Mottinghamprimary School

Mottinghamprimary School

MottinghamPrimary School

Calculation Policy

To be reviewed April 2014

Addition

Mental calculation skills should be taught and developed through all stages of the calculation policy.

These are a selection of mental calculation strategies for addition.

Mental recall of number bonds

6 + 4 = 10 + 3 = 10

25 + 75 = 10019 +  = 20

Use near doubles

6 + 7 = double 6 + 1 = 13

Addition using partitioning and recombining

34 + 45 = (30 + 40) + (4 + 5) = 79

Counting on or back in repeated steps of 1, 10, 100, 1000

86 + 57 = 143 (by counting on in tens and then in ones)

460 - 300 = 160 (by counting back in hundreds)

Add the nearest multiple of 10, 100 and 1000 and adjust

24 + 19 = 24 + 20 – 1 = 43

458 + 71 = 458 + 70 + 1 = 529

Use the relationship between addition and subtraction

36 + 19 = 5519 + 36 = 55

55 – 19 = 3655 – 36 = 19

Use known number facts and place value for mental addition.

47 + 38 = 8574 + 98 = 172

470 + 380 = 8507.4 + 9.8 = 17.2

The following stages should be taught to the children and the children given opportunities to practice and apply them in as many ways as possible. Children should be moved through the stages of the policy as and when they are ready to do so. Children should not move to a higher stage if they are not secure with the calculations in the previous stage. At no time should age be a factor in stopping a child moving to the next stage.

The main aim of the policy is to ensure all children reach stage 4 as a minimum on leaving the school for secondary education.

Stage 1 / To recognise and use operation and numerals up to 10, with picture aids.

Relate addition to counting on; recognise that addition can be done in any order; use practical and informal written methods to support the addition of a one-digit number or a multiple of 10 to a one-digit or two-digit number
Know to start adding from the largest number.
To use numberlines and practical resources to support calculation and teachers demonstrate the use of the numberline.
3 + 2 = 5

______
0 1 2 3 4 5 6 7 8 9
Children then begin to use numbered lines to support their own calculations using a numbered line to count on in ones.
8 + 5 = 13


Children should have access to a wide range of counting equipment, everyday objects, hoops and sorting trays, number tracks and number lines.
Stage 2 / Children will begin to use ‘empty number lines’ themselves starting with the larger number and counting on.
First counting on in tens and ones.
34 + 23 = 57



34 44 54 55 56 57
Then helping children to become more efficient by adding the units in one jump (by using the known fact 4 + 3 = 7).
34 + 23 = 57



34 44 54 57
Children need to understand the concept of equality before using the ‘=’ sign. Calculations should be written either side of the equality sign so that the sign is not just interpreted as ‘the answer’.
2 = 1+ 1
2 + 3 = 4 + 1
3 = 3
2 + 2 + 2 = 4 + 2
Missing numbers need to be placed in all possible places.
3 + 4 =   = 3 + 4
3 +  = 7 7 =  + 4
 + 4 = 7 7 = 3 + 
 +  = 7 7 =  + 
Partition into tens and ones and recombine
12 + 23 = 10 + 2 + 20 + 3
= 30 + 5
= 35
Count on in tens and ones
23 + 12 = 23 + 10 + 2
= 33 + 2
= 35
Stage 3 / Use knowledge that addition can be done in any order to do mental calculations more efficiently. Add three small numbers by putting the largest number first and/or find a pair totalling 10; partition into ‘5 and a bit’ when adding 6, 7, 8 or 9, then recombine partition additions into tens and units, then recombine.
Children will continue to use empty number lines with increasingly large numbers, including compensation where appropriate.
Count on from the largest number irrespective of the order of the calculation.
38 + 86 = 124


86 116 120 124
Develop vocabulary to increase, together and sum of.
Introduce horizontal and vertical expansion.
1. Vertical expansion 2. Horizontal expansion
83 80 + 3
+ _42 + 40 + 2
5 120 + 5 = 125
120
125
Stage 4 / Introduce compensation
49 + 73 = 122



73 122 123
Expand on vertical and horizontal methods with 3 digit numbers.
367 300 + 60 + 7
+185 100 + 80 + 5
12 400 +140+12 = 552
140
400
552
Revert back to number line to introduce decimal work.

