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Cambridge University Press
978-1-107-69787-4 – Cambridge Checkpoint Mathematics
Greg Byrd Lynn Byrd and Chris Pearce
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Greg Byrd, Lynn Byrd and Chris Pearce

Cambridge Checkpoint

Mathematics

8Coursebook

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978-1-107-69787-4 – Cambridge Checkpoint Mathematics
Greg Byrd Lynn Byrd and Chris Pearce
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Cambridge, New York, Melbourne, Madrid, Cape Town,
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Cambridge University Press

e Edinburgh Building, Cambridge CB2 8RU, UK
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is publication is in copyright. Subject to statutory exception
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ISBN 978-1-107-69787-4 Paperback
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Greg Byrd Lynn Byrd and Chris Pearce
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Introduction

Welcome to Cambridge Checkpoint Mathematics stage 8

e Cambridge Checkpoint Mathematics course covers the Cambridge Secondary 1 mathematics
framework and is divided into three stages: 7, 8 and 9. is book covers all you need to know for
stage 8.

ere are two more books in the series to cover stages 7 and 9. Together they will give you a rm
foundation in mathematics.

At the end of the year, your teacher may ask you to take a Progression test to nd out how well you
have done. is book will help you to learn how to apply your mathematical knowledge and to do well
in the test.

e curriculum is presented in six content areas:

t Number t .FBTVSF t Geometry
t Algebra t )BOEMJOH EBUB t 1SPCMFN TPMWJOH

is book has 18 units, each related to one of the rst ve content areas. Problem solving is included in
all units. ere are no clear dividing lines between the ve areas of mathematics; skills learned in one
unit are o en used in other units.
Each unit starts with an introduction, with key words listed in a blue box. is will prepare you for what
you will learn in the unit. At the end of each unit is a summary box, to remind you what you’ve learned.
Each unit is divided into several topics. Each topic has an introduction explaining the topic content,
VTVBMMZ XJUI XPSLFE FYBNQMFT )FMQGVM IJOUT BSF HJWFO JO CMVF SPVOEFE CPYFT "U UIF FOE PG FBDI UPQJD
there is an exercise. Each unit ends with a review exercise. e questions in the exercises encourage you
to apply your mathematical knowledge and develop your understanding of the subject.
As well as learning mathematical skills you need to learn when and how to use them. One of the most
important mathematical skills you must learn is how to solve problems.
When you see this symbol, it means that the question will help you to develop your problem-solving
skills.
During your course, you will learn a lot of facts, information and techniques. You will start to think like
a mathematician. You will discuss ideas and methods with other students as well as your teacher. ese
discussions are an important part of developing your mathematical skills and understanding.
Look out for these students, who will be asking questions, making suggestions and taking part in the
activities throughout the units.

Xavier Mia Dakarai Oditi
Anders Sasha Hassan Harsha
Alicia Shen Tanesha
Jake Maha Ahmad Zalika
Razi

3

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Greg Byrd Lynn Byrd and Chris Pearce
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Contents

