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PL
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CAMBRIDGE
Primary Mathematics
Workbook 6
SA
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Mary Wood, Emma Low, Greg Byrd & Lynn Byrd
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
We are working with Cambridge Assessment International Education towards endorsement of this title.
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First published 2014
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Contents
Contents
5
Thinking and Working Mathemetically
6
1
The number system
8
1.1
1.2
Place value
Rounding decimal numbers
8
13
2
Numbers and sequences
18
2.1
2.2
2.3
Counting and sequences
Special numbers
Common multiples and factors
18
23
27
3
Averages
32
3.1
Mode, median, mean and range
32
4
Addition and subtraction (1)
38
4.1
4.2
Positive and negative integers
Using letters to represent numbers
38
43
5
2D shapes48
5.1
5.2
5.3
Quadrilaterals48
Circles53
Rotational symmetry60
6
Fractions and percentages65
6.1
6.2
6.3
Understanding fractions
Percentages
Equivalence and comparison
7
Exploring measures 77
7.1
7.2
Rectangles and triangles
Time
8
Addition and subtraction (2)89
8.1
8.2
Adding and subtracting decimal numbers
Adding and subtracting fractions
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How to use this book
65
69
73
77
83
89
94
3
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Contents
9
Probability98
9.1
Describing and predicting likelihood
98
10 Multiplication and division (1)107
107
111
114
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10.1 Multiplication
10.2 Division
10.3 Tests of divisibility
11 3D shapes119
11.1 Shapes and nets
11.2 Capacity and volume
119
127
12 Ratio and proportion136
12.1 Ratio
12.2 Direct proportion
136
140
13 Angles146
13.1 Measuring and drawing angles
13.2 Angles in a triangle
146
154
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14 Multiplication and division (2)160
14.1 Multiplying and dividing fractions
14.2 Multiplying decimals
14.3 Dividing decimals
160
164
167
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15 Data171
15.1 Bar charts, dot plots, waffle diagrams and pie charts
15.2 Frequency diagrams, line graphs and scatter graphs
171
182
16 The laws of arithmetic194
16.1 The laws of arithmetic
194
17 Transformations199
17.1 Coordinates and translations
17.2 Reflections
17.3 Rotations
199
208
215
Acknowledgements219
4
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How to use this book
How to use this book
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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 more deeply.
Each exercise is divided into three parts. You might not need to work on all of them. Your
teacher will tell you which parts to do.
You will also find these features:
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Important words that you will use.
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Step-by-step examples
showing a way to solve
a problem.
There are
often many different
ways to solve a
problem.
These questions will help
you to develop your skills
of thinking and
working mathematically.
5
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Thinking and Working Mathematically
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Thinking and Working
Mathematically
There are some important skills that you will develop as you learn
mathematics.
Specialising is
when I give an example
of something that fits a rule
or pattern.
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Characterising
is when I explain how
a group of things are
the same.
SA
Generalising
is when
I explain a rule
or pattern.
Classifying
is when I put things
into groups.
6
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Thinking and Working Mathematically
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Critiquing is
when I think about what
is good and what could
be better in my work or
someone else’s work.
Improving
is when I try to
make my work
better.
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M
Conjecturing
is when I think of
an idea or a question
to develop my
understanding.
Convincing
is when I explain
my thinking to someone
else, to help them
understand.
7
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1
The number
system
1.1 Place value
Worked example 1
compose decimal point decompose
Paulo is thinking of a number.
digit hundredths place value
He says, ‘If I divide my number
by 10 and then by 100, the answer
is 0.375.’
regroup tenths thousandths
M
What number is Paulo thinking of?
0.375 × 100 × 10
10
1
0.1
0.01
0.001
0
3
7
5
SA
100
0
3
7
3
7
5
5
To find Paulo’s number, reverse
the operations.
You could replace × 100 × 10
by × 1000.
× 100
× 10
0.375 × 100 × 10 = 375
Answer: Paulo is thinking of 375.
8
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1.1 Place value
Exercise 1.1
Focus
1
Draw a ring around the expression that is equivalent to 0.67.
6
7
60
7
+ +
10 10
10 100
6
10
7
60
70
+
100
100 100
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+
2
What does the digit 5 in 3.065 represent?
3
Magda regroups 56.079 in different ways but two of her answers
are wrong. Which answers are wrong?
A: 5607 tenths + 9 thousandths
B: 56 ones and 79 thousandths
C: 56 + 0.79
M
D: 50 + 6.079
SA
E: 50 + 6 + 0.07 + 0.009
9
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1 The number system
4
Write the operations to complete these multiplication and division loops.
3.7
0.034
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÷ 10
37
0.37
34
0.34
0.98
5
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0.098
98
Complete the place value diagram.
90
+
+
+
0.06
+
SA
91.969
6
Write the number six tenths, four hundredths and five thousandths as a decimal.
Practice
7
Complete the table to show what the digits in the number 47.506 represent.
4
tens
5
6
7
10
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1.1 Place value
9
Find the missing numbers.
a
5.6 × 100 =
b
0.88 × 1000 =
c
41.28 × 10 =
d
670 ÷ 1000 =
e
191 ÷ 100 =
f
6.3 ÷ 10 =
Draw a ring around the expression that is equivalent to 4.063.
A: 4 + 0.6 + 0.3
C: 4 + 0.06 + 0.03
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8
B: 4 + 0.6 + 0.03
D: 4 + 0.06 + 0.003
10 Petra puts some numbers into a function machine.
in
× 1000
out
in
1.5
M
Complete the table to show her results.
out
1500
SA
937
16.24
490
0.07
11 Write the decimal number that is represented by
–4 – 20 –
7
6
9
–
–
100 1000 10
11
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1 The number system
Challenge
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12 Ingrid says, ‘I can multiply by 100 by adding two zeros.’
Explain why Ingrid is wrong.
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13 Filipe multiplies a number by 10, then again by 10 and again by 10.
His answer is 7.
SA
What number did he start with?
14 Four students Anton, Ben, Kasinda and Anya each think of a number.
The numbers are 45, 4.5, 0.45 and 0.045.
Use these clues to work out which number each student is thinking of.
•
Ben’s number is a thousand times smaller than Kasinda’s number.
•
Anton’s number is ten times smaller than Kasinda’s number.
•
Anya’s number is ten times bigger than Ben’s number.
Anton’s number is
Ben’s number is
Kasinda’s number is
Anya’s number is
12
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1.2 Rounding decimal numbers
15 Leila says, ‘The number represented in the place value grid is the
largest number that can be made with nine counters.’
Do you agree?
Explain your reasoning.
1
10
1s
1
100
1
1000
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10s
SA
1.2 Rounding decimal numbers
Worked example 2
Neve has four number cards.
0.25
1.25
2.25
nearest
round
3.25
She chooses two cards.
She adds the numbers on the cards together.
She rounds the result to the nearest whole number.
Her answer is 4.
Which two cards did she choose?
13
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1 The number system
Continued
1.25 and 2.25
Find two numbers that add to 3.5 as 3.5 rounds to 4
or
0.25 and 3.25
You could choose 1.25 and 2.25 or 0.25 and 3.25
Exercise 1.2
Focus
1
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You are specialising when you choose two numbers and check if the total rounds to 4.
Draw lines to show each number rounded to the nearest tenth.
The first one has been done for you.
to the nearest tenth
8.3
8.52
8.6
8.7
8.8
SA
8.35
8.5
M
8.77
8.4
2
3
Draw a ring around all the numbers which equal 10
when rounded to the nearest whole number.
10.53
10.5
10.35
9.55
10.05
9.5
9.05
9.35
a
Round 7.81 to the nearest tenth.
b
Round 7.81 to the nearest whole number.
Tip
Remember
the numbers
could be less
than 10 or
more than 10.
14
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1.2 Rounding decimal numbers
4
Complete the table.
Number
Number rounded to the
nearest tenth
Number rounded to the
nearest whole number
3.78
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4.45
3.55
4.04
Practice
5
Choose the largest number from the list that gives 100 when rounded
to the nearest whole number.
100.55 99.99 100.9 100.45
Use each of the digits 9, 4, 1 and 2 once to make the decimal number
closest to 20.
SA
6
M
100.5 99.5 99.9
7
Pedro has four number cards.
0.45
1.45
2.45
3.45
He chooses two cards.
He adds the numbers on the cards together.
He rounds the result to the nearest whole number.
His answer is 5.
Which two cards did he choose?
and
15
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1 The number system
8
Huan is thinking of a number. She rounds it to the nearest whole number.
She says, ‘My number is the largest number with 2 decimal places that
rounds to 10.’
What number is Huan thinking of?
9
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Challenge
Here are eight numbers.
3.36
2.71
4.03
3.34
3.29
3.15
2.93
3.44
Use the clues to identify one of the numbers.
The number rounds to 3 to the nearest whole number.
•
The tenths digit is odd.
•
The hundredths digit is even.
•
The number rounds to 3.3 to the nearest tenth.
M
•
10 Write the letters of all the numbers that round to 10.5 to the nearest tenth.
What word is spelt out?
B
C
SA
A
D
E
F
G
H
I
10.81
10.56
10.32
10.65
10.44
10.57
10.44
10.43
19.8
J
K
L
M
N
O
P
Q
R
10.48
10.71
10.51
10.58
10.55
9.24
10.59
10.42
10.57
S
T
U
V
W
X
Y
Z
10.44
10.58
10.54
16.25
10.05
10.35
10.46
10.41
16
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1.2 Rounding decimal numbers
11 Stefan says, ‘When I round 16.51 and 17.49 to the nearest whole number,
the answer is the same. When I round 16.51 and 17.49 to the nearest tenth,
the difference between the answers is one.’
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Explain why Stefan is correct.
12 Draw lines from the containers to the circle that shows each
measurement rounded to the nearest litre.
9459
millilitres
7.65 litres
9.91 litres
SA
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10.5 litres
8 litres
10400
millilitres
9 litres
8.82 litres
10 litres
8100
millilitres
11011
millilitres
11 litres
11.1 litres
9.49 litres
17
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PL
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2
Numbers and
­sequences
2.1 Counting and sequences
Worked example 1
position position-to-term rule
Write a sequence of five terms with steps
of constant size that has first term 1 and
2
second term 1 .
3
2
3
1
3
2 ,
3,
3
2
3
M
1 ,
1,
term term-to-term rule
2
3
The step size is the difference between the 1st and 2nd terms. It is .
You could use a number line to help you with the count.
1
2
1
2
SA
1
13
13
2
23
23
3
1
33
2
33
4
18
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2.1 Counting and sequences
Exercise 2.1
Focus
Here is a rocket made of seven shapes.
PL
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1
Magda draws a sequence using the rockets.
1
2
3
M
She records information about the sequence in a table.
Position
1
2
3
28
SA
Term
(number of shapes)
a
Complete the table.
b
What is the term-to-term rule for the sequence?
c
What is the position-to-term rule for the sequence?
d
What is the 25th term in the sequence?
19
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2 Numbers and ­sequences
2
Felipe counts up in steps of 0.3 starting at 4.
Write the first five terms of Felipe’s sequence.
4
Write the next two terms in each sequence.
a
1.4,
b
1
,
2
1,
c
0,
–0.3,
a
Find the position-to-term rule for the numbers in this table.
1.5,
1.6,
1
2
1 ,
1.7,
2,
–0.6,
Position
1
2
1
2
2 ,
–0.9,
–1.2,
9
18
27
36
What is the 10th term of the sequence 9, 18, 27 …?
SA
5
,
Term
4
b
,
M
3
,
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3
$1 = 100 cents
a
Complete the table.
$ (position)
1
2
5
10
100
cents (term)
b
What is the position-to-term rule for the sequence 100, 200, 300 …?
20
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2.1 Counting and sequences
Practice
6
Given the first term and the term-to-term rule, write down the first six
terms of each sequence. Then find the position-to-term rule and the 50th term.
a
First term: 9, term-to-term rule: add 9
First six terms:
50th term:
b
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Position-to-term rule:
First term: 11, term-to-term rule: add 11
First six terms:
Position-to-term rule:
50th term:
a
What is the 5th number in her sequence?
b
What is the 10th number in her sequence?
a
Follow the instructions in the flow diagram to generate a sequence.
SA
8
Safia counts back in steps of 0.5 starting at 2.9.
M
7
Start
b
Write
down 9
Add 9
What is the position-to-term
rule for the sequence?
Write down
the answer
c
Is the
answer
more than
100?
YES
Stop
NO
Imagine the sequence continues forever.
What is the 60th term in the sequence?
21
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2 Numbers and ­sequences
9
Kiki counts in steps of 0.03 starting at 3.26.
What are the next three numbers in her count?
3.26, 3.29, 3.32,
,
,
,
10 Write a sequence of five terms with steps of constant size which has
2
5
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a first term of 1 and a second term of 1 .
Challenge
11 aWrite the first five numbers in a sequence that starts at 42 and has
a term-to-term rule of add 0.15
b
What is the 10th term in the sequence? 12 A sequence has a position-to-term rule of multiply by 6. Complete the table.
Term
M
Position
1
2
30
SA
36
72
13 The numbers in this sequence increase by equal amounts each time.
a
Write the three missing numbers.
10,
,
,
, 42
b
What is the term-to-term rule for the sequence?
c
Salma says, ‘The position-to-term rule is multiply by 8. Is she correct?
Explain your answer.
22
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2.2 Special numbers
14 Ahmed counts back in steps of
3
4
He counts 3 , 3, 2
1
….
4
3
3
starting at 3 .
4
4
Which of these numbers does Ahmed say?
1
2
1
1
2
–4
1
4
–6
3
4
–8
1
4
PL
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4
2.2 Special numbers
Worked example 2
cube number
Use each of the digits 1, 3, 4, 5, 6 and 8 once to make the
following 2-digit numbers.
square number
A square number
M
A cube number
A multiple of 5
SA
A square number
6
8
The only possible cube number is 64.
5 must be in the ones place for the
multiple of 5.
4
A cube number
5
A multiple of 5
1
A square number
That leaves 1, 3 and 8.
Use 8 and 1 to make a square number
and put 3 in the tens place in the
multiple of 5.
6
4
A cube number
3
5
A multiple of 5
Place these digits first.
23
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2 Numbers and ­sequences
Exercise 2.2
Focus
1
Calculate.
a
b
12 =
c
53 =
d
92 =
13 =
What is the sum of the third square number and the fifth square number?
3
What is the difference between the tenth square number and the
fourth square number?
4
Draw a ring around the expressions that are equal to 6².
PL
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2
6×2
6×6
2×2×2×2×2×2
6+6+6+6+6+6
M
Annie and Heidi play a game of ‘What’s my number?’
Annie says
Heidi replies
Is the number less than 50?
No
Is the number more than 100?
No
Is the number a cube number?
Yes
SA
5
6+6
What is the number?
Practice
6
A number is squared and then 2 is added.
The answer is 6.
What is the number?
24
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2.2 Special numbers
8
Calculate.
a
12 × 1 =
b
5 × 52 =
c
3 × 32 =
d
42 × 4 =
Vincent makes a sequence using patterns of rectangular bricks.
a
Draw the next pattern in the sequence.
1
b
2
Complete the table.
Shape
Number of bricks
1
2
1
4
4
3
4
5
How many bricks are needed for shape 10? How do you know?
Write a number between 0 and 100 in each space on the Carroll diagram.
There are lots of possible answers.
