Facilitatorโs Guide 1/2
10
Mathematics
CAPS
Mathematics FET Phase Grade 10 โข Facilitatorโs Guide 1/2 home classroom college workplace
2210-E-MAM-FG01
9 781990
948374
Mathematics Facilitator's Guide 1/2 โ 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 โ Mathematics
G10 โ Mathematics โ 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 ฮธ, cos ฮธ and tan ฮธ 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 ฮธ, cos ฮธ and tan ฮธ 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 45ยฐ........................................................................ 170
Domain and rage..................................................................................................... 202
The trigonometric ratios of 60ยฐ and 30ยฐ....................................................... 171 The trigonometric ratios of 0ยฐ and 90ยฐ......................................................... 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 ฮธ, sec ฮธ and cot ฮธ, 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 โ Mathematics
G10 โ Mathematics โ 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 โaโon 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 โaโin โy = โaโโโฏ๐ฅโโ on the graph of an exponential function...............................................................................255
6.3.4 The influence of โqโ on the graph of a parabola................................ 227
6.5.3 The influence of โbโin โyโ=โbโ.โโaโโฏ๐ฅโon 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 โq โin โyโ=โโaโโฏ๐ฅโโ +โ qโon 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 โaโ on the graph of a hyperbola.............................. 243
6.6
6.4.3 The influence of โqโ on the graph of a hyperbola.............................. 244
Self-evaluation......................................................................................................... 267
Summary: the effect of โaโand 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 ๐ฅ....................................................................................................................... 285
7.2
7.3
Exercise 7.1.2: Drawing accurate graphs of the cosine function y = cos ๐ฅ....................................................................................................................... 288
8.2
Self evaluation.......................................................................................................... 295
8.3
Exercise 7.1.3: Drawing accurate graphs of the tangent function y = tan ๐ฅ....................................................................................................................... 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 trigoยญnometric 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 โ 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 facilitatorโs guides, which are based on the concepts of Optimiโs GuidEd Learningโข 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 facilitatorโs guides are divided into 15 themes. Study guide 1/2 and facilitatorโs guide 1/2 cover themes 1 to 8 (terms 1 and 2) and study guide 2/2 and facilitatorโs 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. ยฉ Optimi
1
G10 โ Mathematics
G10 โ Mathematics โ 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 ร 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 โdifficultโ 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 facilitatorโs guide.
Prior knowledge
This section tells you what existing knowledge you need to master the theme involved.
The learning activities available in the OLPโs 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 e๐ฅercises 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 oneโs 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 facilitatorโs 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 facilitatorโs 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 โOh, of course!โ 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 โ Mathematics
G10 โ Mathematics โ 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 donโt feel comfortable with the topic at all. I need help. 2. Alarm! I donโt 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, itโs 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 ne๐ฅt 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 Impaqโs 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: โข Maths 4 Africa, available at www.maths4africa.co.za โข The Siyavula textbook, available online for free at www.siyavula.com โข 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 e๐ฅamination papers are subject to change. Always refer to the portfolio book and assessment plan for updated information about the composition of the e๐ฅamination papers. Paper 1
Tip: Use supplementary resources for further e๐ฅplanations, e๐ฅamples, and especially e๐ฅtra e๐ฅercises.
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:
โข 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. โข Formula sheets are not provided during tests and final exams in Grade 10.
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G10 โ Mathematics
Theme 1: Numbers and algebraic expressions
G10 โ Mathematics โ Facilitator's Guide 1/2
THEME 1
Introduction
NUMBERS AND ALGEBRAIC EXPRESSIONS
In this theme learners will learn more about: โข โข โข โข โข โข
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: โข โข โข โข
โข trinomial (more advanced trinomials) โข grouping in pairs โข 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 30ยฑ3 of Paper 1.
6
Different types of numbers
Note
โโโ=โโ{1โ; 2โ; 3โ; 4โ; 5โ; โฆ}โโ =โnatural numbers
โข A rational number is any number that can be written as โ_โฏโฏbaโฏโฏโ, where โaโand โbโ are integers. โข The following are rational numbers:
โโโโฏ0โโ=โโ{0โ; 1โ; 2โ; 3โ; 4โ; 5โ; โฆ}โโ =โwhole numbers
โฆ fractions of which both the numerator and denominator are integers, e.g. 73โ_โฏโโฏ โฆ integers, e.g. โ โ โ5โ โฆ decimal numbers that end, e.g. โ0,125โ
โโคโ=โโ{โฆโ; โโ2โ; โโ1โ; 0โ; 1โ; 2โ; 3โ; โฆ}โโ =โ integers
_____________ โโโ=โโ{numbers that can be written asโฏ โฏa non-zero integer โโฏ โฏ โฏโฏโ}โฏโโ=โrational numbersโ an integer
โฆ decimal numbers that repeat, e.g. โ0,โ151515โฆโ
_____________ โโโฒโฏโโโฏ =โโ{numbersโฏthatโฏcannotโฏbeโฏwrittenโฏasโฏ โฏa โโฏ non-zero integer โฏ โฏโฏ โฏโ}โโ=โirrational numbersโ
an integer
โข Irrational numbers are not rational. They cannot be written with an integer numerator and denominator, e.g. โ0,8672345โฆโ โข If the โnth โ root of a number cannot be written as a rational value, this โnth โ 3 _ root is called a surd, e.g. โโโฏ โฏ5โฏโฏโโ.
(non-terminating and non-repetitive decimal numbers)
โโโ=โโ{rational and irrational numbers}โโ =โreal numbers
โโโฒโฏโโโฏ =โโ{numbers that do not exist in the real number system}โโ =โnon-real numbers
Worked example 1
Summary of the real number system
Real numbers
Rational numbers
Integers
Rewrite โ0,โโ1หโฏโ2หโฏ โฏโโฏas 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 โtailโ.
