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Emmanuel Church of England Primary School

Mathematics Calculation Policy

Updated March 2016

Emmanuel Church of England Primary School Calculation Policy

Nursery

Addition Subtraction Multiplication Division

Children are expected to work with

numbers 0 – 10.

Children are encouraged to use stories,

songs and rhymes to recall simple addition

facts e.g. One, two, buckle my shoe.

Children begin to use appropriate language

to compare two sets of objects (e.g. more).

Children find one more by counting on using

concrete resources e.g. bead strings.


numbers 0 - 10.


songs and rhymes to support their initial

understanding of subtraction e.g. Ten Green

Bottles, Five Little Ducks.


to compare two sets of objects (e.g. fewer)

using counting on.

Children should relate subtraction to

finding one less than a number by counting

back on a number track.

There

are

four balloons. How many are left if one is

removed?


Reception



numbers 0 – 20.


songs and rhymes to recall simple addition

facts e.g. One, two, buckle my shoe.

Children use moveable objects when finding

totals.

Children can add two groups of objects

together by starting with the biggest

number.

Children begin to develop ways of recording

calculations using pictures and objects.

Children

begin to

develop ways of recording calculations using

number lines, pictures and objects.


numbers 0 - 20.


songs and rhymes to support their initial

understanding of subtraction e.g. Ten Green

Bottles, Five Little Ducks.


to compare two sets of objects (e.g. fewer)

using counting on.

Children should relate subtraction to

finding one less than a number by counting

back on a number track.

There are four balloons. How many are left

if one is removed?


calculations using number lines, pictures and

objects.


numbers 0 - 20.

Objects/Pictures/Symbols

Children count equal groups of objects in

real-life contexts.

Count in 2s using socks or shoes.

Count in 5s and 10s using hands.

Count in multiplies using numicon.

Children will work on practical problem-

solving activities involving equal sets or

groups.

Doubling

Multiplication is related to doubling and

counting groups of the same size up to 10.


numbers 0 - 20.


Children will understand equal groups and

will share objects equally into groups.

Grouping and sharing

Children will understand that sharing and

grouping means the same amount in each

group.

Children will work on practical problem-

solving activities involving sharing and

grouping e.g. 8 cars are shared between 4 children. How many cars does each child get?

There are 8 cars. How many children can have two cars each?

Halving

Children to understand

halving as having two equal

groups.


Year 1



numbers 0 – 20.


Children are encouraged to use a variety of

visual resources and pictorial

representations to support their learning of

number facts.

Number Lines:

Record drawing jumps on prepared lines or

constructing own lines. Children should be

encouraged to add units to bridge through

ten.


numbers 0 – 20.


Encourage the use of concrete objects and

pictorial representations. e.g. bead strings,

counters, dienes, number lines, number

tracks and hundred squares.

Number Lines:


calculations using number lines, pictures and

objects.


numbers 0 – 20.


Children will continue to participate in

practical activities that involve counting in

2s, 5s and 10s e.g. counting socks, shoes,

animals’ legs, fingers, gloves, numicon and

coins.

Children to begin to understand

multiplication as repeated addition using

bead strings.

Children to understand multiplication as

groups of objects.


numbers 0 – 20.


Children will continue to participate in

practical activities that involve sharing and

grouping e.g. sharing cards when playing a

game, putting objects into cups, hoops and

onto plates.




group.

Grouping

Children will sort a number of objects into

equal sized groups e.g.

There are 6 coins, how can I group them equally?

Halving

Children to understand halving as having

two equal groups and that it is the inverse

of doubling.


Establish that addition and subtraction are

related operations.

Encourage estimation and the use of inverse

to check answers.

Ensure that children know that addition is

commutative (can be done in any order).

Counting back:

End of Year 1:

Move on to bridging through 10 in

preparation for Year 2.

Counting on:

Establish that addition and subtraction are

related operations.


to check answers.

Ensure that children know that subtraction

is non-commutative (cannot be done in any

order).

Children will describe pictorial arrays and

make their own arrays, counting in groups of

2s, 5s and 10s.

