

√ √ √ ___________ ___________ f(0) = √ ​​ ___ ​ 9 ​ – (0) ​​= ​​√ ___ ​ 9 ​ ​​ > 0 11 11 _____________

_

112 f(11) = ​​ ___ ​ 9 ​ – (11) ​​ = ​​ – _____ ​ ​​< 0 11 11

√

√

_

It is easy to see that ​√ 3 ​ is greater than ​1​and smaller than ​2​. n

𝑥 = –5 gives a positive answer

_

Determine between which two integers the irrational number ​√ 62 ​​ lies.

Solution

Find the two perfect squares to the left of (just smaller than) and to the right of

_

a surd. For example, ​√ 2 ​ and ​√ 3 ​ are surds, but ​√ 4 ​ is not a surd because it can be simplified to the rational number ​2​. n

_

(just bigger than) ​62 ​on the number line.

• The perfect square to the left of ​62​is 4 ​ 9​.

Consider surds of the form ​√ a ​ , where a ​ ​is any positive number, for example​ 3

_

2

• The perfect square to the right of ​62​is 6 ​ 4​.

_

​ ​to be 2 ​ ​, so we usually do not write ​​√ a ​​. √ 7 ​​ or ​​√ 5 ​​. It is very common for n _

Instead, we write the surd as simply ​√ a ​. This is called the square root of a ​ ​.

_

Worked example 4

If the n ​ ​th root of a number cannot be simplified to a rational number, we call it 6

n

For example, ​27​is a perfect cube, because ​3​ 3​ = 27​.

Introduction _

_

For example, ​9​is a perfect square since ​3​ 2​ = 9​.

• A perfect cube is a number which is the cube of an integer.

𝑥 = 0 gives a positive answer

1.2 BETWEEN WHICH TWO INTEGERS DOES A SURD FALL?

_

_

• If ​a​and ​b ​are positive integers and a ​ < b​, then ​√ a ​ < ​√ b ​​.

• A perfect square is the number obtained when an integer is squared.

𝑥 = 11 gives a negative answer

∴ the expression will be non-real if 𝑥 = 11

_

Using a calculator, we know that ​√ 3 ​ = 1,73205… ​

f(𝑥) = ​​ ___ 11 ​ 9 ​ ​​ – 𝑥 and 𝑥∈{–5; 0; 11}? ____________

_

_

For example, let us estimate where ​√ 3 ​ ​lies on the number line:

It is sometimes useful to know the approximate value of a surd without having to use a calculator.

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11

G10 – Mathematics

Theme 1: Numbers and algebraic expressions

G10 – Mathematics – Facilitator's Guide 1/2 Now find the square roots of these perfect squares:

3.

_

• ​√ 49 ​ = 7​ _

• ​√ 64 ​ = 8​

_

_

4.

Do the exercise without using a calculator. _

Find two consecutive integers such that ​√ 26 ​ lies between them.

_

Introduction Rounding numbers makes them simpler and easier to use. It is often just easier to work with rounded numbers.

Rounding numbers means adjusting the digits (up or down) in order to make rough calculations easier. The result will be an estimated answer rather than an exact one.

• The perfect cube to the left of ​49​is 2 ​ 7​.

• The perfect cube to the right of ​49​is 6 ​ 4​. _

​√ 27 ​ = 3​and ​√ 64 ​ = 4​ _

_

1.3 ROUNDING OF REAL NUMBERS

Find the two perfect cubes to the left of (just smaller than) and to the right of (just bigger than) ​49​.

3

_

Estimated value of ​√ 10 ​ = 3,1​

_

3

_

​∴ 3 < ​√ 10 ​ < 4​

Find two consecutive integers such that ​√ 49 ​ lies between them.

_

Find the two perfect squares to the left of (just smaller than) and to the right of (just bigger than) ​10​. _

• The perfect square to the right of ​26​is 3 ​ 6​.

3

_

Estimate ​√ 10 ​ correct to one decimal place.

​√ 9 ​ = 3​and ​√ 16 ​ = 4​

right of (just bigger than) ​26​.