Introduction 3 Unit 6 Planning and collecting data 61
Acknowledgements 6 6.1 Collecting data 62
65
Unit 1 Integers, powers and roots 6.2 Types of data 66
1.1 Arithmetic with integers 7 6.3 Using frequency tables 69
1.2 Multiples, factors and primes 8 End-of-unit review
1.3 More about prime numbers 11 70
1.4 Powers and roots 13 Unit 7 Fractions
End-of-unit review 15 7.1 Finding equivalent fractions, decimals 71
17 and percentages 73
Unit 2 Sequences, expressions and formulae 74
2.1 Generating sequences 7.2 Converting fractions to decimals 75
2.2 Finding rules for sequences 18 7.3 Ordering fractions 77
2.3 Using the nth term 19 7.4 Adding and subtracting fractions 78
2.4 Using functions and mappings 21 7.5 Finding fractions of a quantity 79
2.5 Constructing linear expressions 23 7.6 Multiplying an integer by a fraction 80
2.6 Deriving and using formulae 24 7.7 Dividing an integer by a fraction 82
End-of-unit review 26 7.8 Multiplying and dividing fractions
27 End-of-unit review
Unit 3 Place value, ordering and rounding 30
3.1 Multiplying and dividing by 0.1 and 0.01
3.2 Ordering decimals Unit 8 Shapes and geometric reasoning 83
3.3 Rounding
3.4 Adding and subtracting decimals 31 8.1 Recognising congruent shapes 84
3.5 Dividing decimals 32 8.2 Identifying symmetry of 2D shapes 86
3.6 Multiplying by decimals 34 8.3 Classifying quadrilaterals 88
3.7 Dividing by decimals 36 8.4 Drawing nets of solids 90
3.8 Estimating and approximating 37 8.5 Making scale drawings 92
End-of-unit review 38 End-of-unit review 94
39
Unit 4 Length, mass and capacity 40 Unit 9 Simplifying expressions and 95
4.1 Choosing suitable units 41 solving equations 96
4.2 Kilometres and miles 43 9.1 Collecting like terms 98
End-of-unit review 99
9.2 Expanding brackets 101
Unit 5 Angles 44 9.3 Constructing and solving equations 102
5.1 Parallel lines 45 End-of-unit review
5.2 Explaining angle properties 47
5.3 Solving angle problems 49 Unit 10 Processing and presenting data
End-of-unit review
10.1 Calculating statistics from discrete data 103
50 10.2 Calculating statistics from grouped or
51 continuous data 105
54 10.3 Using statistics to compare two distributions 107
57 End-of-unit review 109
60

4

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Greg Byrd Lynn Byrd and Chris Pearce
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Contents

Unit 11 Percentages 110 Unit 16 Position and movement 158
159
11.1 Calculating percentages 111 16.1 Transforming shapes 161
11.2 Percentage increases and decreases 113 16.2 Enlarging shapes 164
11.3 Finding percentages 115 End-of-unit review 165
11.4 Using percentages 117 166
End-of-unit review 119 Unit 17 Area, perimeter and volume 167
169
Unit 12 Constructions 17.1 e area of a triangle 171
12.1 Drawing circles and arcs 120 17.2 e areas of a parallelogram 173
12.2 Drawing a perpendicular bisector 121 and trapezium 175
12.3 Drawing an angle bisector 122 17.3 e area and circumference 177
12.4 Constructing triangles 124 of a circle 178
End-of-unit review 126 17.4 e areas of compound shapes
128 17.5 e volumes and surface 179
Unit 13 Graphs 182
13.1 Drawing graphs of equations areas of cuboids 184
13.2 Equations of the form y = mx + c 129 17.6 Using nets of solids to work out 186
13.3 e midpoint of a line segment 130 surface areas 188
13.4 Graphs in real-life contexts 132 End-of-unit review 191
End-of-unit review 134 192
136 Unit 18 Interpreting and discussing results 196
Unit 14 Ratio and proportion 139 18.1 Interpreting and drawing frequency
14.1 Simplifying ratios
14.2 Sharing in a ratio diagrams
14.3 Solving problems 140 18.2 Interpreting and drawing pie charts
End-of-unit review 141 18.3 Interpreting and drawing line graphs
143 18.4 Interpreting and drawing stem-and-
145 leaf diagrams
147 18.5 Drawing conclusions

End-of-unit review

Unit 15 Probability 148
15.1 e probability that an outcome does End-of-year review

not happen 149 Glossary and index
15.2 Equally likely outcomes 150
15.3 Listing all possible outcomes 152
15.4 Experimental and theoretical probabilities 154
End-of-unit review 157

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978-1-107-69787-4 – Cambridge Checkpoint Mathematics
Greg Byrd Lynn Byrd and Chris Pearce
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Acknowledgements

e publisher would like to thank Ángel Cubero of the International School Santo Tomás de Aquino,
Madrid, for reviewing the language level.
Cover image © Cosmo Condina concepts/Alamy
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p. 37b Christopher Steer/ iStock; p.38b DAJ/ Getty Images; p. 44mr Chris Ryan/OJO Images/Getty Images;
p. 44b NASA; p. 46tr Lynn Byrd; 46mr Aspen Photo/Shutterstock;
p. 63m dundanim/Shutterstock; p. 83t Diego Cervo/Shutterstock; p. 83mr Francesco Dazzi/Shutterstock;
p. 83br Peter Kirillov/Shutterstock; p. 93mr mbbirdy/iStock; p. 95tr pidjoe/iStock;
p. 95mr Liz Van Steenburgh/Shutterstock; p. 95br Aleksandar Petrovic/iStock; p. 114ml a40757/ Shutterstock;
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l = le , r = right, t = top, b = bottom, m = middle