Fractures
Let โ๐ฅโ=โ0,1212121212โฆโ
โข Negative integers โข Zero โข Positive integers (natural numbers)
โโดโ100๐ฅโ=โ12,1212121212โฆโ โ
REMEMBER
_0 ๐ฅโโฏ โฏโ โฏ= 0
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_
โโโฏโโ3โฏโฏโ is non-real
โโด
_๐ฅโฏโ โฏ is undefinedโ 0
โโดโ 7
โ๐ฅโ = 0,1212121212โฆโ
โรโ 100โto get integer โ+โrecurring โtailโ Subtract
โ 99๐ฅ =โ12โ 12 ๐ฅ =โโ_ โฏโฏ99 โฏโโ
_4 โ๐ฅ = โฏโฏ33 โฏ โโ
Simplify
G10 โ Mathematics
Theme 1: Numbers and algebraic expressions
G10 โ Mathematics โ Facilitator's Guide 1/2
Worked example 2 3
โ
โ
โ 10yโ = 25,121212โฆโ
โโดโ โโ โโด
โ
0,โโ3โฏฬโฏโโ
โรโ 1 000โand ร โ โ 10โto get integer โ+โrecurring โtailโ
_
Remember that the 5 โ โis not recurring
990yโ=โ2487,000000โฆโ 2 487 yโ=โโ_ โฏโโโฏ โฏ990
yโ=โโ_ โฏ829โฏโโ โฏ330
โ
_ โโโฏ02โฏโโ __ 9 โโ โฏ_ โ16 โฏ โฏโโฏโโ
Solution
โโดโ 1 000yโ=โ2512,121212โฆโ
_
โ'
โโ โโฏ โฏ9โฏโโ
Rewrite โ2,5โ1หโฏโ2หโฏ โฏโโฏas a common fraction. Let โyโ=โ2,512121212โฆโ
_ โโโฏ17โฏโโ
โ
Solution
โโโโฏ50โฏโโ
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
_ โโโฏ17โฏโโ
Worked example 3 3
Using your knowledge of the number system, complete the following table by making a ๏ผ in the appropriate block(s):
_
This number is written in the form _ โโโฏabโฏโโ; therefore it is a rational number (โ). It exists (your calculator does not give you a math error message); therefore it is a real number (โ).
โโ โโฏ โฏ9โฏโโโฏ = 0,2080083823โฆ โฆ (non-finite, non-recurring decimal fraction) This number cannot be written in the form โฏ_ โโabโฏโโtherefore it is an irrational number (โ'). It exists (your calculator does not give you a math error message); therefore it is a real number (โ).
8
_ โโโฏ02โฏโโ
__
= 0 (zero divided by any non-zero number = zero)
This number is written in the form _ โโโฏabโฏโโtherefore it is a rational number (โ).
โ
รผ รผ
รผ
รผ
รผ
รผ
_
โโโโฏ50โฏโโ
รผ
รผ
Note that real numbers (โ) are either rational (โ) or irrational (โ') numbers.
1
โ โ โ_ = 3โฏ โโ โฏ This number is written in the form _ โโโฏabโฏโโtherefore it is a rational number (โ).
Exercise 1.1: The number system
It exists (your calculator does not give you a math error message); therefore it is a real number (โ).
โโโโฏ50โฏโโโฏ =3,684031499โฆ (non-finite, non-recurring decimal fraction)
ยฉ Optimi
รผ รผ
0,โโ3โฏฬโฏโโ
0,โโ3โฏฬโฏโโ = 0,333333333โฆ (non-finite, recurring decimal fraction)
_
รผ
_ โโโฏ02โฏโโ __ 9 โโ โฏ_ โ16 โฏ โฏโโฏโโ
โ
It exists (your calculator does not give you a math error message); therefore it is a real number (โ).
โ
รผ
_
โโ โโฏ โฏ9โฏโโ
_ โโโฏabโฏโโtherefore it is a rational
โ'
_ โโโฏ17โฏโโ
3
It exists (your calculator does not give you a math error message); therefore it is a real number (โ).
9 โโ โฏ_ โ16 โฏ โฏโโฏโโ = โ โ โ_ โฏ34โฏโโ This number is written in the form number (โ).
โ
This number cannot be written in the form โฏ_ โโabโฏโโtherefore it is an irrational number (โ'). It exists (your calculator does not give you a math error message); therefore it is a real number (โ).
1.
Is the number zero a positive or negative number?
2.
What type of number is โโโฏ8โฏโ?
Zero is neither positive nor negative. _
_
_
_
โโโโฏ8โฏโฏโ โ =โโโโฏ4โ รโ 2โฏโโ โฏ=โ2โโโโฏ2โฏโโ
โโดโa real, irrational number
9
G10 โ Mathematics
Theme 1: Numbers and algebraic expressions
G10 โ Mathematics โ Facilitator's Guide 1/2 3.
4.
5.
6.
_
What type of number is โโโฏโ8โฏโ?
7.
The square root of a negative number does not exist in the real number system. โด a non-real number
3
7.1
_
_
ร10 to get integer + recurring โtailโ
9๐ฅ = 6 6 ๐ฅ = โโโฏ_ โฏโโ 9
_
7.2
What type of number is โโโฏ โฏโ8โฏโ?โฏ 3
Let ๐ฅ = 0,โโ6โโฏฬโฏ=0,666666โฆ โ ๐ฅ = 0,666666โฆ
โโโโฏ โฏ8โฏโโฏโ=โ2โ โด 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,โโ6โฬ
โด10๐ฅ = 6,666666โฆ
_
What type of number is โโโฏ โฏ8โฏโโฏ?
3
Rewrite the following as common fractions:
โโโฏ โฏโ8โฏโโฏ= โโโ2
2 ๐ฅ = โโโฏ_ โฏโโ 3
3,1โโ5โฏฬโฏโโโโ6โฏโโฬ
Let ๐ฆ = 3,1โโ5ฬโฏโโโโ6โฏโโฬโฬโฏ= 3,1565656โฆ
โด1 000๐ฆ = 3 156,565656โฆ
โด a real number, a rational number and an integer
โ 10๐ฆ = 31,565656โฆ
Without using a calculator, determine all the number types that 2 ___ โโโฏ10โฏโโโฏ 27 belongs to.
ร1 000 to get integer + recurring โtailโ ร10 to get recurring โtailโ
990๐ฆ = 3 125 3โฏ125 โด ๐ฆ = โโโฏ_ โฏ โฏโ 990
2 ___ โโโฏ10โฏโโโฏ= ___ โโโฏ64โฏโโโฏ 27 27
625 โด๐ฆ = โโโฏ_ โฏโฏโ 198
โด a real, rational number (64รท27 gives a non-finite, recurring decimal number as an answer)
REMEMBER Always remember to simplify completely.
10
8.