3

_

​∴ 4 < ​√ 18 ​ < 5​

• The perfect square to the left of ​10​is 9 ​ ​.

• The perfect square to the right of ​10​is 1 ​ 6​.

Find the two perfect squares to the left of (just smaller than) and to the • The perfect square to the left of ​26​is 2 ​ 5​.

2.

_

​√ 16 ​ = 4​and ​√ 25 ​ = 5​

Exercise 1.2: Estimating surds

_

right of (just bigger than) ​18​.

• The perfect square to the right of ​18​is 2 ​ 5​.

_

​∴ 7 < ​√ 62 ​ < 8​

​√ 25 ​ = 5​ and ​√ 36 ​ = 6​ _ ​∴ 5 < ​√ 26 ​ < 6​

Find the two perfect squares to the left of (just smaller than) and to the • The perfect square to the left of ​18​is 1 ​ 6​.

Therefore, ​√ 62 ​ lies between 7 ​ ​and ​8​.

1.

_

Determine between which two consecutive integers ​√ 18 ​​ lies.

​∴ 3 < ​√ 49 ​ < 4​

12

How to round numbers:

Exercise 1.3: Rounding numbers

• When rounding to a required number of places, the next decimal digit is considered, e.g. if you are required to round to three decimal places, consider the 4 ​ th ​ decimal digit.

Round the following numbers to the indicated degree of accuracy:

• If the next digit is less than ​5​, the previous decimal digit stays as it is.

• If the next digit is ​5​or more, the previous decimal digit is increased by one.

• If you are asked to round to three decimal places, you need to have three digits after the decimal comma (even if these are zeros).

Worked example 5

2.

​36,8121212​to the nearest integer

4.

Solution ​= 2,600​

​12,507​to two decimal places

3.

Round ​2,6003​to three decimal places.

5.

Look at the 4th decimal digit: 3<5 3rd digit will stay the same.

​2,6003​

1.

​12,507 ≈ 12,51​

​36,8121212 ≈ 37​

​− 48,2291​to three decimal places ​− 48,2291 ≈ − 48,229​

​28,995​to two decimal places ​28,995 ≈ 29,00​

​− 185,02​to one decimal place ​− 185,02 ≈ − 185,0​

Worked example 6

Round ​473,78​to the nearest integer.

Solution ​473,78​

Look at the 1st decimal digit: 7>5 Units digit will become 1 more.

​= 474​

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13

G10 – Mathematics

Theme 1: Numbers and algebraic expressions

G10 – Mathematics – Facilitator's Guide 1/2

1.4 MULTIPLICATION OF ALGEBRAIC EXPRESSIONS

Self-evaluation

1.4.1 Algebraic expressions

Use the following scale to determine how comfortable you are with each topic in the table that follows:

An algebraic expression contains letter symbols and numbers, e.g. ​2 ​𝑥​ 2​ − 3𝑥y + ​y​ 3​ − 5​.

Note the difference between an algebraic expression and an algebraic equation:

1. Help! I am not at all comfortable with the topic; I need help.

2. Alarm! I am not comfortable with the topic; however, I merely need more time to go through the topic again.

3. OK! I am fairly comfortable with the topic, but do get stuck occasionally.

4. Sharp! I am comfortable with the topic.

5. Whoohoo, it's party time! I am completely comfortable with the topic, and can even answer more difficult questions on this.

Complete the table.

Topic

1

2

3

4

• In an algebraic equation, two expressions are equated (using an equal sign).

Examples: ◦ ​2x − 5 = 0​ ◦ ​3 ​x​ 2​ − 5x = 7x + 2​ • An algebraic expression does not have an equal sign. Examples: ◦ ​2x − 5​ ◦ ​3 ​x​ 2​ − 5x​

5

• Algebraic equations can be solved, i.e. we can find the unique value(s) of the unknown (say ​x​) for which the equation is true.

• Algebraic expressions cannot be solved. In an algebraic expression, the variable(s) (say ​x​) can take on any value that is allowed.

I can classify any real number as rational or irrational.

I can write any recurring decimal fraction as an ordinary fraction.

I can determine between which two integers a given surd falls.

I can round real numbers to a certain degree of accuracy.

