FET Phase Grade 10 • Study Guide 1/2
Mathematics CAPS
Mathematics Study Guide 1/2 – Grade 10
2210-E-MAM-SG01
9 781990
948398
CAPS-aligned
Prof. C Vermeulen, Lead author P de Swardt H Otto M Sherman E van Heerden L Young
1.5.4 Quadratic trinomial....................................................................................22
TABLE OF CONTENTS
Exercise 1.5.2: Quadratic trinomials and mixed factorisation............... 24
1.5.5 Sum and difference of cubes....................................................................25 PREFACE..................................................................................................................... 1 THEME 1: NUMBERS AND ALGEBRAIC EXPRESSIONS........................................ 7 1.1
1.2 1.3
Exercise 1.5.3: Sum and difference of cubes.................................................. 27 Self-evaluation............................................................................................................ 27
Introduction....................................................................................................................7
1.6
Algebraic expressions involving fractions .........................................28
Exercise 1.1: The number system....................................................................... 10
1.6.2 Multiplication and division of algebraic fractions...........................30
Rounding of real numbers........................................................................11
Exercise 1.6.3: Finding the LCM..........................................................................33
The number system...................................................................................... 7
1.6.1 Simplifying single algebraic fractions using factors........................28
Summary of the real number system...................................................... 7
Exercise 1.6.1: Simplifying algebraic fractions using factors.................. 30
Between which two integers does a surd fall?...................................10
Exercise 1.6.2: Multiplication and division of algebraic fractions........ 31
Exercise 1.2: Estimating surds............................................................................. 11
1.6.3 Finding the lowest common multiple (LCM)......................................32
Self-evaluation............................................................................................................ 12
Exercise 1.6.4: Adding and subtracting algebraic fractions.................... 35
Exercise 1.3: Rounding numbers........................................................................ 12
1.6.4 Adding and subtracting algebraic fractions.......................................33
Multiplication of algebraic expressions..............................................13
1.6.5 Simplifying expressions involving fractions within fractions......36
1.4.1 Algebraic expressions................................................................................13
Exercise 1.6.5: Simplifying expressions involving fractions within fractions.......................................................................................................... 38
1.4
1.4.2 Products.........................................................................................................13
Self-evaluation............................................................................................................ 38
Exercise 1.4: Multiplication of algebraic expressions................................ 15
Factorising algebraic expressions.........................................................17
Summary of theme.................................................................................................... 39
1.5.1 Common factor.............................................................................................18
THEME 2: EXPONENTS............................................................................................40
1.5.2 Difference of two squares.........................................................................19
Introduction................................................................................................................. 40
1.5
Self-evaluation............................................................................................................ 17
End of theme exercise.............................................................................................. 39 Revision exercise........................................................................................................ 41
1.5.3 Grouping terms (new work) ...................................................................20 Exercise 1.5.1: Common factor, difference of squares and grouping...... 21 © Optimi
i
G10 – Mathematics
G10 – Mathematics – Study Guide 1/2 2.1
Composite bases .........................................................................................42 Exercise 2.1: Composite bases............................................................................. 44
The formula for the general term of a linear sequence..................61 Exercise 3.3: The formula for the general term of a linear sequence.. 63
Self-evaluation............................................................................................................ 44
3.4
Geometric patterns.....................................................................................64
Rational exponents.....................................................................................47
3.5
Exercise 3.4: Geometric patterns........................................................................ 66
2.2
Simplifying by factorising.........................................................................45
2.3
Exercise 2.2: Simplifying by factorising........................................................... 46
2.4
3.3
Exercise 2.3: Rational exponents........................................................................ 47
Self-evaluation............................................................................................................ 63
Problem-solving involving repeated patterns...................................67
Exercise 3.5: Problem-solving involving repeated patterns.................... 68
Self-evaluation............................................................................................................ 48
Self-evaluation............................................................................................................ 68
Equations.......................................................................................................48
Summary of theme.................................................................................................... 69
2.4.1 Equations with the unknown in the base............................................48
End of theme exercise.............................................................................................. 69
2.4.2 Equations with the unknown in the exponent ..................................49
THEME 4: EQUATIONS AND INEQUALITIES........................................................... 71
Exercise 2.4: Equations........................................................................................... 50
Introduction................................................................................................................. 71
Self-evaluation............................................................................................................ 52
Revision exercise........................................................................................................ 71
Summary of theme.................................................................................................... 52
4.1
Linear equations..........................................................................................74
THEME 3: NUMBER PATTERNS..............................................................................54
4.2
Exercise 4.1: Linear equations............................................................................. 75
End of theme exercise.............................................................................................. 52
3.1
3.2
Introduction................................................................................................................. 54 Revision exercise........................................................................................................ 54
The general term of a number pattern................................................56
4.3
