How to use this book
Each Year 7 topic is presented on a two-page spread
Organise your knowledge with concise explanations and examples
Key points highlight fundamental ideas
Notes help to explain the mathematical steps
Mixed questions further test retrieval skills after all topics have been covered
Key facts and vocabulary section helps to consolidate knowledge of mathematical terms and concepts
Test your retrieval skills by trying the accompanying questions for the topic
Answers are provided to all questions at the back of the book
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ISBN 9780008598648
First published 2023
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British Library Cataloguing in Publication Data.
A CIP record of this book is available from the British Library.
Authors: Leisa Bovey, Ama Dickson and Katherine Pate
Publisher: Clare Souza
Commissioning and Project Management: Richard Toms
Inside Concept Design and Layout: Ian Wrigley and Nicola Lancashire
Cover Design: Sarah Duxbury
Production: Emma Wood
Printed in India
2
1 RETRIEVE 1 ORGANISE 6 7 Place Value and Properties of Number Positive and negative temperatures and numbers 1 Use the thermometer to help answer these questions. a) The temperature is –2°C. It rises by 4°. What is the new temperature? b) The temperature is 3°C. It falls by 7°. What is the new temperature? c) What is the difference in temperature between these two cities? London: 10°C Oslo: –5°C d) Write these temperatures in order, from hottest to coldest: –4°C 8°C –7°C 1°C –9°C 2 Write the correct sign (< or >) between each pair of numbers. a) 5 9 b) –5 –9 c) 6 –27 d) –42 31 Adding and subtracting negative numbers 3 Work out: a) 5 + –5 b) 7 – –4 c) 250 + –100 d) –6 + 4 e) –13 – –3 f) 14 + –6 – 2 Multiplying and dividing negative numbers 4 Work out: a) –30 ÷ 10 b) –9 × 2 c) –100 ÷ –20 d) –6 × 5 e) –5 × –7 f) 28 ÷ –4 5 Work out: a) –4 + 10 ÷ 5 b) –5 × (3 – 7) c) 3 × –6 – 24 ÷ –8 d) 15 ÷ –5 + 4 × –3 6 Fill in the missing numbers in these calculations. a) 8 + 2 – 2 – = 0 b) –5 + 5 – 2 + = 0 c) 12 + 5 – 12 – = 0 d) –270 + 70 + = 0 Positive and negative temperatures and numbers Temperatures below freezing (0°C) are negative. Adding and subtracting negative numbers Adding a negative number is the same as subtracting, e.g. 3 + -2 = 3 - 2 = 1 Subtracting a negative number is the same as adding, e.g. 3 - -2 = 3 + 2 = 5 Negative numbers Negative numbers The temperature is –3°C. It rises by 5°. What is the new temperature? The new temperature is 2°C. Write these temperatures in order, from coldest to hottest: –5°C 7°C –9°C 0°C 3°C Find the numbers on a number line. –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 –9°C –5°C 0°C 3°C 7°C Start at –3°C and count up 5° Write the correct sign (< or >) between each pair of numbers: –4 –8 –10 –2 Find the numbers on a number line. –8 –4 0 –10 –2 0 –4 > –8 –10 < –2 a) Work out 3 + – 5 –3 –2 –1 0 1 2 3 Replace + – with –3 + – 5 = 3 – 5 = –2 b) Work out –3 + 5 –3 –2 –1 0 1 2 3 –3 + 5 = 2 c) Work out 3 – –5 Replace – – with + 3 – – 5 = 3 + 5 = 8 –Multiplying and dividing negative numbers Each counter represents –1. Write two multiplication facts and two division facts for this array. – – – –– – – –4 × –2 = –8 –8 ÷ 4 = –2 2 × –4 = –8 –8 ÷ 2 = –4 Work out: a) 3 × –5 3 × –5 = –15 b) –3 × 5 –3 × 5 = –15 c) –3 × –5 –3 × –5 = +15 d) 12 ÷ –3 12 ÷ –3 = –4 e) –12 ÷ 3 –12 ÷ 3 = –4 f) –12 ÷ –3 –12 ÷ –3 = +4 You don’t need to write the + sign in your positive × or ÷ negative = negative e.g. 6 × -3 = -18 6 ÷ -3 = -2 negative × or ÷ positive = negative e.g. -10 × 2 = -20 -10 ÷ 2 = -5 negative × or ÷ negative = positive e.g. -10 × -5 = 50 -10 ÷ -5 = 2
