Lower Secondary Mathematics LEARNER’S BOOK 9 Sample

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Cambridge Lower Secondary

Mathematics LEARNER’S BOOK 9

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Lynn Byrd, Greg Byrd & Chris Pearce

Second edition

Digital Access

Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.

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Cambridge Lower Secondary

Mathematics LEARNER’S BOOK 9

SA

M

Greg Byrd, Lynn Byrd and Chris Pearce

Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge.

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It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108771436 © Cambridge University Press 2021

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

First published 2014 Second edition 2021

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A catalogue record for this publication is available from the British Library ISBN 978-1-108-77143-6 Paperback + Digital Access (1 year)

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ISBN 978-1-108-xxxxx Digital Edition (1 year) ISBN 978-1-108-xxxxx eBook

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Cambridge University Press 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. Information regarding prices, travel timetables, and other factual information given in this work is correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter.

NOTICE TO TEACHERS IN THE UK It is illegal to reproduce any part of this work in material form (including photocopying and electronic storage) except under the following circumstances: (i) where you are abiding by a licence granted to your school or institution by the Copyright Licensing Agency; (ii) where no such licence exists, or where you wish to exceed the terms of a licence, and you have gained the written permission of Cambridge University Press; (iii) where you are allowed to reproduce without permission under the provisions of Chapter 3 of the Copyright, Designs and Patents Act 1988, which covers, for example, the reproduction of short passages within certain types of educational anthology and reproduction for the purposes of setting examination questions.

Projects and their accompanying teacher guidance have been written by the NRICH Team. NRICH is an innovative collaboration between the Faculties of Mathematics and Education at the University of Cambridge, which focuses on problem solving and on creating opportunities for students to learn mathematics through exploration and discussion https://nrich.maths.org.

Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.

Introduction

Introduction Welcome to Cambridge Lower Secondary Mathematics Stage 9 The Cambridge Lower Secondary Mathematics course covers the Cambridge Lower Secondary Mathematics curriculum framework and is divided into three stages: 7, 8 and 9.

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During your course, you will learn a lot of facts, information and techniques. You will start to think like a mathematician. This book covers all you need to know for Stage 9.

The curriculum is presented in four content areas: • Number • Algebra •

Geometry and measures

•

Statistics and probability.

This book has 15 units, each related to one of the four content areas. However, there are no clear dividing lines between these areas of mathematics; skills learned in one unit are often used in other units. The book encourages you to understand the concepts that you need to learn, and gives opportunity for you to practise the necessary skills.

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Many of the questions and activities are marked with an icon that indicates that they are designed to develop certain thinking and working mathematically skills.

There are eight characteristics that you will develop and apply throughout the course: Specialising – testing ideas against specific criteria;

•

Generalising – recognising wider patterns;

•

Conjecturing – forming questions or ideas about mathematics;

•

Convincing – presenting evidence to justify or challenge a mathematical idea;

•

Characterising – identifying and describing properties of mathematical objects;

•

Classifying – organising mathematical objects into groups;

•

Critiquing – comparing and evaluating ideas for solutions;

•

Improving – Refining your mathematical ideas to reach more effective approaches or solutions.

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•

Your teacher can help you develop these skills, and you will also develop your ability to apply these different strategies.

We hope you will find your learning interesting and enjoyable.

Greg Byrd, Lynn Byrd and Chris Pearce

3 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.