SA
9
3
M
c
PL
E
7
Cube number
Not a cube number
Even number
Not an even number
25
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2 Numbers and ­sequences
10 Write each number in the correct place on the Venn diagram.
1
8
9
10
25
50
64
cube
numbers
PL
E
square
numbers
27
Challenge
11 Find two 2-digit square numbers that have a sum of 130.
+
= 130
12 Draw a ring around all the square numbers in this list.
13
23
33
43
53
M
13 Emma uses small cubes to make a larger cube. She uses 16 cubes
to make the base of her cube.
SA
How many small cubes does Emma use to make the larger cube?
How do you know?
14 Put these values in order starting with the smallest.
52
23
33
32
15 Use each of the digits 2, 3, 4, 6, 7 and 8 once to make these numbers.
A square number
A square number
A cube number
A cube number
26
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2.3 Common multiples and factors
2.3 Common multiples and factors
Worked example 3
Write these numbers in the correct place on
the Venn diagram.
common common factor
1
factor multiple multiple
3
4
factors of
20
5
6
7
PL
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2
factors of
24
What is special about the numbers in the shaded area?
M
1, 2 and 4
are common
factors of 20
and 24
factors of
24 are:
1, 2, 3, 4, 6
SA
factors of
20 are:
1, 2, 4, 5
factors of
20
5
2
4
Make sure you include
every number in the
diagram. You could tick
each number as you place it.
7 is not a
factor of
either 20
or 24
factors of
24
1
Tip
3
6
7
The numbers in the shaded area are common factors of 20 and 24.
27
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2 Numbers and ­sequences
Exercise 2.3
Focus
The multiples of 9 are shaded on the hundred square.
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
M
PL
E
1
93
94
95
96
2
97
98
99
a
Draw a ring around all the multiples of 5.
b
List the common multiples of 5 and 9.
SA
1
100
Sofia is thinking of a number.
My number is a
multiple of 2. My number
is a multiple of 7.
28
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2.3 Common multiples and factors
Tick the number that Sofia could be thinking of.
9
4
28
72
Find all the common factors of these numbers.
a
6 and 8
b
8 and 12
PL
E
3
27
Complete the sentence.
Every number with a factor of 10 must also have factors of
Practice
5
and
M
,
Here are four labels.
multiplies of 2
multiples of 7
not a multiple of 2
not a multiple of 7
SA
Write each label in the correct place on the Carroll diagram.
6
28
56
12
48
63
35
55
47
Faisal is thinking of a number. He says, ‘My number is a multiple of 6.’
What three other numbers must Faisal’s number be a multiple of?
29
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2 Numbers and ­sequences
7
Here is a Venn diagram for sorting numbers.
Write each number in the correct place on the diagram.
8 9 10 11 12
multiples of 2
8
PL
E
multiples of 4
Look at this set of numbers.
13 18 21 36 45
Which two numbers are factors of 90?
b
Which two numbers are multiples of 6 and 9?
SA
M
a
Challenge
9
Write these numbers in the correct place on the Venn diagram.
1 2 3 4 5 6 7 8 9
factors of
30
factors of
24
30
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2.3 Common multiples and factors
What is special about the numbers in the shaded area?
10 A light flashes every 4 minutes and a bell rings every 5 minutes.
PL
E
The light flashes and the bell rings at the same time.
How long will it be until this happens again?
11 Omar packs boxes of mangoes
and boxes of peaches.
Each box contains the same
number of fruits.
Mangoes
Organic & Fresh
goes
Mananic & Fresh
Org
Peaches
Organic & Fresh
Omar packs 56 mangoes and
49 peaches.
Ahmed says, ‘There will be 8 pieces of fruit in each box.’
M
Hassan says, ‘There will be 7 pieces of fruit in each box.’
SA
Who is correct? Explain your answer.
12 Cakes are sold in packs of eight.
Mr Mason wants to buy enough cakes to share equally between six people
with no cakes left over.
What is the smallest number of packs he can buy?
Show your working.
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3
Averages
PL
E
3.1 Mode, median, mean and range
Worked example 1
average mean median
What is the range of these ages?
mode range
43, 54, 67, 22, 43, 18, 19, 61, 59
The highest age is 67.
The lowest age is 18.
Find the highest and lowest ages.
67 – 18 = 49
Subtract the lowest from the highest.
Remember to write the range with the correct units.
M
The range is 49 years.
Exercise 3.1
Focus
Fill in the boxes to work out the mean of the following numbers:
SA
1
a
7, 3 and 2
+
+
=
÷3=
The mean is
b
.
10, 4, 7, 4, 5
+
+
+
+
=
÷5=
The mean is
.
32
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3.1 Mode, median, mean and range
2
Work out the range of these masses.
a
2 kg, 5 kg, 11 kg, 2 kg, 10 kg, 9 kg
highest mass
lowest mass
–
150 g, 103 g, 130 g, 127 g, 144 g
highest mass
lowest mass
–
3
range
=
Draw lines to match the descriptions to the set of data.
The range is 5.
5, 6, 5, 7, 8
The mode is 5.
5, 3, 4, 9, 8
The median is 5.
The mean is 5.
2, 6, 4, 7, 4
5, 6, 1, 6, 7
Practice
Jenny and Carrie took a spelling test each week. These are the scores from 8 tests.
Week
Jenny
1
2
3
4
5
6
7
8
9
10
7
3
19
15
12
13
17
7
5
11
12
7
15
6
SA
Carrie
M
4
=
PL
E
b
range
a
Work out Jenny’s mean score and work out Carrie’s mean score.
b
Work out the range of Jenny’s scores and the range of Carrie’s scores.
33
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3 Averages
c
Tick the true descriptions of Jenny and Carrie’s scores.
Jenny has a higher range of scores, so she scored higher than Carrie.
Carrie’s mean average is lower than Jenny, so Jenny scored better
than Carrie in every test.
PL
E
Jenny’s average score was higher, but her scores were less consistent. Carrie’s range is lower, so her scores were less spread out.
Carrie’s average score was lower than Jenny’s.
Erik and Halima recorded how many minutes they practised playing the
guitar for one week. Here are their times in minutes.
Day
1
Erik
6
Halima
9
2
3
4
5
6
7
5
6
8
5
5
7
8
2
9
8
4
9
Work out the mean average time each person practised for.
b
Work out the range of Erik’s scores and the range of Halima’s scores.
M
a
SA
5
c
Write two sentences to describe the amount of time Erik and
Halima practised for.
34
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3.1 Mode, median, mean and range
6
14
15
16
17
18
14
15
16
17
18
19
19
Make sets of six numbers from these cards that have these mean,
mode and median averages:
mean 16, mode 16, median 16
b
mean 17, mode 18, median 17.5
c
mean 16, modes 15 & 17 (bimodal), median 16
PL
E
a
Challenge
The mean has been calculated for each set of numbers below.
One number in each set is hidden. Work out the missing number.
a
The mean is 6.
7
b
M
7
9
6
The mean is 10.
11
9
7
SA
14
c
The mean is 15.
11
d
18
13
The mean is 19.
16
e
19
31
4
7
23
The mean is 51.
47
63
38
49
35
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3 Averages
Five children have worked out the mode, median and range of their heights,
weights and ages. They have recorded them in this table.
Mode
Median
Range
Height
135 cm
132 cm
16 cm
Weight
33 kg
33 kg
16 kg
Age
10 years & 10 months
PL
E
8
11 years & 5 months
10 months
Find a possible solution for the heights, weights and ages of the five
children and record it here.
Child 1
Height:
Weight:
Age:
Child 2
Height:
Weight:
M
Age:
Child 4
Height:
Height:
Weight:
Weight:
SA
Child 3
Age:
Age:
Child 5
Height:
Weight:
Age:
36
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3.1 Mode, median, mean and range
9
These are the times in seconds of two runners in six 100 m races.
Runner 1: 12.7, 10.4, 11.4, 10.8, 12.2, 10.9
Runner 2: 12.5, 11.9, 10.3, 11.6, 10.8, 11.9
Find the mean and range for each runner.
b
Give reasons for who is the better runner.
SA
M
PL
E
a
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PL
E
4
Addition and
subtraction (1)
4.1 Positive and negative integers
Worked example 1
integer negative number
The temperature in Tallinn is –1 °C and in Moscow
it is –8 °C. What is the difference between these
two temperatures?
temperature
in Moscow
temperature
in Tallinn
–8
M
difference 7
–7
–6
–5
–4
–3
–2
–1
positive number
•
Draw a number line.
•
Mark the temperatures.
•Count the number of degrees
between the two marks.
SA
Difference = 7 °C
Exercise 4.1
Focus
1
The temperature at 8 a.m. is –2 °C.
By midday it is 5° warmer.
What is the temperature at midday?
Tip
Remember you
can draw a
number line to
help you with
these questions.
38
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4.1 Positive and negative integers
2
Use the number line to help you answer these questions.
+3
+3
–15
3
+ 3 = –15
Work out the difference between each pair of numbers.
a
6 and –2
d
–5 and 3
Practice
4
b
–15 – 3 =
PL
E
a
b
–3 and –5
c
–4 and –8
e
–6 and –1
f
0 and –2
The table shows the number of rhinos in the world.
The Black rhino has made a comeback
from the brink of extinction.
Number
M
Rhino
About 5000
Greater
one-horned rhino
More than 3500
Javan rhino
56 – 68
Sumatran rhino
80
White rhino
More than
20 000
SA
Black rhino
Source WWF 2020
Use the information in the table to write an estimate of the total number
of rhinos in the world.
39
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4 Addition and subtraction (1)
6
7
How many more people lived in Tokyo than in New York in 2015?
City
Population in 2015
Tokyo
9 273 000
New York
8 582 000
PL
E
5
The temperature is –15 °C.
a
The temperature rises by 6 °C. What is the new temperature?
b
The original temperature falls by 6 °C. What is the new temperature?
At a ski resort, the morning temperature was –11 °C.
M
In the afternoon, the temperature was 5 °C.
SA
What was the difference in temperature between the
morning and the afternoon?
40
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4.1 Positive and negative integers
8
The table shows the temperatures in some cities and the difference
in their temperature from London on one day. Complete the table.
Difference in temperature
from London
City
Temperature (°C)
London
New York
Oslo
–25
10 degrees colder
13 degrees colder
Rio de Janeiro
9
24 degrees colder
PL
E
Moscow
–1
26
Ola wants to find the answer to 1999 + 1476.
Tick (✓) all the calculations that will give the same answer.
2005 + 1470
Challenge
2000 + 1475
2005 + 1400
M
2000 + 1477
2005 + 1500
SA
10 Petra is thinking of a number.
She adds 4896 to her number, then subtracts 5846. She gets the answer 9481.
What number is Petra thinking of?
11 Meera says, ‘I can work out 79 999 – 19 999 in my head.’
Explain how Meera could do the calculation mentally.
Work out the answer to the calculation.
41
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4 Addition and subtraction (1)
12 Here are ten number cards.
–7
2
0
–5
–3
3
9
–8
–4
–2
Choose one card to complete each number sentence.
–7
+
=
–4
–5
–
3
=
Number of years married
Special anniversary
25
Silver
40
50
60
a
PL
E
13 In some countries, people who have been married for many years have
special anniversaries.
Ruby
Golden
Diamond
Mandy and Derek were married in 1972.
b
M
In what year was their ruby anniversary?
Neve and Sean had their diamond anniversary in 2021.
SA
In what year was their silver anniversary?
14 The difference between two numbers is 3.
One number is –2.
What could the other number be?
Find two different answers.
42
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4.2 Using letters to represent numbers
4.2 Using letters to represent
numbers
constant
variable
Worked example 2
PL
E
a and b each represent a number between 1 and 9 inclusive.
Aba knows that a + 4 = b
Write all the values Aba can use to make the statement true.
a = 1 and b = 5
a = 2 and b = 6
a = 3 and b = 7
a = 4 and b = 8
a = 5 and b = 9
Work systematically.
Start with a = 1:
1+4=5
so the value of b is 5
When a = 5, b = 9
M
9 is the largest possible number, so you have
found all the possible answers.
Exercise 4.2
SA
Focus
1
Hamda plays a board game using a dice. She uses the instructions
together with her dice score to work out how many spaces she moves.
d represents the dice score.
For example:
Score
Instruction
Spaces moved
d+4
9 spaces
43
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4 Addition and subtraction (1)
Work out how many spaces Hamda moves.
Score
Instruction
Spaces moved
a
5+d
PL
E
b
3–d
c
d–2
Mira has 10 more bottles of soda than Noura.
SODA
SODA
SODA
SODA
SODA
b
SODA
SODA
SODA
SODA
SODA
Complete the table where m represents the number of bottles that
Mira has and n represents the number of bottles that Noura has.
SA
a
M
2
m
15
n
5
11
2
21
16
Write a number sentence linking m, n and 10.
44
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4.2 Using letters to represent numbers
3
Olaf and Pierre have 23 toy cars altogether.
Olaf has x toy cars and Pierre has y toy cars.
a
Complete this table to show the number of toy cars each boy has.
x (number of toy cars
Olaf has)
7
11
y (number of toy cars
Pierre has)
Write a number sentence linking x, y and 23.
Practice
The diagram shows a right angle divided into two smaller angles.
M
4
18
PL
E
b
4
14
50°
a
SA
Calculate the size of angle a.
a=
5
°
There are x kiwi fruits and y oranges in a bowl.
Meng knows that x + y = 7.
Write three different pairs of values for x and y.
45
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4 Addition and subtraction (1)
6
x and y each represent a number that is a multiple of 5.
x + y = 50
Write all the possible values of x and y.
x
5
y
45
Challenge
7
PL
E
One is done for you.
The perimeter (p) of a regular pentagon is the sum of the lengths of the sides.
b
M
p=b+b+b+b+b
The perimeter of a regular pentagon is 40 cm.
What is the value of b?
The perimeter (p) of a square and a regular pentagon is the same.
SA
8
b
a
p=a+a+a+a
p=b+b+b+b+b
If the perimeter of each shape is 20 centimetres, what is the value of a and b?
46
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4.2 Using letters to represent numbers
a
If □ = 7 and ○ = 5 what is the value of □ + ○ + ○?
b
If a = 7 and y = 5 what is the value of a + b + b?
c
What is the same and what is different about these two questions?
SA
M
PL
E
9
47
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5
2D shapes
5.1 Quadrilaterals
PL
E
bisect decompose
diagonal justify
Worked example 1
parallel trapezia
I am a quadrilateral. All my sides are equal in length.
None of my angles are 90°. I have two pairs of
equal angles. What shape am I?
Square or rhombus …
Cannot be square …
No angles are 90°.
Must be a rhombus.
Two pairs of equal angles.
Name each of these special quadrilaterals.
All the names are in the box.
SA
1
M
Exercise 5.1
Focus
All sides are equal in length.
a
d
kite
rhombus
parallelogram
rectangle
b
c
e
f
g
square
trapezium
isosceles trapezium
48
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5.1 Quadrilaterals
PL
E
a
It has
pairs of equal sides.
b
It has
pair of equal angles.
c
The diagonals cross each other at
d
It has
°.
line of symmetry.
Complete these properties of a rhombus.
There are some diagrams to help you.
a
It has
equal sides.
b
It has
pairs of equal angles.
c
It has
pairs of parallel sides.
d
The diagonals bisect each other at
e
It has
SA
3
Complete these properties of a kite. There is a diagram to help you.
M
2
°.
lines of symmetry.
49
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5 2D shapes
Practice
4
Write down the name of the shape being described.
a
I am a quadrilateral. All my sides are different lengths.