Learners should know the following terminology regarding algebraic expressions:

14

Consider the algebraic expression ​3a​​ 4​ + 2 ​a​ 2​ − 3a + 8​.

Terminology

Coefficient

Constant Degree of expression Number of terms Variable

Multiply the number outside the bracket by each term inside the bracket:

Description

The number (including its sign) multiplied by the variable The term without any variable (including its sign) The highest exponential value

Terms are separated by plus (​+​) and minus (​−​) signs

The letter symbols that appear in the algebraic expression. Note: Variables can have any permissible value.

Example

2(𝑥 – 4)

Coefficient of ​a ​ ​ 4​​is + ​ 3​ Coefficient of ​a ​ ​ 2​​is + ​ 2​

Worked example 7

Coefficient of a ​ ​is − ​ 3​ Constant is ​+ 8​

1. 𝑥(𝑥 +2) = (𝑥 × 𝑥)+(𝑥 × 2)

Degree of expression is 4 ​​

Using distributive law

= 𝑥² + 2𝑥

Number of terms is 4 ​​

2. – 3𝑦² (𝑥² – 2) = (–3𝑦² × 𝑥²) + (–3𝑦² × –2)

Distributive law

3. 2b(3c + d) – 3(b – c + d)

Distributive law

= – 3𝑥² 𝑦² + 6𝑦²

Variable is ​a​

Algebraic expressions are named according to the number of terms in the expression:

= 2b × 3c + 2b × d – 3 × b – 3 × (– c) – 3 × d = 6bc + 2bd – 3b + 3c – 3d

Monomial – 1 term Binomial – 2 terms Trinomial – 3 terms two or more terms Polynomial –

REMEMBER

(+) × (+) = (+) (–) × (–) = (+)

1.4.2 Products

(+) × (–) = (–)

Multiplying a monomial by a polynomial: • Always use the distributive property to remove the brackets.

• Use exponent laws where required. (Refer to exponent laws in Theme 2 if learners are unsure of these.)

• Simplify like terms.

Multiply the number outside the bracket by each

(–) × (+) = (–)

term inside the bracket: ​2​(​x − 4​)​

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15

G10 – Mathematics

Theme 1: Numbers and algebraic expressions

G10 – Mathematics – Facilitator's Guide 1/2 Multiplying a binomial by a binomial:

Worked example 9

Use FOIL to determine the product: F O I L

First terms Outer terms Inner terms Last terms

multiply the first terms of the binomials multiply 1st term of first binomial by last term of second binomial multiply last term of first binomial by 1st term of second binomial multiply the last terms of the binomials

1. (2𝑥 – 3𝑦)²

= 4𝑥² – 12𝑥𝑦 + 9𝑦²

(2𝑥 × 2𝑥) = 4𝑥2 F

Worked example 8 1.

​(​𝑥 − 2​)​(​𝑥 + 4​)​​

F: (𝑥) × (𝑥) = 𝑥² O: (𝑥) × ( + 4) = 4𝑥 I: ( – 2) × (𝑥) = –2𝑥 L: ( – 2) × ( + 4) = –8

2.

​(3a − 4b)​(​2a + 3b​)​

F: 3𝑎×2𝑎=6𝑎² O: 3𝑎×3b=9ab I: -4b×2𝑎=-8𝑎b L: -4b×3b=-12b2

​= 𝑥​​ 2​ + 4𝑥 − 2𝑥 − 8​ ​= ​𝑥​ 2​ + 2𝑥 − 8​

​= ​6a​​ 2​ + 9ab − 8ab − 12 ​b​ 2​​ ​= 6 ​a​ ​ + ab − 12 ​b​ ​​ 2

2

​​ ​ ​ ​

​​

2 ×(2𝑥 × -3y) = -12𝑥y O+I

2. (2𝑥 + 𝑦)²

= 4𝑥² + 4𝑥𝑦 + 𝑦²

(-3y × -3y) = 9y2 L (i) 2𝑥 × 2𝑥 = 4𝑥² (ii) 2𝑥 × y × 2 = 4𝑥y (iii) y × y = y²

Multiplying polynomials: i)

Work from left to right and multiply every term of first polynomial by every term of second polynomial. ii) Use exponent laws where necessary. iii) Simplify like terms.