Contents 3 Contents 1 Place Value and Properties of Number Place value and rounding 4 Negative numbers 6 Factors and multiples 8 Powers and roots 10 Prime factorisation 12 2 Arithmetic Procedures with Integers and Decimals Addition and subtraction 14 Multiplication and division 16 Order of operations 18 3 Expressions and Equations Algebraic vocabulary and notation 20 Algebraic expressions 22 Simplifying expressions 24 The distributive law 26 4 Coordinates Working with coordinates 28 Solving problems involving coordinates ........................................................................................... 30 5 Perimeter and Area Perimeter 32 Area 34 Solving perimeter and area problems .............................................................................................. 36 6 Fractions Simplifying and converting fractions 38 Equivalent fractions and decimals 40 Ordering fractions and decimals 42 Adding fractions 44 Subtracting fractions 46 Multiplying fractions 48 Dividing fractions ............................................................................................................................ 50 Solving problems with fractions 52 Fractions as a multiplicative relationship 54 7 Ratio Introducing ratio 56 Simplifying ratios 58 Finding parts and wholes from ratios 60 Ratio and proportion 62 8 Transformations Translation 64 Rotation 66 Enlargement 68 Reflection ........................................................................................................................................ 70 Mixed Questions ......................................................................................................................... 72 Key Facts and Vocabulary 76 Answers ........................................................................................................................................ 80
Negative numbers
Positive and negative temperatures and numbers
Temperatures below freezing (0°C) are negative.
The temperature is –3°C. It rises by 5°. What is the new temperature?
Write these temperatures in order, from coldest to hottest:
7°C –9°C 0°C 3°C
Find the numbers on a number line.
Start at –3°C and count up 5° Write the correct sign (< or >) between each pair of numbers:
Find
The new temperature is 2°C.
Adding and subtracting negative numbers
Adding a negative number is the same as subtracting, e.g. 3 + -2 = 3 - 2 = 1
Subtracting a negative number is the same as adding, e.g. 3 - -2 = 3 + 2 = 5
Multiplying and dividing negative numbers
positive × or ÷ negative = negative
e.g. 6 × -3 = -18 6 ÷ -3 = -2
negative × or ÷ positive = negative
e.g. -10 × 2 = -20 -10 ÷ 2 = -5
Each counter represents –1. Write two multiplication facts and two division facts for this array.
negative × or ÷ negative = positive
e.g. -10 × -5 = 50 -10 ÷ -5 = 2
Work out:
a) 3 × –5 3 × –5 = –15
b) –3 × 5 –3 × 5 = –15
c) –3 × –5 –3 × –5 = +15
d) 12 ÷ –3 12 ÷ –3 = –4
4 × –2 = –8 –8 ÷ 4 = –2
2 × –4 = –8 –8 ÷ 2 = –4
e) –12 ÷ 3 –12 ÷ 3 = –4
f) –12 ÷ –3 –12 ÷ –3 = +4
You don’t need to write the + sign in your answer.
1 ORGANISE 6
–5°C
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 –9°C –5°C 0°C 3°C 7°C
–4 –8 –10
–2
the numbers
line. –8 –4 0 –10 –2 0 –4 > –8 –10 < –2 a) Work out 3 + – 5 –3 –2 –1 0 1 2 3 Replace + – with –3 + – 5 = 3 – 5 = –2 b) Work out –3 + 5 –3 –2 –1 0 1 2 3 –3 + 5 = 2
Work out
Replace – – with + 3 – – 5 = 3
+ –
on a number
c)
3 – –5
+ 5 = 8
–
– –
–
– –
– –
Negative numbers
Positive and negative temperatures and numbers
1 Use the thermometer to help answer these questions.
a) The temperature is –2°C. It rises by 4°.