Introduction................................................................................................................. 56 Exercise 3.1: The general term of a number pattern.................................. 59
Using the general term..............................................................................59
4.4
Exercise 3.2: Using the general term................................................................. 60
Quadratic equations...................................................................................76
Exercise 4.2: Quadratic equations...................................................................... 78 Self-evaluation............................................................................................................ 79
Linear equation systems...........................................................................80
Exercise 4.3: Linear equation systems............................................................. 83 Self-evaluation............................................................................................................ 84
Literal equatons (changing the subject of a formula).....................85
Exercise 4.4: Literal equations............................................................................. 86
ii
4.5
Linear inequalities......................................................................................86
The trigonometric ratios of 0° and 90°......................................................... 110
Exercise 4.5: Linear inequalities......................................................................... 88 Self-evaluation............................................................................................................ 89 Summary of theme.................................................................................................... 89 End of theme exercise.............................................................................................. 89
THEME 5: TRIGONOMETRY..................................................................................... 91 5.1
Introduction................................................................................................................. 91
5.2
Exercise 5.1: Basic trigonometric ratios in right-angled triangles ...... 93
5.3
5.4
The definitions of the trigonometric ratios sin θ, cos θ and tan θ in right-angled triangles................................................................91
5.6
Finding the values of trigonometrical ratios using a calculator ....113
5.7
Exercise 5.6: Finding the values of trigonometric ratios using a calculator.................................................................................................... 114
Solving trigonometric equations (calculating sizes of angles) .115
5.8
Exercise 5.7: Solving trigonometric equations for acute angles......... 116
Solving right-angled triangles..............................................................116
Self-evaluation......................................................................................................... 120
Definitions of the inverse trigonometric ratios, cosec θ, sec θ and cot θ, in right-angled triangles.......................................................94
Summary of theme................................................................................................. 121
Exercise 5.2: Inverse trigonometric ratios in right-angled triangles... 96
End of theme exercise .......................................................................................... 121
Self-evaluation............................................................................................................ 97
THEME 6: FUNCTIONS...........................................................................................124
The definitions of the trigonometric ratios sin θ, cos θ and tan θ and their inverses in the Cartesian plane................................98
Exercise 5.3: Applying trigonometric ratios in the Cartesian plane.100
Using the theorem of Pythagoras and trigonometric ratios...... 103
Introduction.............................................................................................................. 124
6.1
Functions and relations.........................................................................125
6.2
Exercise 6.1: Functions and relations............................................................ 127
Linear functions........................................................................................128
6.2.1 Investigating the properties of linear functions............................ 128
Exercise 5.4: Using the theorem of Pythagoras and trigonometric ratios.............................................................................................. 106
Development exercise........................................................................................... 128
Self-evaluation......................................................................................................... 109
Summary of the properties of linear functions.......................................... 132
The trigonometric ratios of 45° ....................................................................... 109
Domain and range.................................................................................................. 133
Direction and steepness of the graph............................................................. 132
Finding the trigonometric ratios of the special angles without using a calculator ....................................................................................109
Intercepts with the axes....................................................................................... 133
The trigonometric ratios of 60° and 30°....................................................... 110 © Optimi
Self-evaluation......................................................................................................... 112
Exercise 5.8 Solving right-angled triangles................................................. 118
Revision exercise..................................................................................................... 103
5.5
Exercise 5.5: Working with special angles without using a calculator....112
iii
G10 – Mathematics
G10 – Mathematics – Study Guide 1/2 6.2.2 Function notation.....................................................................................135
6.4
Hyperbolic functions...............................................................................158
6.2.3 Parallel and perpendicular lines........................................................135
6.4.1 Investigating the properties of hyperbolic functions................... 159
6.2.4 Drawing sketch graphs of linear functions...................................... 136
Development exercise........................................................................................... 159
Exercise 6.2.1: Drawing straight line graphs.............................................. 138
6.4.2 The influence of a on the graph of a hyperbola.............................. 162
6.2.6 Intersecting lines......................................................................................142
6.4.5 Domain and range....................................................................................163
Exercise 6.2.3: Intersecting lines.................................................................... 144
6.4.6 Drawing sketch graphs of hyperbolas............................................... 163
Quadratic functions.................................................................................146
6.4.8 Restrictions on hyperbolas...................................................................166
6.3.1 Investigating the properties of quadratic functions..................... 146
Exercise 6.4.1: Hyperbolas.................................................................................. 167
6.2.5 Finding the equation of a straight line graph................................. 140
6.4.3 The influence of q on the graph of a hyperbola.............................. 163
Exercise 6.2.2: Finding the equation of a straight line graph.............. 142
6.4.4 Asymptotes.................................................................................................163
Self-evaluation......................................................................................................... 145
6.4.7 Finding the equation of a hyperbola.................................................. 165
Development exercise........................................................................................... 146