Contents

Unit

6

How to use this book

9–20

1 Number and calculation 1.1 Irrational numbers 1.2 Standard form 1.3 Indices

Number

21–54

2 Expressions and formulae 2.1 Substituting into expressions 2.2 Constructing expressions 2.3 Expressions and indices 2.4 Expanding the product of two linear expressions 2.5 Simplifying algebraic fractions 2.6 Deriving and using formulae

Algebra

55–81

3 Decimals, percentages and rounding 3.1 Multiplying and dividing by powers of 10 3.2 Multiplying and dividing decimals 3.3 Understanding compound percentages 3.4 Understanding upper and lower bounds

Number

82

Project 1 Cutting tablecloths

83–102

4 Equations and inequalities 4.1 Constructing and solving equations 4.2 Simultaneous equations 4.3 Inequalities

Algebra

103–126

5 Angles 5.1 Calculating angles 5.2 Interior angles of polygons 5.3 Exterior angles of polygons 5.4 Constructions 5.5 Pythagoras’ theorem

Geometry and measure

128–137

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127

Strand

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Page

Project 2 Angle tangle

6 Statistical investigations 6.1 Data collection and sampling 6.2 Bias

Statistics and probability

7 Shapes and measurements 7.1 Circumference and area of a circle 7.2 Areas of compound shapes 7.3 Large and small units

Geometry and measure

161–189

8 Fractions 8.1 Fractions and recurring decimals 8.2 Fractions and the correct order of operations 8.3 Multiplying fractions 8.4 Dividing fractions 8.5 Making calculations easier

Number

190

Project 3 Selling apples

191–211

9 Sequences and functions 9.1 Generating sequences 9.2 Using the nth term 9.3 Representing functions

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138–160

Algebra

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Contents

Unit

Strand

212–233

10 Graphs 10.1 Functions 10.2 Plotting graphs 10.3 Gradient and intercept 10.4 Interpreting graphs

Algebra

234

Project 4 Cinema membership

235–249

11 Ratio and proportion 11.1 Using ratios 11.2 Direct and inverse proportion

Number

250–269

12 Probability 12.1 Mutually exclusive events 12.2 Independent events 12.3 Combined events 12.4 Chance experiments

Statistics and probability

270–299

13 Position and transformation 13.1 Bearings and scale drawings 13.2 Points on a line segment 13.3 Transformations 13.4 Enlarging shapes

Geometry and measure

300

Project 5 Triangle transformations

301–316

14 Volume, surface area and symmetry 14.1 Calculating the volume of prisms 14.2 Calculating the surface area of triangular prisms, pyramids and cylinders 14.3 Symmetry in three-dimensional shapes

Geometry and measure

317–347

15 Interpreting and discussing results 15.1 Interpreting and drawing frequency polygons 15.2 Scatter graphs 15.3 Back-to-back stem-and-leaf diagrams 15.4 Calculating statistics for grouped data 15.5 Representing data

Statistics and probability

Project 6 Cycle training Glossary

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349–351

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348

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5 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.

How to use this book

How to use this book In this book you will find lots of different features to help your learning.

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Questions to find out what you know already.

What you will learn in the unit.

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Important words to learn.

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Step-by-step examples showing how to solve a problem.

These questions help you to develop your skills of thinking and working mathematically.

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How to use this book

Questions to help you think about how you learn.

This is what you have learned in the unit.

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An investigation to carry out with a partner or in groups.

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Questions that cover what you have learned in the unit. If you can answer these, you are ready to move on to the next unit.

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At the end of several units, there is a project for you to carry out, using what you have learned. You might make something or solve a problem.

7 Original material Š Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.

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Acknowledgements

Original material Š Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.

Getting started

2

3

4

3

64

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Write as a number: a 122 b 81 c 53 d 8 2 = 256 Use this fact to work out the value of a 29 b 27 Here is a multiplication: 155 × 152 a Write the correct answer from this list: 157 1510 307 3010 b Write the answer to 155 ÷ 152 in index form.

Look at these numbers: 4 −4.5 3000 17 3 225 20 a Which of these numbers are integers? b Which of these numbers are rational numbers?

Write one million as a power of 10.

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1 Number and calculation

9 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.

1 Number and calculation

1, 4, 9 and 16 are the first four square numbers. They have integer square roots.

