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### Written Calculation Strategies

This page aims to explain how children are taught to carry out written calculations for each of the four number operations (add/subtract/multiply/divide). The methods may look different from those you are familiar with but your child will be using them to learn to calculate at school.

To help support and develop your child’s mathematical understanding, each operation is taught according to a clear progression of stages. Generally, children begin by learning how written methods can be used to support mental calculations. They then move on to learn how to carry out and present calculations horizontally. After this, they start to use vertical methods, first in long form and then in compact form.

In Year 3 children work mentally, using jottings to support them.

In Year 4 children begin to learn more formal methods of recording their calculations. If children are introduced to these too early, before they have mastered the skills of mental Maths, they are less likely to reach national expectations in mental Maths.

Research has shown that two things make a big difference when children are learning to calculate. First, it is important to use the correct words when talking about the numbers in calculations.The numbers should be said using the value of the number, for example:

TU

Second, children find it much easier to grasp new methods when given pictures to look at or concrete apparatus to use. Drawings, counters, objects all help – and, don’t forget, fingers are one of the best maths resources at any age!

1. Mental Methods with Jottings (Year 3 – 6)

A method of adding is to partition the numbers into parts, add the parts and then recombine to find the total.

For example 12 + 26 =

Partition the numbers into tens and units (or ones):

10 + 2 + 20 + 6

10 + 20 = 30

2 +   6 =   8

Recombine the numbers to give the total:

30 + 8 = 38

2. Informal Written Method — ‘Horizontal’ (Year 3+)

This knowledge of partitioning can then be used in an informal calculation where the largest parts of the numbers are added first and the smallest parts of the numbers are added last. This method will start to be used when the numbers being added together get larger. In effect, this is the method older children and adults use when working mentally with larger numbers.

The use of a number line helps children to ‘count on’.

Blank number lines are used to carry out the addition of two 2-digit numbers or 3-digit numbers. These are introduced to the children in Year 3 and Year 4 and are still used by some children in Year 5 and Year 6.

54 + 28 =

54 + 28 = 82

3. Standard Compact Written Method — ‘Vertical & Compact’ (Year 5 onwards)

This can then lead to a more compact method involving carrying between columns where necessary:

e.g. 148 + 286

 148 Add the units: eight plus six is fourteen +286 4 Put 4 in the units column and one ten under the tens column. 1 0 148 Add the tens: forty plus eighty is one hundred and twenty +286 plus ten underneath is one hundred and thirty. Put thirty 34 in the tens column and one hundred under the hundreds 110 column. 148 Add the hundreds: one hundred plus two hundred is three +286 hundred, plus one hundred underneath is four hundred. 434 Put the four hundreds in the hundreds column. 110

SUBTRACTION

1. Mental Methods with Jottings/Informal Written Method

A number line is used again, but this time you begin by only marking the largest number on the line:

331 – 122

You then jump back the amount you are taking away and where you finish gives the answer. Again, it makes it easier if multiples of 10 or 100 are used:

331 - 100 (jump back to 231) - 20 (jump back to 211) - 2 (jump back to 209)

331 - 122 = 209

2. Standard Compact Written Method

This expanded written method then leads to a more compact method:

341-122=

 341 -122

 33411 One take away two can't be done so exchange one ten for ten units. 11 subtract 2. Then, 3 tens subtract 2 tens. -1  2  2 Finally, 3  hundreds subtract 1 hundred. 219

341-122=219

We never use the term ‘borrow’. Instead we use the term ‘exchange’.

MULTIPLICATION

Early multiplication skills begin in Reception by counting in different steps.

• Learning and recalling multiplication tables begins in Year 2. Children in Year 2 are encouraged to count in twos, fives and tens.
• Year 3 children learn their 3 and 4 times tables.
• Year 4 children recap learnt tables and begin to learn 6 and 8s.
• Year 5 children recap all learnt tables and add the 7s and 9s as well.

Helping your child learn their multiplication table facts is one of the best things you can do to help them, not only with multiplication but in virtually all areas of maths.

Little and often is the best way - 5 minutes practice on the way to / from school is ideal! Learning tables is essential for later work on division, fractions and ratio.

Websites such as wmnet hit the button are useful for times tables practice.

1. Mental Methods with Jottings

Children can use partitioning when multiplying larger numbers.

 38 x 7 = (30 x 7) + (8 x 7) multiply the tens     multiply the units Add the totals together: 210 + 56 = 266

2. Informal Written Method — Grid Multiplication

The next step is to use partitioning and organise the calculation as a grid, e.g.

32 x 6

First, thirty two is partitioned into tens and units and put into a grid:

 X 30 2 6 180 12

180 + 12 = 192

This method can be used to multiply combinations of numbers of any size. All that happens is that the size of the grid changes, you multiply across each row, total each row, and add all the row totals together, e.g.

13 x 25 =

 X 20 5 10 3

6 x 134 =

 X 100 30 4 6

The grid method provides an extremely clear and flexible approach to multiplication, which is much easier for children to understand and apply than any vertical methods.

For this reason, vertical methods for multiplying will not be taught until Years 5 and 6.  It should be noted, though, that the grid method can still be used in Years 5 and 6.

3. Expanded Written Method (Year 5 onwards)

When multiplying by a single digit number, another way of setting out multiplication is as a vertical calculation, e.g:

23 x 7

 23 X 7 21 (3 x 7) Multiply the units. 140 (20 x 7) Multiply the tens saying twenty times 7. 161 Total the columns.

This method is particularly good, as it encourages children to check their calculation.

4. Compact Written Method (Year 6)

When multiplying, a more compact method can also be used:-

23 x 7 =

 23 seven times 3 is twenty one x 7 put the twenty under the tens column 1 and the one in the units column. 20 23 Seven times twenty is one hundred and forty plus x 7 the twenty underneath makes one hundred and sixty. 161 20 Put the sixty in the tens column and the one hundred in the hundreds column

DIVISION

Early division begins with sharing in practical activities. However, it is important that children go on to recognise that division has another meaning besides sharing.

For example, 15 ÷ 3 can mean 15 shared between 3

(3 lots of 5)

But it can also mean 15 grouped into 3s

(5 lots of 3)

For written calculations, it is the idea of division as grouping which is used.

1. Mental Methods with Jottings — ‘Chunking’

A method known as ‘chunking’ is introduced when the numbers to be divided start to get larger, e.g.

156 ÷ 7

Take off ten lots of 7 and then another ten lots of 7.  Finally take off 2 lots of 7.  2 is left, this is the remainder.

 7 / 156 70 (10 x 7) 86 70 (10 x 7) 16 14 (2 x 7) 2

Count up the chunks of 7:

10 + 10 + 2 = 22

The answer is 22 remainder 2

156 ÷ 7 = 22r2

 22  r2 7 /156

Or, as another example, with even larger numbers:

363 ÷ 97

How many ninety sevens are there in 363?

What is the biggest ‘chunk’ (lot) of 97 I can get from 363?

 97 / 363 -291 (3 x 97) 72 Remainder

Count up the ‘chunks’ or multiples of 97.

The number left at the end of the calculation, which can not be divided by 97 is the remainder, so:

363 ÷ 97 = 3r72

 3 r72 97 /363

It should be noted that the children learn to use the method of chunking as the only method when dividing larger numbers. Long division does not maintain the value of the numbers involved and, as such, it does not encourage real understanding of the operation of division.

In Year 6, children are taught short division as an efficient means of dividing a 2 or 3-digit number by a single digit number only.