Use “shortcut” method – only applicable if you have a binomial square: i) Square the first term. ii) Multiply the first term and second term, and multiply the answer by​ 2​. iii) Square the second term. iv) Ask learners why this shortcut method works.

(Hint: Use FOIL method.)

Worked example 10

​(𝑥 + y)​​(​2𝑥 − 3y + 8​)​​

​= ​2𝑥​​ 2​ − 3𝑥y + 8𝑥 + 2𝑥y − 3 ​y​ 2​ + 8y​ ​= 2 ​𝑥​ 2​ − 𝑥y + 8𝑥 + 8y − ​3y​​ 2​

16

Multiply 1st term of binomial (i) 𝑥 × 2𝑥 = 2𝑥² (ii) 𝑥 × – 3y = – 3𝑥y (iii) 𝑥 × 8 = 8𝑥 Multiply 2nd term of binomial (iv) y ×2𝑥 = 2𝑥y (v) y × –3y = 3y2 (vi) y × 8 = 8y

1.9

Exercise 1.4: Multiplication of algebraic expressions 1.

1.1

Simplify the following:

1.2 1.3 1.4 1.5 1.6 1.7 1.8

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= −​​𝑥 2​​​​ − 8𝑥 + 3𝑥 + 24 − ​𝑥​​ ​2​ − 4𝑥 − 4

= −𝑥 − 14

1.11 2​​(𝑥 + 1)​​(𝑥 − 1)​​− ​​(𝑥 − 2)​​​ ​2​

= − ​(−3𝑥​​2𝑦​​2)​ = 4​​𝑥​​​ ​− 25 ​​(2𝑥 − 7)​2​​

= 4​​𝑥2​​ ​− 28𝑥 + 49 ​​(𝑥 ​ 2​​ ​ – ​𝑦​​​2​​)2​​​​​ ​

= ​𝑥4​​​​ ​− 2​​𝑥2​​​​ ​𝑦​2​​+ ​𝑦​ 4​ ​​(3 − 𝑥)​2​​

= 2​​(𝑥 ​ ​​ ​2​​ − 1)​​− ​(𝑥 ​ ​​ ​2​​ − 4𝑥 + 4)​​

= 2​​𝑥​​ ​2​​− 2 − ​𝑥​​ ​2​​+ 4𝑥 − 4

1.12 4 − 2 ​(𝑥 + 3)​​ = 4 − 2𝑥 − 6

Note: (2𝑥 – 7)²=(2𝑥 – 7)(2𝑥 – 7)

= −2𝑥 − 2

1.13 ​​{2𝑥 ​(𝑥 − 1)​+ 2𝑥}​​​{−𝑥 ​(𝑥 − 1)​+ 𝑥}​​

​​​(𝑥 + y)​​​ 2 = ​​(𝑥 + y)​​​​(𝑥 + y)​​

= ​​𝑥​​ 2​​ + 𝑥y + 𝑥y + ​​y​​ 2​​

= ​​𝑥​​ 2​​ + 2𝑥y + ​​y​​ 2​​ This is definitely not 𝑥​​ ​​ 2​​ + ​​y​​ 2​​.