Self-evaluation......................................................................................................... 168
6.3
6.3.2 The influence of aon the graph of a parabola ............................... 149
6.5
6.3.3 The turning point of a parabola..........................................................150
6.5.1 Investigating the properties of exponential functions................ 169
6.3.4 The influence of q on the graph of a parabola................................ 150
Development exercise........................................................................................... 169
6.3.5 Domain and range....................................................................................151
Exponential functions.............................................................................168
6.5.2 The influence of ain y = a𝑥 on the graph of an exponential function...............................................................................172
6.3.6 Axis of symmetry......................................................................................151 6.3.8 Drawing sketch graphs of parabolas.................................................. 152
6.5.3 The influence of bin y = b . a 𝑥on the graph of an exponential function.........................................................................173
Exercise 6.3.1: Drawing sketch graphs of parabolas............................... 154
6.5.5 Domain and range....................................................................................173
6.3.7 Intercepts with the axes.........................................................................151
6.5.4 The influence of q in y = a 𝑥 + qon the graph of an exponential function.........................................................................173
6.3.9 Finding the equation of a parabola ................................................... 154
Exercise 6.3.2: Finding the equation of a parabola.................................. 157
6.5.6 Reflections..................................................................................................174
Self-evaluation......................................................................................................... 158
Exercise 6.5.1: Properties of exponential graphs...................................... 174
6.5.7 Drawing sketch graphs of exponential functions.......................... 175
iv
Exercise 6.5.2: Drawing sketch graphs of exponential functions....... 176
6.5.8 Finding the equation of an exponential graph............................... 176
THEME 8: EUCLIDEAN GEOMETRY......................................................................212
Exercise 6.5.3: Finding the equation of an exponential function....... 177
Introduction.............................................................................................................. 212
6.6
8.1
Self-evaluation......................................................................................................... 179
Summary: The effect of aand q on each type of graph................. 179
Lines............................................................................................................................. 215
Revision exercise: Basic geometry: lines and angles............................... 217
Summary of theme................................................................................................. 181
Triangles..................................................................................................................... 218
End of theme exercise........................................................................................... 182
Properties of triangles.......................................................................................... 219
THEME 7: TRIGONOMETRIC FUNCTIONS............................................................186
7.1
Triangles: Similarity.............................................................................................. 219
Introduction.............................................................................................................. 186
Triangles: Congruence.......................................................................................... 221
Revision exercise..................................................................................................... 186 Drawing accurate graphs of the sin, cos and tan function.......... 189
Exercise 7.1.1: Drawing accurate graphs of sine function y = sin 𝑥 . 189
8.2
Exercise 7.1.2: Drawing accurate graphs of the cosine function y = cos 𝑥....................................................................................................................... 192
7.2
7.3
Basic geometry: Lines, angles and triangles................................... 213
Exercise 7.1.3: Drawing accurate graphs of the tangent function y = tan 𝑥....................................................................................................................... 194
8.3
Self-evaluation......................................................................................................... 197
Exercise 8.1: Basic geometry: lines, angles and triangles..................... 222 Self-evaluation......................................................................................................... 226
The mid-point theorem..........................................................................226
Exercise 8.2: The mid-point theorem............................................................. 229
Self-evaluation......................................................................................................... 232
Special quadrilaterals.............................................................................232
Exercise 8.3: Special quadrilaterals................................................................ 239
Drawing and interpreting sketch graphs of trigonometric functions.........................................................................198
Self-evaluation......................................................................................................... 242 Summary of theme................................................................................................. 243
Exercise 7.2: Drawing and interpreting sketch graphs of the trigonometric functions....................................................................................... 203
End of theme exercise........................................................................................... 243
Finding the equations of trigonometric graphs and interpreting graphs.................................................................................204
PAPERS...................................................................................................................245 FORMULAE..............................................................................................................251
Exercise 7.3: Finding the equations of trigonometric graphs and interpreting graphs................................................................................................ 206
EUCLIDIAN GEOMETRY: ACCEPTED REASONS..................................................254
Self-evaluation......................................................................................................... 209
INDEX......................................................................................................................257
Summary of theme................................................................................................. 210 © Optimi
End of theme exercise........................................................................................... 210
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 – Study 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.
Previous 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:
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.
1.
revision of the concepts, definitions and procedures required as previous knowledge, 2. an exercise or activity with solutions so that you can test your previous knowledge yourself, or 3. a combination of the above. 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.
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.
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.
© Optimi
3
G10 – Mathematics
G10 – Mathematics – Study 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: Use supplementary resources for further explanations, examples, and especially extra exercises.
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
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.
