Lyndon Green Junior School

Calculations Policy

Introduction

Children are introduced to the processes of calculation through practical, oral and mental

activities. As children begin to understand the underlying ideas they develop ways of recording

to support their thinking and calculation methods, use particular methods that apply to special

cases, and learn to interpret and use the signs and symbols involved.

Over time children learn how to use models and images, such as empty number lines, to support

their mental and informal written methods of calculation. As children’s mental methods are

strengthened and refined, so too are their informal written methods. These methods become

more efficient and succinct and lead to efficient written methods that can be used more

generally.

By the end of Year 6 children are equipped with mental, written and calculator methods that

they understand and can use correctly. When faced with a calculation, children are able to

decide which method is most appropriate and have strategies to check its accuracy.

At whatever stage in their learning, and whatever method is being used, children’s

strategies must still be underpinned by a secure and appropriate knowledge of number

facts, along with those mental skills that are needed to carry out the process and judge if

it was successful.

Mental methods of calculation

Oral and mental work in mathematics is essential, particularly so in calculation. Early practical,

oral and mental work must lay the foundations by providing children with a good understanding

of how the four operations build on efficient counting strategies and a secure knowledge of

place value and number facts. Later work must ensure that children recognise how the

operations relate to one another and how the rules and laws of arithmetic are to be used and

applied. Ongoing oral and mental work provides practice and consolidation of these ideas.

Written methods of calculation

The aim is that by the end of Key Stage 2, the great majority of children should be able to use

an efficient method for each operation with confidence and understanding. The challenge for

teachers is determining when their children should move on to a refinement in the method and

become confident and more efficient at written calculation.

Addition

Stage 1: Practical experiences

Rote counting and matching

numerals to quantities.

Children should be given

equipment to count items.

Use songs, rhymes and stories.

Children need to count up in ones.

How many cats in total?

Stage 2: Using apparatus and focusing on number bonds to 20

Model how to record addition

number sentences that arise

from practical activities.

Children could draw a picture

to help them work out answers

or use items to count.

Children could use dots or tally

marks to represent objects in

a calculation. This is a lot

quicker than drawing a picture.

Record mental addition in a number sequence using the

+ and = signs.

2 + 3 can be shown as:

At a party, I eat 2 cakes and my friend eats 3 cakes. How

may cakes did we eat altogether?

7 + 3 can be shown as:

There are 7 people on a bus, 3 more get on at the next stop.

How many people are on the bus now?

Stage 3: Using the empty number line

The mental methods that lead

to column addition generally

involve partitioning, e.g. adding

tens and ones separately,

often starting with the tens.

Children need to be able to

partition numbers in ways

other than into tens and ones

to help them make multiples of

ten by adding in steps.

Steps in addition can be recorded on a number line. The

steps often bridge through a multiple of 10.

8 + 7 = 15

6 + ? = 10 ? + 6 = 10

The empty number line helps

to record the steps to record

the steps on the way to

calculate the total.

Children can decide which way

they are most comfortable

with and use this in their

written calculations.

48 + 36 = 84

or:

Stage 4: Partitioning

The next stage is to record

mental methods using

partitioning.

Add the tens and then add the

ones to form partial sums and

then add these partial sums to

get an overall answer.

Partitioning both numbers into

tens and ones mirrors the

column method where ones are

placed under ones and tens

under tens. This also links to

mental methods.

47 + 76 =

The above method can be developed further

by partitioning the tens and ones.

40 + 70 = 110

7 + 6 = 13

110 + 13 = 123

The above method can be developed in

preparation for moving onto the column

method.

Stage 5: Column method

In this method, recording is

reduced. Carrying digits are

recorded above the line, using

the words ‘carry ten’ or ‘carry

one hundred’, not ‘carry one.’

As a school we have decided to

record the digits ‘carried’

above the answer line as

previously the children were

forgetting to add these on.

Later, extend to adding:

a) two two-digit numbers

b) numbers with different

numbers of digits.

c) three two-digit

numbers

Further extend with the use

of the column method to add

decimal numbers with one and

two decimal places.

47 + 76:

Subtraction

Stage 1: Practical experiences

Rote counting and matching

numerals to quantities.

Children should be given

equipment to count and

subtract items.

Use songs, rhymes and stories.

Children need to count backwards (number bonds to 10

and then moving onto number bonds to 20). Move onto

separating a given number of objects. Find own ways of

recording subtraction e.g. cross outs.

Stage 2: Using apparatus and focusing on number bonds to 20

Model how to record

subtraction number sentences

that arise from practical

activities.

Children begin to use dots or

tally marks.

