PL E
Cambridge Lower Secondary
Mathematics LEARNER’S BOOK 8
SA M
Lynn Byrd, Greg Byrd & Chris Pearce
Second edition
Digital access
Original material Š Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
PL E
Cambridge Lower Secondary
Mathematics LEARNER’S BOOK 8
SA
M
Greg Byrd, Lynn Byrd & Chris Pearce
Original material © Cambridge University Press 2020. 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.
PL E
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/9781108746380 © 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
20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Printed in ‘country’ by ‘printer’
A catalogue record for this publication is available from the British Library ISBN 978-1-108-74638-0 Paperback
M
Additional resources for this publication at www.cambridge.org/9781108746380
SA
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 2020. This material is not final and is subject to further changes prior to publication.
Introduction
Introduction This book covers all you need to know for Stage 8. The Cambridge Lower Secondary Mathematics course covers the Cambridge Lower Secondary Mathematics curriculum framework and is divided into three stages: 7, 8 and 9.
PL E
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 8. The curriculum is presented in four content areas: • Number • Algebra •
Geometry and measures
•
Statistics and probability
This book has 16 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.
SA
M
The book encourages you to understand the concepts that you need to learn, and gives opportunity for you to practise the necessary skills. 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. You may be asked to find a generalisation about a concept, or conjecture about why you think a certain mathematical rule works in a specific way. Your teacher can help you develop these skills, and you will also develop your ability to apply these different strategies. Look out for these learners, who will be asking questions, making suggestions and taking part in the activities throughout the units and good luck with your learning.
Greg Byrd, Lynn Byrd and Chris Pearce
3 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
Contents
Unit How to use this book
9–28
1 Integers 1.1 Factors, multiples and primes 1.2 Multiplying and dividing integers 1.3 Square roots and cube roots 1.4 Indices
Number
29–64
2 Expressions, formulae and equations 2.1 Constructing expressions 2.2 Using expressions and formulae 2.3 Expanding brackets 2.4 Factorising 2.5 Constructing and solving equations 2.6 Inequalities
Algebra
65
Project 1 Algebra chains
66–79
3 Place value and rounding 3.1 Multiplying and dividing by 0.1 and 0.01 3.2 Rounding
Number
80–103
4 Decimals 4.1 Ordering decimals 4.2 Multiplying decimals 4.3 Dividing by decimals 4.4 Making decimal calculations easier
Number
104
Project 2 Diamond decimals
105–125
5 Angles and constructions 5.1 Parallel lines 5.2 The exterior angle of a triangle 5.3 Constructions
Geometry and measure
6 Collecting data 6.1 Data collection 6.2 Sampling
Statistics
7 Fractions 7.1 Fractions and recurring decimals 7.2 Ordering fractions 7.3 Subtracting mixed numbers 7.4 Multiplying an integer by a mixed number 7.5 Dividing an integer by a fraction 7.6 Making fraction calculations easier
Number
171–196
8 Shapes and symmetry 8.1 Quadrilaterals and polygons 8.2 The circumference of a circle 8.3 3D shapes
Geometry and measure
197
Project 3 Quadrilateral tiling
198–223
9 Sequences and functions 9.1 Generating sequences 9.2 Finding rules for sequences 9.3 Using the nth term 9.4 Representing simple functions
SA
137–170
M
126–136
Strand of mathematics
PL E
Page 6
Algebra
4 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
Contents
Unit
Strand of mathematics
10 Percentages 10.1 Percentage increases and decreases 10.2 Using a multiplier
Number
235–255
11 Graphs 11.1 Functions 11.2 Plotting graphs 11.3 Gradient and intercept 11.4 Interpreting graphs
Algebra; Statistics and probability
256
Project 4 Straight line mix-up
257–274
12 Ratio and proportion 12.1 Simplifying ratios 12.2 Sharing in a ratio 12.3 Ratio and direct proportion
Number
275–288
13 Probability 13.1 Calculating probabilities 13.2 Experimental and theoretical probabilities
Statistics and probability
289
Project 5 High fives
290–330
14 Position and transformation 14.1 Bearings 14.2 The midpoint of a line segment 14.3 Translating 2D shapes 14.4 Reflecting shapes 14.5 Rotating shapes 14.6 Enlarging shapes
Statistics and probability
331–351
15 Distance, area and volume 15.1 Converting between miles and kilometres 15.2 The area of a parallelogram and a trapezium 15.3 Calculating the volume of triangular prisms 15.4 Calculating the surface area of triangular prisms and pyramids
Geometry and measure
352
Project 6 Biggest cuboid
353–387
16 Interpreting and discussing results 16.1 Interpreting and drawing frequency diagrams 16.2 Time series graphs 16.3 Stem-and-leaf diagrams 16.4 Pie charts 16.5 Representing data 16.6 Using statistics
Statistics and probability
SA
M
PL E
Page 224–234
388–394
Glossary and Index
5 Original material © Cambridge University Press 2020. 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 ďŹ nd lots of different features to help your learning.