= ​{2​𝑥 ​​ 2​− 2𝑥 + 2𝑥}​​​​{−​𝑥​​ 2​+ 𝑥 + 𝑥}​​

= ​{2​𝑥​​ 2​}​​​​{−​𝑥 ​​ 2​+ 2𝑥}​​ = −2​​𝑥 ​​ 4​+ 4​​𝑥​​ 3​

1.14 ​(​𝑥 − 2​)​ 3​

​​(−𝑦 − 3)​2​​

−7 ​​(7𝑥 − 1)​​​​ ​ 2​

= −7 ​(49​𝑥​ ​ − 14𝑥 + 1)​​ 2​

= −343​​𝑥​ ​+ 98𝑥 − 7 2​

2 terms with 3 factors each: First expand brackets in each term

= ​𝑥​​ ​2​​+ 4𝑥 − 6

Difference of squares

= 9 − 6𝑥 + ​𝑥2​​ = ​𝑦2​​ ​+ 6𝑦 + 9

2 terms with 3 factors each: First expand brackets in each term

= −2​​𝑥 2​​​​ − 9𝑥 + 20

​−​(𝑥​𝑦​​ 2)​ ​(−3𝑥)​

2​

= 4​​𝑥2​​ ​− 37𝑥 + 40

= −​​(𝑥 ​ ​​ ​2​ + 8𝑥 − 3𝑥 − 24)​​− ​(𝑥 ​ ​​ ​2​ + 4𝑥 + 4)​​

= 2𝑥 − 2 − 3𝑥 − 12

​(2𝑥 − 5)​​(2𝑥 + 5)​​

= 4​​𝑥2​​​​ − 5𝑥 − 32𝑥 + 40

1.10 −​​(𝑥 − 3)​​(𝑥 + 8)​​− ​​(𝑥 + 2)​​​ ​2​

2​​(𝑥 − 1)​​− 3​​(𝑥 + 4)​​

= 3​𝑥2​​ ​𝑦2​​ ​

​(21​_ ​𝑥 − 4)​

​(8𝑥 − 10)​​

= ​(𝑥 − 2)​​(𝑥 − 2)​​(𝑥 − 2)​​ = ​(𝑥 − 2)​​(𝑥 ​ ​​ 2​​− 4𝑥 + 4)​​

=​​ 𝑥​​ 3​− 4​​𝑥​​ 2​+ 4𝑥 − 2​​𝑥​​ 2​+ 8𝑥 − 8

3 factors: Expand brackets first

= ​𝑥​​ 3 ​− 6​​𝑥​​ 2​+ 12𝑥 − 8

17

G10 – Mathematics

Theme 1: Numbers and algebraic expressions

G10 – Mathematics – Facilitator's Guide 1/2

Simplify the following using the short method:

1.15 ​(𝑎 + 3)​​(𝑎 − 3)​​ = ​𝑎​​ 2​− 9

1.16 ​(2𝑎 − 5)​​(2𝑎 + 5)​​

Do you still remember the difference of two squares?

It is another “short method”.

= 4​​𝑎​​ 2​− 25​

1.17 ​(−𝑎 + 6)​​(−𝑎 − 6)​​

= ​(6 − 𝑎)​​( − )​(𝑎 + 6)​​

Alternative: (–𝑎 + 6)(–𝑎 – 6) = (–𝑎)² – (6)² = 𝑎² – 36

= − ​(6 − 𝑎)​​(6 + 𝑎)​​ = − ​(36 − ​𝑎​​ 2​)​ = ​𝑎​​ 2​− 36

2.2

​​(−𝑥 − 2)​​​ 2​

2.4

= 0,01 − 9​​b​​ 2​

2.5

1.19 ​(30𝑎bc − 1)​​(30𝑎bc + 1)​ = ​​(30𝑎bc)​​​ 2​− 1

= 900​​𝑎 2​​b​2​c​​ 2​− 1 ​

Remind learners of the “shortcut” method:

(​​​​ 𝑥 − 6)​​​ 2​​​​​​​ = ​​​𝑥​​ 2​​​​​​​− 12𝑥 +36

1 term squared st

= ​m​​ 2​− 2m +1 = ​𝑥​​ 2​+ 4𝑥 + 4 ​​(2g − 4m)​​​ 2​

= 4​​g​​ 2​− 16gm + 16​​m​​ 2​ ​​(1​​_2 ​​𝑥 − 2𝑦)​​​ 2​

= 1​​_2 ​​𝑥​​ 2​− 2𝑥𝑦 + 4​​𝑦​​ 2​

The following shows simplification of expressions involving more than two factors. Learners may also use FOIL to find the answer if they are unsure of the shortcut method. They will still get the correct answer.

= ​(0,1)​ 2​− ​​(3b)​​​ 2​

2.