12 = 1 and 1 = 1 22 = 4 and 4 = 2

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32 = 9 and 9 = 3 42 = 16 and 16 = 4

What about 2 ? Is there a rational number n for which n2 = 2?

Remember that you can write a rational number as a fraction.

( ) 11 2

2

= 1 1 × 1 1 = 2 1 so 2 must be a little less than 1 1 .

2

2

A closer answer

4

is 1 5 12

because

( ) 15 12

2

2

=

2 1 .

144

( 408 )

An even closer answer is 1 169 because 1 169 408

2

=2

1 .

166464

Do you think you can find a fraction which gives an answer of exactly 2 when you square it?

A calculator gives the answer 2 = 1.414213562. This is a rational

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number because you can write it as a fraction: 1 414213562 .

1000000000

Is 1.414213562 × 1.414213562 exactly 2?

In this unit, you will look at numbers such as 2 .

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1.1 Irrational numbers

In this section you will …

Key words

•

learn about the difference between rational numbers and irrational numbers

irrational number

•

use your knowledge of square numbers to estimate square roots

surd

•

use your knowledge of cube numbers to estimate cube roots.

rational number

10 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.

1.1 Irrational numbers

Integers are whole numbers. For example, 13, −26 and 100 004 are integers.

You can write rational numbers as fractions. For example, 9 3 , −3 4 and 4 15 18 5 are rational numbers.

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You can write any fraction as a decimal.

Tip The set of rational numbers includes integers.

9 3 = 9.75 −3 4 = −3.26666666... 18 5 = 18.4545454...

4

15

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Tip Square roots of negative numbers do not belong to the set of rational or irrational numbers. You will learn more about these numbers if you continue to study mathematics to a higher level.

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The fraction either terminates (for example, 9.75) or it has recurring digits (for example, 3.266666666666… and 18.45454545454…).

There are many square roots and cube roots that you cannot write as fractions. When you write these fractions as decimals, they do not terminate and there is no recurring pattern. For example, a calculator gives the answer 7 = 2.645751... The calculator answer is not exact. The decimal does not terminate and there is no recurring pattern. Therefore, 7 is not a rational number.

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Numbers that are not rational are called irrational numbers. 7 , 23, 3 10 and 3 45 are irrational numbers. Irrational numbers that are square roots or cube roots are called surds.

There are also numbers that are irrational but are not square roots or cube roots. One of these irrational numbers is called pi, which is the Greek letter π. Your calculator will tell you that π = 3.14159… You will meet π later in the course.

Worked example 1.1

Do not use a calculator for this question. a Show that 90 is between 9 and 10.

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b N is an integer and 3 90 is between N and N + 1. Find the value of N. Answer

a 92 = 81 and 102 = 100 81 < 90 < 100

So 81 < 90 < 100

And so 9 < 90 < 10

This means 90 is between 81 and 100.

b 43 = 64 and 53 = 125 64 < 90 < 125 and so 3

64 < 3 90 < 3 125

So 4 < 3 90 < 5 and N = 4

11 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.

1 Number and calculation

Exercise 1.1 1

5

3 10 000 d 3 1000 e f 10 000 Is each of these numbers rational or irrational? Give a reason for each answer. a 2 + 2 b 2+2

c 4 + 3 4 d 3 4 + 4 Find a two irrational numbers that add up to 0 b two irrational numbers that add up to 2.

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4

99

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Write whether each of these numbers is an integer or an irrational number. Explain how you know. a 9 b 19 c 39 d 49 e 2 a Write the rational numbers in this list.

1 7 5 −38 160 − 2.25 − 35 12 b Write the irrational numbers in this list.

0.3333… −16 200 1.21 23 3 343 8 3 Write whether each of these numbers is an integer or a surd.

Explain how you know.

100 1000 a b 3 100 c

Think like a mathematician 6

a

Use a calculator to find

3 × 12 20 × 5 2 × 18 i ii iii iv 8× 2 What do you notice about your answers?