How many rows? How many columns? How many in each row? How many in each column?

Children will solve counting problems e.g.

How many 2s make 8?

Doubling

Multiplication is related to doubling and

counting groups of the same size up to 10.


to check answers.

Ensure that children understand that

multiplication is commutative (can be done

in any order).


to check answers.


Year 2


Children are expected to work and calculate

with numbers up to 100.


Blank Number Lines:

Children should be encouraged to use empty

number lines to add tens and ones and then

use these to bridge through ten using

number bond knowledge.

When adding two digit numbers, children

should partition the second number into

tens and ones.

Encourage moving on from adding single 10s

and single 1s.

Children are expected to work and calculate

with numbers up to 100.


Blank Number Lines:

Counting Back:

Subtracting a single digit:

42 – 6 =

Children are expected to recall and use the

multiplication facts for 2, 5 and 10

multiplication tables.


Missing box calculations:


repeated addition.

Number lines

5 + 5 + 5 = 15

5 x 3 = 15


grouping.


division facts for 2, 5 and 10 multiplication

tables.



Children should understand division as

repeated subtraction.

Blank Number lines

How many 3s in 15? Children to count forward in 3s to 15 or

count back in 3s from 15.

15 ÷ 3 = 5




group.


100 square

When modelling addition as partitioning and

recombining using a hundred square,

children should be reminded to move down

to count in tens and across to count in ones.

Partitioning in Spring Term

Use place value counters to support

children.

+

68

26

94

60 8

20 6

80 14

Fact Families:

Encourage estimation and the use of

inverse to check answers.


Subtracting a 2 digit number:

Count back on an empty number line by

partitioning the number they are

subtracting into tens and ones.

Counting On:

46 – 33 =

100 square

Children should be encouraged to subtract

9 and 11 on a hundred square by subtracting

10 and then compensating with

understanding.

Partitioning in Spring Term

Use place value counters to support

children.

-

86

33

53

80 6

30 3

50 3

Arrays

Use arrays to group using the multiplication

sign.

2 x 3 = 6

3 x 2 = 6

Children will solve one-step multiplication

problems e.g.

There are 5 oranges in a net. How many oranges in 6 nets?

Doubling

Children to know that doubling is the same

as multiplying by 2.

Children to know the doubles to 30 + 30 and

doubles of multiples of 10 up to 100.





in any order).

Grouping

There are 30 balloons. How can I group them equally?

Sharing

15 sweets are shared between 3 children. How many sweets do they each get?

Arrays

Use arrays to find division facts for 2, 5

and 10 multiplication tables.

o o o o o

o o o o o

o o o o o

o o o o o

o o o o o

o o o o o

We divided 30 into rows of 5. We use the ÷ sign for divide. We write 30 ÷ 5 = 6 as 30 can be divided into 6 rows of 5.


commutative (can be done in any order). Encourage estimation and the use of



subtraction is non-commutative (cannot be

done in any order).

Children will solve one-step division

problems e.g.

Oranges are in nets of 5. How many nets will 30 oranges fill?

Halving

Children to understand halving as the same

as dividing by 2.

Children should be able to halve numbers up

to 30 and halve multiples of 10 up to 100.


to check answers.


division is non-commutative (cannot be done

in any order).


Year 3



numbers up to 1000.

Blank number lines

Children continue to use number lines to

add HTU.

Children continue to use expanded method

to add HTU.

Progressing from:

Using place value counters to support

children.

+

68

26

94

60 8

20 6

80 14

To

Vertical expanded method

Use Place Value Counters to demonstrate

exchange. Record calculations on the right.

235

+ 28

______

13 (5+8)

50 (30+20)

200 (200)

_____

263


numbers up to 1000.

Blank number lines

Counting back:

Children should be able to use an empty

number line to ‘take away’ by partitioning

the second number.

Counting on:

Children are encouraged to use empty

number lines to ‘find the difference’

113-87=

Compensating:

Children need to understand that 178 – 49

is the same as 178 – 50 + 1.

Progressing from:

Using place value counters to support

children.