​​(−m + 1)​​​ 2​

2.3

1.18 ​(0,1 − 3b)​​(0,1 + 3b)​​

​​

2.1

1st term x 2nd term x 2 (including sign)

2.6 last term squared

2.7

18

− ​​(2𝑥 − 7)​​​ 2​

= − ​(4​𝑥​​ 2​ − 28𝑥 +49)​​ = −4​​𝑥​​ ​+ 28​𝑥​− 49​ 2

−2 ​​​​(−𝑥 + 2)​​​ 2​​​

= −2 ​​​(​𝑥​​ 2​ − 4𝑥 + 4)​​​ = −2​​​𝑥​​ 2​​​+ 8𝑥 − 8 ​​​​(−𝑦 − 3)​​​ 2​​​ ​​(−3)​​

= −3 ​​​​(−𝑦 − 3)​​​ 2​​​

= −3 ​​​(​𝑦​​ 2​ + 6𝑦 + 9)​​​

Note: There is a 1 between the – and the bracket.

2.12

REMEMBER Remember the commutative rule. The order of multiplication can be switched: 2×3×4=2×4×3 = 3 × 2 × 4 =4×2×3

2.8

2.9

2.13

= (​​ −4)​​​​​(𝑥 − 4)​​​​​​(𝑥 − 4)​​​ = −4​​​(​𝑥​​ ​ − 8𝑥 + 16)​​​ 2

= −4​​​𝑥​​ ​​​+ 32𝑥 − 64

​(𝑥 ​ ​​ 2​ + 16)​​(𝑥 + 4)​​(𝑥 − 4)​​ = ​(𝑥 ​ ​​ 2​ + 16)​​(𝑥 ​ 2​​ ​ − 16)

2.14

Difference of squares

= ​𝑥​​ ​− 256 4

2.15

2.10 ​(𝑥 − 2)​​(𝑥 + 2)​​​(𝑥 + 3)​​​ 2​

= ​(𝑥 ​ ​​ 2​ − 4)​​− ​(𝑥 ​ ​​ 2​ + 6𝑥 + 9)​​ = ​𝑥​​ 2​− 4 −​​𝑥​​ 2​− 6𝑥 − 9

2.11

= −6𝑥 − 13

​​(𝑥 − 1)​​​ ​​(𝑥 + 1)​​​ ​ 2

2

= ​(𝑥 − 1)​​​(𝑥 − 1)​​​(𝑥 + 1)​​​(𝑥 + 1)​​ = ​(𝑥 − 1)​​​(𝑥 + 1)​​​(𝑥 − 1)​​​(𝑥 + 1)​​ = ​(𝑥 ​ ​​ 2​ − 1)​​​(𝑥 ​ ​​ 2​ − 1)​​ = ​​(𝑥 ​ ​​ 2​ − 1)​​​ 2​

= ​𝑥​​ 2​− 2​​𝑥​​ 2​+ 1​ © Optimi

= ​(𝑎 − b)​​​(𝑎 − b)​​​(𝑎 + b)​​​(𝑎 + b)​

Swap the middle two factors using the commutative rule

= ​(𝑎 − b)​​​(𝑎 + b)​​​(𝑎 − b)​​​(𝑎 + b)​ = (​ ​𝑎​​ 2​ − ​b​​ 2)​ ​​​(​ ​𝑎​​ 2​ − ​b​​ 2)​ ​ = ​(​𝑎​​ 2​ − ​b​​ 2​)​​​ 2​

​​​(𝑥 − 4)​​​​​(−4)​​​​​(𝑥 − 4)​​​ 2

​(𝑎 −b)​​​ 2​​​(​ 𝑎 + b)​​​ 2​

2.16

Swap the middle two factors using the commutative rule

Alternative ​​(𝑥 – 1)​​​​(𝑥 – 1)​​​​(𝑥 – 1)​​​​(𝑥 – 1)​​ = ​​(​𝑥​​ 2​ – 2𝑥 + 1)​​​​(​𝑥​​​​ 2​ + 2𝑥 + 1)​​ = ​​𝑥​​​​ 4​​ + ​​2𝑥​​​​ 3​​ + ​​𝑥​​​​ 2​​ – ​​2𝑥​​ 3​​ – ​​4𝑥​​ 2​​ – 2𝑥 + ​​𝑥​​​​ 2​​ + ​​2𝑥 + 1​​ ​​ = ​​𝑥​​​​ 4​​ – ​​2𝑥​​​​ 2​​ + 1