Two of my sides are parallel.
I am a
b
.
I am a quadrilateral. All my sides meet at right angles.
c
I am a quadrilateral. I have two pairs of parallel sides,
two pairs of equal sides and two pairs of equal angles.
None of my angles are 90°. I am a
5
.
PL
E
My diagonals bisect each other, but not at 90°. I am a
.
Jake draws this rhombus and kite. He labels the lines that make the
shapes a, b, c, d and e, f, g and h. He draws the shapes so that b is
parallel to h, and the angles marked x are the same size.
e
b
X
d
M
a
f
X
h
g
c
Write true or false for each of these statements. Justify your answer.
b is parallel to d
ii
h is parallel to f
SA
i
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5.1 Quadrilaterals
iii d is parallel to h
Draw a diagram to show how an isosceles trapezium can tessellate.
SA
Challenge
M
6
PL
E
iv a is parallel to e
7
a
Describe the similarities between a rectangle and a parallelogram.
b
Describe the differences between an isosceles trapezium and a kite.
51
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5 2D shapes
8
Put the shapes a to f through this classification flow chart and
write down the letter where each shape comes out. For example,
when you start with the square, you end at the letter H.
Start
Yes
One pair of
equal angles?
G
Yes
H
Square
c
Rhombus
e
Kite
No
Yes
All angles 90°?
No
Two lines of
symmetry?
J
No
I
No
One pair of
parallel sides?
K
b
Rectangle
d
Parallelogram
f
Isosceles trapezium
Yes
L
A, B and C are three points shown on this grid. D is another point on the grid.
M
9
a
No
Diagonals meet
at 90°?
PL
E
Yes
A
SA
y
10
9
8
7
6
5
4
3
2
1
0
a
B
C
0 1 2 3 4 5 6 7 8 9 10
x
When D is at (7, 4) is quadrilateral ABDC a square? Explain your answer.
52
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5.2 Circles
b
Point D moves so that quadrilateral ABCD is a parallelogram.
What are the coordinates of point D?
c
Point D moves so that quadrilateral ABDC is a kite.
Write down two possible sets of coordinates for the point D.
Worked example 1
PL
E
5.2 Circles
centre circumference
compasses diameter
Label the parts of this circle.
SA
M
radius
Circumference is
the perimeter.
radius
circumference
Centre is in the
middle.
Radius is the
distance from
the centre to the
circumference.
centre
diameter
Diameter is the
distance across
the circle, going
through the centre.
53
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5 2D shapes
Exercise 5.2
Focus
1
This is how Tami labelled the parts of a circle.
radius
PL
E
circumference
diameter
centre
2
M
Explain the mistakes she has made.
Measure the radius of each of these circles.
SA
a
radius =
cm
b
radius =
mm
54
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5.2 Circles
Draw a circle with a radius of
a
40 mm
M
Practice
4
b
3 cm
PL
E
3
Draw a circle with a radius of
3.7 cm
b
52 mm
SA
a
55
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5 2D shapes
a
A radius of 5 cm is the same as a diameter of 10 cm.
b
A diameter of 6 cm is the same as a radius of 12 cm.
c
A radius of 70 mm is the same as a radius of 7 cm.
d
A radius of 45 mm is the same as a diameter of 9 cm.
a
Draw a dot and label the point C.
Make sure there is about 5 cm of space above, below, to the left
and to the right of your point.
b
Draw the set of points that are exactly 4.2 cm from the point C.
SA
M
6
Write true or false for each of these statements.
PL
E
5
56
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5.2 Circles
Challenge
a
Draw a circle with radius 7 cm. Label the circle A.
SA
M
PL
E
7
57
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5 2D shapes
b
Draw a circle with radius 3 cm, inside circle A so that it touches circle A.
Label the circle B.
Your diagram should look something like this.
A
A
or PL
E
B
B
With a ruler, accurately measure the distance between the centre
of circle A and the centre of circle B.
d
What do you notice about your answer to part c and the radii
measurements of circles A and B?
e
Draw two more circles that touch inside. Choose your own radii
measurements. Measure the distance between the centres of
your two circles. What do you notice?
f
Complete this general rule:
M
c
The distance between the centres of two touching circles that touch
inside is the same as the
Zara wants to draw a pattern made of squares inside circles like this.
SA
8
.
This is the method she uses to
draw one of the squares in a circle.
Step 1: Draw a square.
Step 2: Guess where the centre
of the square is and mark a dot.
Put the point of the compasses on this dot
and open the compasses so that the pencil
is on a corner of the square.
Step 3: Draw a circle. If the corners of the
square don’t touch the circle, rub the
circle out and try again with a
different centre point!
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5.2 Circles
Try to draw a square in a circle using Zara’s method.
What do you think of her method?
SA
M
PL
E
a
b
Can you improve on her method? If you can think of a better method,
write it down.
59
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5 2D shapes
5.3 Rotational symmetry
order
Use tracing paper to work out the order of rotational
symmetry of a rectangle.
rotational symmetry
Step 1: Trace the shape.
PL
E
Worked example 3
SA
M
Step 2: Put your pencil on the centre of
the shape.
Step 3: Turn the tracing paper one full turn and count the number of
times the shape fits on itself.
Start
Once
Twice
The rectangle fits on itself twice, so it has order 2.
60
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5.3 Rotational symmetry
Exercise 5.3
Focus
Use tracing paper to work out the order of rotational symmetry
of these shapes.
a
d
c
e
f
Match each shape to its order of rotational symmetry.
b
c
SA
a
i
3
M
2
b
PL
E
1
Order 4
ii
Order 2
d
iii Order 1
a
Draw the line of symmetry on to the triangle.
b
Write down the order of rotational symmetry of the triangle.
iv Order 3
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5 2D shapes
Practice
4
Write down the order of rotational symmetry of these shapes.
b
c
5
b
c
M
Write down the order of rotational symmetry of these road signs.
a
Challenge
b
c
Write the letter of each shape in the correct space.
Shape A has been done for you.
SA
7
d
Write down the order of rotational symmetry of these patterns.
a
6
PL
E
a
Number of lines of symmetry
0
1
2
3
4
1
Order of rotational
symmetry
2
3
4
A
62
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5.3 Rotational symmetry
C
E
D
F
Mali is making a pattern from grey and white squares.
This is what she has drawn so far.
SA
M
8
B
PL
E
A
a
On this copy shade in one more square so that
the pattern has order 2 rotational symmetry.
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5 2D shapes
c
On this copy shade in seven more squares so that the
pattern has order 2 rotational symmetry.
PL
E
On this copy shade in five more squares so that the pattern
has order 4 rotational symmetry.
Sadik has these nine squares.
M
9
b
SA
He wants to arrange the squares to form a square pattern with two lines of
symmetry and rotational symmetry order 2. Show two ways that he can do this.
64
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PL
E
6
Fractions and
percentages
6.1 Understanding fractions
Worked example 1
denominator improper fraction
Parveen is thinking of a number.
mixed number numerator
She says, ‘Two-thirds of my number is 30.’
operator proper fraction
What number is Parveen thinking of?
1
2
is half of
3
3
M
1
of the number is 30 ÷ 2 = 15
3
3
of the number is 15 × 3 = 45
3
3
1
is three times
3
3
SA
Parveen is thinking of 45.
Exercise 6.1
Focus
1
Represent these divisions as fractions.
a
5 divided by 8
b
4 divided by 3
c
8 divided by 7
d
7 divided by 10
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6 Fractions and percentages
a
Show one way that five children can share 2 pizzas equally between them.
PL
E
2
How much pizza does each child get?
b
Show one way that two children can share 5 pizzas equally between them.
What is
3
of 16?
2
SA
3
M
How much pizza does each child get?
4
Would you rather be given
1
1
of $18 or of $40? Explain your decision.
2
4
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6.1 Understanding fractions
Practice
5
Work out the answer to each question to help you find your way through the maze.
start
5
of 16
2
5
of 18
3
63
27
6
6
of 15
5
5
of 22
2
24
14
4
of 15
3
7
of 16
4
28
20
end
SA
7
of 12
6
55
90
M
30
7
of 9
3
PL
E
40
48
6
Brian reads
1
of a 15-page book.
3
Carlos reads
3
of an 8-page book.
4
Who reads more pages?
Explain how you know.
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6 Fractions and percentages
7
8
Complete this table to show fractions of 36.
Fraction
1
4
Amount
9
3
4
5
4
7
4
9
4
11
4
Find the missing numbers.
of 24 =
3
2
of 24 =
7
2
of 24 =
PL
E
4
3
24
8
3
Challenge
Which is bigger:
3
4
of 32 or of 18?
4
3
M
9
of 24 =
SA
Explain how you know.
10 a
Mandy is thinking of a number.
She says, ‘Five-thirds of my number is 45.’
What number is Mandy thinking of?
b
Ollie is thinking of a different number.
He says, ‘Ten-ninths of my number is 90.’
What number is Ollie thinking of?
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6.2 Percentages
11 Use each of the numbers 2, 3, 4 and 5 once to complete these statements.
of 6 = 4
of 12 = 15
PL
E
6.2 Percentages
Worked example 2
percentage
Leo has two apple trees in his garden. He labels them A and B.
per cent
M
operator
SA
A
B
Tree A produces 40 kg of apples.
Tree B produces 10% more than tree A.
How many kilograms of apples does tree B produce?
10% of 40 =
1
of 40 = 4
10
Find 10% of 40 and add it to 40 to find the
mass of the apples produced by tree B.
40 + 4 = 44
Tree B produces 44 kg of apples.
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6 Fractions and percentages
Exercise 6.2
Focus
Find 10% of the quantity, then use your answer to find 20%, 30% and so on.
Write the answers under the percentages on the number line.
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
PL
E
1
$200
2
3
Find 10% of these quantities.
a
600
c
$40
b
90 cm d
170 kg
Join each box to the correct amount.
M
Amount
SA
50% of 40
10% of 120
100% of 16
10
12
14
16
18
20
Practice
4
a
Find 10% of 80
b
Find 5% of 80
c
Find 15% of 80
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6.2 Percentages
5
Here is a grid of ten squares.
What percentage of the grid is shaded?
a
10
%
10
0%
80
75%
20%
80
M
%
50
%
25
6
%
10
75%
0%
10
%
7
60
30
SA
5%
%
b
PL
E
Complete these percentage diagrams.
50
6
Haibo makes 250 grams of fruit and nut mix.
15% of the mix is raisins. 25% of the mix is cranberries. The rest is almonds.
How many grams of almonds does Haibo use?
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6 Fractions and percentages
Challenge
Huan has two plum trees A and B in her garden.
PL
E
8
A
B
Tree A produces 30 kg of plums.
Tree B produces 10% less than tree A.
9
M
How many kilograms of plums does Huan get from both trees together?
75% of a number is 48.
Tip
The answer is not 36.
SA
What is the number?
10 50% of children in a sports club go swimming.
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6.3 Equivalence and comparison
PL
E
50% of the children who go swimming also dive.
Five children swim and dive.
How many children are in the sports club?
children
M
6.3 Equivalence and comparison
Worked example 3
SA
Write these numbers in order of size,
starting with the smallest.
2
5
0.2 25% 25% = 0.25
3
= 0.3
10
equivalent fraction
simplest form simplify
3
0.23
10
Write the fractions and percentages as
equivalent decimals.
2
4
=
= 0.4
5 10
In order:
0.2, 0.23, 0.25, 0.3, 0.4
0.2, 0.23, 25%,
3 2
,
10 5
Write the decimals in order starting with
the smallest.
Write the original values in order.
73
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6 Fractions and percentages
Exercise 6.3
Focus
1
Write these fractions in their simplest form.
a
b
8
20
c
9
12
What fraction of the shape is shaded?
PL
E
2
5
10
Write your answer in its simplest form.
Write these prices in order starting with the smallest.
M
3
$4.07 74 cents $4.70 $0.47 $7.40
Which numbers are equivalent to
25
0.75 0.25
100
SA
4
0.5 1
?
4
Practice
5
Write these numbers in their simplest form.
a
13
3
15
b
5
14
35
c
10
36
45
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6.3 Equivalence and comparison
7
Use the symbols <, > or = to make these statements correct.
70%
0.65 60%
23%
1
5
1
4
0.06 25%
4
5
0.7
2
5
0.3
Here are three statements about fractions and percentages.
PL
E
6
Tick (✓) the statements that are true.
Cross (✗) the statements that are not true and write the correct statement.
3
is equal to 35%
5
b
7
is equal to 7%
100
c
9
is equal to 9%
10
Which two cards make the number sentence correct?
M
8
a
6
8
0.35
3
6
39%
70%
1
2
SA
>
0.5
Challenge
9
Omar and Hassan do the same test.
Omar scores 70 out of 80. Hassan scores 70%.
Who has the higher score? Explain how you know.
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6 Fractions and percentages
10 Write these numbers in order of size, starting with the largest.
4
5
0.7 0.82 75% 13
20
11 Find two pairs of equivalent fractions from this list.
Circle the fraction in its simplest form in each pair.
PL
E
16
10
9
6
4
2
20
15
12
10
5
3
12 Circle the smaller number in each box.
1
2
1 1.2
1
3
M
1 1.3
1
4
SA
1 1.4
1
5
1 1.5
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PL
E
7
Exploring
measures
7.1 Rectangles and triangles
Worked example 1
area
SA
M
Estimate the area of this triangle.
Centimetre squared paper.
10 whole squares are covered by
the triangle.
Count whole squares that are covered
by the triangle.
There are no squares that are less
than half covered.
Do not count squares that are less than
half covered.
There are 5 half squares covered by
1
the triangle, that makes 2 whole
2
squares covered.
Pair up squares that are half covered to
make whole squares.
I estimate that the area of the
1
triangle is 12 cm2.
Write your estimate using units of area. The
squares are centimetre squares so we use cm2.
2
77
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7 Exploring measures
Exercise 7.1
Focus
1
Multiply the width of the rectangle by its length to calculate each area.
Write your answer using units of area.
7m
PL
E
9 km
8 cm
9 km
4m
Circle the rectangles that have been divided into two equal pieces.
SA
M
2
3 cm
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7.1 Rectangles and triangles
3
a
What is the area of this rectangle?
PL
E
3 cm
4 cm
Divide the area of the rectangle by 2 to find the area of one of the triangles.
What is the area of the triangle?
c
Explain why dividing the area of the rectangle by 2 gives us the
area of the triangle.
Practice
Estimate the area of each triangle to the nearest half centimetre.
SA
4
M
b
79
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7 Exploring measures
5
Circle the grey triangles that have an area of 9 cm2.
You will need to measure the sides of the rectangles.
SA
e
d
M
c
b
PL
E
a
6
Work out the length of the missing side.
m
Area of this triangle is 12 m2
8m
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7.1 Rectangles and triangles
Challenge
Draw some different right-angled triangles on the squares that have
an area of 6 cm2.
M
PL
E
7
SA
How could you test that the triangle has an area of 6 cm2?
81
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7 Exploring measures
8
Chata is painting triangular tiles to create a mosaic.
This is the actual size of one tile.
2 cm
2 cm
PL
E
One 15 ml pot of paint covers 18 cm2.
15 ml
What is the total number of paint pots that he will need to buy to cover 75 tiles?
9
6 cm
M
2 cm
SA
4 cm
6 cm
a
What is the area of the large square?
b
What is the area of each white triangle?
c
What is the area of the grey square?
