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Cambridge Lower Secondary
Mathematics
WORKBOOK 8
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Lynn Byrd, Greg Byrd & Chris Pearce
Second edition
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
WORKBOOK 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.
Contents
Contents
How to use this book
5
Acknowledgements6
6 Collecting data
1 Integers
6.1
6.2
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1.1
1.2
1.3
1.4
Factors, multiples and primes
7
Multiplying and dividing integers 9
Square roots and cube roots 11
Indices12
2 E
xpressions, formulae
and equations
Constructing expressions
14
Using expressions and formulae
18
Expanding brackets
22
Factorising26
Constructing and solving equations
29
Inequalities35
7 Fractions
7.1
7.2
7.3
7.4
7.5
7.6
Fractions and recurring decimals
Ordering fractions
Subtracting mixed numbers
Multiplying an integer by a mixed number
Dividing an integer by a fraction
Making fraction calculations easier
8.1
8.2
8.3
Quadrilaterals and polygons
The circumference of a circle
3D shapes
3 Place value and rounding
9 Sequences and functions
3.1
3.2
9.1
9.2
9.3
9.4
Multiplying and dividing by 0.1 and 0.01
40
Rounding43
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4 Decimals
4.1
4.2
4.3
4.4
Ordering decimals
Multiplying decimals
Dividing by decimals
Making decimal calculations easier
47
50
54
58
Generating sequences
Finding rules for sequences
Using the nth term
Representing simple functions
95
101
104
111
115
119
122
10 Percentages
10.1 Percentage increases and decreases
10.2 Using a multiplier
5 Angles and constructions
11 Graphs
5.1
5.2
5.3
11.1
11.2
11.3
11.4
Parallel lines
62
The exterior angle of a triangle
65
Constructions67
73
76
79
83
87
91
8 Shapes and symmetry
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2.1
2.2
2.3
2.4
2.5
2.6
Data collection
69
Sampling71
129
131
Functions134
Plotting graphs
136
Gradient and intercept
139
Interpreting graphs
142
3
Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
Contents
12 Ratio and proportion
12.1 Simplifying ratios
12.2 Sharing in a ratio
12.3 Ratio and direct proportion
146
150
153
13 Probability
14 Position and transformation
14.1
14.2
14.3
14.4
14.5
14.6
16.1 I nterpreting and drawing frequency
diagrams209
16.2 Time series graphs
213
16.3 Stem-and-leaf diagrams
218
16.4 Pie charts
221
16.5 Representing data
226
16.6 Using statistics
230
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13.1 Calculating probabilities
158
13.2 Experimental and theoretical probabilities 161
16 I nterpreting and
discussing results
Bearings164
The midpoint of a line segment
170
Translating 2D shapes
173
Reflecting shapes
177
Rotating shapes
183
Enlarging shapes
187
15 Distance, area and volume
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15.1 Converting between miles and kilometres 192
15.2 The area of a parallelogram and trapezium 196
15.3 Calculating the volume of
triangular prisms
201
15.4 Calculating the surface area of triangular
prisms and pyramids
205
4
Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1 Record, organiseHow
and to
represent
data
use this book
How to use this book
This workbook provides questions for you to practise what you have learned in class. There is a
unit to match each unit in your Learner’s Book. Each exercise is divided into three parts:
Focus: these questions help you to master the basics
•
Practice: these questions help you to become more confident in using what you
have learned
•
Challenge: these questions will make you think very hard.
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•
You will also find these features:
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Words you need
to know.
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Step-by-step examples
showing how to solve
a problem.
These questions help
you to practise thinking
and working like a
mathematician.
5
Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
Acknowledgements
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TBC
6
Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1
Integers
Exercise 1.1
Focus
2
3
4
factor tree
highest common
factor (HCF)
lowest common
multiple (LCM)
prime factor
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5
Draw a factor tree for 250 that starts with 2 × 125.
Can you draw a different factor tree for 250 that starts with
2 × 125? Give a reason for your answer.
c
Draw a factor tree for 250 that starts with 25 × 10.
d Write 250 as a product of its prime factors.
a
Draw a factor tree for 300.
b Draw a different factor tree for 300.
c
Write 300 as a product of prime numbers.
a
Write as a product of prime numbers
i
6
ii
30
iii 210
b What is the next number in this sequence? Why?