PL E
Questions to find out what you know already.
What you will learn in the unit.
M
Important words to learn.
SA
Step-by-step examples showing how to solve a problem.
These questions help you to develop special Thinking and Working Mathematically skills.
6 Original material Š Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
How to use this book
Questions to help you think about how you learn.
This is what you have learned in the unit.
PL E
An investigation to carry out with a partner or in groups.
M
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.
SA
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 2020. This material is not final and is subject to further changes prior to publication.
PL E SA
M
ACKNOWLEDGEMENTS
Original material Š Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1
Integers
Getting started
2
3
4
M
5
a Find all the prime numbers less than 20. b Show that there are two prime numbers between 20 and 30. a Find all the factors of 18. b Find all the 2-digit multiples of 18. c Find the highest common factor of 18 and 12. d Find the lowest common multiple of 18 and 12. Work out −6 + 3 b −6 − 3 c −6 × 3 a d −6 ÷ 3 e 8 + −10 f −5 − − 9 Write whether each of these numbers is a square number, a cube number or both. a 49 b 27 c 1000 d 64 e 121 f 225 Find 3 125 100 b c 152 − 122 a
PL E
1
SA
Prime numbers have exactly two factors, 1 and the number itself. Some examples of prime numbers are 7, 31, 83, 239 and 953. The number 39 is the product of two prime numbers (3 and 13). It is quite easy to find these two numbers. The number 2573 is also the product of two prime numbers (31 and 83). It is much harder to find the two numbers in this case. It is easy to multiply two prime numbers together using a calculator or a computer.
9 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1 Integers
SA
M
PL E
It is much harder to carry out the inverse operation – that is, to find the two prime numbers that multiply to a given product. This fact is the basis of a system used to encode messages sent across the internet. The RSA cryptosystem was invented by Ronald Rivest, Adi Shamir and Leonard Adleman in 1977. It uses two large prime numbers with about 150 digits each. These numbers are kept secret, but anybody can use their product, N, which has about 300 digits. If someone sends their credit card number to a website, their computer does a calculation using N to encode their credit card number. The computer that receives the coded number does another calculation to decode it. Anyone who does not know the two factors of N will not be able to do this. Your credit card number is protected.