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7.2 Time
7.2 Time
Exercise 7.2
Focus
2
Convert these times into minutes and seconds.
a
2 minutes =
b
2.5 minutes =
minutes and
seconds
c
3.25 minutes =
minutes and
seconds
d
3.75 minutes =
minutes and
seconds
minutes and
seconds
PL
E
1
Draw lines to match the same times.
1.25 days
M
2 hours and 45 minutes
1.25 hours
SA
4 days and 12 hours
5.5 minutes
5 hours and 30 minutes
5.5 hours
1 hour and 15 minutes
2.75 minutes
4.5 days
2.75 hours
2 minutes and 45 seconds
5 minutes and 30 seconds
4 hours and 30 minutes
1 day and 6 hours
4.5 hours
83
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7 Exploring measures
Practice
3
This line graph shows equivalent amounts of time in hours and in minutes.
65
60
55
PL
E
50
45
Minutes
40
35
30
25
20
15
M
10
5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Hours
SA
–5
Use the conversion graph to convert these times from hours to minutes.
a
0.2 hours =
minutes
b
0.7 hours =
minutes
c
0.45 hours =
minutes
d
0.95 hours =
minutes
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7.2 Time
4
2045
Tuesday
Wednesday
Thursday
Friday
Saturday
Sunday
31
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
PL
E
Monday
This is a page from a calendar for the year 2045.
Use what you know about the months of a year and the number
of days in each month to answer these questions.
Suggest two months that this calendar page could be for.
Explain how you know.
b
If the calendar is from the second half of the year,
what is the date circled?
SA
M
a
c
Use what you know about years, months and days to work out
how old each of these people will be on the circled date.
i
Cheng was born on 11th July 2013, he will be
and
ii
iii
years,
months
years,
months
days old.
Magda was born on 27th July 1975, she will be
and
months
days old.
Halima was born on 2nd April 2006, she will be
and
years,
days old.
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7 Exploring measures
iv
Jack was born on 18th October 1972, he will be
months and
v
years,
days old.
Stefan was born on 30th September 1966, he will be
months and
years,
days old.
Challenge
Six people are at a railway station waiting for six trains to
different destinations.
PL
E
5
Use the clues to work out the destination of each train.
Destination
Departure time
11:48
12:18
12:58
13:23
M
13:53
14:28
Clues
SA
The train for Barcelona leaves later than the train for Brussels,
but before the train for Venice.
The train for Brussels leaves between 12 o’clock and 1 o’clock.
The train for Vienna leaves later than the train for Copenhagen,
but before the train for Barcelona.
The train to Vienna leaves 40 minutes before the next train.
The train that leaves at 7 to 2 is going to a place with six
letters in its name.
The train for Warsaw leaves before the train for Venice but later
than the train for Barcelona.
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7.2 Time
Tip
Use this logic table to work out which train
left when. Put a cross in any places that cannot
be the correct time for that train. Tick the time
when you know it is correct for that train.
Barcelona
Venice
Brussels
Vienna
Warsaw
Copenhagen
12:58
13:23
13:53
14:28
Make your own timetable. Use 24-hour clock times.
M
6
12:18
PL
E
11:48
There are five stations on the line. You can name the stations.
There are three trains. They can go back and forth along the line
as many times as you like.
SA
These are the times between stations.
0.95 hours
1.2 hours
0.4 hours
1.15 hours
Trains must stop at each station for 2 minutes.
At the end of the line the train stops for 0.3 hours before going back.
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7 Exploring measures
From
to
:
PL
E
to
:
SA
M
From
Tip
Look at other bus or train timetables for ideas.
In the first table, write the arrival and departure times from
the first station to the fifth station.
In the second table, write the arrival and departure times from
the fifth station back to the first station.
88
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PL
E
8
Addition and
subtraction (2)
8.1 Adding and subtracting
decimal numbers
Worked example 1
decimal place
Add together all the numbers greater than 0.7
trailing zero
0.507 0.8 0.38 0.09 0.747 0.699
Estimate:
Identify the numbers greater than 0.7
M
0.8 + 0.747
Estimate the answer and then add the numbers.
0.8 + 0.7 = 1.5
SA
0.800
+ 0.747
1.547
1
Align the decimal points correctly and add
trailing zeros so all the numbers have the same
number of decimal places.
Compare your answer with the estimate to
check it is reasonable.
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8 Addition and subtraction (2)
Exercise 8.1
Focus
1
Abi pays $2 to buy some cheese that costs $1.35.
Complete this method to show how much change Abi gets.
1.35
2
+
0.9
+
PL
E
2
–
+
+
0.05
+
+
=$
0.007
0.01
A chef is preparing school dinners.
She weighs 6.3 kg of pasta.
She needs 10 kg.
3
M
How much more pasta does she need?
Calculate.
4.504 + 15.096 =
SA
a
b
4
4.985 – 2.347 =
Here are six number cards.
0.003
0.01
0.006
0.004
Use each card once to complete these two calculations.
+
=
+
=
90
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8.1 Adding and subtracting decimal numbers
Practice
A shop has these items for sale.
Coffee maker
$29.95
Toaster
$30.75
Can opener
$14.25
Ice cream maker
$26.80
PL
E
5
Ravinder buys an ice cream maker and a coffee maker.
6
a
How much does he spend altogether?
b
How much change does he get from $60?
Ai cuts a piece of wood measuring 1 metre into three pieces.
0.54 m
?
M
0.2 m
not to scale
SA
How long is the last piece?
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8 Addition and subtraction (2)
7
Decimal pyramids are built like this.
Each number is the total of
the two numbers below.
2.9
Build these pyramids.
b
a
Challenge
4.8
5.9
5.6
6.12
3.6
0.023
Find the missing digits.
a
7
SA
9
–
7
6
3
9
7.323
M
1.07
8
1.7
PL
E
1.2
6
8
b
6
3
+
3
8
1
3
5
7
Find three decimals that add to 1.
Two of the decimals must have three decimal places and
one must have two decimal places.
92
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8.1 Adding and subtracting decimal numbers
10 Draw lines from two bags to each box to make the total mass
written on the box.
4.1 kg
1.19 kg
0.34 kg
2.7 kg
3.8 kg
5.5 kg
4.8 kg
0.49 kg
PL
E
0.27 kg
1.2 kg
8.7 kg
5.99 kg
4.9 kg
1.4 kg
0.92 kg
M
2.9 kg
0.86 kg
Label the last box with the total of the remaining two bags.
SA
Use this space for working.
93
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8 Addition and subtraction (2)
8.2 Adding and subtracting
fractions
common denominator
Worked example 2
5 4
–
2 3
Multiples of 2: 2, 4, 6 …
Multiples of 3: 3, 6 …
Find a common denominator by looking
at multiples of 2 and 3.
Change
5 15
=
2
6
5
4
and to equivalent fractions
2
3
with a denominator of 6.
4 8
=
3 6
7
6
Simplify if possible.
1
6
SA
=1
Subtract the numerators.
M
5 4 15 8
– =
–
2 3
6
6
=
PL
E
Calculate
denominator
Write improper fractions as mixed
numbers.
Exercise 8.2
Focus
1
Olaf draws a diagram to show
3 2
– .
4 3
What answer should Olaf write?
94
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8.2 Adding and subtracting fractions
Calculate.
a
3
1 2
+
4 3
c
3 2
+
8 3
3 1
–
5 2
b
3 2
–
4 3
c
4 2
–
5 3
Write the letters of the calculations in the correct place in the diagram.
A
2 3
+
3 8
B
3 2
+
8 5
Practice
C
2
5
+
3 15
D
3
3
+
4 10
Answer equal to 1
Answer more than 1
M
Answer less than 1
In this diagram the number in each box is the sum of the two
missing numbers below it.
SA
5
b
Calculate.
a
4
1 4
+
4 5
PL
E
2
Write the missing numbers.
1
4
2
3
1
6
Tip
Write all the
fractions in
twelfths.
95
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8 Addition and subtraction (2)
6
Chata solved this calculation.
3 3
6
+ =
5 8 13
What mistake has Chata made?
7
Alana spends
PL
E
Correct Chata’s answer.
3
3
hour preparing an experiment and
hour
4
10
doing the experiment.
8
M
How long did Alana spend on the experiment altogether?
Each student in a club chooses to play cricket, tennis or rounders.
SA
2
of the students play tennis.
5
1
of the students play cricket.
3
What fraction of the students play rounders?
Challenge
9
Write the missing number.
4
10 Find the value of 1 –
7
1
– 10
= 20
3 2
–
8 5
96
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8.2 Adding and subtracting fractions
5
hours, then takes a break.
3
7
She drives another hours to reach her destination.
4
11 Heidi drives for
How long did Heidi spend driving?
PL
E
12 Write the missing numbers to make this calculation correct.
5
+
1
7
= 10
SA
M
Can you find more than one answer?
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9
Probability
Worked example 1
PL
E
9.1 Describing and predicting
likelihood
equally likely outcomes event
If you spun this spinner 20 times,
how many times would you predict
that it would land on ‘1’?
2
4
3
probability probability experiment
M
1
mutually exclusive events outcome
What is the probability of the
spinner landing on ‘1’?
1
or 25%.
4
SA
1 out of 4, or
What is
5
1
of 20?
4
I predict that the spinner will land
on ‘1’ five times.
1 out of 4 equal sections of the spinner is ‘1’.
There is a 1 out of 4 chance of the spinner
landing on ‘1’.
If there is a 1 out of 4 chance of the spinner
landing on ‘1’ you can expect the spinner
to land on ‘1’ for
1
of the spins.
4
Remember, this does not mean that the spinner
will definitely land on ‘1’ five times. Each time
1
the spinner is spun there is a chance that it
4
will land on ‘1’, so it is random. The more trials
you do the more likely it is that the spinner will
1
land on ‘1’ of the times.
4
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9.1 Describing and predicting likelihood
Exercise 9.1
Focus
1
Arun, Marcus, Sofia and Zara have been investigating probability. They are taking
shapes out of bags without looking. Draw a line from each child to a bag.
There is a 4 out of 4
chance of taking
a pyramid from
my bag.
PL
E
A
B
There is a 2 out of 4
chance of taking a
prism from my bag.
There is a 1 out of 4
chance that I shall take
a cube out of my bag.
M
C
D
3
5
SA
There is a 0 out of 4
chance of taking
a pyramid from
my bag.
1
2
4
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9 Probability
One of these bags of shapes does not belong to any of the children.
Complete the statements for this extra bag of shapes.
The probability of taking a prism from the bag is
.
is
.
Remember
that cubes
and cuboids
are types of prism.
PL
E
The probability of taking a 3D shape from the bag
Tip
The probability of taking a pyramid from the bag
is
2
.
Here is a set of cards used in a game.
8
3
2
2
4
2
3
5
M
Tasha is going to investigate the chance of taking a card with a triangle symbol.
Circle the cards with a triangle symbol.
Put an X next to the events below that could not occur at the same time
as taking a card with a triangle symbol.
SA
Taking a card with a square symbol
Taking a ‘2’ card
Taking a card with a value greater than 4
Taking a card with an odd number
3
Tear 10 small pieces of tissue paper.
Draw a circle on a piece of paper and place the paper on the floor.
Drop the tissue paper pieces onto the paper on the floor.
How many pieces landed inside the circle?
out of 10 pieces landed inside the circle.
100
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9.1 Describing and predicting likelihood
Conduct a probability experiment to see how many pieces land
inside the circle in each of 50 trials.
Record your results in this table.
Number of pieces
inside the circle
Tally
Total
1
2
3
4
5
6
7
M
8
PL
E
0
9
10
Describe the results of your experiment.
out of 50 chance of
The experiment showed that there was a
10 pieces of tissue landing in the circle.
out of 50 chance of
SA
The experiment showed that there was a
0 pieces of tissue landing in the circle.
The experiment showed
101
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9 Probability
Practice
4
This is a net of a six-sided dice.
Number the dice so that:
There is a 50% chance of throwing a number
greater than 5.
There is a greater than 50% chance that the number thrown will be even.
PL
E
There is a 0% chance that the number thrown will be a multiple of 3.
The chance of throwing a number less than 0 is greater than 0%,
but less than 50%.
5
Kapil says that rolling a ‘4’ on an ordinary 6-sided dice and rolling an
odd number on a dice are mutually exclusive.
M
Is Kapil correct?
Complete the sentence.
SA
Two events are mutually exclusive when they
6
1
2
3
5
4
What is the chance of spinning a 2?
/
If you spin the spinner 50 times how many 2s would you expect to see?
Use a pencil and paperclip like this to complete the spinner.
1
2
3
5
102
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9.1 Describing and predicting likelihood
SA
M
PL
E
Conduct the experiment by using the spinner 50 times.
Use this space to draw a tally chart for your outcomes.
Describe the results of your experiment.
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9 Probability
Challenge
Under 13 years old
Not under 13 years old
Boys
Cheng
Farrukh
Scott
Girls
Gemma
Sunita
Talia
David
PL
E
Dan
a
How many children are members of the club?
b
One child is chosen at random for a lesson. What is the chance that:
i
The child is a girl under 13 years old?
ii
The child is a boy not under 13 years old?
iii
The child is a boy?
iv
The child is not under 13 years old?
v
The child is not a boy under 13 years old?
Lola has six t-shirts. She takes a t-shirt at random.
SA
8
Some children are members of a tennis club.
This Carroll diagram shows the children in the club.
M
7
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9.1 Describing and predicting likelihood
These are four events that could occur
when Lola takes a t-shirt.
Event A: The shirt has a picture of an animal on it.
Tip
Event B: The shirt has a picture of fruit on it.
If Event A and
Event B are
mutually exclusive
put a tick (✓) in
the grey box.
Event C: The shirt is striped.
Event D: The shirt has a collar.
A
B
PL
E
Tick the pairs of events that are mutually
exclusive for Lola’s set of t-shirts.
C
D
E
Event E:
M
Write your own Event E which is mutually exclusive to Events A and B,
but not mutually exclusive to Events C and D.
SA
Complete the table for your own event.
9
Marco has put 4 different coloured balls into a bag.
He is going to conduct a chance experiment and record the outcomes
when he takes and replaces a ball 100 times from the bag.
These are his predictions:
•
I expect to take a red ball 20 times.
•
I expect that none of the balls I take will be blue.
•
I expect to take a yellow ball 25 times.
•
I expect to take a purple ball 50 times.
•
I expect to take a green ball 5 times.
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9 Probability
a
Colour the picture to show what colour balls Marco put
into the bag.
b
Add 5 more coloured balls to the bag. Write your
predictions for how many of each colour you expect to
take if you repeated Marco’s experiment now.
I expect to take
I expect to take
I expect to take
I expect to take
blue balls.
yellow balls.
purple balls.
green balls.
If you carried out the experiment would you expect your
results to be exactly as you predicted? Why?
SA
M
c
red balls.
PL
E
I expect to take
106
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PL
E
10 Multiplication
and division (1)
10.1 Multiplication
Worked example 1
product
Calculate 256 × 27
Estimate 250 × 20 = 5000 and
250 × 30 = 7500
Start by estimating the size of
the answer.
The answer is between 5000 and 7500.
6
×
2
7
7
9
2
1
2
0
9
1
2
SA
5
5
6
•
Multiply 256 by 7
•
Multiply 256 by 20
•
Add the two answers together
M
1
2
The order you do the multiplications
does not matter.
1
Answer: 6912
Use your estimate to check that your
answer is reasonable.
107
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10 Multiplication and division (1)
Exercise 10.1
Focus
1
Complete the cross number puzzle.