Work out
c
23 × 33 × 73
a
2×3×7
b 22 × 32 × 72
a
Draw a factor tree for 8712.
b Write 8712 as a product of prime numbers.
Write each of these numbers as a product of its prime factors.
a
96
b 97
c
98
d 99
a
b
Key words
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1
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1.1 Factors, multiples and primes
6
Practice
7
8
Write as a product of prime numbers
c
703
a
70
b 702
a
Write each square number as a product of its prime factors.
i
9
ii
36
iii 81
iv 144
v
225
vi 576
vii 625
viii 2401
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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
b
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Challenge
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When a square number is written as a product of prime numbers,
what can you say about the factors?
c
176 400 = 24 × 32 × 52 × 72
Use this fact to show that 176 400 is a square number.
252 = 22 × 32 × 7
660 = 22 × 3 × 5 × 11
9
315 = 32 × 5 × 7
Use these facts to find the highest common factor of
a
315 and 252
b 315 and 660
c
252 and 660
2
3
2
2
72 = 2 × 3
75 = 3 × 5
10 60 = 2 × 3 × 5
Use these facts to find the lowest common multiple of
a
60 and 72
b 60 and 75
c
72 and 75
11 a
Write 104 as a product of its prime factors.
b Write 130 as a product of its prime factors.
c
Find the HCF of 104 and 130.
d Find the LCM of 104 and 130.
12 a
Write 135 as a product of prime numbers.
b Write 180 as a product of prime numbers.
c
Find the HCF of 135 and 180.
d Find the LCM of 135 and 180.
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13 a
Write 343 as a product of prime numbers.
b Write 546 as a product of prime numbers.
c
Find the HCF of 343 and 546.
d Find the LCM of 343 and 546.
14 Find the LCM of 42 and 90.
15 a
Find the HCF of 168 and 264.
b Find the LCM of 168 and 264.
16 a
Show that the LCM of 48 and 25 is 1.
b Find the HCF of 48 and 25.
17 The HCF of two numbers is 6. The LCM of the two numbers is 72.
What are the two numbers?
8
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
1.2 Multiplying and
dividing integers
Key word
Focus
integer
2
3
4
Copy this sequence of multiplications and add three more
multiplications in the sequence.
7 × −4 = −28 5 × −4 = −20 3 × −4 = −12 1 × −4 = −4
Work out
a
−5 × 8
b −5 × −8
c
−9 × −11
d −20 × −6
Put these multiplications into two groups.
A
−12 × −3
D
18 × 2
B
(−6)2
C
−4 × 9
E
9 × −4
F
−4 × −9
Copy and complete this multiplication table.
−4
−9
−45
−16
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×
−6
5
−8
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Exercise 1.2
5
Work out
a
(3 + 4) × 5
b
(3 + −4) × 5
c
(−3 + −4) × −5
d
(3 + −4) × −5
d
(−4.09)2
Practice
6
7
8
Estimate the answers by rounding numbers to the nearest integer.
−2.9 × −8.15
b 10.8 × −6.1
c
(−8.8)2
a
Show that (−6)2 + (−8)2 − (−10)2 = 0
This is a multiplication pyramid.
Each number is the product of the two numbers below.
For example, 3 × −2 = −6
a
Copy and complete the pyramid.
b Show that you can change the order of the numbers on the
bottom row to make the top number 3456.
–6
3
–2
–1
4
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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
The product of two integers is −6.
Find all the possible values of the two integers.
b The product of two integers is 6.
Find all the possible values of the two integers.
10 a
Here is a multiplication: −9 × −7 = 63
Write it as a division in two different ways.
b Here is a different multiplication: 12 × −7 = −84
Write it as a division in two different ways.
11 Work out
a
45 ÷ −7
b −50 ÷ −10
c
27 ÷ −3
d −52 ÷ −4
12 Estimate the answers by rounding numbers to the nearest 10.
92 ÷ −28.5
b −41 ÷ −18.9
c
83.8 ÷ −11.6
a
a
Challenge
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9
e
d
60 ÷ −5
−77 ÷ 19
13 Copy and complete this multiplication pyramid.
270
15
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–3
–3
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14 Find the value of y.
a
−8 × y = 48
b y × −3 = −36
c
−10 × y = 120
d y × −5 = −40
15 Find the value of z.
a
z ÷ −4 = −8
b z ÷ −2 = 20
c
−36 ÷ z = 9
d 30 ÷ z = −6
16 a
Here is a statement: −3 × (−6 × −4) = (−3 × −6) × −4
Is it true or false? Give a reason to support your answer.
b Here is a statement: −24 ÷ (−4 ÷ −2) = (−24 ÷ −4) ÷ −2
Is it true or false? Give a reason to support your answer.