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Greg Byrd Lynn Byrd and Chris Pearce
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1 Integers, powers and roots

e rst primes are 2 3 5 7 11 13 17 19 23 29 ... Key words
Prime numbers have just two factors: 1 and the number itself.
Every whole number that is not prime can be written as a product Make sure you learn and
of prime numbers in exactly one way (apart from the order of understand these key words:
the primes).
8 = 2 × 2 × 2 65 = 5 × 13 132 = 2 × 2 × 3 × 11 integer
2527 = 7 × 19 × 19 inverse
It is easy to multiply two prime numbers. For example, multiple
13 × 113 = 1469. common multiple
It is much harder to do the inverse operation. For example, lowest common multiple (LCM)
2021 is the product of two prime numbers. Can you nd them? factor
common factor
is fact is the basis of a system that is used to encode messages highest common factor (HCF)
sent across the internet. prime number
prime
e RSA cryptosystem was invented by Ronald Rivest, factor tree
Adi Shamir and Leonard Adleman in 1977. It uses two power
large prime numbers with about 150 digits each. ese index (indices)
are kept secret. eir product, N, with about 300 digits, is made square
public so that anyone can use it. cube
If you send a credit card number to a square root
website, your computer performs a cube root
calculation with N and your credit card
number to encode it. e computer
receiving the coded number will do
another calculation to decode it. Anyone
else, who does not know the factors, will
not be able to do this.

Prime numbers more than 200 are 211 223 227 229 233 239 241 251 257 263 269 271 ... ...

1 Integers, powers and roots 7

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1.1 Arithmetic with integers

1.1 Arithmetic with integers 2+3=5
2+2=4
Integers are whole numbers. ey may be positive or negative. Zero is also an integer. 2+1=3
You can show integers on a number line. 2+0=2
2 + −1 = 1
–5 –4 –3 –2 –1 0 1 2 3 4 5 2 + −2 = 0
2 + −3 = −1
Look at the additions in the box to the right. e number added to 2 decreases, or goes 2 + −4 = −2
down, by 1 each time. e answer also decreases, or goes down, by 1 each time.

Now see what happens if you subtract. Look at the rst column. 5−3=2 5 + −3 = 2
e number subtracted from 5 goes down by 1 each time. e answer 5−2=3 5 + −2 = 3
5−1=4 5 + −1 = 4
goes up by 1 each time. Now look at the two columns together. 5−0=5 5+0=5
You can change a subtraction into an addition by adding the inverse 5 − −1 = 6
number. e inverse of 3 is −3. e inverse of −3 is 3. 5 − −2 = 7 5+1=6
For example, 5 – –3 = 5 + 3 = 8. 5 − −3 = 8 5+2=7
5+3=8

Worked example 1.1a

Work these out. a 3 + −7 b −5 − 8 c −3 − −9

a 3 + −7 = −4 Subtract 7 from 3. 3 − 7 = −4
b −5 − 8 = −13 The inverse of 8 is −8. −5 − 8 = −5 + −8 = −13
c −3 − −9 = 6 The inverse of −9 is 9. −3 − −9 = −3 + 9 = 6

Look at these multiplications. 3 × 5 = 15 e pattern continues like this. −1 × 5 = −5
2 × 5 = 10 −2 × 5 = −10
1×5=5 −3 × 5 = −15
0×5=0 −4 × 5 = −20

You can see that negative integer × positive integer = negative answer.

Now look at this pattern. −3 × 4 = −12 e pattern continues like this. −3 × −1 = 3
−3 × −2 = 6
−3 × 3 = −9 −3 × −3 = 9
−3 × −4 = 12
−3 × 2 = −6 −3 × −5 = 15

−3 × 1 = −3

−3 × 0 = 0

You can see that negative integer × negative integer = positive answer.

