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÷

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Written methods of calculations are based on mental strategies. Each of the four operations builds on mental skills which provide the foundation for jottings and informal written methods of recording. Skills need to be taught, practised and reviewed constantly. These skills lead on to more formal written methods of calculation.

Strategies for calculation need to be supported by familiar models and images to reinforce understanding. When teaching a new strategy it is important to start with numbers that the child can easily manipulate so that they can understand the concept.

The transition between stages should not be hurried as not all children will be ready to move on to the next stage at the same time, therefore the progression in this document is outlined in stages. Previous stages may need to be revisited to consolidate understanding when introducing a new strategy.

A sound understanding of the number system is essential for children to carry out calculations efficiently and accurately.

Introduction

3

Progression in written methods

for Addition

Number Track

Number Line

Expanded methodPartitioning and recombining

Formal Compact Method

4

5

Stage 1 – Number Track

• understand addition is combining groups of objects• count on using a number track• use a puppet to accentuate jumps

1 2 3 4 5 87610

90

7 3+and

•understand addition can be done in any order

add 3

and+

+ += =

6

Stage 2a – Introducing the number line

• link number track and number line• count on using a fully numbered number line (start counting on in ones and then move on to larger jumps)

1 2 3 4 5 876 1090

0 1 2 3 4 5 6 7 8 9 10

+ 3

7 3+and

+and

7

+ 1 + 1013 14 24

Stage 2b – Using a number line

13 + 11

• always encourage ESTIMATION first• teach and encourage children to partition numbers in different ways in order to bridge to the nearest multiple of 10• start from the largest number and then count on• progress from not bridging 10 to bridging through 10• progress from fully numbered line to partially numbered line then blank

13 + 18

+ 7 + 10

+ 1

13 20 21 31

Partition the smallest

number.Add the unit(s) first

8

Stage 3 – Partitioning & RecombiningThe Expanded Method

20 8

4 0 + 3

2 0 + 8

6 0 + 1 1 = 7 1

40 3

• Encourage ESTIMATION • reinforce place value by using place value cards to partition alongside place value apparatus (Dienes)• reinforce ‘carrying’ use equipment alongside expanded written method to bridge from concrete to abstract (see appendix a for recording – transition between equipment – pictorial recording and then abstract)

9

4 3

+ 2 8

7 1

1

+

20 8

4 0 + 3

2 0 + 8

6 0 + 1 1 = 7 1

40 3

Expanded method

leading to compact

method

Stage 4 – Expanded Method leading to Formal Compact Method

Add the unit(s) column then the ten(s) column to calculate the final answer

10

0.43

+ 0.28

0.71

1

0.40 + 0.03

0.20 + 0.08

0.60 + 0.11 = 0.71

0.20 0.080.40 0.03

+

Stage 4 – Expanded Method leading to Formal Compact Method (decimal)

Remember to line up decimal points

especially when number of

digits differs

• continue to encourage ESTIMATION• link to money (add more than two amounts) & measurement• link to using a calculator/interpreting calculator display

C M √ ±

AC C % ÷

7

4

1

0

8

5

2

.

9

6

3

=

x

-

+

11

Progression in written methods for Subtraction

Number Track

Number Line(Finding the difference

and counting back)

Expanded methodPartitioning and

recombining

Formal Compact Method

12

13

Stage 1 – Number Track

• understand subtraction is taking away objects

• jump/count back along a number track

• use a puppet to accentuate jumps

1 2 3 4 5 87610

90

7 3seven three

take away 3

14

Stage 2a – Introducing Number Line

• link number track and number line• understand subtraction is taking away objects• jump/count back along a fully numbered number line (start counting back in ones and then move on to larger jumps)

1 2 3 4 5 87610

90

7 3seven three

0 1 2 3 4 5 6 7 8 9 10

- 3

15

Stage 2bi – Using a number line (not bridging through 10)

523 – 18

18 23

• ESTIMATE first• understand ‘finding the difference’ AND ‘counting back’ has the same result• promote finding the difference when the numbers involved are close together• progress from counting back in ones to larger steps.

24 34 37

finding the

difference

counting back 37 – 13

- 3- 10

• Partition the smallest number. Count back the units, then tens.