10 Original material Š Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1.1 Factors, multiples and primes
1.1 Factors, multiples and primes In this section you will …
Key words
•
write a positive integer as a product of prime factors
factor tree
•
use prime factors to find a highest common factor (HCF) and a lowest common multiple (LCM).
highest common factor (HCF)
PL E
index
integer
lowest common multiple (LCM) prime factor
120
12
3
10
4
2
2
5
2
SA
M
Any integer bigger than 1: • is a prime number, or • can be written as a product of prime numbers. Example: 46 = 2 × 23 47 is prime 48 = 2 × 2 × 2 × 2 × 3 49 = 7 × 7 50 = 2 × 5 × 5 You can use a factor tree to write an integer as a product of its prime factors. This is how to draw a factor tree for 120. 1 Write 120. 2 Draw branches to two numbers that have a product of 120. Do not use 1 as one of the numbers. Here we have chosen 12 and 10. 120 = 12 × 10 3 Do the same with 12 and 10. Here 12 = 3 × 4 and 10 = 2 × 5 4 3, 2 and 5 are prime numbers, so circle them. 5 Draw two more branches from 4. 4 = 2 × 2. Circle the 2s. 6 Now all the end numbers are prime, so stop. 7 120 is the product of all the end numbers: 120 = 2 × 2 × 2 × 3 × 5 8 You can check that this is correct using a calculator. You can also write the result like this: 120 = 23 × 3 × 5 23 means 2 × 2 × 2 and the small 3 is an index. Now check that 75 = 3 × 52 You can use products of prime factors to find the HCF and LCM of two numbers.
11 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1 Integers
Worked example 1.1 a Find the LCM of 120 and 75. b Find the HCF of 120 and 75. Answer Write 120 and 75 as products of their prime factors: 120 = 2 × 2 × 2 × 3 × 5 75 = 3 × 5 × 5
PL E
a
Look at the prime factors of both numbers. For the LCM, use the larger frequency of each prime factor. • 120 has three 2s and 75 has no 2s. The LCM must have three 2s. • 120 has one 3 and 75 has one 3. The LCM must have one 3. • 120 has one 5 and 75 has two 5s. The LCM must have two 5s. The LCM is 2 × 2 × 2 × 3 × 5 × 5 = 23 × 3 × 52 = 8 × 3 × 25 = 600 b For the HCF use the smaller frequency of each factor: there are no 2s in 75, and there is one 3 and one 5 in both numbers. Multiply these factors. The HCF is 3 × 5 = 15
M
Exercise 1.1
Think like a mathematician 1
The factor tree for 120 in Section 1.1 started with 12 × 10.
120
Draw a factor tree for 120 that starts with 6 × 20. Compare your answer to part a with a partner’s. Are your trees the same or different? Draw some different factor trees for 120. Can you say 6 how many different trees are possible? Do all factor trees for 120 have the same end points?
SA
a b
c
d
2 a b c d
Complete this factor tree for 108. Draw a different factor tree for 108. Write 108 as a product of its prime factors. Compare your factor trees and your product of prime factors with a partner’s. Have you drawn the same trees or different ones? Are your trees correct?
20
108
2
54
12 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1.1 Factors, multiples and primes
3 a b c
PL E
Draw a factor tree for 200 that starts with 10 × 20. Write 200 as a product of prime numbers. Compare your factor tree with a partner’s. Have you drawn the same tree or different ones? Are your trees correct? d How many different factor trees can you draw for 200 that start with 10 × 20? 4 a Draw a factor tree for 330. b Write 330 as a product of prime numbers. 5 Match each number to a product of prime factors. The first one has been done for you: a and i. a 20 i 2² × 5 b 24 ii 2 × 3 × 7 c 42 iii 2² × 3² × 5 d 50 iv 2 × 5² e 180 v 2³ × 3 6 Work out the product of each set of prime factors. a 32 × 5 × 7 b 23 × 53 c 2 2 × 32 ×11 d 2 4 × 72 e 3 ×172 7 Write each of these numbers as a product of prime factors. a 28 b 60 c 72 d 153 e 190 f 275 8 a Copy the table and write each number as a product of prime numbers.
Tip
M
You can use a factor tree to help you.
Product of prime numbers 5 × 7
SA
Number 35 70 140 280
b 9 a b c 10 a b c
Add more rows to the table to continue the pattern. Write 1001 as a product of prime numbers. Write 4004 as a product of prime numbers. Write 6006 as a product of prime numbers. Use a factor tree to write 132 as a product of prime numbers. Write 150 as a product of prime numbers. 132 × 150 = 19 800. Use this fact to write 19 800 as a product of prime numbers.