What is the new temperature?
b) The temperature is 3°C. It falls by 7°.
What is the new temperature?
c) What is the difference in temperature between these two cities?
London: 10°C Oslo: –5°C
d) Write these temperatures in order, from hottest to coldest: –4°C 8°C –7°C 1°C –9°C
2 Write the correct sign (< or >) between each pair of numbers.
Adding and subtracting negative numbers
3 Work out:
5 + –5
7 – –4 c) 250 + –100
e) –13 – –3
Multiplying and dividing negative numbers
4 Work out:
a) –30 ÷ 10
c) –100 ÷ –20
e) –5 × –7
5 Work out:
a) –4 + 10 ÷ 5
b) –5 × (3 – 7)
c) 3 × –6 – 24 ÷ –8
d) 15 ÷ –5 + 4 × –3
6 Fill in the missing numbers in these calculations.
a) 8 + 2 – 2 – = 0
c) 12 + 5 – 12 – = 0
–9 × 2
–6 × 5
28 ÷ –4
–5 + 5 – 2 + = 0
–270 + 70 + = 0
RETRIEVE 1 7 Place Value and Properties of Number
a)
b) –5 –9 c) 6 –27 d) –42
5 9
31
a)
b)
d)
–6 + 4
f) 14
+ –6 – 2
b)
d)
f)
b)
d)
Multiplication and division
Multiplying and dividing with positive and negative integers
When multiplying or dividing a positive number by a negative number, the result is negative When multiplying or dividing a negative number by a negative number, the result is positive
5 × 1 = 5 –2 × 1 = –2
5 × 0 = 0 –2 × 0 = 0
5 × –1 = –5 –2 × –1 = 2
a) Calculate 6 × 3 + + = 6 × 3 = 18
When multiplying a positive number by a positive number, the result is positive (+ × + = +). 6 × 3 is the same as adding 3 lots of 6.
b) Calculate –8 × 3
–8 × 3 = –24 – × + = –
c) Calculate 8 × –3
8 × –3 = –24 + × – = –
d) Calculate –7 × –4
–7 × –4 = 28 – × – = +
Multiplication is commutative, so a × b = b × a . For example, 4 × 3 = 12 and 3 × 4 = 12
e) Calculate 15 ÷ 5
There are 15 dots with 5 dots in each group. There are 3 groups of dots.
15 ÷ 5 = 3
f) Calculate –20 ÷ 4
–20 ÷ 4 = –5 – ÷ + = –
g) Calculate 20 ÷ –4
20 ÷ –4 = –5 + ÷ – = –
h) Calculate –28 ÷ –7
–28 ÷ −7 = 4 – ÷ – = +
Multiplying and dividing with decimals
When multiplying a number by another number that is less than 1, the value decreases.
Work out 0.09 × 0.7 0.09 × 0.7 × 10 × 100 9 × 7 = 63 ÷ 1000 = 0.063
A dividend is the number that is being divided.
Multiply both numbers so they are whole numbers.
100 × 10 = 1000, so divide 63 by 1000 to get 0.063
A divisor is a number that divides another number and may leave a remainder.
divisor quotient dividend
20 ÷ 4 = 5
A quotient is the result of dividing one number by another.
When dividing with decimals, multiply each number by 10 until the divisor is a whole number.
2 goes into 18 nine times.
also goes into 1.8 nine times.
The calculations are equivalent.
b)
4 1 13 16 Ensure that the decimal points are lined up.
