progression in written calculations

8/8/2019 Progression in Written Calculations

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Introduction

Written methods of calculations are based onmental strategies. Each of the four operations

builds on mental skills which provide thefoundation for jottings and informal written

methods of recording. Skills needs to betaught, practised and reviewed constantly.

These lead on to more formal written methodsof calculations.

This guidance is to support the teaching ofcalculations at Cheam Park Farm Junior School.

The transition between stages should not be

hurried as not all children will be ready to moveon to the next stage at the same time,therefore the progression in this document is

outlined in stages. Previous stages may need tobe revisited to consolidate understanding whenintroducing a new strategy while other children

may need to be moved on to the next step.

Suggestions for year groups have been includedbut this rather depends on the needs of the

class.

These guidelines have been produced using suggestions from 4S. Thank-you

M.Stafford June 2007

Year 3 Year 4 Year 5 Year 6


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40 8

Progression in teaching addition

Words linked to +

add, addition, and,count on, plus, sum,more, altogether,

increase

Mental skills

Recognise the size and position of numbers

Count on in ones and tens

Know number bonds to 10 and 20

Add multiples of 10 to any numberPartition and recombine numbers

Bridge through 10

Models and Images

Counting apparatus

Place value apparatus

Place value cards

Numbered number lines

Marked but unnumbered number lines

Empty number lines

Hundred squareCounting stick

Bead strings

Interactive teaching Programmes


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T U

T U

40 + 8

30 + 6

80 + 4

10

48 + 36

48

+ 36

Expanded method

It is important that the children havea good understanding of place value

and partitioning using concreteresources and visual images to supportcalculations. The expanded method

enables the children to see whathappens to numbers in the standard

written method.

4 8

+ 3 6

8 4

Year 4

Year 5

Year 3 Summer term

Column Method

standard written methodThe previous stages

reinforces what happens tothe numbers when they areadded together using more

formal methods 1

Year 5 Year 6

Extend this method to include

thousands and decimalsdepending on needs of child

Year 4 More able


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Progression in teaching subtraction

Mental skills

Recognise the size and position of numbers

Count on in ones and tens

Know number facts for all numbers to 20

Subtract multiples of 10 from any numberPartition and recombine numbers

Bridge through 10

Models and Images

Counting apparatusPlace value apparatus

Place value cards

Numbered number lines

Marked but unnumbered number lines

Empty number lines

Hundred square

Counting stick

Bead strings

Interactive teaching Programmes

Words linked to -

take away, subtract,subtraction, count back,

minus, less, decrease,difference between

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15 - 7 = 8

Biggest number FIRST

Subtract single digit numbers

often bridging through 10

Year 3

Begin to find the difference by counting up from the smallest number

Year 3

Partition numbers in order to take away

Year 3

Year 4

Year 4

Year 4


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Subtract 10 from a two-digit number

45 - 10

-10

35 45

Year 3

Subtract multiples of 10from any number

45 - 20

453525

-10 -10

Partition the number to be subtracted (no exchanging)

- 10- 10- 3

43 23

43332320

43

43 20 = 23

23 3 = 20

203

Decide whether to count on or count back

74 - 27 = 47

Year 3

Year 3

Year 3 Year 4

Year 5

Year 4

Year 4


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Progression in Teaching Multiplication

Mental Skills

Recognise the size and position of numbersCount on in different steps 2s, 5s, 10sDouble 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 squares

Number tracksNumbered number linesMarked but unnumbered linesEmpty number lines.Multiplication squaresCounting stickBead stringsModels and Images charts

ITPs Multiplication grid, Number Dials, Multiplication Facts

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Words linked to x

multiply, multiplication, multiple,double, array, times, lots of


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Count in tensfrom zero

Count in twosfrom zero

Count in fives fromzero (all multiples)