Find another multiplication similar to the multiplications in part a.

Find similar multiplications using cube roots instead of square roots.

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b c d

7 8 9

Without using a calculator, show that a 7 < 55 < 8 b 4 < 3 100 < 5 Without using a calculator, find an irrational number between a 4 and 5 b 12 and 13.

Without using a calculator, estimate 190 to the nearest integer a b

3

190 to the nearest integer.

12 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.

1.1 Irrational numbers

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10 a Use a calculator to find i ( 2 + 1) × ( 2 −1) ii ( 3 + 1) × ( 3 − 1) iii ( 4 + 1) × ( 4 − 1) b Continue the pattern of the multiplications in part a. c Generalise the results to find ( N + 1) × ( N − 1) where N is a positive integer. d Check your generalisation with further examples.

11 Here is a decimal: 5.020 020 002 000 020 000 020 000 002…

Arun says:

There is a regular pattern: one zero, then two zeros, then three zeros, and so on.

This is a rational number.

a b

Is Arun correct? Give a reason for your answer.

Compare your answer with a partner’s. Do you agree? If not, who is correct?

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In this exercise, you have looked at the properties of rational and irrational numbers. a Are the following statements true or false?

i The sum of two integers is always an integer.

ii The sum of two rational numbers is always a rational number.

iii The sum of two irrational numbers is always an irrational number. b Here is a calculator answer: 3.646 153 846

The answer is rounded to 9 decimal places.

Can you decide whether the number is rational or irrational?

Summary checklist

I can use square numbers and cube numbers to estimate square roots and cube roots.

I can say whether a square root or the cube root of a positive integer is rational or irrational.

13 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.

1 Number and calculation

1.2 Standard form In this section you will …

Key words

•

scientific notation

learn to write large and small numbers in standard form.

standard form

Look at these numbers 4.67 ×10 = 46.7 4.67 ×103 = 4670 4.67 ×106 = 4 670 000

Tip

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4.67 ×10 2 = 467

You can use powers of 10 in this way to write large numbers. For example, the average distance to the Sun is 149 600 000 km. You can write this as 1.496 × 108 km. This is called standard form. You write a number in standard form as a × 10n where 1 ⩽ a < 10 and n is an integer.

You can write small numbers in a similar way, using negative integer powers of 10. For example: 4.67 ×10 −1 = 0.467 4.67 ×10 −2 = 0.0467 −3

Tip

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4.67 ×10 = 0.004 67

4.67 × 102 is the same as 4.67 × 100 or 4.67 × 10 × 10

Think of 4.67 × 10−1 as 4.67 ÷ 10

4.67 ×10 −7 = 0.000 004 67

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Small numbers occur often in science. For example, the time for light to travel 5 metres is 0.000 000 017 seconds. In standard form, you can write this as 1.7 ×10 −8 seconds.

Worked example 1.2

Write these numbers in standard form.

Tip Standard form is also sometimes called scientific notation.

a 256 million b 25.6 billion c 0.000 025 6 Answer

a 1 million = 1 000 000 or 10 So 256 million = 256 000 000 = 2.56 × 108 6

b 1 billion = 1 000 000 000 or 109 So 25.6 billion = 25 600 000 000 = 2.56 × 1010 c 0.000 025 6 = 2.56 × 10−5

Tip Notice that in every case the decimal point is placed after the 2, the first non-zero digit.

14 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.

1.2 Standard form

Exercise 1.2

3

4

5

Planet Mercury Distance (km) 57 900 000

Mars 227 900 000

Uranus 2 870 000 000

Here are the areas of four countries.

Country Area (km2) a b c

China Indonesia Russia Kazakhstan 6 6 7 9.6 × 10 1.9 × 10 1.7 × 10 2.7 × 106

Which country has the largest area?

Which country has the smallest area?

Copy and complete this sentence with a whole number: The largest country is approximately … times larger than the smallest country.