-

86

33

53

80 6

30 3

50 3


multiplication facts for 3, 4, 6 and 8


Pictures/Symbols/Objects:



repeated addition.

Grid Method

46x

4

40x4

6x4

160

24

46x

4 = 184


grouping.

Arrays (until children are secure in using

multiplication facts)

8 x 4 = 32

or

4 x 8 = 32

or

8 + 8 + 8 + 8

or


division facts for 3, 4, 6 and 8




Children to divide two digit numbers by one

digit.

Number lines

How many 6s are there in 32? How many left over? Groups of 6 are taken away and the

numbers of groups are counted. Any left

over are remainders.

32 ÷ 6 = 5 r2

Arrays (until children are secure in using

multiplication facts)

How many 4s in 32?


Compact column addition if children are

ready (modelled side by side)

H T O

2 6 8

3 5 6

1 4

1 1 0

5 0 0

6 2 4

H T 0

2 6 8

3 5 6 1 1 0

6 2 4

+ +

+

Children to leave a line after they have

written the column addition.

“8+6=14. We write 4 in the ones column and a 1 under the numbers we’re adding in the tens column for the 10. We add the 10s, put the 2 in the Tens and 1 for a 100 under the numbers we’re adding in the Hundreds column. Then we add the 100s to get 624”





To

Vertical Expanded Method:

Use Place Value Counters to demonstrate

exchange. Record calculations on the right.

287

- 145

_____

2 (7-5)

40 (80-40)

100 (200-100)

______

142

Progressing to:

Compact column subtraction if children are

ready (modelled side by side.)

287

- 145

_____

2 (7-5)

40 (80-40)

100 (200-100)

______

142





order).

4 + 4 + 4 + 4 + 4 + 4 + 4+ 4

Use known facts and place value

“If I know 3 x 8 = 24, I also know 30 x 8 = 240 3 x 80= 240”

Fact families

4 x 6 = 24

6 x 4 = 24

24 ÷ 4 = 6

24 ÷ 4 = 6

Children will solve multiplication problems.



Ensure that children know that


in any order).

Use known facts and place value

÷ 4 is the same as halving and halving again.

Fact families

4 x 6 = 24

6 x 4 = 24

24 ÷ 4 = 6

24 ÷ 4 = 6

Children will solve division problems.


to check answers.

Ensure that children know that division is

non-commutative (cannot be done in any

order).

287

-145

142


Year 4



numbers up to 10 000.

Children to continue to use the vertical

expanded method to add larger numbers.

Children to leave a line after they have

written the column addition.

Column Addition:

Begin by explicitly writing the carrying of

tens and hundreds.

637

+ 894 1 1 0

______ 1531

Before moving to a more compact method

when appropriate.

637

+ 894 1 1

_____

1531

Further develop this method using decimals

within context e.g. money.

16.37

+ 28.94 1 1 1

_______

45.31




numbers up to 10 000.

Number Lines:

Children should first be encouraged to use

empty number lines to subtract by counting

up (Complementary Addition).

Expanded Method:

Place Value Counters should be used to

reinforce children’s understanding of place

value.

4287 4000 + 200 + 80 + 7

- 2145 2000 + 100 + 40 + 5

_______________________

2000 + 100 + 40 + 2 = 2142

Progressing to:

4287

- 2145

_____

2 (7-5)

40 (80-40)

100 (200-100)

2000 (4000-2000)

______

2142


multiplication facts up to 12 x 12.



Children to multiply two and three digit

numbers by a one digit and two digit

number.

Grid method

Progressing from:

46x4 40x4

6x4

160

24

46x4 = 184

351x9 300x9

50x9

1x9

2700

450

9

351x9 = 3159

To:


division facts up to 12 x 12.



Short division

Supported by use of place value counters.

Dividing by 10 and 100

Move the digits once to the right for ÷10

and twice to the right for ÷100. Use zero

as a placeholder where there are gaps.




Compact Column Subtraction

After exchange, the ‘13’ should be written

next to each other in the square above the

3 in this example.

Further develop this method using decimals

within context e.g. money.





order).