1
2
3
5
PL
E
4
6
7
9
11
8
10
12
13
15
M
14
DOWN
1.
171 × 9
1.
361 × 4
3.
8×9
2.
158 × 6
4.
528 × 8
3.
927 × 8
5.
502 × 9
6.
748 × 2
7.
253 × 5
7.
3×6
9.
732 × 4
8.
628 × 8
11. 224 × 7
9.
513 × 5
12. 128 × 4
10. 956 × 3
13. 157 × 4
12. 117 × 5
SA
ACROSS
14. 774 × 2
15. 6 × 9
108
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10.1 Multiplication
Practice
2
Calculate each product.
3
a
1546 × 7 =
b
2398 × 8 =
c
3594 × 6 =
PL
E
You must write an estimate before doing the calculation.
Write in the missing digits to make this calculation correct.
7
6 ×
1
4
0
3
2
Nailah estimates 2999 × 70 = 210 000
Has she made a good estimate?
Use the digits 0, 1, 5 and 9 to complete this calculation.
SA
5
M
Explain your answer.
6
×
= 1350
Hassan finds the product of two multiples of 10.
The answer is 12 000.
List all the calculations that give his answer.
109
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10 Multiplication and division (1)
Challenge
7
Ella and Roz complete the same multiplication.
Ella
7
Roz
3
×
1
4
1
7
1
5
2
4
×
1
9
3
8
0
6
9
0
0
6
2
8
0
7
3
4
5
2
4
4
6
9
0
0
2
9
2
8
0
7
6
1
8
0
1
1
PL
E
2
4
1
1
Who has the correct answer?
M
What mistake has the other girl made?
Find the product of 6589 × 37
9
The distance from London to Budapest is 1723 kilometres.
SA
8
Mary lives in Budapest and travels to London and back six times.
How far does she travel?
10 The table shows ticket sales during one day at four theatres.
Theatre
Apollo
Lif
Legend
Mani
Number of tickets sold
2108
1935
2245
1649
Cost of ticket
$45
$39
$42
$47
110
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10.2 Division
Which theatre took the most money during the day?
PL
E
You must show all your working.
10.2 Division
Worked example 2
dividend divisor
A coach operator puts a first aid kit on 378 coaches.
quotient remainder
The first aid kits are sold in boxes of 18.
M
How many boxes are needed?
Estimate: 380 ÷ 20 = 19 so the answer
will be approximately 19
SA
21
18 378
– 360
18
–18
18 × 20
18 × 1
0
Start by making an estimate.
There are twenty 18s in 378. Record
2 tens on the answer line. Subtract 360
(18 × 20) from 378 to leave 18.
There is one 18 in 18. Record 1 one on
the answer line. Subtract 18 (18 × 1) from
18 to leave 0. There is no remainder.
Answer: 21 boxes are needed
Exercise 10.2
Focus
1
Calculate 837 ÷ 9
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10 Multiplication and division (1)
2
Four friends have lunch each day at a café. The total cost for the
week is $152. They share the cost equally.
How much does each person pay?
3
Henryk pays $9 each week to go to the gym. He has paid $747 so far.
A team of volunteers made a total of $992 by selling T-shirts for charity.
M
4
PL
E
How many weeks has he been to the gym?
Each T-shirt costs $8.
SA
How many T-shirts did the volunteers sell?
Practice
5
6
Work out the value of the missing digits.
a
49
÷ 5 = 98 remainder 3
b
65
÷ 9 = 72 remainder 6
Calculate 936 ÷ 12
112
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10.2 Division
7
An art gallery collected $600 in entrance fees in 1 hour.
Art Gallery
PL
E
entrance fee
$12 per
person
How many people paid to enter the gallery during the hour?
8
A teacher needs 240 exercise books.
The exercise books are sold in packets of 16.
Challenge
9
M
How many packs must the teacher buy?
Leanne and Carrie complete the same division.
720 ÷ 24
SA
Leanne
720 ÷ 4 ÷ 6
Carrie
720 ÷ 24
720 ÷ 20 ÷ 4
= 180 ÷ 6
= 36 ÷ 4
= 30
=9
Who has the correct answer?
What mistake has the other child made?
113
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10 Multiplication and division (1)
10 Find the missing digits.
2
4
3
4
r2
2
a
59
b
c
÷4=
48
89 ÷ 5 = 77 r
47
PL
E
11 Use the digits 1 to 6 to complete these divisions.
You must use each digit once.
÷3=1
8 r2
M
10.3 Tests of divisibility
Worked example 3
divisible Here is a Venn diagram for sorting numbers.
factor multiple
Write each number in the correct place on the diagram.
test of divisibility
3741
Venn diagram
1679
1569
SA
588
1092
divisible by 3
divisible by 6
114
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10.3 Tests of divisibility
Continued
divisible by 3
Find the numbers that are divisible
by 3 by finding the sum of the digits,
for example:
divisible by 6
3 + 7 + 4 + 1 = 15 so 3741 is divisible
by 3.
1092
1569
588 and 1092 are also divisible by 6
because they are even numbers.
3741
1679
Exercise 10.3
Focus
1679 is not divisible by 3.
Which of these numbers are divisible by 3?
M
1
588, 1092 and 1569 are also divisible
by 3.
PL
E
588
Explain how you know.
54 689
234 567
SA
4563
2
Write a digit in each box so that all the numbers are divisible by 3.
Can you find more than one answer?
a
3
7
23
b
501
7
Here are four labels.
even
divisible by 9
not divisible
by 9
not even
115
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10 Multiplication and division (1)
Write each label in the correct place on the Carroll diagram.
2322
2348
321 426
2331
Practice
4
4867
PL
E
770 679
723 142
126 147
Here are five numbers.
64
128
240
352
424
Tick the statement that describes all the numbers.
A They are all divisible by 6
M
B They are all divisible by 7
C They are all divisible by 8
SA
D They are all divisible by 9
5
Tick in the correct cells to show whether these numbers are divisible by 3, 6 and 9.
3
6
9
21 471
482 211
152 214
116
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10.3 Tests of divisibility
6
Here is a Venn diagram.
Put these numbers on the diagram.
159
204
146
324
222
Challenge
7
divisible
by 9
PL
E
divisible
by 6
189
divisible
by 3
Kojo says, ‘Multiples of 6 can never end in a 3.’ Is he right?
Here are four digits.
SA
8
M
Give a reason for your answer.
1
2
5
7
Use these digits to make three 3-digit numbers. You can use each digit
more than once in any number, but you must use all the digits
at least once and all three numbers must be different.
a number divisible by 3
a number divisible by 6
a number divisible by 9
117
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10 Multiplication and division (1)
9
Here is a Carroll diagram with four sections A, B, C and D.
divisible by 9
not divisible by 9
divisible by 3
A
B
not divisible by 3
C
D
Write a 5-digit number in sections A, B and D.
b
You cannot write a number in Section C. Explain why not.
SA
M
PL
E
a
118
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11
3D shapes
Worked example 1
PL
E
11.1 Shapes and nets
compound shape
prism surface area
M
Describe this compound shape.
Think how you can split the compound
shape into simpler 3D shapes
that you know.
SA
This compound shape is made from a
cone and a cylinder.
119
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11 3D shapes
Exercise 11.1
Focus
1
Complete these descriptions of compound shapes.
a
This compound shape is made from two
.
M
c
and a
This compound shape is made from a
and a
.
The diagram shows four shapes A, B, C and D.
SA
2
.
PL
E
b
This compound shape is made from a
It also shows four sketches of nets i, ii, iii and iv.
A
B
C
D
i
ii
iii
iv
Draw a line to match each shape to the correct net.
120
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11.1 Shapes and nets
3
Write down the smallest number of unit cubes that must be added to
these shapes to make cuboids.
b
Practice
Sketch a compound shape that is made from these simple shapes.
SA
4
M
c
PL
E
a
a
three different cuboids
121
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11 3D shapes
c
two identical pyramids.
PL
E
two identical cones
M
5
b
Describe and sketch a net of these shapes.
triangular prism
SA
a
122
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11.1 Shapes and nets
hexagonal prism.
PL
E
b
Describe how could you work out the surface area of the shapes in question 5.
7
Write down the smallest number of unit cubes that must be
added to these shapes to make cuboids.
b
SA
a
M
6
c
123
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11 3D shapes
Challenge
8
Sketch a net for these shapes.
SA
b
M
PL
E
a
124
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11.1 Shapes and nets
9
Write down the smallest number of unit cubes that must be added to
these shapes to make cubes.
b
SA
M
c
PL
E
a
125
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11 3D shapes
PL
E
10 These two shapes are made from unit cubes.
Fran takes the shapes apart and uses all the unit cubes to make a cuboid.
SA
M
Draw a sketch to show two different cuboids she can make
with all the cubes. Use isometric paper if you have some.
126
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11.2 Capacity and volume
11.2 Capacity and volume
Worked example 2
capacity volume
ml
500
400
300
200
100
0
a
What is the capacity of the jug?
b
What is the volume of water in the jug?
a
500 ml
b
300 ml
500 ml is the maximum the jug can hold
The scale shows the water is at the 300 ml mark.
Focus
M
Exercise 11.2
For each of these jugs write down
i
the capacity of the jug
ii
the volume of water in the jug.
SA
1
PL
E
The diagram shows some water in a jug.
a
ml
500
400
300
200
100
b
ml
100
90
80
70
60
50
40
30
20
10
c
ml
5000
4500
4000
3500
3000
2500
2000
1500
1000
500
i
ii
127
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11 3D shapes
Write the capacity of the container and the volume of the liquid
for each of these diagrams.
A
litres
B ml
litre
200
180
160
140
120
100
80
60
40
20
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Capacity
Capacity
M
C 2
1.75
1.5
1.25
1
0.75
0.5
0.25
1600
1400
1200
1000
800
600
400
200
Capacity
SA
D ml
E
litres
Volume
PL
E
2
Volume
Volume
Capacity
Volume
Capacity
Volume
1.2
1
0.8
0.6
0.4
0.2
128
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11.2 Capacity and volume
3
Show how these bottles should be organised and grouped in the Venn diagram.
Bottle B
Bottle A
Bottle C
Bottle D
0.09 litres
300 ml
200 ml
PL
E
75 ml
400 ml
750 ml
500 ml
1 litres
Bottle E
Bottle F
Bottle G
Bottle H
80 ml
0.2 litres
0.5 litres
M
50 ml
300 ml
80 ml
SA
Capacity less
than 500 ml
0.2 litres
0.75 litres
Volume less
than 100 ml
Draw one more bottle in each of the four sections of the Venn diagram.
Label the bottles with their capacity and the volume of liquid inside.
129
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11 3D shapes
4
Complete this table.
Remember that 1000 ml = 1 litre.
millilitres
litres and millilitres
litres
4100 ml
4 l 100 ml
4.1 l
1500 ml
1 l ml
PL
E
3 l 600 ml
2.5 l
400 ml
0 l 400 ml
9600 ml
What number is the arrow pointing to on each
of these scales? Look at the tip boxes for help.
a
Tip
100 ml
There are four spaces for an increase
of 100 ml, so each increment is worth
100 ÷ 4 = 25 ml.
400 ml
SA
b
200 ml
M
5
300 ml
c
3 l 2l
Tip
There are five spaces for an increase
of 100 ml, so each increment is worth
100 ÷ 5 = 20 ml.
Tip
There are five spaces for an increase
of 1 l, so each increment is worth
1 ÷ 5 = 0.2 l.
130
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11.2 Capacity and volume
Practice
6
ml
Zara and Sofia are looking at this question.
What is the volume of water in this jug?
Read what they say.
100
90
80
70
60
50
40
30
20
10
PL
E
I think the volume
of water is 79 ml.
I think the
volume of water
is 78 ml.
Who is correct? Explain why.
b
Explain the mistake that the other person has made.
SA
M
a
c
What volume of water must be added to the jug to fill it to capacity?
131
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11 3D shapes
7
For each of these jugs write down:
i
the capacity of the jug
ii
the volume of water in the jug.
a
ml
b
c
200
litres
4
800
ml
600
3
PL
E
100
400
2
200
1
i
ii
What volume of water must be added to the jugs in question 7
to fill them to capacity?
a
9
a
M
8
b
What is the total capacity of the 5 cans of oil below?
2.5 litres
a
litres
B
C
D
E
2.5 litres
2.5 litres
2.5 litres
2.5 litres
SA
A
b
c
Estimate the volume of oil needed to fill each can. Give your answers in millilitres.
b
c
d
e
132
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11.2 Capacity and volume
Challenge
10 Dakarai needs 3 litres of water.
He only has the water shown in the measuring jugs.
3
litres
litres
1.8
1.2
0.6
PL
E
2
1
M
Does he have enough water? Explain your answer.
11 Rhian has a bucket with a capacity of 10 litres.
3
full of water.
5
1
Wyn has a bucket which is full of water.
3
The bucket is
SA
He has the same volume of water in his bucket as Rhian.
What is the capacity of Wyn’s bucket?
12 Elin has these four measuring cups, A, B, C and D.
The capacity of each cup, in millilitres, is shown.
240 ml
160 ml
120 ml
60 ml
A
B
C
D
133
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11 3D shapes
40 ml
b
180 ml
c
80 ml
d
100 ml
e
20 ml
SA
M
a
PL
E
Explain how Elin can use the cups to accurately measure out these volumes:
13 Jenny was asked this question:
Is the capacity of a container always greater than the volume of the
liquid inside?
Jenny wrote: The capacity of a container must always be greater than
the volume of the liquid inside because the capacity is the maximum
the container can hold.
Think carefully about capacity, volume and Jenny’s answer.
Write an improved answer of your own.
134
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11.2 Capacity and volume
PL
E
14 Kabir has three fuel containers. One has a capacity of 7 litres,
one has a capacity of 4 litres and one has a capacity of 3 litres.
The 7 litre container is full, the other two containers are empty.
None of the containers have a measurement scale.
SA
M
How can Kabir transfer the fuel so that two of the containers contain
a volume of 2 litres each, and the other contains a volume of 3 litres,
without having to estimate?
135
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PL
E
12 Ratio and
­proportion
12.1 Ratio
equivalent ratio ratio
Worked example 1
simplest form
Write the ratio 6 : 18 in its simplest form.
÷6
6 : 18
÷6
1:3
6 is a factor of both 6 and 18
M
To find an equivalent ratio, we divide both
quantities in the ratio by the same number.
Answer 1 : 3
SA
Exercise 12.1
Focus
1
Look at these shapes.
Write in its simplest form:
a
the ratio of circles to squares
b
the ratio of squares to pentagons
c
the ratio of pentagons to squares.
136
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12.1 Ratio
3
Write these ratios in their simplest form.
a
3 : 15
b
21 : 49
c
24 : 18
d
24 : 6
Write the missing numbers.
a
Practice
4
b
2:3=4 :
5:7=
: 14
c
3 : 5 = 12 :
PL
E
2
Look at this recipe for pasta sauce.
Pasta sauce
300 g tomatoes
120 g onions
SA
M
75 g mushrooms
5
a
Write the ratio of tomatoes to mushrooms in its simplest form.
b
Write the ratio of mushrooms to tomatoes in its simplest form.
Jamie mixes 2 parts of red paint with 3 parts of blue paint to make purple paint.
He uses 12 cans of blue paint.
How many cans of red paint does he use?
137
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12 Ratio and proportion
6
There are 2 milk chocolates for every 3 dark chocolates in a box of chocolates.
There are 8 milk chocolates in the box.