For which value(s) of ๐ฅ will f(๐ฅ) 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 โโโโฏโฏ18โฏโโ,โฏ without using a calculator, learners should first understand the following:
_
โ
_
_
โโฏ 9โฏโฏโโฏโ(โ5) โฏโฏโโโฏ = โโ โฏ___ โฏโ 9โฏโฏโโฏ+5โฏโฏโโโฏ = โโ โฏ___ โฏโ64โฏโโฏ โฏโฏโโโฏ> 0 f(โ5) = โโ โฏ___ 11 11 11
โ โ โ ___________ ___________ f(0) = โ โโ โฏ___ โโฏ 9โฏโฏโโฏโ (0)โฏโฏโโ= โฏโโโโฏ___ โโฏ 9โฏโฏโโฏโฏโฏโโโฏ> 0 11 11 _____________
_
112 f(11) = โโ โฏ___ โโฏ 9โฏโฏโโฏโ (11) โฏโฏโโโฏ = โโ โฏโ _____ โฏโโฏโฏโฏโโ< โฏ0 11 11
โ
โ
_
It is easy to see that โโโฏ3โฏโโฏ is greater than โ1โand smaller than โ2โ. n
๐ฅ = โ5 gives a positive answer
_
Determine between which two integers the irrational number โโโฏ62โฏโโโฏlies.
Solution
Find the two perfect squares to the left of (just smaller than) and to the right of
_
a surd. For example, โโโฏ2โฏโโฏand โโโฏ โฏ3โฏโโฏare surds, but โโโฏโฏ4โฏโโฏis not a surd because it can be simplified to the rational number โ2โ. n
_
(just bigger than) โ62 โon the number line.
โข The perfect square to the left of โ62โis 4 โ 9โ.
Consider surds of the form โโโฏ โฏโฏaโฏโฏโ , where a โ โis any positive number, for exampleโ 3
_
2
โข The perfect square to the right of โ62โis 6 โ 4โ.
_
โ โto be 2 โ โ, so we usually do not write โโโโฏ โฏaโฏโฏโโ. โโฏ7โฏโโโฏ or โโโโฏ โฏ5โฏโโ.โฏ It is very common for n _
Instead, we write the surd as simply โโโฏaโฏโฏโ. This is called the square root of a โ โ.
_
Worked example 4
If the n โ โth root of a number cannot be simplified to a rational number, we call it 6
n
For example, โ27โis a perfect cube, because โ3โโฏ3โโ=โ27โ.
Introduction _
_
For example, โ9โis a perfect square since โ3โโฏ2โโ=โ9โ. โข A perfect cube is a number which is the cube of an integer.
๐ฅ = 0 gives a positive answer
1.2 BETWEEN WHICH TWO INTEGERS DOES A SURD FALL?
_
_
โข If โaโand โb โare positive integers and a โ โ<โbโ, then โโโฏ โฏโฏaโฏโฏโโ<โโโโฏ โฏโฏbโฏโฏโโ. โข A perfect square is the number obtained when an integer is squared.
๐ฅ = 11 gives a negative answer
โด the expression will be non-real if ๐ฅ = 11
_
Using a calculator, we know that โโโฏ3โฏโโ โฏ =โ1,73205โฆ โ
f(๐ฅ) = โโ ___ โฏ11 โ 9โฏโฏโโฏโฏโโโฏโ ๐ฅ and ๐ฅโ{โ5; 0; 11}? ____________
_
_
For example, let us estimate where โโโฏ3โฏโ โฏโlies on the number line:
It is sometimes useful to know the approximate value of a surd without having to use a calculator.
ยฉ Optimi
11
G10 โ Mathematics
Theme 1: Numbers and algebraic expressions
G10 โ Mathematics โ Facilitator's Guide 1/2 Now find the square roots of these perfect squares:
3.
_
โข โโโฏโฏ49โฏโโ โฏ =โ7โ _
โข โโโฏโฏ64โฏโโ โฏ =โ8โ
_
_
4.
Do the exercise without using a calculator. _
Find two consecutive integers such that โโโฏ26โฏโโฏlies 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.
โข The perfect cube to the left of โ49โis 2 โ 7โ.
โข The perfect cube to the right of โ49โis 6 โ 4โ. _
โโโฏ โฏโฏ27โฏโโ โฏ =โ3โand โโโฏ โฏโฏ64โฏโโ โฏ =โ4โ _
_
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) โ49โ.
3
_
Estimated value of โโโฏโฏ10โฏโโ โฏ =โ3,1โ
_
3
_
โโดโ3โ<โโโโฏ10โฏโโ โฏ <โ4โ
Find two consecutive integers such that โโโฏ โฏโฏ49โฏโโฏlies between them.
_
Find the two perfect squares to the left of (just smaller than) and to the right of (just bigger than) โ10โ. _
โข The perfect square to the right of โ26โis 3 โ 6โ.
3
_
Estimate โโโฏโฏ10โฏโโฏcorrect to one decimal place.
โโโฏโฏ9โฏโโ โฏ =โ3โand โโโฏโฏ16โฏโโ โฏ =โ4โ
right of (just bigger than) โ26โ.
3
_
โโดโ4โ<โโโโฏโฏ18โฏโโ โฏ <โ5โ
โข The perfect square to the left of โ10โis 9 โ โ. โข The perfect square to the right of โ10โis 1 โ 6โ.
Find the two perfect squares to the left of (just smaller than) and to the โข The perfect square to the left of โ26โis 2 โ 5โ.
2.
_
โโโฏ16โฏโโ โฏ =โ4โand โโโฏโฏ25โฏโโ โฏ =โ5โ
Exercise 1.2: Estimating surds
_
right of (just bigger than) โ18โ.
โข The perfect square to the right of โ18โis 2 โ 5โ.
_
โโดโ7โ<โโโโฏโฏ62โฏโโ โฏ <โ8โ
โโโฏโฏ25โฏโโ โฏ =โ5โ and โโโฏโฏ36โฏโโ โฏ =โ6โ _ โโดโ5โ<โโโโฏโฏ26โฏโโ โฏ <โ6โ
Find the two perfect squares to the left of (just smaller than) and to the โข The perfect square to the left of โ18โis 1 โ 6โ.
Therefore, โโโฏโฏ62โฏโโฏlies between 7 โ โand โ8โ.
1.
_
Determine between which two consecutive integers โโโฏ18โฏโโโฏlies.
โโดโ3โ<โโโโฏ โฏโฏ49โฏโโ โฏ <โ4โ
12
How to round numbers:
Exercise 1.3: Rounding numbers
โข 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 โ th โ decimal digit.
