Grade 10 • Study Guide 1/2 Mathematics
Owned and published by Optimi, a division of Optimi Central Services (Pty) Ltd.
7 Impala Avenue, Doringkloof, Centurion, 0157 info@optimi.co.za www.optimi.co.za
© Optimi
Apart from any fair dealing for the purpose of research, criticism or review as permitted in terms of the Copyright Act, no part of this publication may be reproduced, distributed, or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system without prior written permission from the publisher.
The publisher has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
There are instances where we have been unable to trace or contact the copyright holder. If notified, the publisher will be pleased to rectify any errors or omissions at the earliest opportunity.
Reg. No.: 2011/011959/07
Mathematics
Study Guide 1/2 − Grade 10
2310-E-MAM-SG01
CAPS-aligned
Prof. C Vermeulen, Lead author P de Swardt H Otto
M Sherman E van Heerden L Young
5.1 The
5.3 The
PREFACE
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.
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.
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.
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.
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.
Time allocation
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.
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.
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 ��o��e e��e����ises ��ou do, the ����
su����ess in ����the����ti
s.
Structure of themes
the
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.
Sample
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
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 CAPS (the curriculum statement).
End of theme exercise
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.
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
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.
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 ��nd 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.
Self-evaluation
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.
Assessment criteria
Visit Impaq’s online platform for the assessment plan and comprehensive information about the compilation and mark allocation of tests, assignments and examinations.
Tip: M��ke su��e th��t ��ou know whi��h the
The the��es ��ove��ed in the e
in
to the
Paper 1
Algebraic expressions, equations and inequalities, exponents (Theme 1, 2 and 4)
Number patterns (Theme 3)
Functions and graphs (Theme 6)
Finance and growth (Theme 10)
Probability (Theme 15)
Note:
Paper 2
Euclidean geometry and measurement (Theme 8, 13 and 14)
Analytical geometry (Theme 9)
Trigonometry (Theme 5 and 7)
Statistics (Theme 11)
• 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.
Supplementary books
Any other books can be used along with this textbook for extra exercises and explanations, including:
• M��ths 4 A����i����, available at www.maths4africa.co.za
Sample
• The Si����vul�� textbook, available online for free at www.siyavula.com
• P��th����o����s, available at www.fisichem.co.za
Tip: Use
Calculator
We recommend the CASIO fx-82ES (Plus) or CASIO fx-82ZA. However, any scientific, non-programmable and non-graphing calculator is suitable.
THEME 1
NUMBERS AND ALGEBRAIC EXPRESSIONS
Introduction
In this theme you will learn more about:
1. different types of numbers
2. how to estimate the values of certain numbers
3. how to round off numbers
4. how to multiply algebraic expressions
5. how to find factors of algebraic expressions
6. how to simplify algebraic fractions.
Prior knowledge
In order to master this theme, you should already know:
• 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
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.
Different types of numbers
ℕ = {1; 2; 3; 4; 5; …} = natural numbers
ℕ 0 = {0; 1; 2; 3; 4; 5; …} = whole numbers
ℤ = {… ; − 2; − 1; 0; 1; 2; 3; …} = integers
ℚ = {numbers that can be written as an integer a non-zero integer } = rational numbers
ℚ ′ = {numbers that cannot be written as an integer a non-zero integer } = irrational numbers
(non-terminating and non-repetitive decimal numbers)
ℝ = {rational and irrational numbers} = real numbers
ℝ ′ = {numbers that do not exist in the real number system} = non-real numbers
Summary of the real number system
Real numbers
Rational numbers
Integers Fractions Irrational numbers
• Negative integers
• Zero
• Positive integers (natural numbers)
Note
• A rational number is any number that can be written as �� �� , where �� and �� are integers with �� ≠ 0 .
• The following are rational numbers:
◦ fractions of which both the numerator and denominator are integers, e.g. 3 7
◦ integers, e.g. − 5
◦ decimal numbers that end, e.g. 0,125
◦ 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 root is called a surd, e.g. 3√ 5 .
Worked example 1
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 ��= 0,1212121212…
∴ 100��= 12,1212121212… × 100 to get integer + recurring “tail” − �� = 0,1212121212… Subtract
99��= 12
∴ ��= 12 99 ∴ ��= 4 33 Simplify
Worked example 2
Rewrite 2,51 2 as a common fraction.