16

Stage 2bii – Using a number line (bridging through 10)

43 – 27

• ESTIMATE first

• encourage children to partition numbers in different ways

• bridge through multiples of 10

• ensure children have the opportunity to solve subtraction problems in a range of different contexts

• encourage use of vocabulary and explanation

36 432616 40

- 10- 10 - 3- 4

17

Stage 3a – Expanded Method (no exchanging)

40 7

- 10 4

30 and 3

take away the units and then take away the

tens

47 - 14 = 33

• use place value apparatus (Dienes) to re-inforce concept of exchanging • move from concrete apparatus to expanded written method (see appendix a for recording – transition between equipment – pictorial recording and then abstract)• continue to encourage ESTIMATION

18

Stage 3b – Expanded Method (with exchanging)

40 3

- 20 7

10 and 6

10 +30

to subtract 7 units we need to exchange a

ten for ten units

43 - 27 = 16

• use place value apparatus (Diennes) to re-inforce concept of exchanging • move from concrete apparatus to expanded written method (see appendix b for recording – transition between equipment – pictorial recording and then abstract)• continue to encourage ESTIMATION

19

Stage 4a – Formal Compact Method

40 3

- 20 7

10 and 6

10 +30

• move from expanded written method to compact method• continue to encourage ESTIMATION

4 3

- 2 7

1 6

13

20

Stage 4 – Formal Compact Method (decimal)

• continue to encourage ESTIMATION

• link to money (giving change) and measurement

• link to using a calculator/interpreting calculator display

4 . 3

- 2 . 7

1 . 6

13 Remember to line up

decimal points especially when

number of digits differs

C M √ ±

AC C % ÷

7

4

1

0

8

5

2

.

9

6

3

=

x

-

+

21

Progression in written methods for multiplication

Repeated addition, arrays

Grid method(with imagery)

Grid method

Long multiplication

22

23

2 + 2 + 2 + 2 = 8

4 x 2 = 8

2 multiplied by 4

4 lots of 2

Stage 1 – Repeated addition, arrays

• understand that multiplication is a shortened form of repeated addition• understand multiplication as arrays and jumps on a number line

24

4 x 13 ‘four lots of thirteen’

4

10 3

40 + 12 = 52

4

10 3

40 12

Stage 2 – Modelling grid method with place value equipment

• use place value apparatus to illustrate grid method, encourage jottings

• use digits of 5 and below to avoid ‘difficult’ tables and ensure method is secure

25

Stage 2a – Modelling grid method with place value equipment (multiples of 10)

4

20 3

80 + 12 = 92

1280

20 ( 2 x 10 ) 3

4

• use place value equipment to illustrate grid method with multiples of 10• reinforce using known facts to multiply e.g. 4 x 20 = 4 x 2 x 10

(4 x 2 x 10) (4 x 3)

4 x 23 ‘four lots of twenty three’

26

Stage 3 – Grid method (no apparatus)

47 x 52

2444

80

200050

40

•continue to reinforce using known facts to multiply e.g. 40 x 50 = 4 x 5 x 10 x 10•progress to using the grid method efficiently to multiply decimals

7350

14

2000

80350

+ 14

45 x 6

36240

40 ( 4 x 10 ) 6

6(6 x 4 x 10) (6 x

6)

240 + 36 = 276

(4 x 5 x 10 x 10)

2

(7 x 5 x 10)

80(4 x 2 x 10)

14(7 x 2 )

27

5 6 × 2 7 1 1 2 0 (56 × 20) 3 9 2 (56 × 7) 1 5 1 2

1

Stage 4 – Long multiplication

• ONLY move on to this method if understanding of grid method is secure

4

1

28

5.6 × 2.7 11.20 (5.6 × 2.0) 3.92 (5.6 × 0.07) 15.12

1

Stage 4a – Long multiplication (decimal)

• continue to encourage ESTIMATION (re-inforce place value)

• link to money and measurement

• link to using a calculator and interpreting display

4

1

C M √ ±

AC C % ÷

7

4

1

0

8

5

2

.

9

6

3

=

x

-

+

29

Progression in written methods

for division÷

Division as sharing and grouping

Grouping on a number line

Link division and multiplication

Vertical recording

Chunking (fact box)

Short/long division

÷

30

31

Stage 1 – Division as sharing and grouping

• understand division as sharing, understand division as grouping• understand remainders

÷

÷15 divided into 3 equal groups

15 divided into 5 equal groups

sharing one at a time

32

Stage 2 – Grouping on a number line

• understand that division is repeated subtraction

• show division as equal groups on a number line

• then begin to understand remainder

33

Vertical recording (teacher model only)

•turn horizontal number line vertical so children can see link to vertical calculation and model recording, use to illustrate need to take ‘chunks’ for efficiency

0 3 6 9 12 15 18

18

15

12

9

6

3

- 3

- 3

- 3

- 3

- 3

0- 3

18 ÷3 = 6

18

- 3 ( 1 x 3 )

1 5

- 3 ( 1 x 3 )

1 2

- 3 ( 1 x 3 )

9

- 3 ( 1 x 3 )

6

- 3 ( 1 x 3 )

3

- 3 ( 1 x 3 )

0

34

• children need to see that when numbers are larger it is more efficient to subtract larger ‘chunks’• building a fact box will help children with the size of the ‘chunks’• children need to work with and without remainders considering if answer needs rounding up or rounding down

Fact Box

1 x 5 = 5

2 x 5 = 10

5 x 5 = 25

10 x 5 = 50

Stage 3 – Linking division & multiplication leading to chunking – introducing fact box

96 5

96 ÷ 5 = 19 r 1

96

- 50 ( 10 x 5 )

46

- 25 ( 5 x 5 )

21

- 20

1

What facts do I know about the 5 times-table?

