It is possible to think of addition and subtraction in a number of ways to support pupils with their mental calculations.

Addition

For example if they are trying to calculate 356 + 38 in their mind, they can use a counting on method by visualising an empty number line in their head or by using their fingers.

Start with 356 and then add on 30 and then add on 8

In a way the mental number line is ‘holding’ the answers as pupils build up from the start number to add on the value required.

It is also possible to use fingers to ‘hold’ the answers as pupils count on the value they require:

For example, 482 + 39 = ?

Using fingers start with 482 and then add on 39

482 + 10 is 492 +10 is 502 add 10 is 512 add 5 is 517 add 4 is 521.

You can expect that pupils will add on in different ways.

Subtraction

When pupils are doing subtraction they can use two methods. They will either count backwards from the first number, or they will count up to the first number from the second number. In both examples they can use an image of a number line in their head or their fingers to ‘hold’ the answers as they build to the required result.

For example 138 – 42

So 138 – 42 = 96

Or …. You can see 138 – 42 as “What do I need to add to 42 to get 138 …

So 138 – 42 = 96

What this might look like in the classroom

It is possible to find many different practical situations that support mental calculation. Encouraging pupils to count on, count back, or count up to find the difference using visible or imaginary number lines or their fingers to keep track of their calculations.

Question:

Luke has measured himself and his sister Sarah on a height chart. He is 123 cm tall. Sarah is 89 cm tall. How much taller is Luke than Sarah?

There are also many different words you can use to describe ‘addition’ and ‘subtraction’

Question

“Tom was 110 cms tall last year, he has now grown 3 more cms. How tall is he now?”

Answer

110 + 3 = 113cm

Question

“The beaker had 345ml of water in it. I tipped out 120ml of water. How much water do I have left in the beaker now?”

Answer

345 – 120 = 225ml

Question

“What is the difference between Jo’s height of 113cm and Tash’s heght of 97cm?”

Answer

113 – 97 = 16cm

For example pupils can also talk about different distances between different locations:

Question

If you walk from my house down the street, you get to the shop first. The shop is 350 m away my house. Then I walk another 130 m to the garage. Another 70m down the road is the library. If I go to the library, how far have I walked?

Answer

350 + 130 + 70 = 550m

Question:
Use an empty number line to work out the answer to 48 + 36 using three different ways. Answer:
Possibility 1

Possibility 2

Possibility 3-

Taking this mathematics further

Ask pupils to write their own addition and subtraction problems.

Pupils should be encouraged to write questions which they can answer, yet at the same time challenge them to write questions with larger numbers.

Ask them to explain how addition and subtraction are related and explain this with the problem they have written.

The history of the use of the empty number line is explained in the document in the following link.

Making connections

Through Key Stage 2 children learn that they can make choices about how to work out subtraction calculations. For example, to work out how much change you would get from a £5 note if you buy goods costing £2.68, many different methods could be used including:

Start with £5, take away or count back £2 then 60p then 8p.

Count up from £2.68 to £5 in steps working out the difference.

Blank number lines also provide an excellent vehicle for keeping track of both subtraction and addition. The exact steps pupils take to solve the problems may vary.

To be able to develop mental calculation strategies for addition, children first need to be able to:

Count on and back in ones, two, fives and tens,

Know or calculate number pairs to familiar numbers such as 10, 20 or 50,

Understand the place value of numbers and partition them into hundreds, tens and ones and,

Understand the concept of addition and its inverse operation of subtraction.

Children need to be able to apply the concept of addition to real−life applications, for example the total cost of two items costing 48p and 36p. They may then need to be able to convert their answer into the appropriate units.

The connection between straightforward addition and subtraction and addition and subtraction of money will help pupils understand the place value system further.