13 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1 Integers
SA
M
PL E
11 a Write each of these numbers as a product of prime numbers. i 15 ii 15² iii 28 iv 28² v 36 vi 36² b What do you notice about your answers to i and ii, iii and iv, v and vi? c If 96 = 25 × 3, show how to find the prime factors of 96 2 . Will your method work for all numbers? 12 40 = 2 × 2 × 2 × 5 and 28 = 2 × 2 × 7 Use these facts to find a the HCF of 40 and 28 b the LCM of 40 and 28. 13 450 = 2 × 3 × 3 × 5 × 5 and 60 = 2 × 2 × 3 × 5 Use these facts to find a the HCF of 450 and 60 b the LCM of 450 and 60. 14 180 = 2² × 3² × 5 and 54 = 2 × 3³ Use these facts to find a the HCF of 180 and 54 b the LCM of 180 and 54. 15 a Write 45 as a product of prime numbers. b Write 75 as a product of prime numbers. c Find the LCM of 45 and 75. d Find the HCF of 45 and 75. 16 a Draw factor trees to find the LCM of 90 and 140. b Compare your answer with a partner’s. Did you draw the same factor trees? Have you both got the same answer? 17 a Write 396 as a product of prime numbers. b Write 168 as a product of prime numbers. c Find the HCF of 396 and 168. d Find the LCM of 396 and 168. 18 a Find the HCF of 34 and 58. b Find the LCM of 34 and 58. 19 Show that the HCF of 63 and 110 is 1. 20 37 and 47 are prime numbers. a What is the HCF of 37 and 47? b What is the LCM of 37 and 47? c Write a rule for finding the HCF and LCM of two prime numbers. d Compare your answer to part c with a partner’s answer. Check your rules by finding the HCF and LCM of 39 and 83.
Tip Use a calculator to help you.
14 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1.2 Multiplying and dividing integers
In this exercise you have: used factor trees to write an integer as a product of prime factors
•
found the HCF of two integers by first writing each one as a product of prime numbers
•
found the LCM of two integers by first writing each one as a product of prime numbers.
a
Which questions have you found the easiest? Explain why.
b
Which questions have you found the hardest? Explain why.
Summary checklist
PL E
•
I can write an integer as a product of prime numbers. I can find the HCF and LCM of two integers by first writing each one as a product of prime numbers.
M
1.2 Multiplying and dividing integers
Key words
•
multiply and divide integers, in particular when both are negative
brackets
understand that brackets, indices and operations follow a particular order.
inverse
SA
In this section you will …
•
conjecture investigate
You can add and subtract any two integers. For example: 2 + −4 = −2 −2 + −4 = −6 −2 − 4 = −6 −2 − −4 = 2 You can also multiply and divide a negative integer by a positive one. For example: 2 × −9 = −18 −6 × 3 = −18 −18 ÷ 3 = −6 In this section you will investigate how to multiply or divide any two integers. You will use number patterns to do this.
15 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1 Integers
Worked example 1.2 Look at this sequence of subtractions.
Answer
3−2 =1 3−0 = 3 3 − −2 = 5 3 − −4 = 7 3 − −6 = 9 3 − −8 = 11 3 − −10 = 13 The first number, 3, does not change. The number being subtracted decreases by 2 each time. The answer increases by 2 each time.
M
a b c
A sequence is a set of numbers or expressions made and written in order, according to some pattern.
PL E
3 − 6 = −3 3 − 4 = −1 3−2 = 3−0 = 3 − −2 = 3 − −4 = a Copy the sequence and fill in the missing answers. b Write the next three lines in the sequence. c Describe any patterns in the sequence.