Round the following numbers to the indicated degree of accuracy:
โข If the next digit is less than โ5โ, the previous decimal digit stays as it is. โข If the next digit is โ5โor more, the previous decimal digit is increased by one. โข 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.
โ36,8121212โto the nearest integer
4.
Solution โ=โ 2,600โ
โ12,507โto two decimal places
3.
Round โ2,6003โto three decimal places.
5.
Look at the 4th decimal digit: 3<5 3rd digit will stay the same.
โ2,6003โ
1.
โ12,507โโโ12,51โ
โ36,8121212โโโ37โ
โโโ48,2291โto three decimal places โโโ48,2291โโโโโ48,229โ
โ28,995โto two decimal places โ28,995โโโ29,00โ
โโโ185,02โto one decimal place โโโ185,02โโโโโ185,0โ
Worked example 6
Round โ473,78โto the nearest integer.
Solution โ473,78โ
Look at the 1st decimal digit: 7>5 Units digit will become 1 more.
โ=โ 474โ
ยฉ Optimi
13
G10 โ Mathematics
Theme 1: Numbers and algebraic expressions
G10 โ Mathematics โ 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. โ2โโ๐ฅโโฏ2โโ โโ 3๐ฅyโ +โ โyโโฏ3โโ โโ 5โ.
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
โข In an algebraic equation, two expressions are equated (using an equal sign). Examples: โฆ โ2xโ โโ 5โ=โ0โ โฆ โ3โโxโโฏ2โโ โโ 5xโ=โ7xโ +โ 2โ โข An algebraic expression does not have an equal sign. Examples: โฆ โ2xโ โโ 5โ โฆ โ3โโxโโฏ2โโ โโ 5xโ
5
โข Algebraic equations can be solved, i.e. we can find the unique value(s) of the unknown (say โxโ) for which the equation is true. โข Algebraic expressions cannot be solved. In an algebraic expression, the variable(s) (say โxโ) 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 โ3aโโโฏ4โโ +โ 2โโaโโฏ2โโ โโ 3aโ +โ 8โ. 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 (โ+โ) and minus (โโโ) signs
The letter symbols that appear in the algebraic expression. Note: Variables can have any permissible value.
Example
2(๐ฅ โ 4)
Coefficient of โa โ โโฏ4โโis + โ โ3โ Coefficient of โa โ โโฏ2โโis + โ โ2โ
Worked example 7
Coefficient of a โ โis โ โ โ3โ Constant is โ+โ8โ
1. ๐ฅ(๐ฅ +2) = (๐ฅ ร ๐ฅ)+(๐ฅ ร 2)
Degree of expression is 4 โโ
Using distributive law
= ๐ฅยฒ + 2๐ฅ
Number of terms is 4 โโ
2. โ 3๐ฆยฒ (๐ฅยฒ โ 2) = (โ3๐ฆยฒ ร ๐ฅยฒ) + (โ3๐ฆยฒ ร โ2)
Distributive law
3. 2b(3c + d) โ 3(b โ c + d)
Distributive law
= โ 3๐ฅยฒ ๐ฆยฒ + 6๐ฆยฒ
Variable is โaโ
Algebraic expressions are named according to the number of terms in the expression:
= 2b ร 3c + 2b ร d โ 3 ร b โ 3 ร (โ c) โ 3 ร d = 6bc + 2bd โ 3b + 3c โ 3d
Monomial โ 1 term Binomial โ 2 terms Trinomial โ 3 terms two or more terms Polynomial โ
REMEMBER
(+) ร (+) = (+) (โ) ร (โ) = (+)
1.4.2 Products
(+) ร (โ) = (โ)
Multiplying a monomial by a polynomial: โข Always use the distributive property to remove the brackets. โข Use exponent laws where required. (Refer to exponent laws in Theme 2 if learners are unsure of these.) โข Simplify like terms. Multiply the number outside the bracket by each
(โ) ร (+) = (โ)
term inside the bracket: โ2โ(โx โ 4โ)โ
ยฉ Optimi
15
G10 โ Mathematics
Theme 1: Numbers and algebraic expressions
G10 โ Mathematics โ 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. (2๐ฅ โ 3๐ฆ)ยฒ
= 4๐ฅยฒ โ 12๐ฅ๐ฆ + 9๐ฆยฒ
(2๐ฅ ร 2๐ฅ) = 4๐ฅ2 F
Worked example 8 1.
โ(โ๐ฅโ โโ 2โ)โ(โ๐ฅโ +โ 4โ)โโ
F: (๐ฅ) ร (๐ฅ) = ๐ฅยฒ O: (๐ฅ) ร ( + 4) = 4๐ฅ I: ( โ 2) ร (๐ฅ) = โ2๐ฅ L: ( โ 2) ร ( + 4) = โ8
2.
โ(3aโ โโ 4b)โ(โ2aโ +โ 3bโ)โ
F: 3๐ร2๐=6๐ยฒ O: 3๐ร3b=9ab I: -4bร2๐=-8๐b L: -4bร3b=-12b2
โ=โ ๐ฅโโโฏ2โโ +โ 4๐ฅโ โโ 2๐ฅโ โโ 8โ โ=โ โ๐ฅโโฏ2โโ +โ 2๐ฅโ โโ 8โ
โ=โ โ6aโโโฏ2โโ +โ 9abโ โโ 8abโ โโ 12โโbโโฏ2โโ โ=โ 6โโaโโฏ โโ +โ abโ โโ 12โโbโโฏ โโ 2
2
โโ โโฏโโฏโฏโ
โโ
2 ร(2๐ฅ ร -3y) = -12๐ฅy O+I
2. (2๐ฅ + ๐ฆ)ยฒ
= 4๐ฅยฒ + 4๐ฅ๐ฆ + ๐ฆยฒ
(-3y ร -3y) = 9y2 L (i) 2๐ฅ ร 2๐ฅ = 4๐ฅยฒ (ii) 2๐ฅ ร y ร 2 = 4๐ฅy (iii) y ร y = yยฒ
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 โshortcutโ method โ 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 byโ 2โ. iii) Square the second term. iv) Ask learners why this shortcut method works. (Hint: Use FOIL method.)