Know doubles andcorresponding halves

Know multiplication tables to 10 x 10

x56 x5 = 30

Use known facts towork out new ones

2 x5 = 10

8 x5 = 40

3 x5 = 15

86420 10

0 5 10 15 20 25 30

0 20 30

0 5010

Year 3

Year 3

Year 3

Year 3 Year 4

Year 4

Year 3

Year 5

Year 6

Year 4

Year 4

Year 4

Double 35

Halve 98Year 5 Year 6


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Understand multiplicationas repeated addition

2 + 2 + 2 + 2 = 84 x 2 = 8

2 multiplied by 4

4 lots of2

Understandmultiplicationas an array

Year 3

Year 3

40 8

3x48=

40

8

X3=120

X3=24

Use place value to support the multiplication of U x TU

3x48=(3x40)+(3x8)

Year 3

Year 4

Understand that

24 x 20 = 24 x 2 x 10

24 x 50 = 24 x 5 x 10

Use factors to multiply

Year 4

Year 5

Year 6

Year 4

Year 4

Year 5


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Use place value apparatus to supportthe multiplication of U x TUalongside the grid method

4

10 3

40 12

40 + 12 = 52

4

10 3

40 12

4 x 13

Year 4

Year 5

Multiplying TU x TU, HTU, ThHTU and decimals

10

4

30 3

300

120

30

12

14 x 33

= 330 +

= 132

462

300120

30

+ 12

462

Year 5

Year 6

56 27

1120 (56 20)

392 (56 7)15121

Standard written method

Year 6Year 5

Year 4


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Progression in Teaching Division

Mental Skills

Recognise the size and position of numbersCount back in different steps 2s, 5s, 10sHalve 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

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Words linked to

group, divide, division, divided by,divisible, factor, share, half,halve, remainder, quotient


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Count back in tens

Count back in twos

Count back in fives

Know halves

Use known multiplication facts to workout corresponding division facts

Half of 6 is 3

of 6 = 3

3020100

151050

10864?

If 2 x 10 = 20

then20 z 10 = 220 z 2 = 10

Year 3

Year 3

Year 3

Year 3

Year 3

Year 4

Year 4

Year 6

Year 5Half of130

Year 5 Year 6

Year 4 All multiples


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Understand divisionas sharing

Understand divisionas grouping

Reinforce division asgrouping through the

use of arrays

12 divided into groupsof 3 gives 4 groups

12 z 3 = 4

12 divided into groupsof 4 gives 3 groups

12 z 4 = 3

Year 3

Year 3

Year 3

Year 4

Year 4

Year 4


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Represent groupsfor division on anumber line using

apparatusalongside the line

0 3 6 9 12 15 18

0 18

18z

6 = 3

18 z 3 = 6

18 divided into groups of 3

18 z 3 = 6

Understand division asrepeated subtraction

using a vertical line andapparatus to make the

links

18

15

12

9

6

3

- 3

- 3

- 3

- 3

- 3

0

18 3 = 6

1 8

- 3 ( 1x

3 )1 5

- 3 ( 1 x 3 )

1 2

- 3 ( 1 x 3 )

9

- 3 ( 1 x 3 )

6

- 3 ( 1 x3 )

3

- 3 ( 1 x3 )

0- 3

Year 3

Year 4

Year 4

Year 5


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Children need to see that as thenumbers get larger, large chunk

subtraction is the more efficientmethod. Multiples of the divisor (largechunks) are taken away. Multiplication

facts are needed to see the size of

the chunk.

100 7 = 14 r 2

100

- 70 ( 10x

7 )30

- 28 ( 4 x7 )

2

Fact Box

1 x7 = 7

2 x 7 = 14

5 x 7 = 35

10 x 7 = 70

20 x7 = 140

50 x 7 = 350

100 x7 = 700

What facts do Iknow about the7 times-table?

518 7 = 74

518

- 350 ( 50 x7 )

168

- 140 ( 20 x 7 )

28

- 28 ( 4 x 7 )

0

Standard written methodLinks directly to large

chunk subtraction

560 24

2 3 r 8

2 4 5 6 0

- 4 8 0

8 0

- 7 2

8

Year 5

Year 6

Year 6

Year 5

Year 4

progression in written calculations

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