SA
Exercise 1.2
Think like a mathematician 1
Here is the start of a sequence of multiplications. −3 × 4 = −12 −3 × 3 = −3 × 2 = a b
Copy the sequence and write six more terms. Use a pattern to fill in the answers. Describe the patterns in the sequence.
16 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1.2 Multiplying and dividing integers
Continued
d e f
3 4
Work out these multiplications. a 5 × − 2 b −5 × 2 Work out these multiplications. a −6 × − 4 b −7 × − 7
× 4 −3 −6 5
c
−5 × − 2
d
−2 × − 5
c
−10 × −6
d
−8 × −11
Copy and complete this multiplication table. −5
3
−8
M
2
Here is the start of another sequence of multiplications. −5 × 4 = −5 × 3 = −5 × 2 = Copy the sequence and write six more terms. Describe any patterns in the sequence. In the sequences in a and c, you have some products of two negative integers. What can you say about the product of two negative integers? Make up a sequence of your own like the ones in a and c. Share your answers to parts d and e with a partner. Are your partner’s sequences correct?
PL E
c
−9
30
SA
Work out b (−3 + −5) × −6 a (3 + 5) × −4 c −4 × (5 − 8) d −6 × (−2 − −7) 6 Round these numbers to the nearest whole number to estimate the answer. b −11.2 × 2.95 a 3.9 × −6.8 c (−6.1)2 d (−4.88)2 7 a Put these multiplications into groups based on the answers. 3 × −4 −6 × −2 12 × 1 −4 × −3 2 × −6 −12 × −1 b Find one more product to put in each group.
Tip Do the calculation in brackets first.
17 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1 Integers
8
These are multiplication pyramids. a b
c
–8 2
–4
–3
–3
5
–1
–4
–5
–2
PL E
Each number is the product of the two numbers below it. For example, in a, 2 × −4 = −8 Copy and complete the multiplication pyramids. 9 a Draw a multiplication pyramid like those in Question 8, with the integers −2, 3 and −5 in the bottom row, in that order. Complete your pyramid.
If you change the order of the bottom numbers, the number at the top of the pyramid is the same.
b
M
Is Zara correct? Test her idea by changing the order of the numbers in the bottom row of your pyramid. 10 Find the missing numbers in these multiplications. a −3 × = −12 b −5 × = 45 c × −6 = 24 d × −10 = 80
Tip
11 A multiplication can be written as a division. For example, 5 × 8 = 40 can be written as 40 ÷ 8 = 5 or 40 ÷ 5 = 8
A conjecture is a possible value based on what you know.
SA
Think like a mathematician
a b c d
Here is a multiplication: −4 × 6 = −24 Write it as a division in two different ways. Write a multiplication of a positive integer and a negative integer. Then write it as a division in two different ways. Here is a multiplication: −7 × −2 = 14 Write it as a division in two different ways. Write a multiplication of two negative integers. Then write it as a division in two different ways.
18 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1.2 Multiplying and dividing integers
Continued e
Can you make a conjecture about the answer when you divide an integer by a negative integer? Test your conjecture. Compare your answer with a partner’s answers. Have you made the same conjectures?
f
6 5
–1
PL E
12 Work out these divisions. a 18 ÷ −6 b −28 ÷ −4 d −30 ÷ −10 e 42 ÷ −6 g 60 ÷ −5 h −25 ÷ −5 13 Here are three multiplication pyramids. a b 12
c f
c
–8
30 ÷ −6 −24 ÷ −4
Tip
–200
–20
–2
SA
M
Copy and complete each pyramid. 14 Work out a (3 × −4) ÷ −2 b (2 − 20) ÷ −3 c (−3 + 15) ÷ −4 d 24 ÷ (2 × −4) 15 Find the value of x. a x ÷ −4 = 8 b x ÷ −3 = −15 c 16 ÷ x = −2 d −15 ÷ x = 3 16 Round these numbers to the nearest whole number to estimate the answer. a −8.75 ÷ 2.8 b 18.1 ÷ −5.9 c −28.2 ÷ −3.8 d −35.2 ÷ −6.9 17 Round these numbers to the nearest 10 to estimate the answer. a −48 × −29 b −18.1 × 61.5 c −71.4 ÷ −11.8 d −99.4 ÷ 19
–4
Remember, division is the inverse of multiplication so you will divide as you work down the pyramid.