Worked example 10
โ(๐ฅโ +โ y)โโ(โ2๐ฅโ โโ 3yโ +โ 8โ)โโ
โ=โ โ2๐ฅโโโฏ2โโ โโ 3๐ฅyโ +โ 8๐ฅโ +โ 2๐ฅyโ โโ 3โโyโโฏ2โโ +โ 8yโ โ=โ 2โโ๐ฅโโฏ2โโ โโ ๐ฅyโ +โ 8๐ฅโ +โ 8yโ โโ โ3yโโโฏ2โ
16
Multiply 1st term of binomial (i) ๐ฅ ร 2๐ฅ = 2๐ฅยฒ (ii) ๐ฅ ร โ 3y = โ 3๐ฅy (iii) ๐ฅ ร 8 = 8๐ฅ Multiply 2nd term of binomial (iv) y ร2๐ฅ = 2๐ฅy (v) y ร โ3y = 3y2 (vi) y ร 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
ยฉ Optimi
= โโโ๐ฅ 2โโโโ โ 8๐ฅ + 3๐ฅ + 24 โ โ๐ฅโโโฏโ2โ โ 4๐ฅ โ 4
= โ๐ฅ โ 14
1.11 2โโ(๐ฅโฏ+โฏ1)โโ(๐ฅโฏโโฏ1)โโโ โโ(๐ฅโฏโโฏ2)โโโโฏโ2โ
= โ โ(โ3๐ฅโโ2๐ฆโโ2)โ = 4โโ๐ฅโโโ โโ 25 โโ(2๐ฅโฏโโฏ7)โ2โโ
= 4โโ๐ฅ2โโ โโ 28๐ฅ + 49 โโ(๐ฅ โ 2โโ โโฏโโฏโ๐ฆโโโ2โโ)2โโโโโ โ
= โ๐ฅ4โโโโ โโ 2โโ๐ฅ2โโโโ โ๐ฆโ2โโ+ โ๐ฆโโฏ4โ โโ(3โฏโโฏ๐ฅ)โ2โโ
= 2โโ(๐ฅ โ โโโฏโ2โโโฏโโฏ1)โโโ โ(๐ฅ โ โโโฏโ2โโโฏโโฏ4๐ฅโฏ+โฏ4)โโ
= 2โโ๐ฅโโโฏโ2โโโ 2 โ โ๐ฅโโโฏโ2โโ+ 4๐ฅ โ 4
1.12 4 โ 2 โ(๐ฅโฏ+โฏ3)โโ = 4 โ 2๐ฅ โ 6
Note: (2๐ฅ โ 7)ยฒ=(2๐ฅ โ 7)(2๐ฅ โ 7)
= โ2๐ฅ โ 2
1.13 โโ{2๐ฅ โ(๐ฅโฏโโฏ1)โ+ 2๐ฅ}โโโ{โ๐ฅ โ(๐ฅโฏโโฏ1)โ+ ๐ฅ}โโ
โโโ(๐ฅโฏ+โฏy)โโโโฏ 2 = โโ(๐ฅโฏ+โฏy)โโโโ(๐ฅโฏ+โฏy)โโ = โโ๐ฅโโโฏ 2โโ + ๐ฅy + ๐ฅy + โโyโโโฏ 2โโ = โโ๐ฅโโโฏ 2โโ + 2๐ฅy + โโyโโโฏ 2โโ This is definitely not ๐ฅโโ โโ โฏ2โโ + โโyโโโฏ 2โโ.
= โ{2โ๐ฅ โโโฏ2โโ 2๐ฅ + 2๐ฅ}โโโโ{โโ๐ฅโโโฏ2โ+ ๐ฅ + ๐ฅ}โโ
= โ{2โ๐ฅโโโฏ2โ}โโโโ{โโ๐ฅ โโโฏ2โ+ 2๐ฅ}โโ = โ2โโ๐ฅ โโโฏ4โ+ 4โโ๐ฅโโโฏ3โ
1.14 โ(โ๐ฅโ โโ 2โ)โโฏ3โ
โโ(โ๐ฆโฏโโฏ3)โ2โโ
โ7 โโ(7๐ฅโฏโโฏ1)โโโโ โ 2โ
= โ7 โ(49โ๐ฅโ โโฏโโฏ14๐ฅโฏ+โฏ1)โโ 2โ
= โ343โโ๐ฅโ โ+ 98๐ฅ โ 7 2โ
2 terms with 3 factors each: First expand brackets in each term
= โ๐ฅโโโฏโ2โโ+ 4๐ฅ โ 6
Difference of squares
= 9 โ 6๐ฅ + โ๐ฅ2โโ = โ๐ฆ2โโ โ+ 6๐ฆ + 9
2 terms with 3 factors each: First expand brackets in each term
= โ2โโ๐ฅ 2โโโโ โ 9๐ฅ + 20
โโโ(๐ฅโ๐ฆโโโฏ2)โ โ(โ3๐ฅ)โ
2โ
= 4โโ๐ฅ2โโ โโ 37๐ฅ + 40
= โโโ(๐ฅ โ โโโฏโ2โโฏ+โฏ8๐ฅโฏโโฏ3๐ฅโฏโโฏ24)โโโ โ(๐ฅ โ โโโฏโ2โโฏ+โฏ4๐ฅโฏ+โฏ4)โโ
= 2๐ฅ โ 2 โ 3๐ฅ โ 12
โ(2๐ฅโฏโโฏ5)โโ(2๐ฅโฏ+โฏ5)โโ
= 4โโ๐ฅ2โโโโ โ 5๐ฅ โ 32๐ฅ + 40
1.10 โโโ(๐ฅโฏโโฏ3)โโ(๐ฅโฏ+โฏ8)โโโ โโ(๐ฅโฏ+โฏ2)โโโโฏโ2โ
2โโ(๐ฅโฏโโฏ1)โโโ 3โโ(๐ฅโฏ+โฏ4)โโ
= 3โ๐ฅ2โโ โ๐ฆ2โโ โ
โ(21โ_โฏโ๐ฅโฏโโฏ4)โ โฏ โ(8๐ฅโฏโโฏ10)โโ
= โ(๐ฅโฏโโฏ2)โโ(๐ฅโฏโโฏ2)โโ(๐ฅโฏโโฏ2)โโ = โ(๐ฅโฏโโฏ2)โโ(๐ฅ โ โโโฏ2โโโโฏ4๐ฅโฏ+โฏ4)โโ
=โโโฏ๐ฅโโโฏ3โโ 4โโ๐ฅโโโฏ2โ+ 4๐ฅ โ 2โโ๐ฅโโ 2โ+ 8๐ฅ โ 8
3 factors: Expand brackets first
= โ๐ฅโโโฏ3 โโ 6โโ๐ฅโโโฏ2โ+ 12๐ฅ โ 8
17
G10 โ Mathematics
Theme 1: Numbers and algebraic expressions
G10 โ Mathematics โ Facilitator's Guide 1/2
Simplify the following using the short method:
1.15 โ(๐โฏ+โฏ3)โโ(๐โฏโโฏ3)โโ = โ๐โโโฏ2โโ 9
1.16 โ(2๐โฏโโฏ5)โโ(2๐โฏ+โฏ5)โโ
Do you still remember the difference of two squares? It is another โshort methodโ.