So, 1.36 ÷ 0.4 = 3.4
2 ORGANISE 16
1.8 ÷
1.8 ÷ 0.2 × 10 × 10 18 ÷ 2 = 9 1.8 ÷ 0.2 = 9
a) Work out
0.2
÷
1.36 ÷ 0.4 × 10 × 10 13.6 ÷ 4 0 3 . 4
Work out 1.36
0.4
2 2 2 2 2 2 2 2 2 0.2
0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
Multiplying and dividing with positive and negative integers
1 Work out:
7 × 8
9 × 6 c) –6 × 2
–5 × 10
2 Work out:
7 × –2
4 × –8 c) –2 × –3
–12 × –8
3 Work out:
4 Work out:
Multiplying and dividing with decimals
RETRIEVE 2 17 Arithmetic Procedures with Integers and Decimals
b)
d)
a)
d)
a)
b)
a)
b)
d)
12 ÷ 2
18 ÷ 9 c) –49 ÷ 7
–63 ÷ 7
a)
b)
c)
d)
108 ÷ –12
30 ÷ –6
–90 ÷ –9
–56 ÷ –7
a) 8 × 0.3 b) 0.5 × 0.7 c) 0.03 × 0.9 d) 32 × 0.1 e) 0.9 × 0.6 f) 0.11 × 0.4 6 Work out: a) 8 ÷ 0.4 b) 12 ÷ 0.6 c) 3.2 ÷ 0.8 d) 8.8 ÷ 0.11 e) 6.4 ÷ 0.08 f) 2.14 ÷ 0.2
5 Work out:
Multiplication and division
3 The distributive law
Using the distributive law
When multiplying a term by a group of terms added together, the result is the same as when multiplying each number separately and then adding them.
Using the distributive law, 4 × 8 can be written as:
4 × (2 + 6) = 4 × 2 + 4 × 6
a (b + c) = a b + a c
a (b – c) = a b – a c
Complete the calculation by finding the missing number.
24 × 36 = 24 × 9 ×
To find the missing number, think about the factors of 36.
9 is a factor of 36 and is multiplied by 4 to make 36. Therefore, 9 and 4 are factor pairs
The missing number is 4.
A factor is a number that divides into another number without leaving a remainder.
3 × 48 can be written as 3 × (40 + 8)
Algebraically, this is written as 3(40 + 8)
This can be solved by multiplying the term outside the brackets with each term inside the brackets.
(3 × 40) + (3 × 8)
= 120 + 24 = 144
Using the distributive law, 3 × 48 can be written as 3 × (50 – 2)
Algebraically, this is written as 3(50 – 2)
Solving:
(3 × 50) + (3 × –2)
= 150 – 6 = 144
The calculation 4 × 12 can be solved in this way using the distributive law:
4(7 + 2 + 3) = (4 × 7) + (4 × 2) + (4 × 3)
= 28 + 8 + 12 = 48
ORGANISE
26
3 × 40 40 3 8 3 × 8 50 3 3 × 2 48 The
law
distributive
3 The distributive law
Using the distributive law
1 Complete each calculation by finding the missing number.
a) 63 × 14 = 63 × 2 ×
b) 54 × 27 = × 2 × 27
2 Using the distributive law, write the calculation that is represented by this area model: 6
4 5
3 Complete the calculation by finding the missing numbers.
5 × 15 = 5(4 + ) = (5 × ) + (5 × )
4 Solve the following by using the distributive law.
a) 5(6 + 2)
b) 8(3 + 4)
c) 6(5 – 2)
5 Solve the following by using the distributive law.
a) 9(5 + 7 + 1)
b) 7(8 + 3 + 2)
RETRIEVE
27 Expressions and Equations
Mixed questions
1 Reflect the shape in the dashed mirror line.
2 What does the digit 3 represent in each of these numbers?
a) 131 279
b) 0.101 348
3 a) Convert 4 3 8 to an improper fraction.
b) Convert 25 7 to a mixed number.
4 Write down the ratio of blue counters to red counters to yellow counters. Give your answer in its simplest form.
5 Calculate:
−17 + −9
12 − −18
0.8 × 0.4
6 Write each fraction in its simplest form.
9 36
R R Y Y Y Y Y Y B B B B B B B B
b)
c)
a)
d) 7.2 ÷ 0.6
a)
b)
c)
72
21 35
15 18
Mixed questions
7 The temperature fell to −7 °C one winter’s night. By 9 am, the temperature was 4 °C.
By how much had the temperature risen?
8 Enlarge this shape by scale factor 3 about centre C.
9 a) Find the highest common factor of 24 and 42.
b) Find the lowest common multiple of 6 and 10.
10 Find the perimeter and the area of this trapezium.
Perimeter =
Area =
11
Calculate:
a) 5 × (8 – 3) + 10 × 3 – 1
b) 5 × 8 – (3 + 10) × 3 – 1
12 Amy and her friends share a birthday cake. Amy eats 1 6 of the cake and her friends eat 2 3 of the remainder.
What fraction of the cake is left?
C
4
4
5
10
cm
cm
cm
cm
73 Mixed Questions
Key facts and vocabulary
Adding and subtracting Line up the digits by place value
Convert fractions to equivalent fractions with the same denominator
–3 12 = 15 24 –6 24 = 9 24 = 3 8
Dividing Use place value to divide by 10, 100, 1000
In fractions, keep the first fraction the same, flip the second fraction, change ÷ to ×
Equivalent Represent the same amount or proportion
Factor A number that divides exactly into another number Fraction 3 4 Numerator Denominator
Highest common factor (HCF)
The highest factor that two numbers share (have in common)
Improper fraction A fraction with a numerator greater than the denominator 13 4 Integer A whole number
Lowest common multiple (LCM)
The lowest multiple that two numbers share (have in common)
Multiples of 12: 12 24 36 48 60 72
Multiples of 18: 18 36 54 72
Mixed number Has a whole number part and a fraction part
Multiple A number in a times table. Multiples of 4 are 4, 8,
76
Number
8
12
– 2 3 6
9
4
. 9 4
5 8
320 ÷ 10 = 32 320 ÷ 100 = 3.2 320 ÷ 1000 = 0.32
C K F 4 9 ÷ 3 5 = 4 9 × 5 3 = 20 27
1 3 = 2 6 = 3 9 6 : 2 3 : 1 =
Factors
1 2 3 4 6
1 2 3 6
of 12:
12 Factors of 18:
9 18
3 1 4
12, 16, 20, …
Key facts and vocabulary
Multiplying Use place value to multiply by 10, 100, 1000 4.5 × 10 = 45 4.5 × 100 = 450
× 1000 = 4500 In fractions, multiply the numerators and multiply the denominators 3
× 4 7 = 12 35
Power Repeated multiplication of a number by itself
Prime numbers have exactly two factors: 1 and itself; 1 is not a prime number
Inverse of a power
= 3.2 (to 1 decimal place)
than 5 so round down
= 9, so √9 = 3 Square root of 9 is 3.
(to 2 decimal places)
77 Key Facts and Vocabulary
4.5
5
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 Increasing Decreasing
( ) x2 ÷ × + – or or Place
Ones tenths hundredths thousandths 1 0.1 1 10 0.01 1 100 0.001 1 1000 7 3 6 4 6 100 = 3 50
Negative numbers
Order of operations B I D M A S
value The value of each digit in a number
85 = 8 × 8 × 8 × 8
8
Base
Prime factor decomposition 70 7 10 2 5 70 = 2 × 5
7 Ratio B B B B B Y Y Y black to yellow = 5 : 3 £6 £6 £6 £6 £6 £6 £30 £24 Sharing in the ratio of 1 : 4
32
33 = 27, so 3√27 = 3 Cube root of 27
3. Rounding
3.235
3.24
5
Simplify Simplify
9 15 = 3 5 ÷ 3 ÷ 3 9 : 3 3 : 1 ÷ 3 ÷ 3
×
Power (or index)
Prime
×
Root
is
3.235
=
Less
or more so round up
fractions and ratios by dividing both numbers by common factors