35

• children need to see that when numbers are larger it is more efficient to subtract larger ‘chunks’• building a fact box will help children with the size of the ‘chunks’• children need to work with and without remainders considering if answer needs rounding up or rounding down

Fact Box

1 x 7 = 7

2 x 7 = 14

5 x 7 = 35

10 x 7 = 70

20 x 7 = 140

50 x 7 = 350

100 x 7 = 700

What facts

do I know about the 7 times-table?

100 ÷ 7 = 14 r 2

100

- 70 ( 10 x 7 )

30

- 28 ( 4 x 7 )

2

518 ÷ 7 = 74

518

- 350 ( 50 x 7 )

168

- 140 ( 20 x 7 )

28

- 28 ( 4 x 7 )

0

Stage 4 – Chunking with a fact box

36

560 ÷ 24

2 3 r 8

2 4 5 6 0

- 5 5 2

8

Stage 5 – Long division

• ONLY move on to this method if understanding is secure

• move on to show remainders as a fraction and decimal

Jottings – Fact Box

20

3

20 4

400 80

60 12

x

400 + 80 + 60 + 12 = 552

37

560 ÷ 24

2 3 r 8/24 ()

2 4 5 6 0

- 5 5 2

8

Stage 5a – Long division (showing remainder as a fraction)

• ONLY move on to this method if understanding is secure

• move on to show remainders as a fraction and decimal

Jottings – Fact Box

20

3

20 4

400 80

60 12

x

400 + 80 + 60 + 12 = 552

38

560 ÷ 24

2 3.333

2 4 5 6 0.00

- 5 5 2

8 0

7 2

8

Stage 5b – Long division (showing remainder as a decimal)

• ONLY move on to this method if understanding is secure

• move on to show remainders as a fraction and decimal

Jottings – Fact Box

20

3

20 4

400 80

60 12

x

400 + 80 + 60 + 12 = 552

39

T - tens U - units

12 + 19

12 + 19

Start with apparatus then show children how to record

pictorially

40

T - tens U - units

19 - 12

19 - 12

Start with apparatus then show children how to record

pictorially

41

Link division and multiplication

• understand that division is the • inverse of multiplication• reinforce division as grouping• emphasise link between times table facts and division facts

12 divided into groups of 3 gives 4 groups

12 3 = 4

12 divided into groups of

4 gives 3 groups

12 4 = 3

3 x 4 = 12 or 4 x 3 = 12

12 4 = 3 or 12 3 = 4

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Understanding the inverse/finding unknowns

(empty boxes)I can work out missing numbers in a number

sentence (year 1 & Year 2)

• Introducing the Inverse – Play Mrs/Mr Opposite. Every instruction the teacher gives the children have to do the opposite e.g. teacher says take one step forward, children take one step backwards or teacher says turn to the right, children turn to the left etc. Explain that in maths we call the opposite the inverse and that we are going to be looking at the inverse (opposite) of adding.

• Addition and Subtraction – Numicon Families (Using the inverse) Ten is the same as/equals nine add/plus one.

• Explain that the children are going to be using Numicon. Show them what it is and explain how it is structured. Model how this can be used to demonstrate the inverse

• Once imagery is secure replace one piece of Numicon with an empty box. Remember to move the = sign!

Ten is the same as/equals nine add/plus

one.

10 = 9 + 1

43

Understanding the inverse/finding unknowns

(empty boxes)I can work out missing numbers in a number sentence including where = sign is used to balance an equation

e.g. 6 + 4 = 3 + ? or 6 x 4 = 3 x ? (Year 3)

• Use Numicon and balance to model and for the children to practise in order to reinforce understanding of equality.

• Once imagery is secure replace one piece of Numicon with an empty box. Remember to move the = sign AND begin to explore balancing different operations.

I can work out missing numbers in a number sentence including those where = balances an equation e.g. 10 – 3 = 3 + ? or 2 x 3 = 60 ?

(Year 4)• Again use Numicon. Emphasis on exploring balancing

equations with different operations.

I can work out missing numbers in more complex calculations e.g. 3?67 – 192? = 1539 or

32500 ? = 325 (Year 5)

I can find the unknown in a calculation such as 0.215 + ? = 0.275 or 5.6 ? = 0.7, drawing on knowledge of number facts and place value, including using a calculator and

inverse operations to solve more complex problems such as 568.1 ? = 24.7 and explain

my reasoning (Year 6)