How many chocolates are in the box altogether?
7
On a school visit, there is 1 teacher for every 8 students.
PL
E
There are 96 students on the visit.
How many teachers are there on the visit?
8
Keon prepares a picnic.
Each person will get:
3 sandwiches
2 cartons of orange juice
1 banana
Ollie packs 45 sandwiches.
How many cartons of orange juice does he pack?
b
How many bananas does he pack?
SA
M
a
Challenge
9
Jodi makes a fruit drink using oranges and lemons.
For every 1 lemon she uses 4 oranges.
She uses 20 pieces of fruit altogether.
How many oranges does she use?
138
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12.1 Ratio
10 Here is a pattern and some statements.
PL
E
One statement does not describe the pattern.
A Ratio of triangles to circles is 1: 4
B 1 in every 5 shapes is a triangle
C 20% of the pattern is triangles
D 1 out of 5 shapes is a triangle
E
F
1
of the pattern is triangles
5
G 2 in every 8 shapes are triangles
H
8
of the pattern is circles
10
I
J
4 out of 20 shapes is a triangle
Ratio of triangles to circles is 4 :16
80% of the pattern is circles
Which statement does not describe the pattern?
b
Correct this statement.
M
a
11 Two numbers are in the ratio 3 : 7.
One of the numbers is 42.
SA
Find two possible values for the other number.
and
12 The sides of an isosceles triangle are in the ratio 2 : 2 : 1.
2
2
Not drawn to scale
1
The shortest side is 5 cm long.
What is the perimeter of the triangle?
cm
139
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12 Ratio and proportion
12.2 Direct proportion
Worked example 2
direct proportion
enlarge proportion
PL
E
This recipe gives the quantities to make 18 doughnuts.
Doughnuts
[makes 18 doughnuts]
300 g flour
60 g margarine
90 ml milk
M
75 g sugar
3 eggs
SA
What quantities should Imran use to make 6 doughnuts?
6 is one-third of 18 so Imran must
divide all the quantities by 3:
100 g flour
300 ÷ 3 = 100
20 g margarine
60 ÷ 3 = 20
30 ml milk
90 ÷ 3 = 30
25 g sugar
75 ÷ 3 = 25
1 egg
3÷3=1
140
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12.2 Direct proportion
Exercise 12.2
Focus
1
Hassan goes shopping at a shop where each item costs $5.
Complete the table.
Number of items
1
A bottle-making machine makes 60 bottles in 5 hours.
a
How many bottles will it make in 20 hours?
b
How long will it take to make 30 bottles?
Here are two squares.
SA
3
4
M
2
3
PL
E
Cost in $
2
Not drawn to scale
3 cm
The ratio of the side length of the small square to the side length of
the large square is 1 : 3.
What is the side length of the large square?
141
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12 Ratio and proportion
Practice
5
A picture of a cat is one-fifth of the size of the real cat.
a
The cat’s body is 9 cm long in the picture.
How long is the body of the real cat?
b
The real cat’s tail is 30 cm long.
What is the length of the tail in the picture?
PL
E
4
This is a recipe for pancakes for 8 people.
Pancakes
[for 8 people]
M
100 g flour
2 eggs
SA
300 ml milk
What quantities should Mira use to make pancakes for 4 people?
142
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12.2 Direct proportion
6
Kojo draws a rectangle 1 cm by 2 cm.
He enlarges the rectangle to make three more rectangles.
Which of these rectangles can he draw? Explain your answer.
B: 5 cm by 20 cm
C: 9 cm by 18 cm
D: 4 cm by 8 cm
PL
E
7
A: 2 cm by 4 cm
Here is a recipe for raspberry ice cream.
Receipe
Receipe
Raspberry ice cream
Serves 8 people
M
1
2 litre cream
SA
1 kg raspberries
250 g sugar
a
Hong makes enough ice cream for 4 people.
How many grams of raspberries does she use?
b
Hassan also makes raspberry ice cream. He uses 1kg sugar.
What mass of raspberries does he use?
143
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12 Ratio and proportion
Challenge
8
Here is rectangle A.
6 cm
A
2 cm
Not drawn to scale
Complete the table.
PL
E
Rectangle A is enlarged to make rectangles B and C.
Ratio of lengths
A
B
C
Length in cm
Perimeter in cm
2
A to B = 1 : 5
A to C = 1 : 10
Here is a recipe for chocolate ice cream.
M
9
Width in cm
Receipe
Receipe
SA
Chocolate ice cream
Serves 8 people
420 ml cream
400 ml milk
4 egg yolks
120 g chocolate
100 g sugar
144
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12.2 Direct proportion
a
Pierre has only 40 g of chocolate to make his ice cream.
How much cream should he use?
b
Maxine makes chocolate ice cream for 12 people.
PL
E
How much milk does she use?
10 Saif makes milkshakes using 1 part syrup to 7 parts milk.
a
How much syrup should he add to these glasses of milk?
A
ml milk
D
630 ml milk
350 ml milk
490 ml milk
ml milk
ml milk
ml milk
What proportion of each milkshake is syrup?
SA
b
C
M
700 ml milk
B
145
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13
Angles
PL
E
13.1 Measuring and drawing angles
Worked example 1
a
protractor
Measure angle x.
X°
a
Draw an angle of 135°
M
b
0
03
33
40
SA
0
22
350
340
10
20
30
40
40
0
0 2
1
01
15
13
0
0
23
0
260 270 280 29
0
250
30
0
0
00 90 80
7
0
31
60
0
50
12
24
0
32
Place your protractor over angle x so
that the centre of your protractor is
at the point of your angle. Make sure
the horizontal arm of your angle is
lined up exactly with 0°. As the angle
opens clockwise, use the numbers
on the outside circle.
170 1
60
0
33
170 180 190 20
90
00 1
160
02
X°
350 3
0
21
10
13
30
0
20
0
02
15
30
12
24
22
20
30
10
280 270 260 250
0
0
31
90
02
90 100
1
14
40
50
70
60
80
01
10 1
x = 60°
146
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13.1 Measuring and drawing angles
Continued
0
40
24
Now take away the protractor and
join your mark to the right end of the
horizontal line. Draw an angle arc
and write 135° onto you diagram.
10 1
250
00 90 80
40
50
60
70
0
0 3
0
260 270 280 29
0
31
M
0
01
350
340
10
20
30
Start by drawing a horizontal line.
Place the centre of your protractor
on the right end of the line, with the
left end of the line at 0°. Measure
135° clockwise, using the outside
numbers. Mark a small line as this
point.
PL
E
12
0
32
0
0
33
0
0
13
135°
13
23
350 3
20
12
02
40
3
80 270 260 25
90 2
02
30
10
0
10
10
90 100
1
170 180 190 20
0
160
2
10
0
170
90
15
1
6
22
00 1
01
0
02
50
0
14
21
14
0
0
22
30
4
33
03 0
20
50
80
0
60
70
23
b
Exercise 13.1
SA
Focus
1
Measure the size of each of these acute angles.
Circle the correct answer for each one.
a°
a = 20° a = 30°
b°
b = 55° b = 65°
147
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PL
E
13 Angles
c°
c = 70° c = 78°
2
Measure the size of each of these obtuse angles.
SA
M
Circle the correct answer for each one.
d°
d = 100° d = 110°
e°
e = 160° e = 170°
f°
f = 132° f = 142°
148
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13.1 Measuring and drawing angles
3
Measure the size of each of these reflex angles.
Circle the correct answer for each one.
PL
E
g°
g = 210° g = 220°
SA
M
h°
h = 260° h = 270°
i°
i = 340° i = 330°
149
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13 Angles
Practice
Measure the size of each of these angles.
PL
E
4
SA
M
x°
y°
z°
150
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13.1 Measuring and drawing angles
5
Draw angles of the following sizes.
10°
b
c
165°
230°
d
330°
SA
M
PL
E
a
6
a
b
Measure the angles x and y in this diagram.
x°
y°
Which calculation can you do to check that your answers to part a are correct?
151
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13 Angles
Challenge
The diagram shows a ski jump ramp.
PL
E
7
x
y
In a competition, the angle marked x must be between 36° and 38°,
and the angle marked y must be between 7° and 12°.
a
Measure the angles a, b and c in this diagram.
SA
8
M
Can this ramp be used in a competition? Explain your answer.
a=
b
b=
a
b
c
c=
Explain why the angles should add up to 360°.
Use this fact to check your accuracy.
152
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13.1 Measuring and drawing angles
Show that angle b is three times the size of angle a.
d
Write down the missing numbers in these statements:
i
angle c is
times the size of angle b.
ii
angle c is
times the size of angle a.
The diagram shows two triangles.
PL
E
9
c
j
g
d
a
Triangle 1
c
h
i
f
Triangle 2
l
Complete these tables which show the sizes of all the angles.
Also fill in the totals of the angles shown.
M
a
e
b
k
Triangle 1
d=
a+d=
b=
e=
b+e=
c=
f=
c+f=
a+b+c=
d+e+f=
a+b+c+d+e+f=
g=
j=
g+j=
h=
k=
h+k=
i=
l=
i+l=
g+h+i=
j+k+l=
g+h+i+j+k+l=
SA
a=
Triangle 2
153
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13 Angles
What do you notice about the totals that you have found in both tables?
c
Do you think that these totals will be the same for any triangle that you draw?
Explain your answer.
PL
E
b
13.2 Angles in a triangle
Worked example 2
equilateral triangle
Work out angle y in this triangle.
75°
SA
y
scalene triangle
M
43°
isosceles triangle
43 + 75 = 118°
Add together the two angles that you know.
180 – 118 = 62°
The angles in a triangle add to 180°,
so subtract the total so far from 180°.
y = 62°
Write down the value of y.
154
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13.2 Angles in a triangle
Exercise 13.2
Focus
Complete the workings to find angle x in each of these triangles.
a
b
80°
x
40°
65°
80 + 40 = c
65 + 45 =
180 –
180 –
x=
=
°
45°
°
=
°
°
x=
°
Complete the workings to find angle y in each of these triangles.
M
2
x
PL
E
1
a
y
b
35°
SA
30°
90 + 30 =
°
180 –
y=
=
°
°
y
90 + 35 =
°
180 –
y=
=
°
°
155
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13 Angles
3
Complete the workings to find angle z in these triangles.
a
b
z
80°
30°
z
°
180 –
z=
=
°
=
°
°
÷2=
y=
Practice
4
180 –
PL
E
30 × 2 =
°
°
Work out the lettered angle in each of these triangles.
a
b
120°
M
m
n
35°
48°
SA
56°
m=
5
n=
This is part of Juan’s homework.
Question
a
This triangle is isosceles.
Work out angles a and b.
Solution
a = 62°
62 × 2 = 124
180 – 124 = 56
b = 56°
b
62°
156
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13.2 Angles in a triangle
Explain the mistake that Juan has made.
b
Write down the correct values of a and b.
a=
6
PL
E
a
b =
Helmut builds a swing in his garden.
The diagram shows the largest angle that his swing can turn through.
115°
SA
M
m
Helmut finds out that the swing is safe when the angle marked m is
greater than 30°.
Is Helmut’s swing safe? Explain your answer.
157
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13 Angles
Challenge
7
The diagram shows a right-angled triangle.
y
x
a
b
PL
E
Zara is investigating different values for x and y.
Complete the table for these different values for x and y.
x
40°
y
50°
30°
55°
Tip
24°
72°
61°
When x = 40°,
y = 180 – 90
– 40 = 50°
M
In a right-angled
triangle, the sum of x
and y is always 90°.
SA
Show that Zara is correct using the values in the table.
158
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13.2 Angles in a triangle
c
Explain why the sum of x and y is always 90°.
Is it possible for x or y to be greater than 90°? Explain your answer.
PL
E
d
8
Is it possible to draw a triangle with angles of 48°, 72° and 50°?
Explain your answer.
9
In a right-angled triangle, the angles are d, e and f.
Work out the values of d, e and f when:
M
d is the largest angle and e and f are the same size.
SA
a
b
f is the smallest angle, e is two times the size of f and
d is three times the size of f.
159
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PL
E
14 Multiplication
and division (2)
14.1 Multiplying and dividing
fractions
Worked example 1
Calculate
4
×4
5
denominator numerator operator
proper fraction unit fraction
Use a diagram in your answer.
4
16
×4=
5
5
1
5
M
=3
Multiply the numerator by the whole number.
Change the improper fraction to a mixed number.
4
5
SA
4
5
4
5
4
5
= 16
5
You can use other types of diagram, including
a number line.
This shows
+ 45
0
4
4
× 4 as repeated addition of
5
5
+ 45
4
5
+ 45
8
5
+ 45
12
5
16
5
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14.1 Multiplying and dividing fractions
Exercise 14.1
Focus
Write an addition sentence and a multiplication sentence for the
shaded parts in this diagram.
2
Calculate.
a
b
Find the missing numbers.
a
1
of 20 =
5
b
3
× 20 =
4
Write a division sentence for this diagram.
SA
4
3
÷4
5
M
2
÷3
3
3
PL
E
1
Practice
5
Calculate.
a
7
÷3
9
b
3
÷4
7
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14 Multiplication and division (2)
6
My family eats
3
of a box of cereal each week.
4
How much cereal do they eat in 4 weeks?
a
Complete the multiplication table.
1
8
×
3
4
5
b
8
3
8
5
8
7
8
PL
E
7
Which two calculations give the same answer?
Leila cuts a
3
metre length of ribbon into 5 equal pieces.
4
M
What is the length of each piece of ribbon?
Give your answer as a fraction of a metre.
SA
Challenge
9
3
8
Omar says, ‘ multiplied by 4 equals
12
.’
32
Is Omar correct?
Draw a diagram to explain your answer.
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14.1 Multiplying and dividing fractions
10 Write the letter of each expression in the correct cell in the table below.
A
2
÷8
3
B
1
÷2
3
C
3
÷9
4
D
5
÷ 10
6
E
2
÷2
3
F
2
÷4
3
1
6
Answer
11 Here are six numbers.
14
16
18
1
3
Answer
1
12
PL
E
Answer
21
24
40
Use each of these numbers once to make these statements correct.
2
of
3
c
2
of
5
=
b
3
of
4
=
M
a
=
SA
12 Use the digits 1, 2, 3 and 4 to complete this calculation.
÷
=
6
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14 Multiplication and division (2)
14.2 Multiplying decimals
Worked example 2
decimal decimal place
Mia is thinking of a number.
decimal point product
PL
E
She says, ‘If I divide my number by 5 the
answer is 73.45’
What number is Mia thinking of?
÷5
Multiplication and division are inverse operations,
so you need to calculate 73.45 × 5.
73.45
×5
73.45 × 5 = ?
Estimate:
Start with an estimate.
7
3
4
5
5
SA
×
3
You can round 73.45 down to 70 and up to 80,
then multiply both numbers by 5. The correct
answer will be between these two values.
M
70 × 5 = 350 and 80 × 5 = 400
The answer is between 350
and 400.
6
7
2
1
2
2
Answer: 367.25
5
Check the answer is between the two
estimated values.
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14.2 Multiplying decimals
Exercise 14.2
Focus
1
Calculate.
a
b
24.1 × 6
c
18.2 × 4
One bracelet costs $4.65.
How much do five bracelets cost?
3
Heidi helps in a shop for 8 hours.
She is paid $7.55 for each hour she works.
PL
E
2
40.9 × 5
How much does she earn?
4
Draw a line from each calculation to the correct label for the answer.