Record mental subtraction in a number sequence using the

– and = signs.

7 – 3 can be shown as:

Mum baked 7 biscuits. I ate 3. How many are left?

Take away

Lisa has 7 coloured pencils and Tim has 3. How many more

does Lisa have than Tim?

Find the difference

Stage 3: Using the empty number line

The empty number line helps to

record or explain the steps in

mental subtraction.

The ‘jumps’ go underneath the

line for subtraction.

A calculation like 74 – 27 can be

recorded by counting back 27

from 47.

Steps in subtraction can be

recorded on a number line. The

steps often bridge through a

multiple of 10.

15 – 7 = 8 can be shown as:

74 – 27 = 47 worked out by counting back:

8 10 15

- 2 - 5

- 3 - 4 - 20

47 50 54 74

10 – 6 = ? 10 – 4 = 6

Work backwards

from here!

The steps may be recorded in a different order:

Or combined:

Stage 4: Partitioning

Subtraction can be recorded

using partitioning to write

equivalent calculations that can

be carried out mentally.

For 74 – 21 this involves

partitioning the 21 into

20 and 1, and then subtracting

from 74 the 20 and the 1 in

turn.

Subtraction can be recorded using partitioning:

74 – 20 = 54

54 – 1 = 53

This requires children to subtract a single-digit number

or a multiple of 10 from a two-digit number mentally. The

method of recording links to counting back on the number

line.

Stage 5: Column method

At this stage the children’s

understanding of place value is

vital. They must recognise that

932 is 900, 30 and 2

(9 hundreds, 3 tens and 2 ones).

This is a formal written method

that can only be used if the

children’s knowledge of place

value is secure.

Also continue to use counting up

method, with informal notes or

jottings when appropriate e.g.

subtraction from 1000 or 100.

932 – 427

Step 1: Set out the calculation correctly.

The larger number must be placed above the smaller

number.

The ones, tens and hundreds should be in line).

- 20 - 3 - 4

47 67 70 74

47 70 74

- 23 - 4

HTO

HTO

Step 2: Start by subtracting the ‘ones’ column.

Step 3: Now move onto the ‘tens’ column.

Step 4: Finally, move onto the ‘hundreds’ column.

Step 5: Check the answer to see if it is ‘reasonable.’

Estimating and rounding skills can be used for this OR add

the answer back onto the small number to see if you get

back to the large number.

The answer to 932 – 457 = 475

HTO

HTO

HTO

The term

‘exchanging’ should

be used here to

show how numbers

in the columns are

effected.

Multiplication

Stage 1: Pictures and symbols and role play

Solve practical problems in a

real life role play.

Oral counting in twos, fives and

tens.

Carry out activities in context, for example:

How many shoe lace holes are there on this shoe?

Put 5 cherries on each cake. How many cherries do

you need?

Stage 2: Times table facts (2, 5 and 10)

Children will need to learn to

count in twos, fives and tens.

Count in 2s, 5s, 10s and derive

the multiples of 2, 5 and 10.

Solve practical problems that

involve combining groups of 2, 5

and 10.

Again, pictures can be useful to

illustrate the problem to the

children.

How many fingers are there on 4 hands?

Draw around hands and write numbers underneath.

Each child has 2 eyes. How many eyes do 4 children have?

2 + 2 + 2 + 2

Stage 3: Grouping

Dots or tally marks are often

drawn in groups.

5 x 3 =

There are 5 cakes in a pack. How many cakes in 3 packs?

Stage 4: Arrays

Most children will understand

multiplication as repeated

addition at this stage.

Drawing an array gives the

children an image of the

answer.

4 x 3 =

A chew costs 4p. How much do 3 chews cost?

3 rows of 4 or 3 columns of 4.

This helps the children understand that 4 x 3 is the same

as 3 x 4.

Stage 5: Times table facts (3, 4 and 8)

Children must recall and use

multiplication facts for the 3, 4

and 8 multiplication tables.

Carry out practical tasks and encourage children to use

their 3, 4 and 8 multiplication facts in context. This must

be done at every opportunity possible.

Stage 6: Repeated addition (number lines)

Children could count on in equal

steps, recording each jump on

an empty number line.

When numbers get bigger, it is

inefficient to do lots of small

jumps. This introduces

partitioning but still with use of

the number line.

6 x 4

There are 4 cats. Each cat has 6 kittens. How many

kittens are there altogether?

This shows 4 jumps of 6.

13 x 7

There are 13 biscuits in a packet. How many biscuits in 7

packets?

13 is split (partitioned) into parts (10 and 3). This gives

two jumps (10 x 7 and 3 x 7).