Solution Let ��= 2,512121212…
∴ 1 000��= 2512,121212… × 1 000 and × 10 to get integer + recurring “tail” 10y = 25,121212… Remember that the 5 is not recurring 990��=2 487,000000…
��= 829 330 Always remember to simplify completely
Worked example 3
Using your knowledge of the number system, complete the following table by making a in the appropriate block(s):
2 = 0 (zero divided by any non-zero number = zero)
This number is written in the form �� �� therefore it is a rational number (ℚ).
Solution
In order to determine where these numbers fit into the number system, you can use your calculator to find the decimal fraction where applicable:
1 7
This number is written in the form �� �� ; 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). 3√ 9 = 0,2080083823… [non-finite, non-recurring (not-repeating) decimal fraction]
Sample
This number cannot be written in the form �� �� 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).
It exists (your calculator does not give you a math error message); therefore it is a real number (ℝ).
9 16 = 3 4
This number is written in the form �� �� 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).
0,3 = 0,333333333… (non-finite, recurring decimal fraction) = 1 3
This number is written in the form �� �� 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).
√ 50 = 3,684031499… (non-finite, non-recurring decimal fraction)
This number cannot be written in the form �� �� 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).
Note that real numbers (ℝ) are either rational (ℚ) or irrational (ℚ') numbers.
Exercise 1.1: The number system 1. Is the number zero a positive or negative number?
What type of number is
What type of number is
What type of number is
What type of number is 3√ −8 ?
6. Without using a calculator, determine all the number types that 2 10 27 belongs to.
7. Rewrite the following as common fractions:
Sample
8. For which value(s) of �� will f(��) be non-real if:
and��
{−5; 0; 11}?
1.2 BET WEEN WHICH TWO INTEGERS DOES A SURD FALL?
Introduction
If the nth root of a number cannot be simplified to a rational number, we call it a surd. For example, √ and 6√ 3 are surds, but √ 4 is not a surd because it can be simplified to the rational number 2.
Consider surds of the form n√ �� , where �� is any positive number, for example √ 7 or 3√ 5 . It is very common for n to be 2, so we usually do not write 2√ .
Instead, we write the surd as simply √ . This is called the square root of ��.
It is sometimes useful to know the approximate value of a surd without having to use a calculator.
For example, let us estimate where √ 3 lies on the number line:
Using a calculator, we know that √ = 1,73205…
It is easy to see that √ 3 is greater than 1 and smaller than 2.
But to estimate the values of other surds, such as √ 18 , without using a calculator, you should first understand the following:
• If �� and �� are positive integers and �� < ��, then n√ �� < n√ �� .
• A perfect square is the number obtained when an integer is squared.
For example, 9 is a perfect square since 32 = 9.
• A perfect cube is a number which is the cube of an integer.
For example, 27 is a perfect cube, because 33 = 27.
Worked example 4
Determine between which two integers the irrational number √ 62 lies.
Solution
Find the two perfect squares to the left of (just smaller than) and to the right of (just bigger than) 62 on the number line.
• The perfect square to the left of 62 is 49.
• The perfect square to the right of 62 is 64.
Now find the square roots of these perfect squares:
• √ 49 = 7
• √ 64 = 8
Therefore, √ 62 lies between 7 and 8.
Exercise 1.2: Estimating surds
Do the exercise without using a calculator.
1. Find two consecutive integers such that √ 26 lies between them.
2. Find two consecutive integers such that 3√ 49 lies between them.
3. Determine between which two consecutive integers √ 18 lies.
4. Estimate √ 10 correct to one decimal place.
1.3 R OUNDING OF REAL NUMBERS
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.
How to round numbers:
• When rounding to a required number of places, the next decimal digit is considered, e.g. if you are required to round to three decimal places, consider the 4th decimal digit.
• If the next digit is less than 5, the previous decimal digit stays as it is.
• If the next digit is 5 or more, the previous decimal digit is increased by one.
• If you are asked to round to three decimal places, you need to have three digits after the decimal comma (even if these are zeros).
• Revision exercises to refresh prior knowledge.
• Detailed explanations of concepts and techniques.
• Worked examples help learners to better understand new concepts.
• Varied exercises to entrench theory and practise mathematical skills.
• Test papers and memorandums for exam preparation
• Formula sheets and accepted geometrical reasons for quick reference.
• Index of mathematical terms.
• The facilitator’s guide contains step-by-step calculations and answers.
• Use in school or at home.
home classroom college workplace