Summary checklist
I can multiply two negative integers. I can divide any integer by a negative integer.
19 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1 Integers
1.3 Square roots and cube roots In this section you will …
Key words
•
find the squares of positive and negative integers and their corresponding square roots
cube root
find the cubes of positive and negative integers and their corresponding cube roots
rational numbers
•
square root
PL E
•
natural numbers
learn to recognise natural numbers, integers and rational numbers.
Tip
The natural numbers are the counting numbers and zero.
SA
M
52 = 25 This means that the square root of 25 is 5. This can be written as 25 = 5. This is the only answer in the set of natural numbers. However (−5)2 = −5 × −5 = 25 This means that the integer −5 is also a square root of 25. Every positive integer has two square roots, one positive and one negative. 5 is the positive square root of 25 and −5 is the negative square root. No negative number has a square root. For example, the integer −25 has no square root because the equation x2 = −25 has no solution. 53 = 125 This means that the cube root of 125 is 5. This can be written as 3 125 = 5. You might think −5 is also a cube root of 125. However (−5)3 = −5 × −5 × −5 = (−5 × −5) × −5 = 25 × −5 = −125 So 3 −125 = −5 Every number, positive or negative or zero, has only one cube root.
Worked example 1.3 Solve each equation. a x2 = 64 b x3 = 64 c x3 + 64 = 0
20 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1.3 Square roots and cube roots
Continued Answer
Exercise 1.3 2 3
4
5
Work out a 72 b (−7)2 c 73 Find a 3 125 b 3 −27 c 3 −1 Solve these equations. a x2 = 100 b x2 = 144 c 2 2 d x = 0 e x + 9 = 0 Solve these equations. a x3 = 216 b x3 + 27 = 0 c x3 + 1 = 0 d x3 + 125 = 0 272 = 93 = 729 Use this fact to find 729 −729 a b c 3 729
M
1
PL E
a 64 has two square roots. One is 64 = 8 and the other is − 64 = −8 So the equation has two solutions: x = 8 or x = −8 b 3 64 = 4. This means 43 = 4 × 4 × 4 = 64 and so x = 4 c If x3 + 64 = 0 then x3 = −64. So x = 3 −64 = −4
d
(−7)3
d
3
−8
3
−729
x2 = 1
d
SA
6 a A calculator shows that 82 − ( −8)2 = 0 Explain why this is correct. b Find the value of 3 43 − 3 ( −4 )3 . Show your working. 7 The square of an integer is 100. What can you say about the cube of the integer? 8 The integer 1521 = 32 × 132 Use this fact to a find 1521 b solve the equation x2 = 1521 9 a How is −52 different from (−5)2? b What is the difference between −53 and (−5)3?
21 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1 Integers
10 a b
PL E
Show that 32 + 42 = 52 Are these statements true or false? Give a reason for your answer each time. i (−3)2 + (−4)2 = (−5)2 ii (−13)2 = 122 + (−5)2 iii 82 = −102 − 62 c Show your work to a partner. Do they find your explanation clear?
Think like a mathematician
11 a Here is an equation: x2 + x = 6 i Show that x = 2 is a solution of the equation. ii Show that x = −3 is a solution of the equation. b
Here is another equation: x2 + x = 12
i Show that x = 3 is a solution of the equation. ii Find a second solution to the equation.
e 12 a
Find two solutions to this equation: x2 + x = 20 What patterns can you see in the answers to a, b and c? Find some more equations like this and write down the solutions. Compare your answers with a partner’s.