© Optimi
5
G10 – Mathematics
THEME 1
Different types of numbers ℕ = {1 ; 2 ; 3 ; 4 ; 5 ; …} =natural numbers
NUMBERS AND ALGEBRAIC EXPRESSIONS
ℕ0 = {0 ; 1 ; 2 ; 3 ; 4 ; 5 ; …} =whole numbers ℤ = {… ; − 2 ; − 1 ; 0 ; 1 ; 2 ; 3 ; …} = integers
Introduction
ℚ = { numbers that can be written as _____________ = rational numbers a non-zero integer} an integer
ℚ′ = { numbers that cannot be written as _____________ a non-zero integer} = irrational numbers (non-terminating and non-repetitive decimal numbers)
In this theme you will learn more about: • • • • • •
an integer
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.
ℝ = {rational and irrational numbers} =real numbers
ℝ′ = { numbers that do not exist in the real number system} = non-real numbers
Prior knowledge
Summary of the real number system
In order to master this theme, you should already know: • • • •
Real numbers
what types of numbers there are how we classify numbers how to multiply simple algebraic expressions how to simplify simple fractions.
Rational numbers
Integers
1.1 THE NUMBER SYSTEM
• • •
Introduction This sub-theme is a summary of work covered in Grades 8 and 9. If you should find it difficult to complete this section, you should first revise the work covered in these grades.
© Optimi
Fractures
Negative integers Zero Positive integers (natural numbers)
REMEMBER
_0 𝑥 = 0
7
Irrational numbers
_
√ − 3 is non-real
_𝑥 is undefined 0 G10 – Mathematics
Theme 1: Numbers and algebraic expressions
G10 – Mathematics – Study Guide 1/2 Note • •
◦ • •
Worked example 2
A rational number is any number that can be written as _ab, where aand b are integers. The following are rational 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
Rewrite 2,51˙ 2˙ as a common fraction.
Solution
Let y = 2,512121212…
decimal numbers that repeat, e.g. 0, 151515…
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 .
∴ 1 000y = 2512,121212… − 10y = 25,121212…
∴
Rewrite 0, 1˙ 2˙ as a common fraction.
Solution
In order to rewrite a recurring fraction as a common fraction, you need to manipulate the recurring fraction to lose the recurring “tail”. Let 𝑥 = 0,1212121212… ∴ 100𝑥 = 12,1212121212… −𝑥 = 0,1212121212…
× 100to get integer +recurring “tail” Subtract
99𝑥 = 12 ∴
𝑥= _ 12 99
4 ∴ 𝑥 = _33
Simplify
8
Remember that the 5 is not recurring
990y = 2487,000000…
2 487 ∴ y = _ 990
Worked example 1
× 1 000and × 10to get integer +recurring “tail”
y = _ 829 330
Always remember to simplify completely
_ 02
Worked example 3 Using your knowledge of the number system, complete the following table by making a in the appropriate block(s):
3
1 _ 7
_
ℚ
ℚ'
ℝ
√
0 _ 2
__
9 _ 16
1
√ 50 _
_
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)
In order to determine where these numbers fit into the number system, you can use your calculator to find the decimal fraction where applicable:
3
It exists (your calculator does not give you a math error message); therefore it is a real number (ℝ). _ = 3 This number is written in the form _ab therefore it is a rational number (ℚ).
_
_ 17
It exists (your calculator does not give you a math error message); therefore it is a real number (ℝ).
0,3̇ = 0,333333333… (non-finite, recurring decimal fraction)
0,3̇
Solution
This number is written in the form _ab therefore it is a rational number (ℚ).
9 _ 16 = _ 34 This number is written in the form _ab therefore it is a rational number (ℚ).
√ 9
√
__
= 0 (zero divided by any non-zero number = zero)
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).
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).
√ 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). © Optimi
9
G10 – Mathematics
Theme 1: Numbers and algebraic expressions
G10 – Mathematics – Study Guide 1/2
ℚ
ℚ'
ℝ
ü
7.
Rewrite the following as common fractions:
ü
ü
7.2
3,15̇6 ̇
1 _ 7
ü
_ 02 __ 9 _ 16
ü
ü
ü
ü
ü
ü
3
_
√ 9
√
0,3̇
_
√ 50
ü
7.1 8.
0,6̇
For which value(s) of 𝑥 will f(𝑥) be non-real if: _
f(𝑥) = ___ 9 – 𝑥 and 𝑥∈{–5; 0; 11}? 11
√
1.2 BETWEEN WHICH TWO INTEGERS DOES A SURD FALL?
ü
Introduction
Note that real numbers (ℝ) are either rational (ℚ) or irrational (ℚ') numbers.
If the n th root of a number cannot be simplified to a rational number, we call it _
6
Exercise 1.1: The number system
be simplified to the rational number 2. n
1.
Is the number zero a positive or negative number?
3.
What type of number is √ −8 ?
2. 4. 5. 6.
_
_
3
_
2
_
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 .
It is sometimes useful to know the approximate value of a surd without having to use a calculator.
