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Cambridge Lower Secondary
Mathematics
LEARNER’S BOOK 8
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Lynn Byrd, Greg Byrd & Chris Pearce
Second edition
Digital access
Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
PL
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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.
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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.
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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 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.
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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
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137–170
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126–136
Strand of mathematics
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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
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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 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 special Thinking and
Working Mathematically skills.
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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
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 © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
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ACKNOWLEDGEMENTS
Original material © 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
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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
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1
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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.
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Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1 Integers
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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 © 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)
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index
integer
lowest common
multiple (LCM)
prime factor
120
12
3
10
4
2
2
5
2
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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
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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
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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
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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
5
6
7
8
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?
a
Draw a factor tree for 330.
b Write 330 as a product of prime numbers.
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
Work out the product of each set of prime factors.
32 × 5 × 7
a
b 23 × 53
c
2 2 × 32 ×11
d 2 4 × 72
e 3 ×172
Write each of these numbers as a product of prime factors.
a
28
b 60
c
72
d 153
e
190
f
275
aCopy the table and write each number as a product of prime
numbers.
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4
a
b
c
Tip
You can use a
factor tree to help
you.
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3
Product of prime numbers
5×7
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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
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1 Integers
11 a
14
15
16
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17
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13
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12
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?
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.
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.
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.
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.
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?
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.
a
Find the HCF of 34 and 58.
b Find the LCM of 34 and 58.
Show that the HCF of 63 and 110 is 1.
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.
18
19
20
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
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•
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.
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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
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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.
A sequence is a set of
numbers or expressions
made and written in
order, according to some
pattern.
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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.
Answer
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a 3−2 =1
3−0 = 3
3 − −2 = 5
3 − −4 = 7
b 3 − −6 = 9
3 − −8 = 11
3 − −10 = 13
c The first number, 3, does not change.
The number being subtracted decreases by 2 each time.
The answer increases by 2 each time.
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.
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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
3
6
7
−5 × − 2
d
−2 × − 5
c
−10 × −6
d
−8 × −11
−8
−9
30
Work out
b (−3 + −5) × −6
a
(3 + 5) × −4
c
−4 × (5 − 8)
d −6 × (−2 − −7)
Round these numbers to the nearest whole number to estimate
the answer.
b −11.2 × 2.95
3.9 × −6.8
a
c
(−6.1)2
d (−4.88)2
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.
SA
5
c
Copy and complete this multiplication table.
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?
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c
Tip
Do the calculation
in brackets first.
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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
–3
–3
5
–1
–4
–5
–2
Each number is the product of the two numbers below it. For example,
in a, 2 × −4 = −8
Copy and complete the multiplication pyramids.
aDraw 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.
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9
–4
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
15
c
–8
30 ÷ −6
−24 ÷ −4
Tip
–200
–20
–2
Copy and complete each pyramid.
Work out
a
(3 × −4) ÷ −2
b (2 − 20) ÷ −3
c
(−3 + 15) ÷ −4
d 24 ÷ (2 × −4)
Find the value of x.
a
x ÷ −4 = 8
b x ÷ −3 = −15
c
16 ÷ x = −2
d −15 ÷ x = 3
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
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.
SA
16
12
c
f
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14
–1
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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
17
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
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•
natural numbers
learn to recognise natural numbers, integers and rational
numbers.
Tip
The natural
numbers are the
counting numbers
and zero.
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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
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
Exercise 1.3
2
3
4
5
d
(−7)3
d
3
−8
3
−729
x2 = 1
d
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.
The square of an integer is 100.
What can you say about the cube of the integer?
The integer 1521 = 32 × 132
Use this fact to
a
find 1521
b solve the equation x2 = 1521
a
How is −52 different from (−5)2?
b What is the difference between −53 and (−5)3?
a
SA
6
Work out
a
72
b (−7)2
c
73
Find
3
3
3
−1
−27
a
125
b
c
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
3
729
−729
729
a
b
c
M
1
PL
E
a
7
8
9
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
c
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
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
ii
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
Show that x = 3 is a solution of the equation.
Find a second solution to the equation.
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
−8 3 3 0 6.3 − 10
7
5
3
Give your diagram to a partner to check.
M
e
Tip
Integers and
fractions are
included in the
set of rational
numbers.
PL
E
13 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.
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
1
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.
30
31
3
32
33
27
36
37
2187
38
PL
E
Power
Number
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
b
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
4
5
6
7
Write the answers to these calculations as powers of 6.
a
6 2 × 63
b 64 × 6
c
65 × 6 2
d 63 × 63
Write the answers to these calculations in index form.
a 103 × 102
b 205 × 20
c
153 × 153
d 55 × 53
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.
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
PL
E
Is Sofia correct? Give a reason for your answer.
9
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:
You can write this using indices:
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
32 ÷ 4 = 8
25 ÷ 22 = 23
13 Write the answers to these calculations in index form.
a
2 7 ÷ 25
b 106 ÷ 103
c
10
9
15
11
d 8 ÷8
e
f
2 ÷2
14 Write the answers to these calculations in index form.
a
b 95 ÷ 92
c
(95)2
95 × 92
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
2
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.
Copy and complete this multiplication table.
×
−6
−10
3 −18
−7
4
5
Are these calculations correct? If not, correct them.
2
a
b −9 × −11 = −99
( −5 ) = −25
3
c
d ( −10 ) = −1000
45 ÷ −9 = −6
Work out
a
b −36 ÷ −6
40 ÷ −5
100 ÷ ( 2 – 7 )
c
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
a
( −5 )2 − ( −4 )2
Here is an expression: x3 + x 2
Find the value of the expression when
a
x=3
Write as a single power of 8
a
82 × 83
c
1
a
Write 46 as a power of 2.
b Write 94 as a power of 3.
SA
6
7
M
3
−5
PL
E
1
7
8
9
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.