Maths Frameworking Step 2 Intervention Workbook

Step 2

Developed in partnership with

Name Class

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How to use this book Organise your learning The Contents table at the start of the Workbook shows the topics you are going to cover.

Your teacher or tutor can set a s Y date for you to complete each topic by. s You can give a traffic light colour for each topic to show how you feel it went.

Y your teacher and your s You, parent or carer can write comments.

Work through each topic step by W step For each topic, there are: s Clear learning objectives s Worked examples to show you how to answer the questions s Practice questions to help you consolidate what you have learnt. A glossary and answers are available on the Collins website.

At the end of each chapter, there’s a comments box for your teacher or tutor to fill in on how you did.

Practise your mental maths Try the mental maths questions at the end of the Workbook as a warm up or to see what you have learned.

Celebrate your progress When you finish the Workbook, your teacher or tutor can fill in the Record of achievement certificate for you to keep.

How to use this book

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Step 2 Contents Strand/topic

Page Hours Due date

1 Number 1.1

Using place value to multiply and divide whole numbers by 10 or 100

Feedback Student

6

1

1.2 Putting decimals in order

8

1

1.3 Using mental methods

11

1

1.4 Multiplication and division facts

14

1

1.5 Written methods

16

1

1.6 Multiplying decimals

20

1

1.7 Solving problems

22

1

1.8 Number patterns

24

1

1.9 Number relationships

26

1

1.10 Square numbers and prime numbers

28

1

2.1 Coordinates

31

1

2.2 Formulae

36

1

3.1 Fractions and percentages

40

1

3.2 Ratio

43

1

Teacher/ tutor/mentor

2 Algebra Parent/carer

3 Ratio, proportion and rates of change

4

Contents

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Strand/topic

Page Hours Due date

4 Geometry and measures

Feedback Student

4.1 Properties of 2D and 3D shapes

46

1

4.2 Drawing 2D shapes

49

1

4.3 Reflections and rotations

53

1

4.4 Measurement

56

1

4.5 Reading measuring instruments

59

1

4.6 Perimeter and area

62

1

5.1 Recording data

66

1

5.2 Grouping data

69

1

5.3 Carroll diagrams and Venn diagrams

72

1

5.4 Frequency diagrams and line graphs

76

1

5.5 Statistical measures

80

1

Teacher/ tutor/mentor

5 Statistics

Parent/carer

Mental maths warm-ups Certificate of achievement

Contents

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1

Number

1.1 Using place value to multiply and divide whole numbers by 10 or 100 I can • multiply a whole number by 10 or 100 • divide a whole number by 10 or 100

Example a Karen has £65. How many pence is that? b A plank is 3200 mm long. How many centimetres is that?

Solution a There are 100 pence in one pound. Multiply by 100. Add two zeros.

65 × 100 = 6500

£65 is 6500p.

b One centimetre is 10 mm. Divide by 10. Remove a zero.

3200 ÷ 10 = 320

3200 mm is 320 cm.

Practice questions 1

Complete these multiplication grids.

×

10

100

8

80

800

10

100

59 40 71

× 5 70 200 19

6

1 Number

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×

10

100

6 30 700 240

×

10

100

20 400 800 9000

2

3

Write down the answer to each of the following. a 100 ÷ 10 =

b 470 ÷ 10 =

c 600 ÷ 10 =

d 500 ÷ 100 =

e 2700 ÷ 100 =

f 21500 ÷ 100 =

g 6000 ÷ 100 =

h 7000 ÷ 100 =

i 8000 ÷ 10 =

Write these amounts of money in pence. a £5 =

4

pence

c £40 =

pence

mm

b 32 cm =

mm

c 60 cm =

mm

Write these amounts of money in pounds. a 400 pence =

6

pence

Write these lengths in millimetres. a 9 cm =

5

b £18 =

£

b 5200 pence = £

c 19000 pence = £

d 6000 pence = £

Write these lengths in centimetres. a 300 mm =

cm

b 1450 mm =

cm

c 7000 mm =

1.1 Using place value to multiply and divide whole numbers by 10 or 100

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cm

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1.2 Putting decimals in order I can • order decimals to three decimal places

Example Order these three numbers from smallest to largest.