_ _
What type of number is √ 8 ? 3
_
Consider surds of the form √ a , where a is any positive number, for example
What type of number is √ 8 ? 3
_
a surd. For example, √ 2 and √ 3 are surds, but √ 4 is not a surd because it can
_
For example, let us estimate where √ 3 lies on the number line:
_
_
Using a calculator, we know that √ 3 = 1,73205…
What type of number is √ −8 ?
_
It is easy to see that √ 3 is greater than 1and smaller than 2.
Without using a calculator, determine all the number types that 2 ___ 10 27 belongs to.
10
_
But to estimate the values of other surds, such as √ 18 , without using a calculator, you should first understand the following: • • •
n
_
n
Exercise 1.2: Estimating surds
_
If aand b are positive integers and a < b, then √ a < √ b . A perfect square is the number obtained when an integer is squared.
Do the exercise without using a calculator.
For example, 9is a perfect square since 32 = 9. A perfect cube is a number which is the cube of an integer.
Determine between which two integers the irrational number √ 62 lies.
The perfect square to the left of 62is 4 9.
How to round numbers:
The perfect square to the right of 62is 6 4.
•
Now find the square roots of these perfect squares: _
√ 49 = 7
• • •
√ 64 = 8
_
Therefore, √ 62 lies between 7 and 8. _
∴ 7 < √ 62 < 8
© Optimi
_
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.
(just bigger than) 62 on the number line.
_
_
Determine between which two consecutive integers √ 18 lies.
Rounding numbers makes them simpler and easier to use. It is often just easier to work with rounded numbers.
Find the two perfect squares to the left of (just smaller than) and to the right of
•
_
Introduction
Solution
•
Estimate √ 10 correct to one decimal place.
3
Find two consecutive integers such that √ 49 lies between them.
1.3 ROUNDING OF REAL NUMBERS
_
•
4.
3.
Worked example 4
•
Find two consecutive integers such that √ 26 lies between them.
2.
For example, 27is a perfect cube, because 33 = 27.
_
1.
11
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.
If the next digit is less than 5, the previous decimal digit stays as it is. If the next digit is 5or 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).
G10 – Mathematics
Theme 1: Numbers and algebraic expressions
G10 – Mathematics – Study Guide 1/2
Worked example 5
Self-evaluation Use the following scale to determine how comfortable you are with each topic in the table that follows:
Round 2,6003to three decimal places.
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.
Solution
Look at the 4 decimal digit: 3<5 3rd digit will stay the same. th
2,6003
= 2,600
Worked example 6
Complete the table.
Round 473,78to the nearest integer.
Topic
I can classify any real number as rational or irrational.
Solution
Look at the 1st decimal digit: 7>5 Units digit will become 1 more.
473,78 = 474
I can write any recurring decimal fraction as an ordinary fraction. I can determine between which two integers a given surd falls.
Exercise 1.3: Rounding numbers
I can round real numbers to a certain degree of accuracy.
Round the following numbers to the indicated degree of accuracy: 1.
12,507to two decimal places
4.
28,995to two decimal places
2. 3. 5.
36,8121212to the nearest integer − 48,2291to three decimal places − 185,02to one decimal place
12
1
2
3
4
5
Consider the algebraic expression 3a4 + 2 a2 − 3a + 8.
1.4 MULTIPLICATION OF ALGEBRAIC EXPRESSIONS
Terminology
Introduction
Description
Coefficient
In this sub-theme we deal with multiplication of algebraic expressions.
1.4.1 Algebraic expressions
An algebraic expression contains letter symbols and numbers, e.g. 2 𝑥 − 3𝑥y + y − 5. 2
3
Constant
Note the difference between an algebraic expression and an algebraic equation: • • • •
Degree of expression
In an algebraic equation, two expressions are equated (using an equal sign). Examples: ◦ 2x − 5 = 0 ◦ 3 x2− 5x = 7x + 2
Coefficient of a 4is + 3
The term without any variable (including its sign)
Constant is + 8
Terms are separated by plus (+) and minus (−) signs
Variable
An algebraic expression does not have an equal sign. Examples: ◦ 2x − 5 ◦ 3 x2− 5x
The number (including its sign) multiplied by the variable
The highest exponential value
Number of terms
Example
The letter symbols that appear in the algebraic expression. Note: Variables can have any permissible value.
Coefficient of a 2is + 2 Coefficient of a is − 3
Degree of expression is 4 Number of terms is 4 Variable is a
Algebraic expressions are named according to the number of terms in the expression:
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.