Column multiplication (expanded)

T U

5 7

0 x 3

2 1 (3 x 7)

1 5 0 (3 x 150)

1 7 1

When moving to formal methods:

-begin with calculations that children can do

mentally.

-demonstrate informal and formal methods

side by side and step by step e.g.

partitioning and grid method, grid method

and column method.

Multiplying by 10 and 100

Move the digits once to the left for x10

and twice to the left for x100. Use zero as

a placeholder where there are gaps.





in any order).


to check answers.



order).


Year 5



numbers up to 10 000, including decimals

with up to two places.

Column Method:

1637

+ 2894

1 1 1

_______

4531

Consolidate method using decimals.

16.37

+ 28.94 1 1 1

_______

45.31






numbers up to 10 000, including decimals

with up to three places.

Expanded Method:

4287

- 2145

_____

2 (7-5)

40 (80-40)

100 (200-100)

2000 (4000-2000)

______

2142

Progressing to:

Compact Column Subtraction



7 in this example.

Children are expected to multiply numbers

using known facts e.g.

7 x 6 = 42 Therefore 70 x 60 = 4200

Children are expected to multiply numbers

up to four digits by a one or two digit

number using an efficient written method,

including long multiplication for two digit

numbers.

Column method (expanded)

3 2 6 8

x 4

3 2 (4 x 8)

2 4 0 (4 x 60)

8 0 0 (4 x 200)

1 2 0 0 0 (4 x 3000)

1 3 0 7 2

1

Column method

Short multiplication to multiply by a 1-digit

number

Children to leave a line after writing the

calculation:

T O HT O ThHT O

8 7 3 8 7 4 3 8 7

x 6 x 6 x 6 4 5 4 2 5 4

5 2 2 2 3 2 2 2 6 3 2 2

Children are expected to divide numbers

using known facts e.g.

63 ÷ 9 = 7 Therefore 630 ÷ 9 = 70

Children are expected to divide up to 4

digit numbers by a 1 or 2 digit number using

the efficient written method of short

division and interpret remainders

appropriately.

Dividing by a 1-digit number

Progressing from:

T O

2 9

3 8 7

6 0

2 7

2 7

0

To work out 87÷3 we start with the 80. Count out the 10s of 3:10x3=30; 20x3=60; 30x3=90. Stop too big. We can only get 20 3s out of 80. We write the 2 in the tens column, as 20 is 2 tens, and write the 60 below the 87 and subtract. This leaves 27. Now we find the 3s in 27 which is 9. This means 87÷3=29.


Children should identify when numbers are

close together and finding the difference

on an empty number line would be more

appropriate.





order).

Long multiplication to multiply by a 2-digit

number


calculation.

Children to cross out smaller numbers so

they know not to add them.

8 7 3 8 7

x 3 4 x 3 4 2 3 2 3 4 8 1 5 4 8 2 2 2 2 6 1 0 1 1 6 1 0

2 9 5 8 1 3 1 5 8 1

Multiply 3 and 4-digit whole numbers and

decimals with up to 2 decimal places, by 1-

and 2-digit numbers

9 .5 3 8. 7 6

x 2 7 x 3 4____ 3 3 3 2

6 6. 5 1 5 5. 0 4 1 2 2 1

1 9 0. 0 1 1 6 2. 8 0

2 5 6. 5 1 3 1 7. 8 4 1

Children to solve multi-step word problems.

To:

Instead of writing down each step we do an

extra calculation in our heads.

T U

1 4

7 9 28

We work out how many 7s will go into the 9; a short cut to working out how many 10s of 7 go into 90. The answer is 1. The 2 left over we carry over to the 8 to make 28. 7s into 28? 4, we put the 4 in the 1s column.


H T U

5 4 r 11

1 6 8 7 5

8 0

7 5

6 4

1 1

Since we are dividing by 16 it is useful to derive the 16 times table we can refer to as we do the division.

1x16 16

2x16 32

3x16 48

4x16 64

5x16 80

6x16 96

: :


Multiplying by 10, 100 and 1000

Children will multiply whole and decimal

numbers by 10, 100 and 1000 where answers

have up to 3 decimal places

Move the digits once to the left for x10,

twice to the left for x100 and three times

to the left for x1000. Use zero as a

placeholder where there are gaps.