When secure give work a context through money and units of measurements.
Stage 5 / Develop the column method and begin to carry below the line.
625783367
+ 48 + 42 + 85
673 825 452
1 1 1 1
Using similar methods, children will:
add several numbers with different numbers of digits
begin to add two or more three-digit sums of money, with or without adjustment from the pence to the pounds
know that the decimal points should line up under each other, particularly when adding mixed amounts, e.g. £3.59 + 78p
Extend to up to two places of decimals (same number of decimals places) and adding several numbers (with different numbers of digits).
72.8
+54.6
127.4
1 1

Subtraction

Mental calculation skills should be taught and developed through all stages of the calculation policy.

These are a selection of mental calculation strategies for subtraction.

Mental recall of addition and subtraction facts

10 – 6 = 417 -  = 11

20 - 17 = 310 -  = 2

Find a small difference by counting up

82 – 79 = 3

Counting on or back in repeated steps of 1, 10, 100, 1000

86 - 52 = 34 (by counting back in tens and then in ones)

460 - 300 = 160 (by counting back in hundreds)

Subtract the nearest multiple of 10, 100 and 1000 and adjust

24 - 19 = 24 - 20 + 1 = 5

458 - 71 = 458 - 70 - 1 = 387

Use the relationship between addition and subtraction

36 + 19 = 5519 + 36 = 55

55 – 19 = 3655 – 36 = 19

The following stages should be taught to the children and the children given opportunities to practice and apply them in as many ways as possible. Children should be moved through the stages of the policy as and when they are ready to do so. Children should not move to a higher stage if they are not secure with the calculations in the previous stage. At no time should age be a factor in stopping a child moving to the next stage.

The main aim of the policy is to ensure all children reach stage 4 as a minimum on leaving the school for secondary education.

Stage 1 / Understand the vocabulary associated with subtraction: take away, less than, counting back....
Drawing groups of objects, using physical resources and physically moving objects away.
6 – 2 = 4

13 – 5 = 8



Stage 2 / Find a 'difference' by counting up.
I have saved 5p. The socks that I want to buy cost 11p. How much more do I need in order to buy the socks?

15 – 7


815
Start to use addition as a way of checking answer to subtraction sum and building the understanding of the relationship between the 2 processes.
Develop language further by introducing difference, fewer than, subtract and through the context of money, change.
47 – 23 = 24
-1 -1 -1 -10 -10


Children need to understand the concept of equality before using the ‘=’ sign. Calculations should be written either side of the equality sign so that the sign is not just interpreted as ‘the answer’.
Stage 3 / 15 – 7 - 2 - 5


8 10 15
38 – 13 - 3 - 10


25 28
‘Finding the difference’ (particularly when numbers are in near proximity or close to a boundary)
By counting back from the largest number and then by counting up from the smallest (children need to see that the answer is the same, but the counting-up is easier to calculate)
74 – 38 -4 -10 -10 -10 -4 = 36


36 40 50 60 70 74
Develop the understanding of the relationship between addition and subtraction and introduce the term inverse.
13-7=6 7+6=13
Apply knowledge of single digit subtraction to 2 and 3 digit numbers.
7-3=4 70-30=40 700-300=400
Complementary addition
754 – 86 = 668

For those children with a secure mental image of the number line they could record the jumps only:
754 – 86 = 668
14 (100)
600 (700)
54 (754)
668
Stage 4 /

Subtract 9 or 11. Begin to add/subtract 19 or 21.

35 – 9 = 26
Use compensation

Use known number facts and place value to subtract.
6.1 – 2.4 = 3.7

Construct subtraction problems in columns using clear columns for tens and units.
89=80 + 9
- 5750 + 7
30 + 2 = 32
Stage 5 / Expanded partitioning layout
77-24 74– 27

707 60 70 14
20 4 - 20 7 -
50 3 = 50+3 = 53 40 7 = 47
Extend to larger numbers, decimals and decomposition needed in tens column etc.
741 – 367
600 130
700 40 11
300 60 7 -
300 70 4
Compact method
74 – 27 67 14
2 7 –
4 7
Extend compact method to using 3 digit numbers and decimals to 2 places.
614 1
754
- 86
668

Multiplication

Mental calculation skills should be taught and developed through all stages of the calculation policy.

These are a selection of mental calculation strategies for multiplication.

Doubling and halving

Applying the knowledge of doubles and halves to known facts.

e.g. 8 x 4 is double 4 x 4

Using multiplication facts

Tables should be taught from stage 2 onwards, either as part of the mental oral starter or other times as appropriate within the day.

Stage 2 2 times table

5 times table

10 times table

Stage 3 2 times table

3 times table

4 times table

5 times table

6 times table

10 times table

Stage 4 Derive and recall all multiplication facts up to 10 x 10

Stage 5Derive and recall quickly all multiplication facts up to 10 x 10.