Monomial Binomial Trinomial Polynomial
You should know the following terminology regarding algebraic expressions:
– – – –
1.4.2 Products
1 term 2 terms 3 terms two or more terms
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 you are unsure of these.) • Simplify like terms. © Optimi
13
G10 – Mathematics
Theme 1: Numbers and algebraic expressions
G10 – Mathematics – Study Guide 1/2
Multiplying a binomial by a binomial:
Multiply the number outside the bracket by each term inside the bracket:
Use FOIL to determine the product:
2(𝑥 – 4)
F O I L
Worked example 7
1. 𝑥(𝑥 +2) = (𝑥 × 𝑥)+(𝑥 × 2)
Worked example 8
Using distributive law
= 𝑥² + 2𝑥
2. – 3𝑦² (𝑥² – 2) = (–3𝑦² × 𝑥²) + (–3𝑦² × –2)
Distributive law
3. 2b(3c + d) – 3(b – c + d)
Distributive law
= – 3𝑥² 𝑦² + 6𝑦²
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
First terms Outer terms Inner terms Last terms
= 2b × 3c + 2b × d – 3 × b – 3 × (– c) – 3 × d = 6bc + 2bd – 3b + 3c – 3d
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
= 𝑥2+ 4𝑥 − 2𝑥 − 8 = 𝑥2+ 2𝑥 − 8
= 6a2+ 9ab − 8ab − 12 b2
= 6 a + ab − 12 b 2
2
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.
REMEMBER
(+) × (+) = (+) (–) × (–) = (+) (+) × (–) = (–) (–) × (+) = (–)
Why do you think this shortcut method works? (Hint: Use FOIL method.)
14
Worked example 9
Exercise 1.4: Multiplication of algebraic expressions
1. (2𝑥 – 3𝑦)²
= 4𝑥² – 12𝑥𝑦 + 9𝑦²
(2𝑥 × 2𝑥) = 4𝑥2 F
2 ×(2𝑥 × -3y) = -12𝑥y O+I
2. (2𝑥 + 𝑦)²
= 4𝑥² + 4𝑥𝑦 + 𝑦²
1.
1.1
(-3y × -3y) = 9y2 L
i)
ii) iii)
= 2 𝑥2− 𝑥y + 8𝑥 + 8y − 3y2
(2𝑥 − 7)2
1.7 1.8
Worked example 10
= 2𝑥2− 3𝑥y + 8𝑥 + 2𝑥y − 3 y2 + 8y
1.4 1.6
Work from left to right and multiply every term of first polynomial by every term of second polynomial. Use exponent laws where necessary. Simplify like terms.
(𝑥 + y)(2𝑥 − 3y + 8)
−(𝑥𝑦2)(−3𝑥)
1.5
Multiplying polynomials:
2(𝑥 − 1)− 3(𝑥 + 4)
1.2
1.3
(i) 2𝑥 × 2𝑥 = 4𝑥² (ii) 2𝑥 × y × 2 = 4𝑥y (iii) y × y = y²
Simplify the following:
1.9
(2𝑥 − 5)(2𝑥 + 5)
(𝑥 2 – 𝑦2)2 (3 − 𝑥)2
(−𝑦 − 3)2
−7 (7𝑥 − 1)2
(21_𝑥 − 4)(8𝑥 − 10)
1.10 −(𝑥 − 3)(𝑥 + 8)− (𝑥 + 2)2
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
1.11 2(𝑥 + 1)(𝑥 − 1)− (𝑥 − 2)2 1.12 4 − 2 (𝑥 + 3)
1.13 {2𝑥 (𝑥 − 1)+ 2𝑥}{−𝑥 (𝑥 − 1)+ 𝑥} 1.14 (𝑥 − 2)3
1.15 (𝑎 + 3)(𝑎 − 3) © Optimi
15
G10 – Mathematics
Theme 1: Numbers and algebraic expressions
G10 – Mathematics – Study Guide 1/2 1.16 (2𝑎 − 5)(2𝑎 + 5)
2.12
(𝑎 −b)2(𝑎 + b)2
1.18 (0,1 − 3b)(0,1 + 3b)
2.14
(𝑎 + b + c)(𝑎 + b − c)
1.17 (−𝑎 + 6)(−𝑎 − 6)
2.13 2.15
1.19 (30𝑎bc − 1)(30𝑎bc + 1)
2.
2.1
2.2 2.3 2.4
(−m + 1)2
3.2
(1_2𝑥 − 2𝑦)2
3.3
−2 (−𝑥 + 2)2
3.5
2.7
(−𝑦 − 3)2 (−3)
2.9
3.1
(2g − 4m)2 − (2𝑥 − 7)2
2.8
3.
