Understanding of the commutative principle that a + b = b + a halves the number of addition facts to be remembered but remember also that subtraction does not have this property.

Addition and subtraction facts to 20 can be developed on an empty number line by bridging through 10.

8 + 7 = 15

The symmetry of an addition table can help learners to learn addition facts.

Addition and subtraction facts can also be learned by applying inverse operations and getting learners to develop families of four facts, e.g.

8 + 5 = 13 5 + 8 = 13 13 – 8 = 5 13 – 5 = 8

and to remember trios of numbers such as 5, 8, 13.

What this might look like in the classroom

Children should begin practically, adding quantities of beads, for example, to make numbers to 20. They should talk about what they are doing and the results obtained. As they make each number they should check by taking away the number added to ensure they have the number they started with

Before adding and subtracting on a number line, children need to be confident with counting along one, identifying numbers and saying one more or less than the given number by moving their finger accordingly. Children should then be able to identify missing numbers on a partially labelled number line.

In the transition from the practical activity of counting out the beads to using a number line, they should complete the practical, and then record on a number line and with a number sentence. Once they are confident with this, they move on to record on an empty number line placing the appropriate numbers where they think they should go as in the example.

Children develop this method for adding and subtracting 2− and 3−digit numbers e.g. 67 + 34

Each time they do this they check with the inverse operation i.e. 67 + 34 = 101, 101 − 34 = 67

Taking this mathematics further

Links to fractions and decimals Once confident with addition and subtraction of whole numbers on a number line try decimals and fractions e.g.
1.6 + 2.7

3

4

+ 1

1

2

Using the number line to demonstrate multiplication as repeated addition and division as repeated subtraction is also effective e.g.
4 × 3 = 3 + 3 + 3 + 3 = 12

12 ÷ 3 = 12 − 3 − 3 − 3 − 3, four groups of 3

Developing trios for all these is a great reinforcement e.g.
2.4, 3.5 and 6.9

1

1

2

, 2

1

4

and 3

3

4

3, 5, 15

Other types of number line To reinforce and consolidate work with number lines, it is important to make links with other types of number line such as an analogue clock face, vertical axis on a graph, measuring equipment e.g. cylinder, scales, thermometer, ruler. This will enable children to be familiar with them in different contexts and to apply the skills for basic number lines to these others.

Making connections

Children should begin practically, adding quantities of beads, for example, to make numbers to 20. They should talk about what they are doing and the results obtained. As they make each number they should check by taking away the number added to ensure they have the number they started with

Before adding and subtracting on a number line, children need to be confident with counting along one, identifying numbers and saying one more or less than the given number by moving their finger accordingly. Children should then be able to identify missing numbers on a partially labelled number line.

In the transition from the practical activity of counting out the beads to using a number line, they should complete the practical, and then record on a number line and with a number sentence. Once they are confident with this, they move on to record on an empty number line placing the appropriate numbers where they think they should go as in the example.

Children develop this method for adding and subtracting 2− and 3−digit numbers e.g. 67 + 34
+ 30 + 4
67 97 101
I I I
Each time they do this they check with the inverse operation i.e. 67 + 34 = 101, 101 − 34 = 67

Related information and resources from the NCETM

Number line troubles: This Mathemapedia entry gives suggestions of how to help children who struggle to use the empty number line