19

= ​𝑎​​ 4​− 2​𝑎​​ 2​​b​​ 2​+ ​b​​ 4​

​(𝑎 ​ ​​ 2​​b​​ 2 + ​𝑎​​ 2​)​(​ 𝑎b − 𝑎)​​(𝑎b + 𝑎)​​

= ​(𝑎 ​ ​​ 2​​b​​ 2​ + ​𝑎​​ 2​)​​ ​(​𝑎​​ 2​​b​​ 2​ – ​𝑎​​ 2​​​​)​​

= ​​𝑎​ 4 ​​b​ 4​– ​𝑎​ 4​

​(𝑎 + b + c)​​(𝑎 + b − c)​​ = ​[​(𝑎 + b)​+ c]​​​[​(𝑎 + b)​− c]​​ = ​​(𝑎 + b)​​ 2​− ​c​​ 2​ = ​𝑎​​ 2​+ 2𝑎b + ​b​​ 2​− ​c​​ 2​

​𝑥​​ 2​(𝑥 ​ ​​ 2​​ 𝑦​​ 2​ + ​𝑥​​ 2​)​(​ 𝑥𝑦 −𝑥 )​​(𝑥𝑦 + 𝑥)​​ = ​𝑥​​ 2​. ​(𝑥 ​ ​​ 2​​​​ 𝑦​​ 2​ + ​𝑥​​ 2​)​(​ 𝑥 ​ ​​ 2​​​​ 𝑦​​ 2​ − ​𝑥​​ 2​)​​ =​​𝑥 ​​ 2​​(𝑥 ​ ​​ 4​​​​ 𝑦​​ 4​ − ​𝑥​​ 4​)​​ 6 4 = ​𝑥 ​​ ​𝑦 ​​ ​− ​𝑥​​ 6​

2𝑎b​​(𝑎b − ​𝑎2​b)​​− ​𝑎2​​ ​b2​− ​(𝑎b − b)​​(𝑎b + b)​​ = 2​​𝑎2​​ ​b2​− 2​​𝑎3​b2​− ​𝑎2​b2​− ​​(​𝑎2​b2​+ ​b2​​)​​ = 2​​𝑎2​b2​− 2​​𝑎3​b2​− ​𝑎2​​ ​b2​​ ​− ​𝑎2​​ ​b2​+ ​b2​ = −2​​𝑎3​b2​+ ​b2​​ ​

Last two brackets form a difference of square

Manipulate the factors to get a difference of squares

Last two brackets form a difference of squares

Beware of the signs in this question!

G10 – Mathematics

Theme 1: Numbers and algebraic expressions

G10 – Mathematics – Facilitator's Guide 1/2 3.

3.1

Simplify the following using the short method: 2

​​​(_ ​ 3𝑥 ​− ​_ 3𝑥 ​)​​​ ​​ 2

3 ​​​(_ ​ 3𝑥 ​− ​ _ ​ ​​​ ​​ 𝑥)

LCM of 3 and 𝑥 is 3𝑥

2

​ − 9 __ = (​​​ ​ ​𝑥​​ ​ ​​​ ​​ 3𝑥 ) 2

​​(​𝑥​​ 2​− 9)​​​ 2​ ​​ 3𝑥 ​​​ ​

= ​​________

​​ ( )2

3.3

1​_ ​ 14 ​​= ​_​ 54 ​​

Also correct: 1 _ ​​ 9 ​​​𝑥​​ 2​​– 2 + ​__ ​ ​𝑥9​​ 2 ​​​

3.4

2

9​𝑥​​ ​

REMEMBER • Proper fraction:

e.g.​_ ​ 14 ​ 15

• Mixed fractions:

e.g. 2​_ ​ 3 ​​ 8 Consists of an integer + a common fraction

Numerator < denominator

• Improper fraction: e.g. _ ​​ 15 ​​ Numerator > denominator 14 3.2

​​(𝑎− ​2_1 )​ ​​(𝑎+ ​4_1 )​ ​ =​​(_ ​ 2𝑎− 1 ​ ​​ _ ​ 4𝑎+ 1 ​ ​​ 2 )( 4 ) =​​___ 8𝑎 + 2𝑎− 4𝑎− 1