Fractions

Multiply proper fractions and mixed

numbers by whole numbers, supported by

materials and diagrams.

e.g. 2/3 x 3 = 6/3 = 2





in any order).

We cannot divide the 8 by 16 so we work out how many 16s will go into 87. There are 5 as 5x16=80 so we write 5 in the 10s and subtract to get 7. Now we involve the 5 and use 75. There are 4 16s in 75 as 4x16=64 and 5x16=80 is too big. We write 4 in the 1s column and subtract; the remainder is 11, we write: 874÷16 = 54 r

11, or 54 11

16.

Children are expected to divide whole and

decimal numbers by 10, 100 and 1000 where

answers have up to 3 decimal places

Dividing by 10 and 100 and 1000

Move the digits once to the right for ÷10,

twice to the right for ÷100 and three times

to the right for ÷ 100. Use zero as a



to check answers.



order).


Year 6



numbers with more than four digits.

Column Addition:

Extend to numbers with any number of

digits and decimals with 1 and 2 decimal

places. E.g. 124.9 + 117.25 = 242.15.

124.9

+ 117.25 1 1

________

242.15






numbers with more than four digits.

Compact Column Subtraction:



7 in this example.

Extend to numbers with any number of

digits and decimals with 1, 2 and 3 decimal

places. e.g. 327.67 – 143.25 =

Children should identify when numbers are

close together and finding the difference

on an empty number line would be more

appropriate.

Find the difference by counting up. e.g. 0.5

Children are expected to identify common

factors, common multiples and prime

numbers.

Children are expected to multiply multi-

digit numbers, up to 4 digits, by a two digit

whole number using the efficient written

method of long multiplication.


calculation.

Children to cross out smaller numbers so

they know not to add them.

3 8 7 5

x 3 4 3 3 2

1 5 5 0 0 2 2 1

1 1 6 2 5 0

1 3 1 7 5 0 1

3 8. 7 6

x 3 4____ 3 3 2

1 5 5. 0 4 2 2 1

1 1 6 2. 8 0

1 3 1 7. 8 4 1

Fractions

Children are expected to identify common

factors, common multiples and prime

numbers.

Children are expected to divide up to 4

digit numbers by 1 and 2 digit numbers

using the efficient written method of long

division and interpret remainders as whole

number remainders, fractions or decimals.


T U

1 4

7 9 28

We work out how many 7s will go into the 9; a short cut to working out how many 10s of 7 go into 90. The answer is 1. The 2 left over we carry over to the 8 to make 28. 7s into 28? 4, we put the 4 in the 1s column.


H T U

5 4 r 11

1 6 8 7 5

8 0

7 5

6 4

1 1


– 0.31 = 0.19

This can be modelled on an empty number

line.





order).

Multiply simple pairs of proper fractions,

writing the answer in its simplest form e.g.

¼ x ½ = 1/8





in any order).

Since we are dividing by 16 it is useful to derive the 16 times table we can refer to as we do the division.

We cannot divide the 8 by 16 so we work out how many 16s will go into 87. There are 5 as 5x16=80 so we write 5 in the 10s and subtract to get 7. Now we involve the 5 and use 75. There are 4 16s in 75 as 4x16=64 and 5x16=80 is too big. We write 4 in the 1s column and subtract; the remainder is 11, we write: 874÷16 = 54 r

11, or 54 11

16.

Fractions

Children are expected to divide proper

fractions by whole numbers e.g.

1/3 ÷ 2 = 1/6


to check answers.



order).

1x16 16

2x16 32

3x16 48

4x16 64

5x16 80

6x16 96

: :


Children are expected to divide whole and

decimal numbers by 10, 100 and 1000 where

answers have up to 3 decimal places

Dividing by 10 and 100 and 1000

Move the digits once to the right for ÷10,

twice to the right for ÷100 and three times

to the right for ÷ 100. Use zero as a



to check answers.



order).

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