8 1 Integers, powers and roots

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Greg Byrd Lynn Byrd and Chris Pearce
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1.1 Arithmetic with integers

Here is a simple rule, which also works for division. When you multiply two integers:
if they have same signs positive answer
if they have different signs negative answer

Worked example 1.1b

Work these out. a 12 × −3 b −8 × −5 c −20 ÷ 4 d −24 ÷ −6

a 12 × −3 = −36 12 × 3 = 36 The signs are different so the answer is negative.
b −8 × −5 = 40 8 × 5 = 40 The signs are the same so the answer is positive.
c −20 ÷ 4 = −5 20 ÷ 4 = 5 The signs are different so the answer is negative.
d −24 ÷ −6 = 4 24 ÷ 6 = 4 The signs are the same so the answer is positive.

Warning: is rule works for multiplication and division. It does not work for addition or subtraction.

Exercise 1.1

1 Work out these additions.
a 3 + −6 b −3 + −8 c −10 + 4 d −10 + −7 e 12 + −4

2 Work out these additions. d −30 + −70 e 45 + −40
a 30 + −20 b −100 + −80 c −20 + 5

3 Work out these subtractions. e −2 − 10
a 4−6 b −4 − 6 c 6−4 d −6 − 6

4 Write down additions that have the same answers as these subtractions. Then work out the answer

to each one. d −4 − −6 e 12 − −10
a 4 − −6 b −4 − −6 c 8 − −2

5 Work out these subtractions. c 12 − −4 d −6 − −6
a 7 − −2 b −5 − −3 e −2 − −10

6 Here are some addition pyramids. Each number is the sum of the two in In part a, 3 + −5 = −2
the row below it.
Copy the pyramids. Fill in the missing numbers. e

a bc d –7–7–7–7–7

3 333 3

–2–2–2–2–2 2 222 2 –6–6–6–6–6

3 3 3 3–53–5–5–51–51 1 1 1–2–2–2–2––32–3–3–35–35 5 5 5 2 2 2 2–42–4–4–4––64–6–6–6–6–3–3–3–3–3 2 222 2

7 Here is a subtraction table. Two answers have already been filled in: 4 − −4 = 8 and −4 − 2 = −6.
Copy the table and complete it.

second number
− −4 −2 0 2 4
48
2
first 0
number −2

−4 −6

1 Integers, powers and roots 9

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1.1 Arithmetic with integers

8 Work out these multiplications.
a 5 × −4 b −8 × 6 c −4 × −5 d −6 × −10 e −2 × 20

9 Work out these divisions. c −12 ÷ −4 d −50 ÷ −5 e 16 ÷ −4
a 20 ÷ −10 b −30 ÷ 6

10 Write down two correct division expressions.
a 4 × −10 b −20 ÷ 5 c −20 × 5 d −40 ÷ −8 e −12 × −4

11 Here are some multiplications. In each case, use the same numbers to write down

two correct division expressions. c −6 × 7 = −42
a 5 × −3 = −15 b −8 × −4 = 32

12 Here is a multiplication table. Three answers have already been filled in.

× −3 −2 −1 0 1 2 3
36
2

1 −2
0

−1 3
−2

−3

a Copy the table and complete it.
b Colour all the 0 answers in one colour, for example, green.
c Colour all the positive answers in a second colour, for example, blue.
d Colour all the negative answers in a third colour, for example, red.

13 These are multiplication pyramids. Each number is the product of the The product is the result of
two in the row below it. multiplying two numbers
Copy each pyramid. Fill in the missing numbers. In part a, 2 × −3 = −6

a bcd

4848 48486464 6464

–6–6 –6–6 –1–212 –1–212 –1–616 –1–616

2 2 –3–3 –2–2 2 2–4–4–3–35 5–2–2–1–1 –4–4–3–35 5–3–3–1–13 3 –3–3–3–3–3–3 3 3 –3–3 22 22

14 a What integers will replace the symbols to make this multiplication correct? × ∆ = −12
b How many different pairs of numbers can you find that give this answer?