(−𝑥 − 2)2
2.5 2.6
2.16
Simplify the following using the short method:
3.4
(𝑥 − 4)(−4)(𝑥 − 4)
(𝑥 + 16)(𝑥 + 4)(𝑥 − 4)
2.10 (𝑥 − 2)(𝑥 + 2)(𝑥 + 3) 2.11
2
𝑥 2(𝑥 2 𝑦 2+ 𝑥 2)( 𝑥𝑦 −𝑥 )(𝑥𝑦 + 𝑥)
2𝑎b(𝑎b − 𝑎2b)− 𝑎2 b2− (𝑎b − b)(𝑎b + b)
Simplify the following using the short method: 3 (_ 3𝑥 − _ (_ _)(__)________ ( ) __ __ _ __ 𝑥)
(𝑎− _ 12 )(𝑎+ _ 14 )
()
1 1 (2_ b − 3e)(1_ 4) 2
(_ 1a − _ 1 ( 2a − 3) 2 3)
(_ a− 2)(__ a4 + a + 4) 2 2
3.6
(_ 1𝑥 − _ 13 𝑦) 2
3.8
m _ (_ 1− m)(_ 1+ m)− ___ 1 + mn2) 2 (m n n
3.7
2
(𝑎 2b 2 + 𝑎 2)( 𝑎b − 𝑎)(𝑎b + 𝑎)
2
(_ 1+ _ 1 ) − 2(_ 1 )(_ 1)− (_ 1− _ 1 )(_ 1+ _ 1 ) n m m n n m n m 2
3.9 Using a calculator:
(𝑥 − 1)2 (𝑥 + 1)2
n
(2,3 − 1,7m)(3,5m − 0,07)
3.10 (0,2𝑥 − 0,3𝑦)2 16
Without a calculator:
(_ 23– _ 17m _ 35m– _ 7 10 10 )( 10 100)
1.5 FACTORISING ALGEBRAIC EXPRESSIONS
Self-evaluation
Introduction
Use the following scale to determine how comfortable you are with each topic in the table that follows: 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
It is important to understand the difference between factors and products.
What is a factor? • The factors of a number are what we can multiply to get the number. • We can also think of a factor of a number as something that we can divide into that number without leaving a remainder. What is a product? • This is the answer when two or more factors are multiplied. Examples:
5
24 = 4 × 6
I know the difference between an algebraic expression and an algebraic equation. I know the following terminology regarding algebraic expressions, and can apply it: • Coefficient • Constant • Variable • Degree of an expression • Number of terms
9=3×3
[4 and 6 can be divided into 24 without leaving a remainder.] 24 is the product of 4 and 6. 3 is a factor of 9.
[3 can be divided into 9 without leaving a remainder.]
Multiplication and factorisation are opposite (inverse) procedures.
If we have the product, we can find the factors that have been multiplied to give the product. Factorise = find the factors
9 is the product of 3 and 3 (32).
I can multiply algebraic expressions.
© Optimi
4 and 6 are the factors of 24.
Types of factorisation:
• • • • • 17
Common factor Difference of squares Quadratic trinomial Grouping in pairs Sum and difference of cubes G10 – Mathematics
Theme 1: Numbers and algebraic expressions
G10 – Mathematics – Study Guide 1/2 Let us first revise Grade 9 factorisation, namely taking out a common factor and the difference of squares. We shall also revise quadratic trinomials later on.
Worked example 11
1.5.1 Common factor What is a common factor? A common factor is a number that can be divided into two or more different numbers without leaving a remainder. Numbers often share more than one common factor.
Factorise:
–3𝑥³ + 6𝑥𝑦 – 12𝑥²
= –3.𝑥.𝑥.𝑥 + 2.3.𝑥.𝑦 – 3.4.𝑥.𝑥
Determine the highest common factor (3𝑥)
For example, 2is a common factor of 4and 12.
= –3.𝑥(𝑥.𝑥 – 2.𝑦 + 4.𝑥)
In this case, 4is the highest common factor (HCF).
You should always check your answer by multiplying the factors to see if they get the original expression, i.e. the given product.
= –3𝑥(𝑥² – 2𝑦 + 4𝑥)
However, 4is also a common factor of 4and 12. 𝑥 is a common factor of 𝑥2and 𝑥3.
This should become a habit.
However, 𝑥2is also a common factor of 𝑥2and 𝑥3. In this case, 𝑥2is the highest common factor.
–3𝑥(𝑥² – 2𝑦 + 4𝑥) = –3𝑥³ + 6𝑥𝑦 – 12𝑥²
Factorise using a common factor of an algebraic expression such as 4 𝑥 + 12 𝑥 : i) ii)
iii)
Take out the negative sign if the first term is negative
2
3
Determine the highest common factor of the terms (4 𝑥 2in this case). The second factor is obtained by dividing the common factor into each term of the original algebraic expression. This gives 1 + 3𝑥. Therefore, 4 𝑥 2+ 12 𝑥 3is factorised as 4 𝑥 2(1 + 3𝑥). Note that the second factor is written in brackets. So, the factors of 4 𝑥 2 + 12 𝑥 3are 4 𝑥 2and 1 + 3𝑥. Therefore, if we multiply 4 𝑥 2and 1 + 3𝑥, we will obtain the original expression, 2 4 𝑥 + 12 𝑥 3.