Write these numbers in standard form. a 0.000 007 b 0.000 812 c 0.000 066 91 d 0.000 000 205 These numbers are in standard form. Write each number in full. a 1.5 × 10−3 b 1.234 × 10−5 c 7.9 × 10−8 d 9.003 × 10−4 The mass of an electron is 9.11 × 10−31 kg.

This is 0.000…911 kg. a How many zeros are there between the decimal point and the 9? b Work out the mass of 1 million electrons.

Give the answer in kilograms in standard form.

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Write these numbers in standard form. a 300 000 b 320 000 c 328 000 d 328 710 Write these numbers in standard form. a 63 000 000 b 488 000 000 c 3 040 000 d 520 000 000 000 These numbers are in standard form. Write each number in full. a 5.4 × 103 b 1.41 × 106 c 2.337 × 1010 d 8.725 × 107 Here are the distances of some planets from the Sun. Write each distance in standard form.

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1

7

8

15 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.

1 Number and calculation

9

Here are four numbers:

w = 9.81 × 10−5 x = 2.8 × 10−4 y = 9.091 × 10−5 z = 4 × 10−4 a Which number is the largest? b Which number is the smallest?

10 a Explain why the number 65 × 104 is not in standard form. b Write 65 × 104 in standard form. c Write 48.3 × 106 in standard form.

11 Write these numbers in standard form. a 15 × 10−3 b 27.3 × 10−4 c 50 × 10−9 12 Do these additions. Write the answers in standard form. a 2.5 × 106 + 3.6 × 106 b 4.6 × 105 + 1.57 × 105 c 9.2 × 104 + 8.3 × 104 13 Do these additions. Write the answers in standard form. a 4.5 × 10−6 + 3.1 × 10−6 b 5.12 × 10−5 + 2.9 × 10−5 c 9 × 10−8 + 7 × 10−8 14 a Multiply these numbers by 10. Give each answer in standard form. i 7 × 105 ii 3.4 × 106 iii 4.1 × 10−5 iv 1.37 × 10−4 b Generalise your results from part a. c Describe how to multiply or divide a number in standard form by 1000.

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What are the advantages of writing numbers in standard form?

Summary checklist

I can write large and small numbers in standard form.

16 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.

1.3 Indices

1.3 Indices In this section you will … use positive, negative and zero indices

•

use index laws for multiplication and division.

This table shows powers of 3.

32 9

33 27

34 81

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•

35 243

36 729

Tip

When you move one column to the right, the index increases by 1 and the number multiplies by 3.

9 × 3 = 27 27 × 3 = 81 81 × 3 = 243, and so on.

When you move one column to the left, the index decreases by 1 and the number divides by 3. You can use this fact to extend the table to the left: 3−3

3−2

3−1

30

31

32

33

34

35

36

1 81

1 27

1 9

1 3

1

3

9

27

81

243

729

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3−4

9 ÷ 3 = 3 3 ÷ 3 = 1 1 ÷ 3 = 1 1 ÷ 3 = 1 1 ÷ 3 = 1 , and so on.

3

3

9

9

27

You can see from the table that 3 = 3 and 3 = 1.

1

0

3−2 = 12 3−3 = 13 , and so on.

Also: 3−1 = 1 3

3

3

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In general, if n is a positive integer then 3− n = 1n . These results are not

The index is the small red number.

Tip 30 = 1 seems strange but it fits the pattern.

3

only true for powers of 3. They apply to any positive integer.

8−3 = 13 = 1 60 = 1 For example: 5−2 = 12 = 1 5

25

8

512

In general, if a and n are positive integers then a0 = 1 and a − n = 1n . a

Exercise 1.3 1

2

Write each number as a fraction. a 4−1 b 2−3 c −3 −4 d 6 e 10 f Here are five numbers: 2−4 3−3 4−2 5−1 60 List the numbers in order of size, smallest first.

9−2 2−5

17 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.

