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 children to memorise and reason with addition facts.

+

0

1

2

3

4

0

0

1

2

3

4

1

1

2

3

4

5

2

2

3

4

5

6

3

3

4

5

6

7

4

4

5

6

7

8

This can be used to develop an understanding of the commutative principlea + b = b + a. Realising that the number of addition facts to be remembered is halved aids fluency. Subtraction does not have this property.

Children can reason with trios of numbers such as 5, 8 and 13: Addition and subtraction facts can be learned by applying inverse operations and developing families; for example,

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

What this might look like in the classroom

Problem 1

Use the numbers 6, 7 and 13 to make four different number sentences.

Solution:

6 + 7 = 13; 13 - 7 = 6
7 + 6 = 13; 13 - 6 = 7

Problem 2

Each of these trios uses numbers up to 20. What is the missing number in each problem?

6, 19, ?
11, 18, ?
2, 16, ?
4, 9, ?

Solution:

13
7
14 or 18
5 or 13

Taking this mathematics further

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

1.6 + 2.7

34 + 112

Using the number line to demonstrate multiplication as repeated addition and division as repeated subtraction helps children to make connections, e.g.

4 × 3 = 3 + 3 + 3 + 3 = 12

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

Children can solve problems involving trios of decimals or fractions, e.g.
2.4, 3.5 and 6.9

112, 214 and 334

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 reason 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

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

Once an understanding of addition and subtraction has been developed, children progress to using columnar addition and subtraction with increasingly large numbers.

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