Original expression given
Worked example 12
Factorise:
3𝑥(𝑥 – 1) – 2(𝑥 – 1) = (𝑥 – 1)(3𝑥 – 2)
Check your answer.
18
In this case, the common factor is a binomial
Conversely, to find the factors of a difference of squares, we follow this procedure:
Worked example 13 Factorise:
3𝑥(𝑥 – 1) – 2(1 – 𝑥)
In this case it might seem as if the common factor is (𝑥 – 1), but the factors (𝑥 – 1) and (1 – 𝑥) are not exactly the same. However, we can write (1 – 𝑥) as –(𝑥 – 1). [Check this by removing the brackets.]
REMEMBER
Add +in one bracket and −in the other.
_
_
_
Worked example 14
= (4𝑥 + 5𝑦)(4𝑥 – 5𝑦)
_ _ _ _ (√ 16 𝑥 2 √ 25y 2)(√ 16𝑥 2 √ 25y 2)
Worked example 15
3𝑥(𝑥 – 1) – 2(1 – 𝑥)
= 3𝑥(𝑥 – 1) + 2(𝑥 – 1) – 2(1 – 𝑥) = 2(𝑥 – 1) = (𝑥 – 1)(3𝑥 + 2)
Factorise completely:
Check your answer by multiplying the factors
8𝑎² – 32b²
1.5.2 Difference of two squares A difference of two squares is a squared number/expression subtracted from another squared number/expression. Examples: 52− 42or 𝑥2 − y2
= 2(4𝑎² – 16b²)
= 2(2𝑎 + 4b)(2𝑎 – 4b)
"Difference" means "minus".
(𝑥 − y)(𝑥 + y) = 𝑥 + 𝑥y − 𝑥y − y 2
(8 = 2 × 4) and (32 = 2 × 16) Difference of squares
Take out common factor first Factorise difference of squares
REMEMBER
Let us see what happens when we find the product of (𝑥 − y)and (𝑥 + y):
© Optimi
ii)
_
(√ 1stterm √ 2ndterm )(√ 1stterm √ 2ndterm )
16𝑥2− 25 y 2
Therefore
= 𝑥2− y2
Create two brackets:
Factorise:
Since variables such as x represent numbers, the same principles that apply to numbers also apply to algebraic expressions and variables. Example: 5 = (7 – 2) = – (2 – 7) = – ( – 5) = 5
2
i)
Always factorise until it is impossible to factorise any further.
Multiply a binomial by a binomial
Product is the difference of two squares
19
G10 – Mathematics
Theme 1: Numbers and algebraic expressions
G10 – Mathematics – Study Guide 1/2
Worked example 16 Factorise completely: (𝑥 – 1)² – 4
= [(𝑥 – 1) + 2][(𝑥 – 1) – 2] = (𝑥 – 1 + 2)(𝑥 – 1 – 2) = (𝑥 + 1)(𝑥 – 3)
Worked example 17
First term is a binomial _ √( 𝑥 − 1) 2
Solution
_
= (𝑥 − 1) and √ 4 = 2
a − b − a𝑥 + b𝑥
Simplify
Group together (a − b) and ( − a𝑥 + b𝑥)
= (a − b)− 𝑥(a − b)
Take out − 𝑥as a common factor from − a𝑥 + b𝑥
a b Remember that _ a − − b = 1
= (a − b)(1 − 𝑥)
1.5.3 Grouping terms (new work) When you are doing basic factorisation of algebraic expressions, you usually start by taking out a common factor from every term in the expression. What happens if the given expression has four or five terms, but there is no common factor?
Worked example 18
When you have four or five terms with no common factor for all of them, consider factorising by grouping the terms.
Check your answer by e𝑥panding the brackets
Factorise: a2+ a(3 + b) + 3b
To factorise by grouping terms: i) ii)
Note: –a𝑥 + b𝑥 = –(a𝑥 – b𝑥) = –𝑥(a – b)
Factorise: a − b − a𝑥 + b𝑥
Solution
First split the terms in the expression into two groups. Then factorise each group of terms separately. If set up correctly, you should get a common factor in binomial form.
𝑎² + 𝑎(3 + b) + 3b
= 𝑎² + 3𝑎 + 𝑎b + 3b = (𝑎² + 𝑎b) + (3𝑎 + 3b) = 𝑎(𝑎 + b) + 3(𝑎 + b)
= (𝑎 + b)(𝑎 + 3)
20
Remove brackets using distributive law to get four terms Group (𝑎² + 𝑎b) and (3𝑎 + 3b) together
Take out a as a common factor from the 1st pair; take out 3 as a common factor from the 2 nd pair. (a + b)is the common factor in each binomial Take out ( a + b)as the common factor