= 4โโ๐โโโฏ2โโ 25โ
1.17 โ(โ๐โฏ+โฏ6)โโ(โ๐โฏโโฏ6)โโ
= โ(6โฏโโฏ๐)โโ(โฏโโฏ)โ(๐โฏ+โฏ6)โโ
Alternative: (โ๐ + 6)(โ๐ โ 6) = (โ๐)ยฒ โ (6)ยฒ = ๐ยฒ โ 36
= โ โ(6โฏโโฏ๐)โโ(6โฏ+โฏ๐)โโ = โ โ(36โฏโโฏโ๐โโโฏ2โ)โ = โ๐โโโฏ2โโ 36
2.2
โโ(โ๐ฅโฏโโฏ2)โโโโฏ2โ
2.4
= 0,01 โ 9โโbโโโฏ2โ
2.5
1.19 โ(30๐bcโฏโโฏ1)โโ(30๐bcโฏ+โฏ1)โ = โโ(30๐bc)โโโโฏ2โโ 1
= 900โโ๐ 2โโbโ2โcโโโฏ2โโ 1 โ
Remind learners of the โshortcutโ method:
(โโโโ ๐ฅโฏโโฏ6)โโโโฏ2โโโโโโโ = โโโ๐ฅโโโฏ2โโโโโโโโ 12๐ฅ +36
1 term squared st
= โmโโโฏ2โโ 2m +1 = โ๐ฅโโโฏ2โ+ 4๐ฅ + 4 โโ(2gโฏโโฏ4m)โโโโฏ2โ
= 4โโgโโโฏ2โโ 16gm + 16โโmโโโฏ2โ โโ(1โโ_2โฏโโ๐ฅโฏโโฏ2๐ฆ)โโโโฏ2โ
= 1โโ_2โฏโโ๐ฅโโโฏ2โโ 2๐ฅ๐ฆ + 4โโ๐ฆโโโฏ2โ
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.
= โ(0,1)โโฏ2โโ โโ(3b)โโโโฏ2โ
2.
โโ(โmโฏ+โฏ1)โโโโฏ2โ
2.3
1.18 โ(0,1โฏโโฏ3b)โโ(0,1โฏ+โฏ3b)โโ
โโ
2.1
1st term x 2nd term x 2 (including sign)
2.6 last term squared
2.7
18
โ โโ(2๐ฅโฏโโฏ7)โโโโฏ2โ
= โ โ(4โ๐ฅโโโฏ2โโฏโโฏ28๐ฅโฏ+49)โโ = โ4โโ๐ฅโโโฏ โ+ 28โ๐ฅโโ 49โ 2
โ2 โโโโ(โ๐ฅโฏ+โฏ2)โโโโฏ2โโโ
= โ2 โโโ(โ๐ฅโโโฏ2โโฏโโฏ4๐ฅโฏ+โฏ4)โโโ = โ2โโโ๐ฅโโโฏ2โโโ+ 8๐ฅ โ 8 โโโโ(โ๐ฆโฏโโฏ3)โโโโฏ2โโโ โโ(โ3)โโ
= โ3 โโโโ(โ๐ฆโฏโโฏ3)โโโโฏ2โโโ
= โ3 โโโ(โ๐ฆโโโฏ2โโฏ+โฏ6๐ฆโฏ+โฏ9)โโโ
Note: There is a 1 between the โ and the bracket.
2.12
REMEMBER Remember the commutative rule. The order of multiplication can be switched: 2ร3ร4=2ร4ร3 โ= 3 ร 2 ร 4 =4ร2ร3
2.8
2.9
2.13
= (โโ โ4)โโโโโ(๐ฅโฏโโฏ4)โโโโโโ(๐ฅโฏโโฏ4)โโโ = โ4โโโ(โ๐ฅโโโฏ โโฏโโฏ8๐ฅโฏ+โฏ16)โโโ 2
= โ4โโโ๐ฅโโโฏ โโโ+ 32๐ฅ โ 64
โ(๐ฅ โ โโโฏ2โโฏ+โฏ16)โโ(๐ฅโฏ+โฏ4)โโ(๐ฅโฏโโฏ4)โโ = โ(๐ฅ โ โโโฏ2โโฏ+โฏ16)โโ(๐ฅ โ 2โโ โโฏโโฏ16)
2.14
Difference of squares
= โ๐ฅโโโฏ โโ 256 4
2.15
2.10 โ(๐ฅโฏโโฏ2)โโ(๐ฅโฏ+โฏ2)โโโ(๐ฅโฏ+โฏ3)โโโโฏ2โ
= โ(๐ฅ โ โโโฏ2โโฏโโฏ4)โโโ โ(๐ฅ โ โโโฏ2โโฏ+โฏ6๐ฅโฏ+โฏ9)โโ = โ๐ฅโโโฏ2โโ 4 โโโ๐ฅโโโฏ2โโ 6๐ฅ โ 9
2.11
= โ6๐ฅ โ 13
โโ(๐ฅโฏโโฏ1)โโโ โโ(๐ฅโฏ+โฏ1)โโโ โ 2
2