3.8

3.64

3.675

Solution Write the numbers in a column with the decimal points in line. units

tenths

hundredths thousandths

3

.

8

0

0

3

.

6

4

0

3

.

6

7

5

You can put in extra 0s to make all the numbers the same length if you want.

Look at the digits in each column separately, starting on the left.

They all have the same units digit, 3.

The first number has the largest tenths digit (8) so this is the largest number.

Of the other two, the last one has the largest hundredths digit (7) so this is the next largest number.

Here are the three numbers from smallest to largest.

3.64

3.675

3.8

Practice questions 1

8

Order these groups of numbers from the smallest to the largest. a 6.3

7.3

2.6

5.6

b 6.8

9.2

9.7

5.2

c 14.8

13.6

14

13

d 78.1

87

78.3

87.4

e 22.6

26.7

25.1

23

f 22.6

22.9

21

20.5

1 Number

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2

3

4

5

Order these groups of numbers from the smallest to the largest. a 4.3

4.1

3.45

4.25

b 0.6

6.0

6.6

1.6

c 19.3

13.3

13.9

11.3

d 63.4

34.6

46.4

34.4

Order these amounts of money from the largest to the smallest. a £1.08

£1.80

£0.88

£8.81

b £3.41

£3.14

£3.61

£4.13

c £61.42

£60.24

£64.04

£60.40

d £0.85

£0.87

£0.78

£0.58

Circle the larger number in each pair of decimal numbers. a 0.6

0.7

b 4.3

4.25

c 12.6

11.9

d 10.9

11.1

e 4.4

3.75

f 7.7

8.5

g 9.35

9.4

h 4.7

4.71

i 5

4.43

j 0.7

0.17

k 0.3

0.03

l 0.89

0.9

Circle the larger number in each pair of decimal numbers. a 0.625

0.652

b 4.125

4.711

c 8.675

9.005

d 3.25

3.195

e 4.42

4.125

f 7.77

7.111

g 9.325

9.4

h 4.7

4.815

i 5

4.435

1.2 Putting decimals in order

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6

10

Order these groups of decimals from the smallest to the largest. a 7.512

7.251

7.505

b 9.125

8.885

9.075

c 4.500

4.425

4.615

d 7.275

7.25

7.35

e 2.125

2.5

2.25

f 6.125

5.875

5.9

1 Number

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1.3 Using mental methods I can • use a range of mental methods with all operations

Example 1 Work out the following in your head. Do not use a calculator. a 56 + 27

b 56 – 27

c 56 × 2

d 56 ÷ 2

Solution Here is one way to answer each question. You might have different methods. Use the method you prefer. a 56 + 27 = 50 + 6 + 20 + 7

Split the numbers into tens and units.

= 70 + 13

Add 50 and 20; add 6 and 7.

= 83

70 + 10 = 80, then add 3.

b 56 – 27 = 56 – 20 – 7 = 36 – 7

56 – 20 is 36.

= 29

Count 7 back from 36.

c 56 × 2 = 50 × 2 + 6 × 2

Multiply the 50 and the 6 separately.

= 100 + 12 = 112 d 56 ÷ 2 = 50 ÷ 2 + 6 ÷ 2

Find half of 50 and half of 6.

= 25 + 3 = 28

Example 2 Work out the following in your head. Do not use a calculator. a £8.23 + £3.99

b £8.23 – £3.99

Solution £3.99 is 1p less than £4.00. a £8.23 + £3.99

b £8.23 – £3.99

= £8.23 + £4.00 – 1p

= £8.23 – £4.00 + 1p

= £12.23 – 1p

= £4.23 + 1p

= £12.22

= £4.24 1.3 Using mental methods

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Practice questions Do not use a calculator for these questions.