15.4 × 6
answer less than 100
M
12.45 × 9
answer equal to 100
13.84 × 7
SA
12.5 × 8
answer more than 100
Practice
5
Write the three missing numbers in this multiplication grid.
×
6
0.56
3.36
0.27
0.69
4
3.92
1.08
4.14
7
1.89
2.76
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14 Multiplication and division (2)
Which calculation is the odd one out? Explain why.
41.5 × 5
7
Calculate.
a
8
32.7 × 4
46.3 × 11
16.25 × 8
13.35 × 6
14.2 × 9
PL
E
6
b
39.3 × 23
c
23.8 × 35
Trudy buys 4.25 kg fish to make fish pies.
If the fish costs $13 per kilogram, how much does she pay?
Challenge
Pedro sells rugs for $24.75 each.
M
9
In one week he sells 26 rugs.
SA
How much money does he make in that week?
10 Salma is thinking of a number.
She says, ‘If I divide my number by 16 the answer is 15.54.’
What number is Salma thinking of?
11 Hassan has 18 crates each with a mass of 0.3 tonnes.
The maximum load on his lorry is 5 tonnes.
Can Hassan load the crates safely? Show your working to explain your answer.
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14.3 Dividing decimals
12 A school chef has 75 kg of flour.
She uses an average of 5.35 kg flour each week.
There are 13 weeks in the school term.
Is she likely to have enough flour for the term?
PL
E
Show calculations to explain your answer.
14.3 Dividing decimals
Worked example 3
Two different shops sell modelling clay.
dividend
divisor
quotient
M
Shop A sells 4 packets of clay for $10.28
Shop B sells 3 packets of clay for $7.80
SA
Which shop has the better deal? Explain your answer.
1 packet in shop A costs:
2.5 7
2 2
4 1 0.2
8
1 packet in shop B costs:
2.6 0
To compare the prices, work
out how much 1 packet costs in
each shop.
Divide the total cost by the
number of packets.
1
3 7.8 0
Answer:
The cost of 1 packet in shop A is cheaper, but you
must buy 4 packets at a time.
Remember you will also need to
think about how many packets
you need.
If you only need 3 packets you would go to shop B.
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14 Multiplication and division (2)
Exercise 14.3
Focus
1
Calculate.
a
c
22.4 ÷ 7
34.4 ÷ 8
Work out the answer to each question to help you find your way
through the maze.
PL
E
2
b
14.4 ÷ 3
start
12.4 ÷ 4
3.1
4.1
3.6
3.4
SA
31.8 ÷ 6
3
5.4
5.7
44.8 ÷ 8
M
30.6 ÷ 9
17.1 ÷ 3
5.5
5.6
11.2
5.2
17.4 ÷ 6
12.5 ÷ 5
36.4 ÷ 7
5.2
3.4
end
Nine notebooks cost $14.58.
If each notebook costs the same, what is the price of one notebook?
4
A regular pentagon is marked out on the playground.
The perimeter of the pentagon is 24.95 metres.
What is the length of one side of the pentagon?
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14.3 Dividing decimals
Practice
Work out the answer to each calculation and write it in the correct part of the table.
76.32 ÷ 8 =
56.2 ÷ 5 =
24.15 ÷ 7 =
61.2 ÷ 3 =
Answer less
than 10
6
Answer between
10 and 20
Answer more
than 20
PL
E
5
Pierre is thinking of a number.
He multiplies his number by 9 and his answer is 147.6.
What number is Pierre thinking of?
Find the odd one out
M
7
15.6 ÷ 6 16.8 ÷ 7 20.8 ÷ 8 23.4 ÷ 9
SA
Explain your answer.
8
Work out the answers to these calculations.
Five answers are on the grid. Which answer is missing?
6.12
6.04
6.15
6.23
6.07
6.17
a
91.05 ÷ 15
b
73.44 ÷ 12
c
87.22 ÷ 14
d
78.52 ÷ 13
e
111.24 ÷ 18
f
98.72 ÷ 16
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14 Multiplication and division (2)
Challenge
9
A chef needs lots of bags of rice.
Deal A: buy 4 bags for $5.08
Deal B: buy 5 bags for $6.25
Which deal should he choose?
10 Find the missing digit.
PL
E
Explain your answer by showing your calculations.
3 •6
7 2
•2
11 Look at these four calculations.
One of them is wrong.
359.1 ÷ 9 = 39.9
M
281.7 ÷ 3 = 93.9
939.7 ÷ 9 = 93.3
117.9 ÷ 3 = 39.3
Identify the incorrect calculation without working out the answers.
SA
Explain your answer.
12 A dress maker cuts pieces of ribbon 15 centimetres
long from a roll of ribbon that is 5.625 metres long.
What is the greatest number of 15-centimetre
pieces she can cut from the roll of ribbon?
Tip
Change metres
to centimetres.
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15
Data
Worked example 1
PL
E
15.1 Bar charts, dot plots,
waffle diagrams and pie charts
bar chart data dot plot
How many of the people represented
by this pie chart chose ‘mountains’
as their favourite place?
city
M
forest
pie chart waffle diagram
ocean
SA
mountains
Pie chart showing the favourite
places of a group of 20 people
A quarter of the pie chart is labelled
1
‘mountains’, so of the people chose the
4
Can you identify the fraction of the
pie chart for mountains?
mountains as their favourite place.
1
= 25%. 25% of the people chose the
4
mountains.
Can you identify the percentages of
the pie chart for mountains?
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15 Data
Continued
5 people chose mountains as their
favourite place.
There are 20 people represented in
the pie chart.
25% of the people chose mountains.
Exercise 15.1
Focus
1
PL
E
Work out 25% of 20.
This is a table of how many umbrellas a shop sold in one week.
Day
Number of Umbrellas
a
1
2
3
4
5
6
7
0
2
6
4
0
1
0
Draw a dot plot of the data.
SA
M
Remember to label the horizontal axis and the vertical axis.
b
On which day were the most umbrellas sold?
c
How many umbrellas were sold in total during the week?
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15.1 Bar charts, dot plots, waffle diagrams and pie charts
This is a table of how many umbrellas the shop sold later in the year in one week.
Day
1
2
3
4
5
6
7
Number of Umbrellas
3
4
6
6
7
5
6
Draw a dot plot of the data.
e
Write two sentences to describe how the umbrella sales were
different in the two weeks represented by your dot plots.
M
PL
E
d
SA
1.In the first week
but in the second week
.
2.In the first week
but in the second week
f
.
Why do you think there might be differences in the umbrella sales
in the two weeks?
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15 Data
Four groups of children were asked to say their
favourite flavour of ice-cream. The frequency table
and the pie charts show the data. Draw lines to match
the frequency tables to the correct pie charts.
Favourite flavour
Frequency
Vanilla
1
Strawberry
1
Chocolate
Lemon
Favourite flavour
Vanilla
Strawberry
Chocolate
1
1
Frequency
2
0
0
2
M
Lemon
PL
E
2
Favourite flavour
Frequency
0
Strawberry
Chocolate
1
Lemon
0
SA
Vanilla
3
Vanilla
4
Strawberry
0
Chocolate
0
Lemon
0
Favourite flavour
Frequency
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15.1 Bar charts, dot plots, waffle diagrams and pie charts
Practice
This table shows the number of children in each household along a street.
a
Complete the table with the percentages of households in each category.
Number of children
0
1
2
3
4
Frequency
3
2
3
1
1
Percentage
b
PL
E
3
Complete the pie chart and waffle diagrams to represent the data.
SA
M
Key
c
What percentage of the households had children?
d
Is it easier to see the percentage of households that have children
from the table, the pie chart or the waffle diagram? Explain your answer.
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15 Data
4
20 people were asked to respond to this survey question – Are the school
holidays too short? This is the pie chart made from the data collected.
Mostly disagree
20%
Mostly agree
55%
Strongly disagree
10%
PL
E
Don’t know 5%
Strongly agree
10%
SA
M
Convert the data from the pie chart into a bar graph.
Use this space for your working.
Use the data from the pie chart or your bar graph to write three true statements.
1.
2.
3.
176
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15.1 Bar charts, dot plots, waffle diagrams and pie charts
Challenge
5
Juma surveyed people outside a swimming pool and outside a cinema about their
favourite activities. These waffle diagrams represent her data.
Waffle Diagram A
Key
PL
E
reading
watching TV
playing computer games
swimming
M
running
SA
Waffle Diagram B
Key
reading
watching TV
playing computer games
swimming
running
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15 Data
Which waffle diagram do you think represents the survey taken
outside the swimming pool? Why?
b
What percentage of people in waffle diagram B prefer playing
computer games?
c
What percentage of people in waffle diagram A do not prefer
watching TV?
d
PL
E
a
If 50 people are represented in waffle diagram A.
How many of those people prefer watching TV?
Describe one thing that is similar between the two sets of data
represented in the waffle diagrams.
f
If you carried out the same survey with people in your class what do
you predict the data would look like in a waffle diagram? Why?
Describe how you predict it might be similar or different to another
class in your school.
SA
M
e
g
Describe how you would conduct an investigation to find out the favourite
activities of two classes in your school to check your predictions.
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15.1 Bar charts, dot plots, waffle diagrams and pie charts
6
The Tornadoes and the Hurricanes football teams played in a tournament.
The mode of the Tornadoes goals scored was 1.
The range of the goals scored by the Hurricanes was 4.
Sort these 6 different graphs and charts. There are 3 representations of
the games won and goals scored by each of the two different teams.
PL
E
A Bar chart of goals scored
Number of games
6
5
4
3
2
1
0
0
1
2
3
4
5
M
Number of goals
SA
B Waffle diagram of goals scored
Key
0
1
2
3
4
5
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15 Data
C
Dot plot of goals scored
6
4
3
2
1
0
0
PL
E
Number of games
5
1
2
3
4
5
Number of goals
M
D Pie chart of the outcomes of the games
Won
SA
Lost
E
Drew
Tally chart of the outcomes of the games
Game outcome
Tally
Total
Won
IIII IIII I
11
Lost
IIII II
7
Drew
II
2
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15.1 Bar charts, dot plots, waffle diagrams and pie charts
F
Venn Diagram
Games where
the team scored
more than 1 goal
5
7
4
PL
E
4
Games where
the team won
Write the 3 letters of the representations that match each team.
The Tornadoes
SA
M
The Hurricanes
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15 Data
15.2 Frequency diagrams, line
graphs and scatter graphs
Worked example 2
frequency diagram
30
PL
E
20
18
16
14
12
10
8
6
4
2
line graph
35
40
45
50
55
Time to run 10 km (min)
scatter graph
60
M
Hours playing sport
This scatter graph shows how many hours 10 people
play sport in a week and how quickly they run 10 km.
Hours playing sport
SA
Use the line of best fit to estimate how many hours of sport someone
does if they run 10 km in 55 minutes.
20
18
16
14
12
10
8
6
4
2
30
35
40
45
50
55
Time to run 10 km (min)
Time to run 10 km is on the
horizontal axis.
Find 55 minutes on the horizontal
axis and follow the find up.
60
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15.2 Frequency diagrams, line graphs and scatter graphs
20
18
16
14
12
10
8
6
4
2
30
Where the line from 55 minutes
meets the line of best fit, look
along to find how high it is on the
vertical axis. It is at 5 hours on
the vertical axis.
PL
E
Hours playing sport
Continued
35
40
45
50
55
Time to run 10 km (min)
60
We can estimate that someone who runs 10 km
in 55 minutes plays sport for 5 hours a week.
Focus
These are the heights of a group of children in centimetres.
129
145
135
146
128
151
140
136
141
142
125
134
147
150
144
131
152
133
136
146
141
147
137
148
153
145
128
149
150
147
SA
1
M
Exercise 15.2
Complete the table below.
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15 Data
Tally the heights of the children and write the totals.
Height (cm)
Tally
Total
125 – less than 130
130 – less than 135
140 – less than 145
145 – less than 150
150 – less than 155
PL
E
135 – less than 140
SA
M
Draw a frequency diagram on the squared paper to show the children’s heights.
184
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15.2 Frequency diagrams, line graphs and scatter graphs
2
Sofia’s height was measured every year from when she was born
until she was 12 years old. This line graph represents the data collected.
160
140
100
80
60
40
20
0
0
1
2
3
PL
E
Height (cm)
120
4
7
5
6
Age (years)
8
9
10
11
12
How tall was Sofia when she was 4 years old?
b
What age was Sofia when she was 130 cm tall?
c
Use the line graph to estimate Sofia’s height when she was
5
1
years old.
2
What happened to Sofia’s height between the ages of 10 and 11 years?
How do you know?
SA
d
M
a
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15 Data
3
Look at the lines of best fit on these scatter graphs. Are they correct?
Write wrong direction, too high, too low, too steep, not steep enough,
or just right to describe the line.
a
PL
E
b
d
SA
M
c
e
f
186
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15.2 Frequency diagrams, line graphs and scatter graphs
Practice
4
These two frequency diagrams show the mass of a group of boys
and a group of girls.
Boys
5
4
3
2
1
0
PL
E
Frequency
6
25
30
35
40
45
50
55
Mass (kg)
6
Frequency
M
5
Girls
4
3
2
SA
1
0
25
30
35
40
45
50
55
Mass (kg)
a
How many boys are between 45 kg and 50 kg?
b
How many girls have a mass 40 kg or greater?
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15 Data
c
Tom says: ‘The child with the greatest mass is a boy.’
Circle your reply.
•
Tom is correct.
•
Tom is not correct.
•
Tom could be correct, but we cannot tell from the graphs.
Imagine you investigate the mass of fruit and vegetables and put the
data into two frequency diagrams.
SA
M
5
PL
E
Explain your reply.
a
What equipment would you need to conduct the investigation?
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15.2 Frequency diagrams, line graphs and scatter graphs
Draw a table that you could use to collect the data.
c
What do you predict that the data would show? Why?
PL
E
6
b
The ages and heights of 24 palm trees are recorded in this table.
a
Complete the scatter graph to represent the data.
Remember to label the axes.
Height of
tree (m)
M
Age of tree
(years)
Age of tree
(years)
Height of
tree (m)
3.5
17
6
9
2
21
9
1
0.5
21
8.5
15
5
2
0.5
3
1
8
4
6
3
17
7
5
2
14
6
5
1.5
12
3
17
5
24
9.5
9
3.5
13
5
22
7
15
4.5
25
11
15
7
SA
10
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15 Data
12
10
8
6
PL
E
4
2
0
0
5
10
15
20
25
30
b
Draw a line of best fit on the scatter graph.
c
Use your line of best fit to estimate the height of a tree
that is 11 years old.
Use your line of best fit to estimate the age of a tree that is 8 m tall.
Challenge
7
M
d
This line graph shows temperatures recorded every two hours over a 24 hour period.
6
Temperature (°C)
SA
4
2
0
–2
–4
12
2
midnight
4
6
8
10 12
2
4
6
8
noon
10 12
midnight
Time
a
What happened to the temperature between 6pm and 8pm?
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15.2 Frequency diagrams, line graphs and scatter graphs
b
Use the line graph to estimate the temperatures at these times.
i
3pm
ii
9pm
iii 9am
Why can we only use the graph to estimate the temperature at 3pm,
and not know the temperature precisely?
d
This table also shows temperatures recorded every two hours over a
24 hour period.
8
10
2
4
6
8
10
12 midnight
6
14 14 15 16 18 20 21 22 20 18 16 15 14
M
Temperature (oC)
4
12 noon
2
12 midnight
Time
PL
E
c
SA
Represent the data in a line graph.