Using and applying division facts

Children should be able to utilise their tables knowledge to derive other facts.

e.g. If I know 3 x 7 = 21, what else do I know?

30 x 7 = 210, 300 x 7 = 2100, 3000 x 7 = 21 000, 0.3 x 7 = 2.1 etc

Use closely related facts already known

13 x 11 = (13 x 10) + (13 x 1)

= 130 + 13

= 143

Multiplying by 10 or 100

Knowing that the effect of multiplying by 10 is a shift in the digits one place to the left.

Knowing that the effect of multiplying by 100 is a shift in the digits two places to the left.

Partitioning

23 x 4 = (20 x 4) + (3 x 4)

= 80 + 12

= 102

Use of factors

8 x 12 = 8 x 4 x 3

The following stages should be taught to the children and the children given opportunities to practice and apply them in as many ways as possible. Children should be moved through the stages of the policy as and when they are ready to do so. Children should not move to a higher stage if they are not secure with the calculations in the previous stage. At no time should age be a factor in stopping a child moving to the next stage.

The main aim of the policy is to ensure all children reach stage 4 as a minimum on leaving the school for secondary education.

Stage 1 /

Relate multiplication to doubling and counting groups of the same size.


Looking at columns Looking at rows
2 + 2 + 2 3 + 3
3 groups of 2 2 groups of 3
Counting using a variety of practical resources
Counting in 2s e.g. counting socks, shoes, animal’s legs…
Counting in 5s e.g. counting fingers, fingers in gloves, toes…
Counting in 10s e.g. fingers, toes…
Pictures / marks
There are 3 sweets in one bag.
How many sweets are there in 5 bags?

Develop vocabulary associated with multiplication: Total, pairs, double.
Stage 2 / Repeated addition can be shown easily on a number line:
5 x 3 = 5 + 5 + 5



and on a bead bar:
5 x 3 = 5 + 5 + 5

Commutativity
Children should know that 3 x 5 has the same answer as 5 x 3. This can also be shown on the number line.




Children should be able to model a multiplication calculation using an array. This knowledge will support with the development of the grid method.




Children need to understand the concept of equality before using the ‘=’ sign. Calculations should be written either side of the equality sign so that the sign is not just interpreted as ‘the answer’.
Introduce missing numbers
3x?=15
Children use inverse of multiplication to rearrange and solve.
15÷3=5 3x5=15
Stage 3 / Repeated addition
4 times 6 is 6 + 6 + 6 + 6 = 24 or 4 lots of 6 or 6 x 4
Children should use number lines or bead bars to support their understanding.



0 6 12 18 24


Arrays
Children should be able to model a multiplication calculation using an array. This knowledge will support with the development of the grid method.





Extend missing numbers
?x5=20 3x?=18 ?x?=32
Introduce 2 digit number multiplication with grid method.
72 x 38 is approximately 70 x 40 = 2800

2100+560+60+16=2736
Extend to simple decimals with one decimal place.
Stage 4 / Grid method
HTU x U
(Short multiplication – multiplication by a single digit)
346 x 9
Children will approximate first
346 x 9 is approximately 350 x 10 = 3500
x / 300 / 40 / 6
9 / 2700 / 360 / 54 / 2700
+ 360
+ 54
3114
1 1
Using similar methods, they will be able to multiply decimals with one decimal place by a single digit number, approximating first. They should know that the decimal points line up under each other.
e.g. 4.9 x 3
Children will approximate first
4.9 x 3 is approximately 5 x 3 = 15
x / 4 / 0.9
3 / 12 / 2.7 / 12
+ 2.7
14.7
Stage 5 / Children will approximate first
4346 x 8 is approximately 4346 x 10 = 43460
x / 4000 / 300 / 40 / 6
8 / 32000 / 2400 / 320 / 48 / 32000
+ 2400
+ 320
+ 48
34768
Using similar methods, they will be able to multiply decimals with up to two decimal places by a single digit number and then two digit numbers, approximating first. They should know that the decimal points line up under each other.
For example:
4.92 x 3
Children will approximate first
4.92 x 3 is approximately 5 x 3 = 15
x / 4 / 0.9 / 0.02
3 / 12 / 2.7 / 0.06 / 12
+ 0.7
+ 0.06
12.76

Division

Mental calculation skills should be taught and developed through all stages of the calculation policy.

These are a selection of mental calculation strategies for Division.