1

2

3

Work out these additions. a 53 + 26 =

b 25 + 42 =

c 29 + 43 =

d 87 + 32 =

e 75 + 52 =

f 69 + 63 =

a 84 – 23 =

b 72 – 43 =

c 91 – 54 =

d 60 – 47 =

e 89 – 35 =

f 114 – 82 =

Work out these subtractions.

Double each of these numbers.

24

4

6

12

34

66

55

28

72

80

144

194

156

178

182

45

Halve each of these numbers.

128

5

62

136

150

Work out these multiplications. a 14 × 3 =

b 3 × 32 =

c 3 × 41 =

d 24 × 4 =

e 22 × 5 =

f 3 × 62 =

g 4 × 17 =

h 15 × 6 =

i 54 × 3 =

j 4 × 62 =

k 5 × 31 =

l 36 × 4 =

a 84 ÷ 4 =

b 93 ÷ 3 =

c 128 ÷ 4 =

d 155 ÷ 5 =

e 64 ÷ 3 =

f 89 ÷ 4 =

Work out these divisions.

1 Number

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7

8

Work out these additions and subtractions. a 34 + 98 =

b 65 + 99 =

c 98 + 99 =

d 225 – 98 =

e 175 – 97 =

f 523 – 96 =

Work out the total cost of each of the following.

a Total cost =

b Total cost =

1.3 Using mental methods

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1.4 Multiplication and division facts I can • recall multiplication facts up to 10 × 10 and use them to do divisions

Example Work out the following. a 50 ÷ 5

b 50 ÷ 6

c 50 ÷ 7

Solution Use your knowledge of the multiplication tables to think of the closest numbers in each case. a The answer is 10 because 5 × 10 = 50. b 6 × 8 = 48 so the answer is 8 remainder 2. c 7 × 7 = 49 so the answer is 7 remainder 1.

Practice questions 1

Fill in the missing number in these calculations. a

2

14

× 6 = 30

b

× 9 = 36

d 7×

= 42

e 3×

= 21

g 24 ÷

=6

h 48 ÷

=6

c 9×

= 81

f 20 ÷

=4

i

× 10 = 70

Fill in the missing numbers to make these calculations correct. a

×

= 30

b

×

= 40

c

×

= 25

d

×

= 63

e

×

= 28

f

×

= 90

g

×

= 35

h

×

= 64

1 Number

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3

4

5

Work out these multiplications. a 7×9=

b 9×8=

c 6×8=

d 7×5=

e 5×9=

f 7 × 10 =

Work out these divisions. There may be remainders. a 41 ÷ 5 =

b 38 ÷ 6 =

c 60 ÷ 8 =

d 54 ÷ 6 =

e 54 ÷ 7 =

f 54 ÷ 8 =

In a multiplication grid, you multiply the numbers along the top by the numbers down the side to fill in each square. Complete the grids with the missing numbers. a

×

3

6

5

42

2

16

20

27

b

×

6

3

8

48

21

24

8

80

1.4 Multiplication and division facts

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1.5 Written methods I can • use efficient written methods of addition and subtraction • use efficient written methods of short multiplication and division

Example Work out the following. a £12.62 – £3.45

b 56 × 7

c 263 ÷ 5

Solution a Here is a written method for subtraction. Put the numbers in a column. Then take away the numbers, working from right to left.

1l 1 2 . 56l 12 – 3 . 45 9 . 1 7

The answer is £9.17. b Here are two methods for multiplication: the grid method 50 7 350

6 42

350 + 42 = 392

the column method 56 x 7 392 4

The answer is 392.

Use the method you prefer. c Here is a written method for division. Try dividing each number below the line by 5, working from left to right.

52 r 3 5 2 6 13

5 does not divide into 2, so divide 5 into 26.

The answer is 52 remainder 3.