​​ 8 2 =​​___ 8​a ​​ ​ − 2𝑎− 1

​​ 8 2 =​​___ 8​a ​​ ​ − 2𝑎 −1

​​ 8

Also correct:

Work with improper fractions

=​​(_ ​ 5b ​ − ​_ 3e ​ ​​ _ ​ 5 ​ ​​ 2 1 )(4)

25 _ ​​ 8 ​ b​– ​_ ​ 15 4 ​ e​ 25b – 30e = ​ ​___________ ​ ​ 8

=​​(_ ​ 5b − 6e ​ ​​ _ ​ 5 ​ ​​ 2 )(4)

and

4 ​𝑥 ​​ 2​ + 81 1 = _______________ ​ ​​𝑥 ​​ ​ – 18

​​ _ ​​ 2 ​𝑥 ​ = _​​ 2𝑥 ​​ ​9𝑥​​ 2 ​

​ − 18​𝑥​​ ​ + 81 ______________ = ​​​𝑥​​

​​ 2 4

​​(2​_ 1 ​b − 3e)​​(1​_ 1 ​ ​​ 4) 2

3.5

=​​_ 25b − 30e

​​ 8

​​(_ ​ 1 a ​ − ​_ 1 ​ ​​(2a − 3)​​ 2 3)

=​​(_ ​ a ​− ​_ 1 ​ ​​ _ ​ 2a ​ − ​_ 3 ​ ​​ 2 3)( 1 1)

Also correct: 13

​​a ​​ 2​​– ​_ ​ 6 ​ a + 1​

=​​(_ ​ 3a − 2 ​ ​​ _ ​ 2a − 3 ​ ​​ 6 )( 1 ) 2 a​​ ​ − 9a − 4a + 6 =​​ 6​___

​​ 6 ​​ 2​ − 13a + 6 =​​ 6​a___

​​ 6

2 + 6 = _______________ ​ ​6​ a​​ ​– 13a

​ ​ 6

​​(_ ​ a ​− 2)​​(__ ​ ​a4​​ ​ ​+ a + 4)​​ 2 2

=​​(_ ​ a − 4 ​ ​​​​ __ ​ ​a​​ ​ + 4a +16

​)

​​ 4 2 )( 2

= __________________ ​​ ​a​​ ​ + 4​a​​ ​ + 16a − 4​

a​​ ​ − 16a − 64

​​ 8 3

Also correct:

2

= __ ​​ ​a​​ ​ − 64

​​ 8

2

3

​​a​​ ​​– ​_ ​ 41 a​ ​– ​_​ 81 ​​ 2 = __ ​ ​​8a​​ ​ – 2a – 1

​​ 8 2

3.6

20

1 ​​​(_ ​ 1 𝑥 ​ − ​ _ ​ 𝑦 ​​​ ​​ 2 3 )

2

𝑦 2 =( ​​​ _ ​ 𝑥 ​− ​_ ​ ​​​ ​​ 2 3) 2 3𝑥 − 2𝑦 =​​​(_ ​ 6 ​) ​​​ ​​ 9​𝑥​​ 2​ − 12𝑥𝑦 + 4​𝑦​​ 2​ =​​ _______________

​​ 36

Also correct: Difference of cubes a

​​​(_​ 2 )​ ​​​ 3​​– ​​​(2)​​​ 3​​ 3 = __ ​ ​a​8 ​​ ​​ ​– 8

3 = __ ​ ​a​ ​​ ​– 64 ​​ 8

Also correct: 1 2 _ ​​ _ ​​ ​​– ​​ 13 𝑥​ 𝑦 + _​ 19 𝑦​​​​ 2​​ 4 𝑥​​ ​9𝑥​​ 2​ – 12𝑥𝑦 + ​4𝑦​​ 2​ = ​ ​_______________

​​ 36