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Abbey Primary School – December 2012

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Written methods of calculations are reliable and efficient procedures which can be used in many different contexts. However, they are of no use to someone who applies them inaccurately or who cannot judge whether the answer is reasonable.

The progression towards a written method is crucial, since it is based on steps which are done mentally and which need to be secured first.

A structured approach to developing written methods relies on:

1) Apparatus

2) Apparatus + picture

3) Apparatus + picture + recording

4) Picture + recording

5) Recording

Developing a common whole school approach enhances pupils’ confidence and aids continuity across the school.

It can be hard for a pupil to hold all the steps of a calculation in their heads, so informal jottings should be encouraged at all times.

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.

Introduction

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Progression in Teaching Addition

Mental Skills

Recognise the size and position of numbersCount on in ones and tensKnow number bonds to 10 and 20Add multiples of 10 to any numberPartition and recombine numbersBridge through 10

Models and Images

Counting apparatus Place value apparatusPlace value cardsNumber tracksnumiconNumbered number linesMarked but unnumbered number linesEmpty number linesHundred squareCounting stickBead stringModels and Images chartsITPs – Number Facts, Ordering Numbers, Number Grid, Counting on and back in ones and tens

Key Vocabulary

add additionplus andcount ongreatermoreMostsumtotal altogetherincrease

40 8

4

1, 2, 3, 4, 5, 6 … there

are 6 teddies

Recognise numbers 0 to 10

Cardinality: Knowing that 6 objects is the number 6.

Count reliably up to 10 everyday objects

Find one more than a number One more than three is

four

Count all

Count in ones and tens

Count on (keeping number in your head / hiding the first

pattern) 3 + 2 =

5

Begin to use the + and = signs to record mental calculations in a number

sentence6 + 4 =

10

Know doubles of numbers and their corresponding halves

5

Know by heart all pairs of numbers with a total of 20

Know that addition can be done in any order

1 9 2 8 3 7 4 6 5 5

Put the biggest number first and count on

Begin to partition and recombine numbers in

order to add

3 +

5 5 8

+ 3

6

Know which digit changes

when adding 1s or 10s to any

number

15 + 1 = 16

15

15

15 + 10 = 25

15 + 20 = 35

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Adding two two-digit numbers (bridging through tens boundary)

Using a number line 48 8478

+30

80

+2 +4

48 + 36 = 84

Adding two two-digit numbers (without bridging)

Counting in tens and ones

15 + 13 = 28

15 + 10 + 3 = 28

15 25 28

15 16 17 18

25 26 27 28

25 35

25

16

48 + 30 + 6 = 84

7

48 + 36 = 84

T U

40 + 8

+ 30 + 6

80 + 4

10

It is important that the children have a good understanding of

place value and partitioning using concrete resources and visual

images to support calculations. The expanded method enables children to see what happens to numbers in the standard written

method.

T U

Remember to add the least significant digits (e.g. units in this case)

first

48 + 36 = 84

Adding two two-digit numbers (bridging through tens boundary)

Using place value cards and place value apparatus to partition

numbers and recombine

48

+ 36

14 (8 + 6)

70 (40 + 30)

84

40 8 30 6

8

225 + 156 = 381

200 + 20 + 5

+ 100 + 50 + 6

300 + 80 + 1

10

2 2 5 +1 5 6 3 8 1 1

4 8

+ 3 6

8 41

Standard written methodThe previous stages reinforce what happens to the numbers when they are added together

using more formal written methods.

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Progression in Teaching Subtraction

Mental Skills

Recognise the size and position of numbers Count back in ones and tensKnow number facts for all numbers to 20Subtract multiples of 10 from any numberPartition and recombine numbers (only partition the number to be subtracted)Bridge through 10

Models and Images

Counting apparatusPlace value apparatusPlace value cardsNumber tracks Numbered number linesMarked but unnumbered linesHundred squareEmpty number lines.Counting stickBead stringsModels and Images ChartsITPs – Number Facts, Counting on and back in ones and tens, Difference

Key Vocabulary

Subtract take away (specific use)minuscount backless fewerdifference between

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Begin to count backwards in familiar contexts such

as number rhymes or stories

Continue to count back in ones from any given

number

Take away

Find one less than a number

Count back in tens

Ten green bottles hanging on the wall …

Five fat sausages frying

in a pan …

Count backwards along a number

line to ‘ take away

If I take away four shells there are six

left

Three teddies take away two teddies leaves

one teddy

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Begin to use the – and = signs to record mental

calculations in a number sentence

6 - 4 = 2

Maria had six sweets and she ate four. How

many did she have left?