16

1 Number

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Practice questions 1

2

3

Work out the totals of these amounts of money. a 451 + 385 =

b 565 + 478 =

c 197 + 262 =

d 96 + 39 =

e 23 + 86 =

f 865 + 472 =

Work out the totals of these amounts of money. a £6.43 + £2.56 =

b £2.67 + £1.15 =

c £4.56 + £9.60

d £2.64 + £6.45 =

e £4.15 + £7.49

f £4.56 + £8.71

Work out these additions. a 6.5 + 6.7 =

b 10.7 + 11.9 =

c 21.3 + 66.7 =

d 78.4 + 52.3 =

e 7.26 + 4.55 =

f 88.5 + 44.7 =

1.5 Written methods

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4

5

6

18

Work out the totals of these amounts of money. a £6.12 – £3.08 =

b £7.43 – £2.27 =

c £6.43 – £1.85 =

d £18.42 – £6.56 =

e £19.42 – £3.45 =

f £13.40 – £10.26 =

Work out these subtractions. a 263 – 135 =

b 143 – 86 =

c 783 – 462 =

d 286 – 59 =

e 934 – 445 =

f 462 – 275 =

a 23.4 – 12.6 =

b 29.0 – 11.3 =

c 33.2 – 20.8 =

d 71.2 – 40.5 =

e 90.8 – 87.6 =

f 33.3– 24.5 =

Work out these subtractions.

1 Number

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7

8

Use any method to work out these multiplications. a 55 × 3 =

b 46 × 5 =

c 32 × 6 =

d 24 × 8 =

e 29 × 9 =

f 42 × 7 =

Work out these divisions. There may be remainders. a 207 ÷ 3 =

b 342 ÷ 6 =

c 826 ÷ 7 =

d 121 ÷ 9 =

e 307 ÷ 4 =

f 461 ÷ 8 =

1.5 Written methods

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1.6 Multiplying decimals I can • multiply a simple decimal by a single digit whole number

Example Multiply 3.45 × 7.

Solution First multiply 345 × 7 and then insert a decimal point.

3 . 45 x 7 24 . 1 5 3

3

Keep the decimal point in the same place, counting from the right.

In this case, it is two places from the right.

The answer is 24.15.

Practice questions 1

20

Work out these multiplications. a 4.3 × 4 =

b 7.8 × 4 =

c 5.6 × 4 =

d 9.9 × 5 =

e 5.8 × 4 =

f 6.2 × 5 =

g 6.3 × 7 =

h 4.9 × 6 =

i 0.9 × 5 =

1 Number

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2

3

Work out these multiplications. a 3.5 × 7 =

b 42.8 × 4 =

c 76.8 × 3 =

d 4.04 × 6 =

e 73.8 × 4 =

f 3.93 × 5 =

g 3.51 × 9 =

h 4.08 × 6 =

i 2.64 × 5 =

Work out these multiplications. a 5.7 × 4 =

b 6.9 × 2 =

c 2.37 × 6 =

d 5 × 2.56 =

e 18.4 × 5 =

f 6.8 × 7 =

g 42.8 × 3 =

h 8 × 12.7 =

i 4 × 10.8 =

1.6 Multiplying decimals

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1.7 Solving problems I can • decide on the correct calculation to solve a problem

Example Books cost £6 each. How many can you buy with £100?

Solution This is a division problem.

100 ÷ 6 = 16 remainder 4.

The money will buy 16 books. There is £4 left over.

1

Pairs of trainers cost £47 and shirts cost £24.

£4

7

Practice questions 4

£2

a Work out the cost of a pair of trainers and two shirts. b Work out the change from £60 if you buy a pair of trainers. c Work out the cost of six shirts.

2

Look at this price list. a Find the cost of six loaves of bread. b Find the cost of buying milk, cheese and butter. c Serena pays for cheese with a £20 note.

Work out how much change she receives.

3

A teacher has 100 sheets of card.

To how many pupils can she give six sheets of card?

22

1 Number

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4

A minibus holds nine people. a Work out how many people seven minibuses hold. b How many minibuses are needed to transport 120 people?