1 Number and calculation

3

4

Write these numbers as powers of 2. a 1 b 1

c

64

d

f

8−1

c f

1 0.000 001

2 1 64

4

e

1

Write each number as a power of 10. a 100 b 1000 d 0.1 e 0.001

Write 1 64 a as a power of 64 b as a power of 8 c as a power of 4 d as a power of 2.

1 6 a Write as a power of a positive integer.

81 b How many different ways can you write the answer to part a?

7 When x = 6, find the value of a x2 b x−2 c x0 d x−3 8 Write m−2 as a fraction when a m = 9 b m = 15 c m = 1 d m = 20 2 −2 9 y = x + x and x is a positive number. a Write y as a mixed number when i x = 1 ii x = 2 iii x = 3 b Find the value of x when i y = 25.04 ii y = 100.01 10 a Write the answer to each multiplication as a power of 3. i 32 × 33 ii 34 × 35 iii 36 × 34 iv 3 × 35 b In part a you used the rule 3a × 3b = 3a + b when the indices are positive integers.

In the following multiplications, a or b can be negative integers.

Show that the rule still gives the correct answers. i 32 × 3−1 ii 3−2 × 3 iii 33 × 3−1 iv 3−1 × 3−1 v 3−2 × 3−1 c Write two examples of your own to show that the rule works. d Give your work to a partner to check.

11 Write the answer to each multiplication as a power of 5. a 54 × 52 b 54 × 5−2 c 5−4 × 52 d 5−4 × 5−2

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Tip Write out the numbers and multiply.

18 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.

1.3 Indices

12 Write the answer to each multiplication as a single power. a 6 −3 × 62 b 75 × 7 −2 c 11−4 ×11−6 d 4 −6 × 42 13 Find the value of x in each case. a 25 × 2 x = 29 b 3x × 3−2 = 34 c 4 x × 4 −3 = 4 −5 d 12 −3 ×12 x = 12 2

14 a

b

c

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Think like a mathematician Write as a single power

i 25 ÷ 23

ii 45 ÷ 42

iii 56 ÷ 55

iv 210 ÷ 27 The rule for part a is that na ÷ nb = na −b when the indices a and b are positive integers.

Write some examples to show that this rule also works for indices that are negative integers.

Give your examples to a partner to check.

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15 Write the answer to each division as a single power. a 62 ÷ 65 b 93 ÷ 94 c 152 ÷156 d 103 ÷108 16 Write the answer to each division as a single power. a 22 ÷ 2 −3 b 85 ÷ 8−2 c 5−4 ÷ 52 d 12 −3 ÷12 −5 17 Write down a 83 as a power of 2 b 8−3 as a power of 2 c 272 as a power of 3 d 27−2 as a power of 3 e 272 as a power of 9 f 27−2 as a power of 9.

Summary checklist

I can understand positive, negative and zero indices.

I can use the addition rule for indices to multiply powers of the same number.

I can use the subtraction rule for indices to divide powers of the same number.

19 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.

1 Number and calculation

Check your progress

2 3 4 5

6 7

c

6.25

62.5 625 d e Write whether each number is rational or irrational. Give a reason for each answer.

32 + 42 9+ 7 a b Without using a calculator, find an integer n such that n < 3 50 < n + 1.

Write each number in standard form. a 86 000 000 000 b 0.000 006 45 Write these numbers in order of size, smallest first.

A = 9 × 10−4 B = 6 × 10−3 C = 8 × 10−5 D = 7.5 × 10−4 Write each number as a fraction. a 7−2 b 3−4 c 2−7 Write each number as a power of 5. a 125 b 1 c 0.04 Write the answer to each calculation as the power of a single number. a 68 × 6 −3 b 12 −2 ×12 −3 c 42 ÷ 48 d 15−4 ÷15−6

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Write whether each number is rational or irrational.

5 a 4 b

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1

20 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.