Know by heart subtraction facts for numbers up to 10 and 20

Begin to find the difference

40 – 27 = 13

27 4030

+ 1 + 10+ 1 + 1

28 29

Find the difference using the number line

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Find the difference using the number line

74 - 27 = 47

Extend to 3 digit numbers and decimals

40 – 27 = 13

27 30 40

+ 3 + 10

Find the difference using the number line

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Expanded method

It is important that the children have a good understanding of

place value and partitioning using concrete resources and visual

images to support calculations. The expanded method enables children to see what happens to numbers in the standard written

method.

40 + 3

- 20 + 7

10 + 6

10 +30

4 3

- 2 7

1 6

1 3

to subtract 7 units we need to

exchange a ten for ten units

43 - 27 = 16

Standard written methodThe previous stages reinforce

what happens to numbers when they are subtracted using more

formal written methods. It is important that the children have a good understanding of place

value and partitioning.

43 – 32 = 11

4 3- 3 2 1 1

40 + 3- 30 + 2 10 + 1

Extend to 3 digit numbers without exchanging.

column subtraction (without exchanging)

column subtraction (with exchanging)

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Progression in Teaching Multiplication

Mental Skills

Recognise the size and position of numbers Count on in different steps 2s, 5s, 10s Double numbers up to 10Recognise multiplication as repeated additionQuick recall of multiplication factsUse known facts to derive associated factsMultiplying by 10, 100, 1000 and understanding the effectMultiplying by multiples of 10

Models and Images

Counting apparatusPlace value apparatusArrays100 squaresNumber tracks Numbered number linesMarked but unnumbered linesEmpty number lines.Multiplication squaresCounting stickBead stringsModels and Images chartsITPs – Multiplication grid, Number Dials, Multiplication Facts

Vocabulary

lots ofgroups oftimesmultiplymultiplicationmultipleproductonce, twice, three timesarray, row, columndoublerepeated addition

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Count in tens from zero

Count in twos from zero

Count in fives from zero

Know doubles and corresponding halves

Know multiplication tables to 10 x 10

x 5 6 x 5 =

30

Use known facts to work out new ones

2 x 5 = 10

8 x 5 = 40

3 x 5 = 15

864 20 10

0 5 10 15 20 25 30

0 20 30 40 5010

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Understand multiplication as repeated addition 2 + 2 + 2 + 2 = 8

4 x 2 = 8

2 multiplied by 4

4 lots of 2

Understand multiplication as an array

Understand how to represent arrays on a number line

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Use place value apparatus to support the multiplication of U x TU

13 x 4 = 52

Use place value apparatus to support the multiplication of U x TU alongside the grid method

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10 3

4

10 3

40 12

40 + 12 = 524

10 3

40 12

4

10 310

23 x 4 = 92

Use place value apparatus to represent the

multiplication of U x TU alongside the grid method

80 + 12 = 92

13 x 4 = 52

1280

20 3

4

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Multiplying TU x TU

104

30 3

300

120

30

12

33 x 14 = 462

300

120

30

+ 12

462

Use same method to multiply HTU X U

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Progression in Teaching Division

Mental Skills

Recognise the size and position of numbers Count back in different steps 2s, 5s, 10s Halve numbers to 20Recognise division as repeated subtractionQuick recall of division factsUse known facts to derive associated factsDivide by 10, 100, 1000 and understanding the effectDivide by multiples of 10

Models and Images

Counting apparatusArrays100 squaresNumber tracks Numbered number linesMarked but unnumbered linesEmpty number lines.Multiplication squaresModels and Images chartsITPs – Multiplication grid, Number Dials, Grouping, Remainders

Vocabulary

lots ofgroups ofsharegrouphalvehalfdividedivisiondivided byremainderfactorquotientdivisible

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Count on in tens

Count on in twos

Count on in fives

Know halves

Use known multiplication facts to work out corresponding division

facts

Half of 6 is 3

½ of 6 = 3

If 2 x 10 = 20

then20 10 = 220 2 = 10

0 20 30 40 5010

864 20 10

201510 50 25

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Understand division as sharing

Understand division as grouping

Reinforce division as grouping through the

use of arrays

12 divided into groups of 3 gives 4

groups

12 3 = 4

12 divided into groups of 4 gives 3

groups

12 4 = 3

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18 4 = 4 r 2

Represent ‘groups’ for division on a number line

using apparatus alongside the line

0 18

18 divided into groups of 3

18 3 = 6

Remember to ask the question – how many groups of 3 in 18?

Introduce remainders

0 3 6 9 12 15 18

0 4 8 12 16 18

1 x 4 1 x 4 1 x 4 1 x 4 r 2

How many groups of 4s in 18? There are 4 groups of 4 with 2 left over.

3 6 9 12 15

1 x 3 1 x 31 x 31 x 31 x 3 1 x 3

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Children need to see that as the numbers get larger, chunking groups is the more efficient

method.

65 5 = 13

0 50 65

10 x 5 3 x 5

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How many groups of 5 in 65?There are 13 groups of 5 in 65?