5

Katie has £23.28 and Jack has £8.63. a How much more than Jack does Katie have? b How much do they have all together? c Katie gives half her money to her sister. How much is that?

6

A teacher always uses four pins to put posters up on the wall. She has 50 pins. a How many posters can she put up? b How many pins will she have left over?

7

Tony buys two t-shirts. They cost £12.90 and £15.49. a Work out the total cost. b Work out the difference in cost between the two t-shirts.

8

There are 73 people on a train.

At a station 15 people get off and 34 get on.

How many people are on the train now?

9

Adult tickets for the zoo cost £14.25. Child tickets cost £9.75.

Find the total cost for two adults and two children.

1.7 Solving problems

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1.8 Number patterns I can • recognise and describe number patterns

Example Here is a sequence of numbers:

9

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

24

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

1 Number

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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

1.8 Number patterns

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1.9 Number relationships I can • recognise and work out multiples of a number • recognise and work out factors of a number

Example a Write down two multiples of 16. b Write down all the factors of 16.

Solution a Multiples of 16 are 1 × 16, 2 × 16, 3 × 16, 4 × 16, and so on.

So two multiples of 16 are two from 16, 32, 48, 64, and so on. b The factors of 16 are numbers that multiply to make 16.

16 = 1 × 16 so 1 and 16 are factors.

16 = 2 × 8 so 2 and 8 are factors.

16 = 4 × 4 so 4 is a factor.

There are no other factors of 16. There are five all together.

Practice questions 1

Write down two factors that multiply to make each boxed number.

1

a

b

c

18

2

32

21

42

Fill in the missing multiples. a Multiples of 2:

2

4

6

b Multiples of 3:

3

6

9

c Multiples of 5:

5

10

d Multiples of 10: 10 20

26

d

20

35 50

100

1 Number

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3

4

Circle the numbers in each row that are multiples of: a 2

4

8

10

15

18

20

21

27

35

b 3

4

8

10

15

18

20

21

27

35

c 5

4

8

10

15

18

20

21

27

35

d 10

4

8

10

15

18

20

21

27

35

Write each number as different pairs of factors. a ____ ⫻ ____

5

6

b

8

21 ____ ⫻ ____

____ ⫻ ____

____ ⫻ ____

Write down all of the factors of each number. a 15

b 18

c 20

d 21

e 25

f 28

g 30

h 31

Find the common factors of both the numbers in each pair. a 15 and 24 b 18 and 24 c 10 and 25

7

Here is a list of numbers:

21

22

23

24

25

26

27

From the list, write down a all the multiples of 3

b a multiple of 6

c a multiple of 7

d a number that has 12 as a factor

e a number that has 9 as a factor 1.9 Number relationships

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1.10 Square numbers and prime numbers I can • recognise and work out square numbers • recognise and work out prime numbers

Example Here is a list of numbers:

14

15

16

17

18

19

20

21

a Find the prime numbers in the list. b Write down the square number on the list.

Solution a Prime numbers have exactly two factors: 1 and the number itself.

The only factors of 17 are 1 and 17. The only factors of 19 are 1 and 19.

17 and 19 are the only prime numbers in the list. b A square number is formed by multiplying a whole number by itself, such as 9 (3 × 3) or 49 (7 × 7).

The only square number on the list is 16. It is 4 × 4.

Practice questions

28

1

Write down the square numbers shown in these diagrams.

1

a

b

c

d

e

1 Number

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2

The first square number is 1 Ă— 1 = 1.

The second square number is 2 Ă— 2 = 4.

Write down a the 3rd square number. b the 6th square number. c the 9th square number.

3

Write down all the prime numbers between 10 and 20.

4

Here is a list of numbers.

32

33

34

35

36

37

38

39

a Write down the prime numbers on the list.

b Write down the square number on the list.

5

Here is a list of numbers.

121

132

144

156

169

180

196

Work out all the square numbers on the list.

6

Work out the prime numbers between 90 and 100.

1.10 Square numbers and prime numbers

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29

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Comments, next steps, misconceptions

30

1 Number

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