Stage 7: Partitioning

Begin to develop informal ways

of calculating at recording

multiplication problems.

Partitioning will help the

children to solve multiplication

problems involving a two-digit

number multiplied by a one-digit

number.

13 x 3

13 x 3

10 x 3 3 x 3

30 + 9

39

13 x 3 = 39

Once the children understand how partitioning can be

used, they can move onto recording it in this way:

13 x 3 = (10 x 3) + (3 x 3) = 30 + 9 = 39

Stage 8: The grid method (1-digit x 2-digit)

The grid method takes the idea

of partitioning one step further

and allows children to multiply

larger numbers.

Initially children will work with

1-digit x 2-digit numbers.

7 x 35

Stage 9: The grid method (2-digit x 2-digit moving up to 4-digits by 2-digits)

Once the children are confident

with using the grid method and

partitioning, they will be able to

use this method to multiply

larger numbers.

Children can partition the

numbers to use in the grid in

different ways, using

multiplication facts they know

well.

26 x 35

26 x 35

Stage 10: Short multiplication (multiplying by a 1-digit number)

This is a compact method that

moves on from the grid method.

If, after practice, children

cannot use the compact method

without making errors, they

should return to the grid

method of multiplication and

look at developing that method

into expanded short

multiplication.

7 x 64

Stage 11: Long multiplication (multiplying by more than 1-digit number)

If children are secure with the

short method of multiplication,

they may move onto using the

method of long multiplication

when multiplying larger

numbers. Again, if this method

confuses the children they may

go back to the grid method.

598 x 463

Division

Stage 1: Sharing (practical tasks)

Understand sharing as giving

everyone the same amount.

Solve practical problems in a

real role play context.

Drawing pictures can help the

children solve division problems

at this stage.

6 ÷ 2

6 Easter eggs are shared between 2 children. How many

eggs do they get each?

Stage 2: Arrays (using drawings)

Dots or tally marks can either

be shared out one at a time or

split into groups.

12 ÷ 4

4 apples are packed into a basket. How many baskets can

you fill with 12 apples

Grouping in fours

12 ÷ 4 = 3

Stage 3: Grouping

Grouping is repeated

subtraction and should be

linked to arrays.

Grouping can also be modelled

on a number line. Use prepared

number lines.

15 ÷ 3

There are 15 apples in a box. How many bags of 5 apples

can be filled?

This explores how many groups of 5 can you make from 15.

28 ÷ 7

A chocolate bar costs 7p. How many can I buy with 28p?

To work out how many 7’s there are in 28, draw jumps of

7 along a number line. This shows you need 4 jumps of 7

to reach 28.

Stage 4: Looking at relationships between multiplication and division

Division is the inverse

(opposite) of multiplication.

Ensure grouping continues to be

modelled.

How many 7s make 28? (28 ÷ 7)

Four 7s make 28.

Children can count forwards or backwards.

If… 28 ÷ 7 = 4

28 ÷ 4 = 7

So… 7 x 4 = 28

4 x 7 = 28

Stage 5: Informal written methods (to support larger numbers)

Use practical and informal

written methods to support

division of larger numbers to

encourage chunking.

Many children can partition and

multiply with confidence at this

stage.

Children should also be

encouraged to find a remainder

mentally, for example the

remainder when 34 is divided

by 6.

84 ÷ 6

I need 6 drawing pins to put up a picture. How many

pictures can I put up with 84 pins?

It would take a long time to jump in sixes to 84 so

children can jump on in bigger ‘chunks’. A jump of 10

groups of 6 takes you to 60. Then you need another 4

groups of 6 to reach 84. Altogether, that is 14 sixes.

192 ÷ 8

8 pencils fit into each packet. If you have 192 pencils,

how many packets can be filled?

It is helpful to split 192 into sensible ‘chunks’ before

diving. As you are dividing by 8, the ‘chunks’ chosen must

also be multiples of 8. Divide each ‘chunk’ (how many

groups of 8?) and then add the answers together.

Stage 6: Chunking method

This method is based on

subtracting multiples of the

divisor, or ‘chunks’. Initially

children subtract several

chunks, but with practice they

should look for the biggest

multiples of the divisor.

Start off with questions with

no remainders and then move

onto numbers with remainders

once children have understood

the method and process.

276 ÷ 25

Stage 7: Long division

Long division is the next stage.

This should be taught before

moving onto short division.

2461 ÷14

Stage 8: Short division

Once children are able to

‘chunk’ they may be introduced

to short division. This method

is a formal written method. For

children to move onto this

stage, their understanding of

partitioning and place value

need to be sound.

964 ÷ 7

Lyndon Green Junior School

April 2016