Using multiplication facts

Tables should be taught from stage 2 onwards, either as part of the mental oral starter or other times as appropriate within the day.

Stage 2 2 times table

5 times table

10 times table

Stage 3 2 times table

3 times table

4 times table

5 times table

6 times table

10 times table

Stage 4 Derive and recall all multiplication facts up to 10 x 10

Stage 5Derive and recall quickly all multiplication facts up to 10 x 10.

Using and applying division facts

Children should be able to utilise their tables knowledge to derive other facts.

e.g. If I know 3 x 7 = 21, what else do I know?

30 x 7 = 210, 300 x 7 = 2100, 3000 x 7 = 21 000, 0.3 x 7 = 2.1 etc

Dividing by 10 or 100

Knowing that the effect of dividing by 10 is a shift in the digits one place to the right.

Knowing that the effect of dividing by 100 is a shift in the digits two places to the right.

Use of factors

378 ÷ 21378 ÷ 3 = 126378 ÷ 21 = 18

126 ÷ 7 = 18

Use related facts

Given that 1.4 x 1.1 = 1.54

What is 1.54 ÷ 1.4, or 1.54 ÷ 1.1?

The following stages should be taught to the children and the children given opportunities to practice and apply them in as many ways as possible. Children should be moved through the stages of the policy as and when they are ready to do so. Children should not move to a higher stage if they are not secure with the calculations in the previous stage. At no time should age be a factor in stopping a child moving to the next stage.

The main aim of the policy is to ensure all children reach stage 4 as a minimum on leaving the school for secondary education.

Stage 1 / Recognise differences in quantities when counting objects.
Sort or match objects and talks about sorting.
Grouping
Sorting objects into 2s / 3s/ 4s etc
How many pairs of socks are there?

There are 12 crocus bulbs. Plant 3 in each pot. How many pots are there?
Jo has 12 Lego wheels. How many cars can she make?
Stage 2 / Develop vocabulary of division: Sharing, grouping, each, equally, dividing.
12 ÷ 2 sharing



12 ÷ 2 grouping

Understand division as sharing and grouping

18 ÷ 3 can be modelled as:
Sharing – 18 shared between 3
OR
Grouping - How many 3’s make 18?

0 3 6 9 12 15 18
Model number line and allow children to experiment.
Children need to understand the concept of equality before using the ‘=’ sign. Calculations should be written either side of the equality sign so that the sign is not just interpreted as ‘the answer’.
Stage 3 / Understand that division is the inverse of multiplication and known multiplication facts can be applied to division.
Number lines – to support ‘grouping’
12 ÷ 2 -2 -2 -2 -2 -2 -2


0 2 4 6 8 10 12
and prove the answer is the same as

0 2 4 6 8 10 12
Relate repeated subtraction to the grouping model of division.
Know that halving is dividing by 2 and the inverse of doubling.
Using arrays
Can be used to demonstrate 12 grouped in 3 rows of 4 and in 4 rows of 3.
Link to repeated subtraction on a number line




0 4 8 12
Begin to relate to fraction as ½ is dividing by 2, ¼ is dividing by 4.....
Stage 4 / Working towards ‘chunking’ for TU ÷ U
84 ÷ 7 2 x 10 x = 12



0 14 84
Build up from single jumps of 7, then 2 jumps of 7 at one time, then reinforcing that 10 jumps of 7 is more efficient.
‘Chunking’ for TU ÷ U

78 4
- 7 0 10 x 7
1 4
- 1 4 2 x 7
0
= 12
72 ÷ 3

3 ) 72
- 30 10x
42
- 30 10x
12
- 6 2x
6
- 6 2x
0
Answer : 24
Stage 5 / Children will continue to use written methods to solve short division TU ÷ U.
Children can start to subtract larger multiples of the divisor, e.g. 30x
Short division HTU ÷ U
196 ÷ 6
32 r 4
6 ) 196
- 180 30x
16
- 12 2x
4
Answer : 32 remainder 4 or 32 r 4
Any remainders should be shown as integers, i.e. 14 remainder 2 or 14 r 2.
Children need to be able to decide what to do after division and round up or down accordingly. They should make sensible decisions about rounding up or down after division. For example 240 ÷ 52 is 4 remainder 32, but whether the answer should be rounded up to 5 or rounded down to 4 depends on the context.
Extend to decimals with up to two decimal places. Children should know that decimal points line up under each other.
87.5 ÷ 7
12.5
7 ) 87.5
- 70.0 10x
17.5
- 14.0 2x
3.5
- 3.5 0.5x
0

Answer : 12.5