15 Work these out. b 5 + −3 d −60 ÷ −10
a 5 × −3 c −4 − −5 e −2 + 18 f −10 − 4

16 Write down the missing numbers.
a 4 × = −20 b ÷ −2 = −6 c − −5 = −2 d × −3 = 12
e −2 + = 2 f − 4 = −3

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1.2 Multiples, factors and primes

1.2 Multiples, factors and primes 6 × 1 = 6 6 × 2 = 12 6 × 3 = 18 … …

e multiples of 6 are 6, 12, 18, 24, 30, 36, …, … 9 × 1 = 9 9 × 2 = 18 9 × 3 = 27 … …

e multiples of 9 are 9, 18, 27, 36, 45, 54, …, …

e common multiples of 6 and 9 are 18, 36, 54, 72, …, … 18 36 54 … … are in both lists of multiples.
e lowest common multiple (LCM) of 6 and 9 is 18.

e factors of a number divide into it without a remainder. 3 × 6 = 18 so 3 and 6 are factors of 18
e factors of 18 are 1, 2, 3, 6, 9 and 18.

e factors of 27 are 1, 3, 9 and 27.

e common factors of 18 and 27 are 1, 3 and 9.

e highest common factor (HCF) of 18 and 27 is 9.

Some numbers have just two factors. Examples are 7 (1 and 7 are factors), 13 (1 and 13 are factors)
and 43. Numbers with just two factors are called prime numbers or just primes. e rst ten primes
are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.

Worked example 1.2a

a Find the factors of 45. b Find the prime factors of 48.

a The factors of 45 are 1, 3, 5, 9, 15 45 = 1 × 45 so 1 and 45 are factors. (1 is always a factor.)
and 45. Check 2, 3, 4, … in turn to see if it is a factor.
2 is not a factor. (45 is an odd number.)
45 = 3 × 15 3 and 15 are factors.
4 is not a factor.
45 = 5 × 9 5 and 9 are factors.
6, 7 and 8 are not factors. The next number to try is 9 but we
already have 9 in the list of factors. You can stop when you
reach a number that is already in the list.

b The prime factors of 48 are 2 and 3. You only need to check prime numbers.
48 = 2 × 24 2 is a prime factor. 24 is not.
48 = 3 × 16 3 is a prime factor. 16 is not.
5 and 7 are not factors.
Because 7 × 7 is bigger than 48, you can stop there.

Worked example 1.2b

Find the LCM and HCF of 12 and 15.

The LCM is 60. The multiples of 12 are 12, 24, 36, 48, 60, …, ….
The multiples of 15 are 15, 30, 45, 60, 75, …, …
60 is the first number that is in both lists.

The HCF is 3. The factors of 12 are 1, 2, 3, 4, 6 and 12.
The factors of 15 are 1, 3, 5 and 15.
3 is the largest number that is in both lists.

1 Integers, powers and roots 11

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1.2 Multiples, factors and primes

Exercise 1.2

1 Find the factors of each number.
a 20 b 27 c 75 d 23 e 100 f 98

2 Find the first four multiples of each number.
a8 b 15 c7 d 20 e 33 f 100

3 Find the lowest common multiple of each pair of numbers.
a 6 and 8 b 9 and 12 c 4 and 14
d 20 and 30 e 8 and 32 f 7 and 11

4 The LCM of two numbers is 40. One of the numbers is 5. What is the other number?

5 Find: b the factors of 32 c the common factors of 24 and 32
a the factors of 24
d the highest common factor of 24 and 32.

6 List the common factors of each pair of numbers.
a 20 and 25 b 12 and 18 c 28 and 35
d 8 and 24 e 21 and 32 f 19 and 31

7 Find the HCF of the numbers in each pair.
a 8 and 10 b 18 and 24 c 40 and 50
d 80 and 100 e 17 and 33 f 15 and 30

8 The HCF of two numbers is 8. One of the numbers is between 20 and 30. The other number is
between 40 and 60. What are the two numbers?

9 31 is a prime number. What is the next prime after 31?

10 List the prime numbers between 60 and 70.

11 Read what Xavier and Alicia say about the number 91.

91 is a prime number. 91 is not a prime number.

Who is correct? Give a reason for your answer.

12 73 and 89 are prime numbers. What is their highest common factor?

13 7 is a prime number. No multiple of 7, except 7 itself, can be a prime number. Explain why not.

14 List the prime factors of each number. f 77
a 12 b 15 c 21 d 49 e 30

15 a Write down three numbers whose only prime factor is 2.
b Write down three numbers whose only prime factor is 3.
c Write down three numbers whose only prime factor is 5.