= โ(๐ฅโฏโโฏ1)โโโ(๐ฅโฏโโฏ1)โโโ(๐ฅโฏ+โฏ1)โโโ(๐ฅโฏ+โฏ1)โโ = โ(๐ฅโฏโโฏ1)โโโ(๐ฅโฏ+โฏ1)โโโ(๐ฅโฏโโฏ1)โโโ(๐ฅโฏ+โฏ1)โโ = โ(๐ฅ โ โโโฏ2โโฏโโฏ1)โโโ(๐ฅ โ โโโฏ2โโฏโโฏ1)โโ = โโ(๐ฅ โ โโโฏ2โโฏโโฏ1)โโโโฏ2โ
= โ๐ฅโโโฏ2โโ 2โโ๐ฅโโโฏ2โ+ 1โ ยฉ Optimi
= โ(๐โฏโโฏb)โโโ(๐โฏโโฏb)โโโ(๐โฏ+โฏb)โโโ(๐โฏ+โฏb)โ
Swap the middle two factors using the commutative rule
= โ(๐โฏโโฏb)โโโ(๐โฏ+โฏb)โโโ(๐โฏโโฏb)โโโ(๐โฏ+โฏb)โ = (โ โ๐โโโฏ2โโฏโโฏโbโโโฏ2)โ โโโ(โ โ๐โโโฏ2โโฏโโฏโbโโโฏ2)โ โ = โ(โ๐โโโฏ2โโฏโโฏโbโโโฏ2โ)โโโโฏ2โ
โโโ(๐ฅโฏโโฏ4)โโโโโ(โ4)โโโโโ(๐ฅโฏโโฏ4)โโโ 2
โ(๐โฏโb)โโโโฏ2โโโ(โ ๐โฏ+โฏb)โโโโฏ2โ
2.16
Swap the middle two factors using the commutative rule
Alternative โโ(๐ฅโฏ โโฏ 1)โโโโ(๐ฅโฏ โโฏ 1)โโโโ(๐ฅโฏ โโฏ 1)โโโโ(๐ฅโฏโโฏ1)โโ = โโ(โ๐ฅโโโฏ 2โโฏโโฏ2๐ฅโฏ +โฏ 1)โโโโ(โ๐ฅโโโโโฏ 2โโฏ+โฏ2๐ฅโฏ+โฏ1)โโ = โโ๐ฅโโโโโฏ 4โโ + โโ2๐ฅโโโโโฏ 3โโ + โโ๐ฅโโโโโฏ 2โโ โ โโ2๐ฅโโโฏ 3โโ โ โโ4๐ฅโโโฏ 2โโ โ 2๐ฅ + โโ๐ฅโโโโโฏ 2โโ + โโ2๐ฅโฏ +โฏ 1โโ โโ = โโ๐ฅโโโโโฏ 4โโ โ โโ2๐ฅโโโโโฏ 2โโ + 1
19
= โ๐โโโฏ4โโ 2โ๐โโโฏ2โโbโโโฏ2โ+ โbโโโฏ4โ
โ(๐ โ โโโฏ2โโbโโโฏ2โฏ+โฏโ๐โโโฏ2โ)โ(โ ๐bโฏโโฏ๐)โโ(๐bโฏ+โฏ๐)โโ
= โ(๐ โ โโโฏ2โโbโโโฏ2โโฏ+โฏโ๐โโโฏ2โ)โโ โ(โ๐โโโฏ2โโbโโโฏ2โโฏโโฏโ๐โโโฏ2โโโโ)โโ
= โโ๐โโฏ4โฏโโbโโฏ4โโ โ๐โโฏ4โ
โ(๐โฏ+โฏbโฏ+โฏc)โโ(๐โฏ+โฏbโฏโโฏc)โโ = โ[โ(๐โฏ+โฏb)โ+โฏc]โโโ[โ(๐โฏ+โฏb)โโโฏc]โโ = โโ(๐โฏ+โฏb)โโโฏ2โโ โcโโโฏ2โ = โ๐โโโฏ2โ+ 2๐b + โbโโโฏ2โโ โcโโโฏ2โ
โ๐ฅโโโฏ2โ(๐ฅ โ โโโฏ2โโ ๐ฆโโโฏ2โโฏ+โฏโ๐ฅโโโฏ2โ)โ(โ ๐ฅ๐ฆโฏโ๐ฅโฏ)โโ(๐ฅ๐ฆโฏ+โฏ๐ฅ)โโ = โ๐ฅโโโฏ2โ. โ(๐ฅ โ โโโฏ2โโโโ ๐ฆโโโฏ2โโฏ+โฏโ๐ฅโโโฏ2โ)โ(โ ๐ฅ โ โโโฏ2โโโโ ๐ฆโโโฏ2โโฏโโฏโ๐ฅโโโฏ2โ)โโ =โโ๐ฅโฏโโ 2โโ(๐ฅ โ โโโฏ4โโโโ ๐ฆโโโฏ4โโฏโโฏโ๐ฅโโโฏ4โ)โโ 6 4 = โ๐ฅโฏโโ โ๐ฆโฏโโ โโ โ๐ฅโโโฏ6โ
2๐bโโ(๐bโฏโโฏโ๐2โb)โโโ โ๐2โโ โb2โโ โ(๐bโฏโโฏb)โโ(๐bโฏ+โฏb)โโ = 2โโ๐2โโ โb2โโ 2โโ๐3โb2โโ โ๐2โb2โโ โโ(โ๐2โb2โ+ โb2โโ)โโ = 2โโ๐2โb2โโ 2โโ๐3โb2โโ โ๐2โโ โb2โโ โโ โ๐2โโ โb2โ+ โb2โ = โ2โโ๐3โb2โ+ โb2โโ โ
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 โ Mathematics
Theme 1: Numbers and algebraic expressions
G10 โ Mathematics โ Facilitator's Guide 1/2 3.