12
2
midnight
4
6
8
10 12
noon
2
4
6
8
10 12
midnight
Time
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15 Data
Describe what is similar and what is different about the two line graphs.
f
Give two possible explanations for the difference between the graphs
PL
E
e
1.
2.
8
If you recorded the temperatures outside where you live every
two hours for 24 hours what do you predict would be similar and
different about your line graph to the line graphs above. Why?
M
g
Some people attended a course. At the end of the course there was an exam.
This table shows how many days of the course a person did not attend and
the percentage of correct marks they scored on the exam.
0
1
0
2
5
3
1
4
7
2
Percentage scored
85
88
92
74
59
42
67
50
48
57
Days absent
5
3
7
1
4
3
0
2
3
5
Percentage scored
36
61
23
85
64
65
70
70
48
42
SA
Days absent
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15.2 Frequency diagrams, line graphs and scatter graphs
Draw scatter graph of the data in the table, including a line of best fit.
b
Use your line of best fit to estimate the percentage that would
PL
E
a
Describe the link between the number of days absent and the
percentage scored on the test.
SA
c
M
be score by someone who was absent for 6 days of the course.
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PL
E
16 The laws of
­arithmetic
16.1 The laws of arithmetic
Worked example 1
associative rule brackets
commutative rule distributive rule
Put brackets in the calculation to make it
correct.
4 + 5 × 3 × 2 = 54
4 + 5 × 3 × 2 = 4 + 30
The order of operations is multiplication
before addition.
M
= 34
order of operations
Which is not the required answer.
(4 + 5) × 3 × 2 = 9 × 3 × 2
SA
= 54
Brackets are worked out before
multiplication.
Answer: (4 + 5) × 3 × 2 = 54
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16.1 The laws of arithmetic
Exercise 16.1
Focus
1
Work out the answer to each calculation.
2
Answer
(12 – 3) × 8
18
10 × 8 + 1
27
6 × (5 – 2)
36
7 × (4 + 5)
45
6 × (3 + 6)
54
(9 – 4) × 9
63
(8 + 4) × 3
72
(12 – 3) × 3
81
PL
E
Join each calculation to the correct answer.
The first one has been done for you.
Calculation
Which expression has the same value
as 9 × (6 – 1)?
54 – 5 54 – 1 9 × 5 9 × 7
3
Work out the answer to each question to help you find your way
through the maze.
M
start
3 × (14 – 6)
24
SA
36
10 ÷ (8 – 3)
15
2
9 × (18 – 9)
81
20 ÷ (2 + 3)
13
4
5 × (3 + 4)
25
35
19
20 ÷ (7 – 3)
6 × (13 + 2)
24 ÷ (3 + 5)
3
3
end
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16 The laws of ­arithmetic
4
Safiya is learning to use brackets.
She writes down four calculations but one of her calculations is wrong.
A: 10 + (2 + 8) × 3 = 40
B: (10 + 2 + 8) × 3 = 60
C: (10 + 2) + 8 × 3 = 58
D: 10 + 2 + 8 × 3 = 36
Practice
5
PL
E
Which calculation is wrong? What is the correct answer?
Complete the calculation.
36 × 97 = 36 × (100 –
)
= (36 × 100) – (36 ×
=
–
M
=
6
)
Mandy writes 4 + 9 × 5 × 2 = 130
Is she correct?
SA
Explain your answer.
7
Write the missing number to make the calculation correct.
(10 –
) × 10 = 10
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16.1 The laws of arithmetic
8
Which number sentence is equivalent to 8 × 12?
A: (8 × 10) + (8 × 2)
B: (8 × 1) + (8 × 2)
C: (8 × 10) + 2
D: 8 + (10 × 2)
Write the sign <, > or = to make each expression correct.
a
2 × (3 + 4)
b
(10 + 6) ÷ 2
5+6÷2
c
(13 + 5) ÷ 9
(13 + 8) ÷ 7
Challenge
2×3+4
M
9
PL
E
Explain why the other sentences are wrong.
10 Use the numbers 2, 3, 4 and 5 to complete the calculations.
20 – (
+ 9) = 7
SA
a
c
11 – (
– 3) = 9
b
9+
–3=8
d
11 –
+ 2 = 10
11 Draw brackets to make each expression equal to 10.
a
14 – 12 × 5
b
11 – 6 – 5
c
20 – 15 – 5
d
20 ÷ 4 – 2
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16 The laws of ­arithmetic
12 Write the missing numbers.
3×(
c
(6 + 7) ×
e
3×(
+ 4) = 27
b
4 × (8 –
) = 20
= 39
d
16 ÷ (
– 3) = 16
– 6) = 6
f
(21 – 12) ÷
13 Here are six cards.
7
8
25
=3
PL
E
a
(
)
=
Use all these cards together with any of the operation signs +, –, ×
and ÷ to make a number sentence with an answer of 10.
14 Use these numbers together with brackets and operation signs to make
the target number.
Example: 3, 4, 6
Target 42
a
2, 5, 6
b
2, 3, 5
Target 4
c
3, 4, 6
Target 12
Answer (3 + 4) × 6
SA
M
Target 40
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17
Transformations
Worked example 1
PL
E
17.1 Coordinates and translations
axes axis
Translate rectangle ABCD 2 squares right and
4 squares up. Label it A’B’C’D’. Write down the
coordinates of the vertices A’, B’, C’ and D’.
y
5
4
3
M
2
corresponds translate
1
1
2
3
4
5
x
SA
0
–5 –4 –3 –2 –1
–1
A
B
–2
D
–3
C
–4
–5
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17 Transformations
Continued
Move each vertex of the
rectangle 2 squares right and
4 squares up.
y
5
4
3
2
D'
1
0
–5 –4 –3 –2 –1
–1
A
B
–2
D
–3
C
–4
Aˊ (-2, 3)
Bˊ (1, 3)
Cˊ (1, 1)
C'
1
2
3
4
5
x
Remember that when you
write coordinates the x-axis
number is first and the y-axis
number is second.
SA
Dˊ (–2, 1)
Vertex A on the original
corresponds with vertex
Aˊ (you say A dash) on
the translated rectangle.
B corresponds with Bˊ, C
corresponds with Cˊ and D
corresponds with Dˊ.
M
–5
B'
PL
E
A'
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17.1 Coordinates and translations
Exercise 17.1
Focus
1
Match each point on the grid to its correct coordinates.
y
5
4
B
2
1
A
PL
E
3
0
–5 –4 –3 –2 –1
–1
1
2
3
4
5
x
–2
–3
i
–4
Tip
–5
M
D
C
(2, –3)
(–3, 1)
iv (–3, –4)
(+, -) (right, down)
(-, +) (left, up)
(-, -) (left, down)
SA
iii (2, 3)
ii
(+, +) (right, up)
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17 Transformations
2
Here is an orienteering map. It shows the start and finish,
and checkpoints A, B, C and D on the route.
y
4
START/
FINISH
3
D
PL
E
2
1
–5
–6
–4
–3
–2
–1
A
0
1
–1
2
3
4
5
6
x
–2
B
–3
–4
M
C
Write down the coordinates of checkpoints A, B, C and D.
B
SA
A
3
C
D
The diagram shows four triangles.
Tip
y
6
5
B
4
3
T
2
1
0
Start with
triangle T and
work out how
you can move
T to A, T to B,
and T to C.
A
C
1
2
3
4
5
6
x
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17.1 Coordinates and translations
Triangle T has been translated to triangles A, B and C.
Match each translation i, ii and iii to the correct triangle A, B and C.
i
1 square right and 2 squares down
ii
2 squares right and 3 squares up
Practice
4
PL
E
iii 1 square left and 2 squares up.
The diagram shows line segments PQ and RS.
Write down the coordinates of the points P, Q, R and S.
y
5
4
3
2
P
Q
M
1
0
–5 –4 –3 –2 –1
–1
1
2
3
4
5
x
R
SA
–2
P
Q
S
–3
–4
–5
R
S
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17 Transformations
Draw axes from –5 to +5 on squared paper. Draw a parallelogram
with vertices at E (–2, –2), F (2, –2), G (4, –4) and H (0, –4).
SA
M
PL
E
5
a
Translate parallelogram EFGH 1 square right and 6 squares up.
Label the parallelogram EˊFˊGˊHˊ and write down the coordinates
of its vertices.
Eˊ
b
Fˊ
Gˊ
Hˊ
Translate parallelogram EFGH 3 squares left and 1 square down.
Label the parallelogram EˊˊFˊˊGˊˊHˊˊ and write down the coordinates
of its vertices.
Eˊˊ
Fˊˊ
Gˊˊ
Hˊˊ
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17.1 Coordinates and translations
Draw axes from –6 to +6 on squared paper. Plot the points A (0, 1),
B (2, 1) and C (4, –2).
SA
M
PL
E
6
a
rite down the coordinates of D so that A,
W
B, C and D are the vertices of an isosceles
trapezium.
b
rite down two possible coordinates of D
W
so that D is a point on the line segment AB.
Tip
You can use
fractions or
decimals in
coordinates.
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17 Transformations
c
Write down two possible coordinates of D so that A, B, C and D
are the vertices of a parallelogram.
Is it possible to find coordinates for D so that A, B, C and D are
the vertices of a rectangle? Explain your answer.
Challenge
PL
E
d
(-1, 3) and (3, 1) are the coordinates of two vertices of a square.
What could the other vertices of the square be?
Find all the possible solutions.
8
The diagram shows shape P on a coordinate grid.
M
7
y
SA
6
P
–5 –4 –3 –2 –1
5
4
3
2
1
0
1
2
3
4
5
x
Erin translates shape P 2 squares right and 3 squares up.
She labels the shape Q.
206
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17.1 Coordinates and translations
What translation should Erin do to take shape Q back to shape P?
Explain how you worked out your answer.
b
Erin translates shape Q 3 squares right and 1 square up.
She labels the shape R.
PL
E
a
Erin thinks that she could use the single translation 6 squares
right and 4 squares up to take shape P to shape R. Is Erin correct?
Explain your answer.
ii
What do you notice about the single translation P to R, and the two
translations P to Q and Q to R?
SA
M
i
9
Rectangle K has vertices as the points (-2, -1), (-5, -1), (-5, -3) and (-2, -3).
Shen translates K four times, using these four different translations
A, B, C and D.
A 3 squares right
and 2 squares up
B 5 squares right
and 6 squares up
C 3 squares right
D 1 square left and
1 square down
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17 Transformations
After which translation will K and the new rectangle be:
touching end to end
b
touching corner to corner
c
overlapping
d
not touching or overlapping?
PL
E
a
17.2 Reflections
Worked example 2
diagonal mirror line
SA
M
Reflect this triangle in the diagonal mirror line.
Take one vertex of the triangle at a time.
Draw arrows (black) to the mirror line, then draw
the same length arrows (grey) the other side of the
mirror line. Join the vertices with straight lines to
complete the reflected triangle.
208
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17.2 Reflections
Exercise 17.2
Focus
1
Which drawings show correct reflections of triangle A?
a
b
A
c
PL
E
A
d
A
SA
M
A
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17 Transformations
2
Reflect each shape in the mirror lines. They have all been started for you.
b
PL
E
a
3
d
M
c
Is A, B or C the correct reflection for each of these?
SA
i
A
B
C
210
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17.2 Reflections
ii
SA
A
C
PL
E
iii
B
M
A
B
C
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17 Transformations
Practice
4
Reflect the shape in the horizontal and vertical mirror lines.
5
d
M
c
b
PL
E
a
This is part of Jose’s homework.
Question: Reflect shape A in the diagonal line of symmetry.
SA
Label your answer shape B.
B
A
Has Jose drawn shape B correctly? Explain your answer.
212
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17.2 Reflections
6
Reflect the shape in the diagonal mirror lines.
b
d
c
7
a
PL
E
a
Describe the mirror line for each of these reflections.
ii
SA
M
i
iii
b
Tip
Is the mirror line
horizontal, vertical
or diagonal?
Draw in the correct mirror line for each reflection.
213
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17 Transformations
Challenge
Draw in the correct mirror line for each reflection.
9
a
b
Reflect the shapes in the mirror lines to complete the pattern.
i
b
c
PL
E
a
ii
What is the order of rotational symmetry of the completed pattern?
i
M
8
ii
10 The diagram shows shape A on a coordinate grid.
Reflect shape A in the mirror line. Label the new shape B.
SA
a
A
b
Translate shape B 2 squares left and 1 square down.
Label the new shape C.
c
Reflect shape C in the mirror line. Label the new shape D.
d
Describe the translation that takes shape D back to shape A.
e
What do you notice about your answer to part d and the translation
you carried out in part b?
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17.3 Rotations
17.3 Rotations
Worked example 3
anticlockwise centre of rotation
A
C
clockwise corresponding rotate
PL
E
Rotate triangle A 90° anticlockwise
about the centre of rotation marked C.
Label your answer triangle B.
Step 2
Step 1
A
A
M
C
C
Start turning the tracing paper 90°
(a quarter turn) anticlockwise.
Step 3
Step 4
SA
Trace the shape, then put your point of
your pencil on the centre of rotation.
A
A
B
C
C
Once the turn is completed make a note of
where the new triangle is.
Draw the new triangle onto the grid and
label it B.
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17 Transformations
Exercise 17.3
Focus
In each of these diagrams, shape A has been rotated to shape B around
centre C. Write down if the rotation is clockwise or anticlockwise.
a
b
A
B
C
c
C
B
d
B
C
Clockwise: A
B
A
C
Anticlockwise:
Complete these rotations of 90° clockwise
about the centre C.
a
M
2
Tip
A
PL
E
1
b
C
SA
C
3
Complete these rotations of 90° anticlockwise about the centre C.
a
C
b
C
Practice
4
Rotate the shapes 90° clockwise about the centre C.
a
b
C
c
C
C
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17.3 Rotations
5
Rotate the shapes 90° anticlockwise about the centre C.
a
b
c
C
C
C
6
This is part of Alysha’s homework. The centre of rotation is shown by a dot (•).
PL
E
Question:
Rotate shape A 90° clockwise about the centre of rotation (•).
Label the shape B.
Answer:
A
B
SA
M
Has Alysha got her homework correct?
Use diagrams to help you explain your answer.
Challenge
7
Rotate the shapes 90° about the centre of rotation C, using the direction shown.
a
anticlockwise
b
C
8
a
clockwise
C
Follow these instructions to make a pattern. 1. Rotate the shape 90° clockwise about C.
2. Draw the new shape.
C
3. Rotate the new shape 90° clockwise about C.
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17 Transformations
4. Draw the new shape.
5. Rotate the new shape 90° clockwise about C.
6. Draw the new shape.
9
b
What is the order of rotational symmetry of your completed pattern?
a
Rotate triangle A1B1C, using the same
instructions as question 8a. C
B1
PL
E
Label the vertices of the three new
triangles A2, B2, C then A3, B3, C then
A4, B4, C.
A1
On your completed diagram, join A1 to A2 to A3 to A4 to A1 with
straight lines. What shape have you just drawn?
c
On your completed diagram, join B1 to B2 to B3 to B4 to B1 with
straight lines. What shape have you just drawn?
d
Do you think that whatever shape you rotate, if you rotate it 90° clockwise
or anticlockwise three times, then the shape you get when you join
corresponding vertices will always be the same? Explain your answer.
You can use diagrams to help your explanation.
SA
M
b
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17.3 Rotations
SA
M
PL
E
Acknowledgements
219
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