10
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
1.3 Square roots and cube roots
Exercise 1.3
Key words
Focus
natural numbers
rational numbers
2
3
Work out
a
142
b
Work out
a
43
If possible, work out
a
−64
Practice
5
6
(−30)2
b
(−6)3
c
(−10)3
d
(−1)2 + (−1)3
b
3
−64
c
3
−125
d
3
d
x2 + 121 = 0
d
x3 + 8000 = 0
d
x3 + 12 167 = 0
Solve each equation.
a
x2 = 25
b x2 = 225
c
x2 − 81 = 0
Solve each equation.
a
x3 = 216
b x3 = −216
c
x3 + 1000 = 0
232 = 529 and 233 = 12 167
Use these facts to solve the following equations.
a
x2 = 529
b x2 + 529 = 0
c
x3 = 12 167
Write whether each statement is true or false.
a
9 is a rational number
b −9 is a natural number
c
99 is an integer
d −999 is both an integer and a rational number
e
9999 is both a natural number and a rational number
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d
(−20)2
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c
(−14)2
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1
−729
Challenge
8
a
Copy and complete this table.
x
x2 + x
x3 + x
b
−3
−2
−1
0
0
0
1
Use the table to solve these equations.
i
x2 + x = 2
2
ii
x3 + x = 2
11
Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1 Integers
9
x6 = 64
o
o
x3 = 8
x=2
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Here is an equation: x3 − x = 120
a
Is x = 5 a solution? Give a reason for your answer.
b Is x = −5 a solution? Give a reason for your answer.
10 a
Write 64 as a product of its prime factors.
b Show that 64 is a square number and a cube number.
c
Write 729 as a product of prime numbers.
d Show that 729 is both a square number and a cube number.
e
Find another integer that is both a square number and a cube number.
11 Look at the following solution of the equation x6 = 64
There is an error in this solution. Write a corrected version.
1.4 Indices
Focus
Write as a single power
b 7 × 73
a
32 × 3
Write as a single power
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Exercise 1.4
Key words
2
3
a
a
b
c
c
12 × 125
index
power
d
154 × 15
b 105 × 102
c
36 × 33
d 143 × 144
63 × 63
Show that 20 + 21 + 22 + 23 = 24 – 1
Can you find a similar expression for 20 + 21 + 22 + 23 + 24 + 25?
Read what Zara says:
30 + 31 + 32 + 33 = 34 − 1
Is she correct? Give a reason for your answer.
12
Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.
1.4 Indices
Practice
Write as a single power
c
(7 )
a
Write 4 as a power of 2.
c
Write 93 as a power of 3.
54 = 625
Write as a power of 5
a
6252
b 6253
Find the missing power.
b
Write 43 as a power of 2.
a
b
a
5
6
7
8
3 2
b
42 × 4
= 45
(15 )
3 2
3 3
d
c
74 × 7
c
153 × 15
= 156
d 15
Work out and write the answer in index form.
a
83 ÷ 8
b 56 ÷ 52
d 36 ÷ 33
e
124 ÷ 124
Find the missing power of 6.
a
65 ÷ 6
c
6
= 62
÷ 62 = 66
(3 )
4 5
6254
= 76
× 154 = 154
c
b
68 ÷ 6
d
6
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9
(5 )
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4
210 ÷ 22
= 64
÷ 63 = 63
Challenge
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10 Work out and write the answer in index form.
a
45 ÷ 23
b 94 ÷ 35
c
322 ÷ 26
11 Write as a power of 5
a
125
b 1252
c
4
12 12 = 20 736
Write as a power of 12
a
20 7362
b 20 7363
c
13 Read what Marcus says:
d
272 ÷ 36
1254
20 736
24 = 42 and so 34 = 43
Is Marcus correct? Give a
reason to support your answer.
13
Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication.