• Build understanding with clear objectives, step-by-step worked examples and practice questions • Strengthen and practise mental maths with warm-ups and fact pages • Use alongside KS3 Maths Now Learn and Practice Book with topics covered in the same order
Support and Practice Workbook
Download a free PDF version for printing and copying (available in colour or black and white): www.collins.co.uk/KS3MathsNow/downloads
KS3 Maths Now
Support pupils to access KS3 Maths with tailored and scaffolded practice.
Support and Practice Workbook
ISBN 978-0-00-837833-2
KS3 Maths Now Learn and Practice Book 978-0-00-836286-7
KS3 Maths Now Teacher Handbook 978-0-00-836285-0
9 780008 378332
Boost confidence with worked examples and practice questions
Contents How to use this book
5
Correlation 179
Factors and multiples
6
Congruence and scaling
183
Sequences 10
Manipulating algebraic expressions 186
Perimeter and area
23
Working with fractions
Negative numbers
35
Circles 191
187
Averages 40
Finding probabilities
193
Equivalent fractions
60
Equations and formulae
202
Algebraic expressions
65
Proportion 204
Angles 80
Applications of graphs
213
Decimals 90
Comparing sets of data
216
Linear graphs
Percentage changes
219
106
Percentages 117
Polygons 222
3D shapes
122
Prisms and cylinders
224
Introduction to probability
132
Compound units
227
Ratio 135
Solving equations graphically
230
Symmetry 143
Pythagoras’ theorem
236
Using data
157
Manipulating brackets
240
Pencil and paper calculations
160
Review sheet
242
Transformations 169
Mental warm-ups
243
Working with numbers
175
Maths Facts
253
Percentage changes
177
Answers 258
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How to use this book Work through each topic step by step Clear learning objectives Worked examples to show you how to answer the questions Practice questions to help you consolidate what you have learned.
Be in charge of your learning You, your teacher or tutor can note down how you feel each topic went and write down any comments.
Practise your mental maths
Remember the key maths facts
Check your own answers
Remind yourself of the key maths facts at any point.
Answers are provided at the end of the Workbook.
Try the mental maths questions at the end of the Workbook to see what you have learned.
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Sequences 1 I can
• recognise patterns in sequences of numbers
Example Here is a sequence of numbers:
9
13
17
21
25
Work out the next two numbers. Solution The numbers increase by 4 each time. The next number is 25 + 4 = 29. The number after 29 is 29 + 4 = 33.
Practice questions 1
Write the next two numbers in each of these number sequences using the rule shown. a The rule is add on 2 each time. 22
24
26
28
b The rule is add on 5 each time. 7
12
17
22
c The rule is subtract 3 each time. 30
27
24
21
d The rule is subtract 10 each time. 95
85
75
65
e The rule is double each time. 1
10
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2
4
8
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2
3
Write the next two numbers in each of these number sequences. a 3
5
7
9
b 42
44
46
48
c 30
35
40
45
d 1
7
13
19
e 40
37
34
31
f 50
44
38
32
g 10
21
32
43
h 120
116
112
108
Write in the missing numbers in each of these number sequences. a 12
16
b 17
22
27
c 99
96
93
d 46
42
24
28
32
40 42
87
47
52
62
81 78 26
72 22
69
18
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Sequences 2 I can
• recognise and describe number patterns
Example 9
Here is a sequence of numbers:
13
17
21
Work out the next two numbers in the sequence. Solution Look at the differences between the numbers. They increase by 4 each time. ‘Add 4’ is the term-to-term rule. The next two numbers are 21 + 4 = 25 and 25 + 4 = 29.
Practice questions 1
12
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Write down the next two terms in these sequences. a 1
2
3
4
5
b 1
3
5
7
9
c 3
6
9
12
15
d 10
9
8
7
6
e 20
18
16
14
12
f 100
95
90
85
80
g 4
14
24
34
44
h 50
48
46
44
42
i 3
2
1
0
–1
j –10
–20
–30
–40
–50
–2
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2
3
4
Work out the next four terms in these sequences using the term-to-term rule shown. a Rule: add 4
3
b Rule: subtract 2
19
c Rule: add 5
8
d Rule: subtract 6
50
e Rule: add 7
10
f Rule: subtract 10
83
g Rule: subtract 4
27
h Rule: add 1
–4
Work out the next two terms in these sequences using the rule shown. a Rule: multiply by 2
1
2
b Rule: divide by 2
40
20
c Rule: multiply by 3
1
3
d Rule: divide by 4
64
16
4
9
Work out the next two numbers in these sequences. a 1
2
4
8
16
b 3
6
12
24
48
c 64
32
16
8
4
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Sequences 3 I can • recognise number patterns
Example Here is a sequence of numbers.
1
2
4
7
11
16
Work out the next two numbers in the sequence. Solution Look at the differences between the numbers. 1
2 +1
4 +2
7 +3
11 +4
16 +5
+6
+7
The differences increase by one each time. The next number is 16 + 6 = 22. Then the one after that is 22 + 7 = 29.
Practice questions 1
Work out the next two numbers in each of these sequences. a 17
20
23
26
29
32
3
4
6
9
13
18
c 10
12
16
22
30
40
1
2
5
10
17
e 80
76
72
68
f 50
49
47
44
40
b
d
14
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g
3
4
7
12
19
28
h
9
9.5
11
13.5
17
21.5
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2
Here is a sequence of patterns.
1+2=3
1+2+3=6
1 + 2 + 3 + 4 = 10
Work out the next two sums in the sequence.
3
Here is a different sequence of patterns.
1+3=4
1+3+5=9
1 + 3 + 5 + 7 = 16
Work out the next two sums in the sequence.
4
Fill in the missing numbers in these sequences. a
9
14
19
b
40
37
34
31
c
4
5
7
10
19
d
1
2
5
10
26
29
44 19 32 37
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65
15
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Sequences 4 I can
• recognise sequences and find further terms
Example Work out the next two terms in each sequence. a 3 6 12 24 48 __ __ b 2 8 18 32 __ __ c 2 5 7 12 19 31 __ __ Solution a You have already met arithmetic sequences, where the difference between terms is constant. This is not an arithmetic sequence. In this sequence each term is double the previous one. The next two terms are 48 × 2 = 96 and 96 × 2 = 192. b Look at the differences. 2 8 18 32 ____
6 10 14
The next differences will be 18 and 22. The next two terms are 32 + 18 = 50 and 50 + 22 = 72. You might also notice that each term is double a square number: 2 × 1 = 2, 2 × 4 = 8, and so on. The next two will be 2 × 25 = 50 and 2 × 36 = 72 as before. c This is an example of a Fibonacci sequence. Each term is the sum of the previous two. 2 + 5 = 7, 5 + 7 = 12 and so on. The next two are 19 + 31 = 50 and 31 + 50 = 81.
16
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Practice questions 1
Work out the next term in each sequence. a 10
16
4
34
____
4
8
16
32
____
c 1
3
9
27
81
____
60
30
15
____
Work out the next term in each sequence. a 3
3
28
b 2
d 120
2
22
12
27
48
75
3
____
b 24
12
6
c 0.5
2
4.5
d 640
160
8
____
12.5
40
10
____
____
Work out the missing terms in each sequence. a 3
____
11
15
19
____
b 5
____
20
40
80
____
6
3
c ____
24
d ____
600
12
300
150
27
____ ____
37.5
a Work out the number of dots in each of the next two patterns.
1
3
6
10
_____
_____
b Work out the next two numbers in each sequence. (Hint: Use the sequence in part a.) i 3
5
8
12
____
____
ii 3
9
18
30
____
____
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5
6
7
Work out the next two terms in each sequence. a 1
1
2
3
b 1
5
6
11
5
8
17
13 28
____ ____
____ ____
Work out the next two fractions in each sequence. a
2 3
3 5
4 7
5 9
b
1 2
3 4
5 8
7 16
6 11
9 32
Work out the number of dots in each of the next two patterns. a
2
6
12
20
_____
_____
_____
_____
b
1
18
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7
19
37
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8
Here is a sequence of growing patterns. 1
2
3
4
There is one small triangle in pattern 1 and four small triangles in pattern 2.
9
a How many small triangles are there in patterns 3 and 4?
________ and ________
b Work out the number of small triangles in pattern 10.
____________________
Here is a sequence of rectangles. Rectangle 1
Rectangle 2
4 6
Rectangle 3
5 7
Rectangle 4
6 8
Work out the area of the next rectangle in the pattern.
7
9
____________________
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Sequences 5 I can
• use a term-to-term definition to find the terms of a sequence • use a position-to-term definition to find the terms of a sequence • write an expression for the nth term of an arithmetic sequence
Example a This is the start of an arithmetic sequence. 20 24 28 32 … Work out the 10th term. b The nth term of a sequence is given by the formula n(n + 1). Work out the 10th term. c This is the start of an arithmetic sequence. 7 16 25 34 … Find a formula for the nth term. Solution a In an arithmetic sequence the difference between successive terms is always the same. In this case the term-to-term rule is ‘add 4’. To find the 10th term, add 4 nine times to the first term. The 10th term is 20 + 4 × 9 = 20 + 36 = 56. b To find the 10th term, replace n in the formula with 10. The 10th term is 10 × (10 + 1) = 10 × 11 = 110. c The term-to-term rule is ‘add 9’. Compare the sequence with the 9 times table. 7 16 25 34 … 9 18 27 36 … The nth term of the lower sequence is 9n. So the nth term of the upper sequence is 9n − 2.
20
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Practice questions 1
Find the term-to-term rule for each of these arithmetic sequences. a 5
8
c 53
2
73
14 93
…
____________
113
…
____________
10
b 15 c 6
e 7
14
20 16
d 13
25 26
19 9
18 30 36
25
11
22 35 46
31
13
…
…
37
15
…
… …
26
37
48
…
____________
d 50
46
42
38
…
____________
10th term = ___________________ 12th term = ___________________ 15th term = ___________________ 11th term = ___________________ 50th term = ___________________
…
f 40
37
34
31
28
…
…
10th term = ___________________
g 72
77
82
87
92
…
…
20th term = ___________________
h 72
66
60
54
48
…
…
11th term = ___________________
The nth term of a sequence is 5n − 3. Work out: b the 3rd term ________
c the 12th term ________
The nth term of a sequence is 10n + 8. Work out: b the 8th term ________
c the 15th term ________
The nth term of a sequence is 100 − 2n. Work out: a the 5th term _________
6
…
…
a the 1st term _________
5
…
…
a the 1st term _________
4
b 15
Work out the given term for each sequence. a 6
3
11
b the 10th term ________
c the 24th term ________
The nth term of a sequence is n2 + 4. Work out the first five terms.
_______________________
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7
The nth term of a sequence is 100 − 2n. Work out the first five terms.
8
_______________________
The nth term of a sequence is n2 − 1. Work out: a the 1st term _________
9
b the 5th term ________
The nth term of a sequence is n(n + 3). Work out: a the 1st term _________
10
11
a 5
10
15
20
25
30
…
nth term = ___________
b 8
13
18
23
28
33
…
nth term = ___________
c 1
6
c the 7th term ________
11
16
21
26
nth term = ___________
…
Work out a formula for the nth term for each of these arithmetic sequences. nth term = ___________
a 7
9
b 8
14
20
26
32
…
nth term = ___________
c 6
13
20
27
34
…
nth term = ___________
e 7 f 12
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b the 3rd term ________
Work out a formula for the nth term for each of these arithmetic sequences.
d 14
22
c the 9th term ________
11
17 16 14
13
20 25 16
15
23 34 18
…
26 43 20
… … …
nth term = ___________ nth term = ___________ nth term = ___________
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Mental warm-up 1: Number
Mental warm-up 2: Number
1 Round 4719 to the nearest hundred. 2 48 × 9 = 432. What is 432 ÷ 9? 3 Work out 299 + 399. 4 Work out 35 ÷ 4. 5 Tables seat five people. How many tables are needed for 27 people? 6 What is the hundreds digit in 3456? 7 The temperature is 2 °C. It falls by five degrees. What is the new temperature? 8W hat fraction of this shape is filled? Write your answer as simply as possible.
1 Round 18.4 to the nearest 10. 2 13 × 15 = 195. What is 195 ÷ 13? 3 Work out 97 – 79. 4 Work out 73 ÷ 10. 5 Work out 43 + 69. 6 Write in figures six hundred and six. 7 What fraction of this shape is filled?
3 18 4 7 remainder 3 3 7 10 8 Down 4 degrees 11 2000 12 14:15
Answers to Mental warm-up 2: Number
Write down the next number in the sequence. 11 What units should you use to measure the mass of a pen? 12 How many minutes is 212 hours? 13 Write 6.45 am in the 24-hour clock.
1 20 2 15 5 112 6 606 9 £30.05 10 101
9 Write 83 mm in centimetres. 10 Here is a sequence of numbers: 30 27 24 21 18
8 The temperature changes from –6 °C to –10 °C. Has it gone up or down? How many degrees? 9 Write thirty pounds and five pence in figures. 10 Write down the next odd number after 99. 11 A bottle holds 2 litres of water. How many millilitres is that? 12 The time now is 13:45. Write down the time 30 minutes later.
1 4800 5 6 10 15
2 48 3 698 6 4 7 –3 °C 11 grams
4 8 remainder 3 1 8 4 9 8.3 cm 12 150 13 06:45
Answers to Mental warm-up 1: Number
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Maths Facts
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