16 Find a number bigger than 10 that has an odd number of factors.

17 Find a number that has three prime factors.

12 1 Integers, powers and roots

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1.3 More about prime numbers

1.3 More about prime numbers

Any integer bigger than 1, that is not prime, can be written as a product of prime numbers.
Here are some examples.

84 = 2 × 2 × 3 × 7 45 = 3 × 3 × 5 196 = 2 × 2 × 7 × 7

You can use a factor tree to nd and show factors.

is is how to draw a factor tree for 120. 120 5
12 10
1 Draw branches to two numbers that multiply to make 120. Here 12 and 3 42
10 are chosen. 22

2 Do the same with 12 and 10. 12 = 3 × 4 and 10 = 2 × 5
3 3, 2 and 5 are prime, so stop.
4 4 = 2 × 2 so draw two branches.
5 Stop, because all the end numbers are prime.
6 Multiply all the numbers at the ends of the branches.

120 = 2 × 2 × 2 × 3 × 5 120
2 60
You can draw the tree in di erent ways.
Here is a di erent tree for 120. 2 30
56
e numbers at the ends of the branches are the same.
You can write the result like this. 120 = 23 × 3 × 5.

e small number ³ next to the 2 is called an index. 23 means 2 × 2 × 2.

Check that these are correct. 23
60 = 2² × 3 × 5 75 = 3 × 5²
You can use these expressions to nd the LCM and HCF of 60 and 75. 60 = 22 × 3 × 5
For the LCM, take the larger frequency of each prime factor and multiply 75 = 3 × 52
them all together. Two 2s, one 3, two 5s
LCM = 2² × 3 × 5² = 4 × 3 × 25
60 = 22 × 3 × 5
= 300 75 = 3 × 52
For the HCF, take the smaller frequency of each prime factor that occurs in No 2s, one 3, one 5
both numbers and multiply them all together.
HCF = 3 × 5

= 15

1 Integers, powers and roots 13

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1.3 More about prime numbers

Exercise 1.3

1 Copy and complete each of these factor trees.
a 48 b 100 c 108

6 8 10 10 2 54

2 a Draw a different factor tree for each of the numbers in question 1.
b Write each of these numbers as a product of primes.
i 48 ii 100 iii 108

3 Match each number to a product of primes.
One has been done for you.

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1.4 Powers and roots

1.4 Powers and roots

A number multiplied by itself several times is called a power of The plural of index is indices:
that number. You use indices to show powers. one index, two indices.

Here are some powers of 5. means square root.
3 means cube root.
5² = 5 × 5 = 25 is is ve squared or the square of ve.

53 = 5 × 5 × 5 = 125 is is ve cubed or the cube of ve.

54 = 5 × 5 × 5 × 5 = 625 is is ve to the power four.

55 = 5 × 5 × 5 × 5 × 5 = 3125 is is ve to the power ve.

e square of 5 is 5² = 25.

erefore the square root of 25 is 5 and you write this as 25 = 5.

e cube of 5 is 53 = 125.
erefore the cube root of 125 is 5 and you write this as 3 125 = 5.

5 is not the only square root of 25.
(−5)2 = −5 × −5 = 25 so 25 has two square roots, 5 and −5.

25 means the positive square root.

125 only has one integer cube root. −5 is not a cube root because −5 × −5 × −5 = −125.

Square numbers have square roots that are integers.
Examples: 132 = 169 so 169 = 13 192 = 361 so 361 = 19

Try to memorise these ve cubes and their corresponding cube roots:

13 = 1 so 3 1 = 1 23 = 8 so 3 8 = 2 33 = 27 so 3 27 = 3

43 = 64 so 3 64 = 4 53 = 125 so 3 125 = 5

Exercise 1.4

1 Find the value of each power. c 34 d 35
a 3² b 33

2 Find the value of each power.
a 10² b 10³ c 104

3 106 is one million and 109 is one billion. Over one billion
Write down these two numbers in full. people live in India.

4 In each pair, which of the two numbers is larger?
a 35 or 53 b 26 or 62 c 54 or 45

5 a N 3 is 27. What number is N ?
b 6M is 1296. What number is M ?