M
c d
Copy and complete this table. x−1 1 2
x3 − 1 7
x2 + x + 1
SA
x 2 3 4 5
b c d e
13
What pattern can you see in your answers? Add another row to see if the pattern is still the same. Add three rows where x is a negative integer. Is the pattern still the same if x is a negative integer? Compare your answers with a partner’s.
22 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1.3 Square roots and cube roots
Any number that can be written as a fraction is a rational number. Examples are 7 3 , −12 18 , 6, 1 , −2 9 4 25 10 15 Here is a list of six numbers: 5 − 1 −500 16 −4.8 99 1 5 2 Write a all the integers in the list b all the natural numbers in the list c all the rational numbers in the list. I 14 This Venn diagram shows the relationship N between natural numbers and integers. N stands for natural numbers and I for integers. a Copy the Venn diagram. b Write each of these numbers in the correct part of the diagram. 1 −3 7 −12 41 −100 2 1 2 c Add another circle to your Venn diagram to show rational numbers. d Add these numbers to your Venn diagram.
−8 3 3 0 6.3 − 10
e
Give your diagram to a partner to check.
5
3
M
7
Tip Integers and fractions are included in the set of rational numbers.
PL E
13
Tip
Remember, all integers are included in the rational numbers.
Summary checklist
SA
I can find and recognise square numbers and their two corresponding square roots. I can find and recognise positive and negative cube numbers and their cube roots. I can recognise natural numbers, integers and rational numbers.
23 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1 Integers
1.4 Indices In this section you will …
Key words
•
generalise
use positive and zero indices to represent numbers and in multiplication and division.
power
n 5n
0
1
2 25
PL E
In this section you will investigate numbers written as powers. Look at these powers of 5 3 125
4 625
5 3125
M
So 53 = 5 × 5 × 5 = 125 and 54 = 5 × 5 × 5 × 5 = 625 and so on. As you move to the right the numbers in the bottom row multiply by 5. As you move to the left the numbers in the bottom row divide by 5. 3125 ÷ 5 = 625, 625 ÷ 5 = 125, 125 ÷ 5 = 25 If you continue to divide by 5, 25 ÷ 5 = 5 so 51 = 5 There is another number missing in the table. What is 50? Divide by 5 again: 50 = 51 ÷ 5 = 5 ÷ 5 = 1 So 50 = 1 If n is any positive integer then n0 = 1.
Worked example 1.4
SA
a Show that 73 = 343 b Work out 4 i 7 ii 70 Answer
a 73 = 7 × 7 × 7 = 49 × 7 = 343 b i 7 4 = 73 × 7 = 343 × 7 = 2401 ii 70 = 1
24 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1.4 Indices
Exercise 1.4 Copy and complete this list of powers of 2. Power Number 2
20 1
21 2
22
23 8
24
25
26 64
34
35
27
28
29 512
210
Copy and complete this list of powers of 3. Power Number
30
31 3
32
33 27
36
37 2187
38
PL E
1
Think like a mathematician 3
Look at this multiplication: 4 × 16 = 64 You can write all the numbers as powers of 2: 22 × 24 = 26
c
Tip
‘Generalising’ means using a set of results to come up with a general rule.
SA
d
Write each of these multiplications as powers of 2. i 8 × 4 = 32 ii 16 × 8 = 128 iii 4 × 32 = 128 iv 2 × 128 = 256 v 16 × 32 = 512 Can you see a pattern in your answers? Make a conjecture about multiplying powers of 2. Test your conjecture on some more multiplications of your own. Make a conjecture about multiplying powers of 3. Use some examples to test your conjecture. Generalise your results so far.
