Why is this skill important?
Division is the most complex of the four mathematical operations and the hardest for children to understand, and therefore to perform in calculations. Part of children’s difficulty with multiplication often arises from a characterisation of division as ‘sharing’ stemming from the language used in relation to this when they were very young. Unfortunately, seeing division as ‘sharing’ is largely unhelpful in mathematical calculation and can lead to some very slow and inefficient strategies or else to a complete blockage in the child’s mind about the whole thing.
Division needs to be seen as the inverse of multiplication (see below) if children are to be encouraged to develop comfortable and efficient mental strategies for dividing one number into another, where the numbers are appropriately handled in their heads without writing anything down. They will also (in the upper juniors) need to understand division and its relation to multiplication when they are taught the harder written procedures for long division.
What is the skill?
With children of 7, 8 and 9 years of age, multiplication is often most helpfully conceived as ‘repeated addition’ (see Multiplication and Division: Multiplication as repeated addition). This means that children learn to interpret 3 x 8 as three lots of eight, or 8 + 8 + 8. They also understand that multiplication can be done either way round and so ‘three lots of eight’ will give the same total as ‘eight lots of three’, 3 x 8 ≡ 8 x 3, both give an answer of 24.
Division needs to be understood as the inverse of this process. Thus, we can think of division as ‘how many lots of eight make twenty-four?’ or as ? x 8 = 24.
With children, we often refer to this as a multiplication with a ‘hole’ in it.
- How many sets of four make twenty? x 4 = 20
- How many towers of 3 make 15? x 3 = 15
- How many fives are there in thirty-five? x 5 = 35
These are all ways in which division can be presented to children so as to make its relation to multiplication very clear. The clear benefit here is that this way of presenting division also suggests a way of finding the answer. If we are asked how many sets of 4 make 20, then we should count in steps of 4 to find out:
- 4 (1st step), 8 (2nd step), 12 (3rd step), 16 (4th step) and 20 (5th step).
- Five steps, five sets of 4 make 20.
- So the answer is 5.
In x 4 = 20 it is ‘5’ that goes in the ‘hole’. Similarly, to find how many towers of 3 make 15, I count in threes to 15, or make a large tower of 15 and find out how many little towers of 3 I can break it into.
None of these interpretations refer to ‘sharing’ which is a wholly different concept. In sharing, we ask not ‘how many sets of four make twenty? But instead ‘if I share 20 cubes between 4 children, how many will they get each?’ In order to work this out, I would have to share one by one to each of four children: one for you, one for you, one for you and one for you. Then: a second for you, a second for you, and so on. This is not a good way to approach 20 ÷ 4, and it is an even worse way to try to answer 78 ÷ 13. It would take a long time! By contrast, counting in 13s will be a far more efficient strategy, especially if we can double up (three 13s are 39, so six 13s are double 39 or 78).
So how is this skill taught?
This skill depends upon being taught multiplication well, and upon the children having developed a good understanding of this operation. They will also be greatly aided by a good memory for their times tables facts.
Children should be able to explain that 7 x 5 is another, rather quicker way of writing 5 + 5 + 5 + 5 + 5 + 5 + 5. The number sentence 7 x 5 can be read as seven lots of 5 or seven times 5 or seven multiplied by 5 or even as 5, seven times. Once children are fully aware of these ways of saying this calculation, they should be able to read
x 5 = 35 as ‘what times five is thirty-five?’ or as ‘how many lots of 5 give me thirty-five?’. They need to be shown that it is by counting in fives – or better still, by knowing our 5x table – that we are able to answer this question. We can count seven 5s or simply recall the fact 7 x 5 = 35. Either way, children can then deduce that the answer to this question is 7.
It is then a short step to be shown that 35 ÷ 5 = ? which can be read as ‘how many fives in thirty-five?’ and is another way of writing our multiplication with a ‘hole’ in it, x 5 = 35.
Once children learn to read division as ‘how many sets of this number are there in this larger number?’, they have a method of tackling larger divisions which arise beyond the tables facts.
76 ÷ 4 = ? is read as how many 4s in 76? This means the child has to find out how many fours they need to count to reach seventy-six. Since they do not learn the tables facts above 12 x 4 = 48, they will need to do a little more hard thinking here.
x 4 = 76
10 x 4 = 40
9 x 4 = 36
So 19 x 4 = 76, which means that 76 ÷ 4 = 19
In the lower juniors, we stress the interpretation of division as ‘How many of these 4s in 24?’ Children are encouraged to re-write divisions as multiplications with ‘holes’ in.
24 ÷ 4 = ? 15 ÷ 3 = ?
x 4 = 24 x 3 = 15
They are then asked to count in steps of the multiple in each case ~ i.e. in steps of 4 for the first calculation or in steps of 3 for the second. If they can use their tables facts, that is excellent!
Turning division round in this way is often a revelation for children who will tell you ‘I can’t do division!’ to which the response is ‘Ok, then do a multiplication with a hole in!’
Practise Together: These activities are intended to be shared. Read the Explanation of the skill being practised and then play the game or share the task. Watch out for the points highlighted in the Explanation and if necessary, help your child, following the advice in ‘How this skill is taught’ section. Shared activities are not only more fun – they enable you to actively support your child’s learning.
Explanation & Worksheets:
Test: Take a test, questions from this area
Multiplication as repeated addition
Counting in steps
Division as inverse of multiplication
The grid method
Divisions with remainders
Division beyond tables
Multiplication as repeated addition:
Understand that multiplication is a way of doing repeated additions, e.g. 4 x 5 is the same as 5 + 5 + 5 + 5
Counting in steps:
Count confidently in twos, , threes, fours fives and tens: 3, 6, 9, 12, etc.
Perform multiplications involving 9 using finger-method
Know the 2x, 3x, 4x, 5x and 10x tables off by heart
Division as inverse of multiplication:
Understand that division is the reverse of multiplication, so that we read 24 ÷ 6 as how many sixes in twenty-four or ? x 6 = 24
The grid method:
Use the grid method to find an answer to larger multiplications, e.g. 4 x 27 or 13 x 8
Divisions with remainders:
Perform divisions with remainders, e.g. 38 ÷ 4
Division beyond tables:
Perform divisions with remainders, e.g. 38 ÷ 4
Number Concepts: Count in different ways, understand how numbers work, become fluent in the ways of numbers
Adding and Subtracting: Mentally add or subtract numbers with confidence and develop written ways of adding and subtracting larger numbers or more of them!
Multiplying and Dividing: Know the times tables and use these to perform mental multiplication and divisions; develop written methods for multiplication and division.