1 Integers, powers and roots 15

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978-1-107-69787-4 – Cambridge Checkpoint Mathematics
Greg Byrd Lynn Byrd and Chris Pearce
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1.4 Powers and roots

6 Can you find two different integers, A and B, so that AB = BA?

7 Write down two square roots for each of these numbers. e 225 f 400
a 9 b 36 c 81 d 196

8 Read what Maha says about I am thinking of a number. It is between 250 and
her number. What could her 350. Its square root is an integer.
number be?

9 Read what Hassan says about his I am thinking of a number. It is less than 500.
number. What is the largest possible Its cube root is an integer.
value of his number?

10 Find the value of each of these.

a 100 b 400 c 3 27 d 3 125 e 3 1000

11 Read what Oditi says about her number. I am thinking of a number. Its square root
Find a possible value for her number. is an integer. Its cube root is an integer.

12 210 = 1024. Use this fact to find: c 29
a 211 b 212

13 a Find the value of each expression. i 1³ + 2³ ii 13 + 23

b Find the value of 13 + 23 + 33 . c Find the value of 13 + 23 + 33 + 43 .

d Can you see an easy way to work out the value of 13 + 23 + 33 + 43 + 53 ? If so, describe it.

Summary You should be able to:

You should now know that: Add, subtract, multiply and divide integers.

You can multiply or divide two integers. If they Identify and use multiples and factors.
have the same sign the answer is positive
(−5 × −2 = 10). If they have different signs the Identify and use primes.
answer is negative (−5 × 2 = −10).
Find common factors and highest common factors
You can subtract a negative number by adding the (HCF).
corresponding positive number.
Find lowest common multiples (LCM).
You can find multiples of a number by multiplying
by 1, 2, 3, etc. Write a number in terms of its prime factors, for
example, 500 = 22 × 53.
Prime numbers have just two factors.
Calculate squares, positive and negative square
You can write every positive integer as a product roots, cubes and cube roots.
of prime factors.
Use index notation for positive integer powers.
You can use the products of prime factors to find
the lowest common factor and highest common Calculate accurately, choosing operations and
multiple. mental or written methods appropriate to the
numbers and context.
54 means 5 × 5 × 5 × 5.
Manipulate numbers and apply routine
Positive integers have two square roots. algorithms, for example, to find the HCF and LCM
49 = 7 and 3 64 = 4. of two numbers.

16 1 Integers, powers and roots

© in this web service Cambridge University Press www.cambridge.org

Cambridge University Press
978-1-107-69787-4 – Cambridge Checkpoint Mathematics
Greg Byrd Lynn Byrd and Chris Pearce
Excerpt
More information

End-of-unit review

End-of-unit review

1 Work these out. c −8 + −7
a 5 + −3 b −3 − 5 d 3 − 13 e −7 − 7
d −5 − −12 e −9 − −9
2 Work these out. b −3 − −4 c 12 − −5 d −30 ÷ −5 e −16 ÷ 8
a 2 − −5

3 Work these out. c −20 × 4
a −3 × −9 b 8 ÷ −4

4 Copy and complete this multiplication table.

× −2 3 5

−4

−3

6 30

5 Here is a number chain. Each number is the product of the previous two numbers.
−1 −2 2 −4 ☐ ☐
Write down the next two numbers in the chain.

6 Find all the factors of each number.
a 42 b 52 c 55 d 29 e 64 f 69

7 a Find two prime numbers that add up to 40.
b Find another two prime numbers that add up to 40.
c Are there any more pairs of prime numbers that add up to 40? If so, what are they?

8 Write each of these numbers as a product of its prime factors.
a 18 b 96 c 200 d 240 e 135 f 175

9 Use your answers to question 8 to find: b the highest common factor of 135 and 175
a the highest common factor of 200 and 240 d the lowest common multiple of 200 and 240.
c the lowest common multiple of 18 and 96

10 Find the square roots of each number.
a 25 b 81 c 169 d 256

11 Find the value of each number.

a 64 b 3 64

12 In computing, 210 is called 1K. Write down as a number:
a 1K b 2K c 4K.

13 a Read Shen’s comment. What mistake has he made?
b Correct the statement.

35 and 53 are both equal to 15.

14 The HCF of two numbers is 6. The LCM is 72. One of the numbers is 24.
Find a possible value of the other number.

1 Integers, powers and roots 17

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