3.1
Simplify the following using the short method: 2
โโโ(_ โโฏ3๐ฅโฏโโโฏโ_ โฏ3๐ฅโฏโ)โโโโฏ โโ 2
3 โโโ(_ โโฏ3๐ฅโฏโโโฏโโฏ_ โฏโโฏ โโโโฏ โโ ๐ฅ)
LCM of 3 and ๐ฅ is 3๐ฅ
2
โโฏโโฏ9 __ = (โโโ โโฏโ๐ฅโโ โฏ โโฏโโโโฏ โโ 3๐ฅ ) 2
โโ(โ๐ฅโโโฏ2โโโฏ9)โโโโฏ2โ โโ 3๐ฅ โโโโฏ โ
= โฏโโ________ โฏโฏ โโ ( )2
3.3
1โ_ โโฏ14โฏโโ= โ_โโฏ54โฏโโ
Also correct: 1 _ โโโฏ9โฏโโโ๐ฅโโโฏ2โโโ 2 + โ__ โโฏโ๐ฅ9โโโฏ2โฏโฏโโโ
3.4
2
9โ๐ฅโโโฏ โ
REMEMBER โข Proper fraction:
e.g.โ_ โโฏ14 โฏโ 15
โข Mixed fractions:
e.g. 2โ_ โโฏ3โฏโโ 8 Consists of an integer + a common fraction
Numerator < denominator
โข Improper fraction: e.g. _ โโโฏ15โฏโโโฏ Numerator > denominator 14 3.2
โโ(๐โโฏโ2_1โฏ )โโฏโฏโโ(๐+โฏโ4_1โฏ )โโฏโฏโ =โโ(_ โโฏ2๐โโฏ1 โฏ โฏโโฏ โโ _ โโฏ4๐+โฏ1 โฏ โฏโโฏ โโ 2 )( 4 ) =โโ___ โฏ8๐โฏ+โฏ2๐โโฏ4๐โโฏ1 โฏ โฏ โโโฏ 8 2 =โโ___ โฏ8โaโฏโโ โโฏโโฏ2๐โโฏ1 โฏ โฏ โฏ โโ 8 2 =โโ___ โฏ8โaโฏโโ โโฏโโฏ2๐โฏโ1 โฏ โฏ โฏ โโ 8
Also correct:
Work with improper fractions
=โโ(_ โโฏ5b โฏโโฏโโฏโ_ โฏ3e โฏโโฏ โโ _ โโฏ5โฏโ โโ 2 1 )(4)
25 _ โโโฏ8โฏโโฏbโโ โ_ โโฏ15 4โฏโโฏeโ 25bโฏโโฏ30e = โโฏโ___________ โโฏ โ 8
=โโ(_ โโฏ5bโฏโโฏ6e โฏ โโฏโฏโโ _ โโฏ5โฏโ โโ 2 )(4)
and
4 โ๐ฅโฏโโ 2โโฏ+โฏ81 1 = _______________ โโฏโโ๐ฅโฏโโ โโฏโโฏ18 โฏ โโโฏ _ โโโฏ2โฏโ๐ฅ โ = _โโโฏ2๐ฅโฏโโ โ9๐ฅโโโฏ2โฏโ
โโฏโโฏ18โ๐ฅโโโฏ โโฏ+โฏ81 ______________ = โฏโโโ๐ฅโโ โฏ โฏ โฏ โโ 2 4
โโ(2โ_ โฏ1โฏโbโฏโโฏ3e)โโ(1โ_ โฏ1โฏโ โโ 4) 2
3.5
=โโ_ โฏ25bโฏโโฏ30e โฏ โฏ โโโฏ 8
โโ(_ โโฏ1โฏa โ โฏโโฏโ_ โฏ1โฏโ โโ(2aโฏโโฏ3)โโ 2 3)
=โโ(_ โโฏaโฏโโโฏโ_ โฏ1โฏโ โโ _ โโฏ2aโฏโโฏโโฏโ_ โฏ3โฏโ โโ 2 3)( 1 1)
Also correct: 13
โโaโฏโโ 2โโโ โ_ โโฏ6โฏโโฏa + 1โ
=โโ(_ โโฏ3aโฏโโฏ2 โฏ โโฏโฏโโ _ โโฏ2aโฏโโฏ3 โฏ โโฏโฏโโ 6 )( 1 ) 2 aโโโฏ โโฏโโฏ9aโฏโโฏ4aโฏ+โฏ6 =โโโฏ6โ___ โฏโฏ โฏโโ 6 โโโฏ2โโฏโโฏ13aโฏ+โฏ6 =โโโฏ6โa___ โฏ โฏ โฏโโ 6
2 + 6โฏ = _______________ โโฏโ6โ aโโโฏ โโ 13a โฏ โโฏ โ 6
โโ(_ โโฏaโฏโโโฏ2)โโ(__ โโฏโa4โโโฏโฏโโฏโ+โฏaโฏ+โฏ4)โโ 2 2
=โโ(_ โโฏaโฏโโฏ4 โฏ โโฏโฏ โโโโ __ โโฏโaโโโฏ โโฏ+โฏ4aโฏ+16 โฏ โ) โฏ โฏ โโ 4 2 )( 2
= __________________ โโโฏโaโโโฏ โโฏ+โฏ4โaโโโฏ โโฏ+โฏ16aโฏโโฏ4โ โฏ โฏ aโโโฏ โโฏโโฏ16aโฏโโฏ64 โฏ โฏ โโโฏ 8 3
Also correct:
2
= __ โโโฏโaโโโฏ โโฏโโฏ64 โฏ โฏ โโโฏ 8
2
3
โโaโโโฏ โโโ โ_ โโฏ41โฏaโ โโ โ_โโฏ81โฏโโ 2 = __ โโฏโโ8aโโโฏ โโฏโโฏ2aโฏโโฏ1 โฏ โโโฏ 8โฏ 2
3.6
20
1 โโโ(_ โโฏ1โฏ๐ฅ โ โฏโโฏโโฏ_ โฏโโฏ๐ฆ โโโโฏ โโ 2 3 )
2
๐ฆ 2 =( โโโ _ โโฏ๐ฅโฏโโโฏโ_ โฏ โฏโ โโโโฏ โโ 2 3) 2 3๐ฅโฏโโฏ2๐ฆ =โโโ(_ โโฏ 6โฏ โ) โฏโฏโโโโฏ โโ 9โ๐ฅโโโฏ2โโฏโโฏ12๐ฅ๐ฆโฏ+โฏ4โ๐ฆโโโฏ2โ =โโโฏ_______________ โฏโฏ โฏ โฏ โโ 36
Also correct: Difference of cubes a
โโโ(_โโฏ2โฏ)โ โโโโฏ3โโโ โโโ(2)โโโโฏ3โโ 3 = __ โโฏโaโ8โฏโโ โฏโโโฏโโ 8
3 = __ โโฏโaโ โโโฏ โฏโโโฏ64 โโโฏ 8
Also correct: 1 2 _ โโโฏ_ โโ โฏ โโโ โโโฏ13โฏ๐ฅโ ๐ฆ + _โโฏ19โฏ๐ฆโโโโ โฏ2โโ 4โฏ๐ฅโโ โ9๐ฅโโโฏ2โโฏโโฏ12๐ฅ๐ฆโฏ+โฏโ4๐ฆโโโฏ2โ = โโฏโ_______________ โฏ โโโฏ 36