M
a b
4
Write the answers to these calculations as powers of 6. a 62 × 63 b 6 4 × 6 c 65 × 62 d 63 × 63 5 Write the answers to these calculations in index form. a 103 × 102 b 205 × 20 c 153 × 153 d 55 × 53 6 a 38 = 6561 Use this fact to find 39 and show your method. b 56 = 15 625 Use this fact to find 57 and show your method. 7 Find the missing power. a
33 × 3
c
124 × 12
= 35 = 126
b
93 × 9
d
15
= 98 × 153 = 1510
25 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1 Integers
8
Read what Sofia says. 42 is equal to 24 and 43 is equal to 34
9
PL E
Is Sofia correct? Give a reason for your answer. A million is 106. A billion is 1000 million. Write as a power of 10 a one billion b 1000 billion 10 Write in index form a 22 × 23 × 2 b 33 × 34 × 32 c 5 × 53 × 53 d 103 × 102 × 104 11 a (32)3 = 32 × 32 × 32 Write (32)3 as a single power of 3. b Write in index form i (23)2 ii (53)2 iii (42)3 iv (152)4 v (104)3 c N is a positive integer. Write in index form i ( N 2 )3 ii ( N 4 )2 iii ( N 5 )3 d Can you generalise the results of part c?
M
Think like a mathematician
12 Here is a division: 32 ÷ 4 = 8 You can write this using indices: 25 ÷ 22 = 23 Write each of these divisions using indices. All the numbers are powers of 2 or 3. i 64 ÷ 4 = 16 ii 81÷ 3 = 27 iii 512 ÷ 16 = 32 iv 729 ÷ 9 = 81 v 9 ÷ 9 = 1 Write some similar divisions using powers of 5. Can you generalise your results from a and b? Check with some powers of other positive integers. Compare your results with a partner’s.
SA
a b c
d
13 Write the answers to these calculations in index form. a 27 ÷ 25 b 106 ÷ 103 c 10 9 15 11 d 8 ÷ 8 e 2 ÷ 2 f 14 Write the answers to these calculations in index form. a 95 × 92 b 95 ÷ 92 c (95)2 e 128 ÷ 123 f (73)3 g (100 )4
158 ÷ 156 25 ÷ 25 d
55 × 54
26 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1.4 Indices
15 Read what Zara says.
I think that ( 52 ) = ( 53 ) 3
2
Summary checklist
PL E
a Is Zara correct? Give a reason for your answer. b Is a similar result true for other indices? 16 15 = 3 × 5 Use this fact to write as a product of prime factors b 153 c 155 a 152 17 a Write 56 ÷ 54 as a power of 5. b Write 56 ÷ 56 as a power of 5. c Is it possible to write 54 ÷ 56 as a power of 5?
d
158
SA
M
I can use index notation for positive integers where the index is a positive integer or zero. I can multiply and divide numbers written as powers of a positive integer.
27 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1 Integers
Check your progress
× −6 −10 3 −18 −7
4
5
6
7
Are these calculations correct? If not, correct them. 2 a ( −5 ) = −25 b −9 × −11 = −99 3 c 45 ÷ −9 = −6 d ( −10 ) = −1000 Work out a 40 ÷ −5 b −36 ÷ −6 c 100 ÷ ( 2 – 7 ) d (12 − −18 ) ÷ −3 Solve these equations. a x 2 = 36 b x 2 + 16 = 0 c x3 = 8 d x3 + 27 = 0 Work out
M
3
−5
PL E
1 a Draw a factor tree for 350. b Write 350 as a product of prime factors. c Write 112 as a product of prime factors. d Find the HCF of 350 and 112. e Find the LCM of 350 and 112. 2 Copy and complete this multiplication table.
SA
a ( −5 )2 − ( −4 )2 7 Here is an expression: x3 + x 2 Find the value of the expression when a x = 3 8 Write as a single power of 8 a 82 × 83 c 1 9 a Write 46 as a power of 2. b Write 94 as a power of 3.
64 + 3 −64
b
3
b
x = −3
b d
86 ÷ 82 (83)3
28 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.