Why is this skill important?
Division is the hardest of the four mathematical operations (addition, subtraction, multiplication and division) for several reasons. It is the inverse of multiplication, and inverses are always that bit more tricky. It can be expressed or presented in many ways:
- as grouping (how many sets of 4 are there in 28?)
- as sharing (how many sweets will four children have each if there are 28 to be shared?)
- as scaling down (how long will a 24cm picture become if it is scaled down by a factor of 4?)
In addition to these complications, which make division quite difficult for teachers to teach and for children to learn, there is also the fact that it doesn’t always give an exact whole number answer. The following are examples of this potential problem:
- 33 ÷ 6 = 5 remainder 3
- 33 ÷ 6 = 5½
- 33 ÷ 6 = 5.5
- 33 ÷ 6 = 6 if the question was: ‘There are 33 children and 6 fit in a people-carrier automobile, so how many cars will we need to take them to the zoo?’
- 33 ÷ 6 = 5 if the question was: ‘6 candles fit in a box and we have 33 candles so how many boxes can we fill?’
It is easy to see, therefore, how important it is that children learn to deal with remainders in division from the very start of their mathematical careers, and that they are helped to develop a robust understanding of the concept.
What is the skill?
Children need to understand division as repeated subtraction or ‘How many groups of this number are there in this larger number?’ (For a full explanation of this see Multiplication and Division Skill 5: Division as the inverse of X) For example, children need to be clear that 28 ÷ 7 = ? can equally well be written ? x 7 = 28 or how many sevens in twenty-eight?
Once children fully comprehend the relationship between division and multiplication in this way, they will recognise the need to learn the division facts that are associated with every multiplication fact in the times tables. Thus, in learning …
1 x 3 = 3
2 x 3 = 6
3 x 3 = 9
4 x 3 = 12
5 x 3 = 15, etc,
children need to understand that they are also learning…
3 ÷ 3 = 1
6 ÷ 3 = 2
9 ÷ 3 = 3
12 ÷ 3 = 4
15 ÷ 3 = 5, etc.
And also that, since multiplication can be done either way round, if they know that 5 x 3 = 15, they also know that 3 x 5 = 15, and therefore that 15 ÷ 5 = 3.
These two aspects of children’s learning – that division is the inverse of multiplication and that the times tables facts are also division facts that we should know off by heart, are both very important when it comes to understanding that division can leave remainders.
134 ÷ 10 = ? is a division where it is clear that the answer is not a whole number. There are not a whole number of 10s in 134. But this calculation is complicated in that it has so many possible correct answers.
134 ÷ 10 = 13.4
134 ÷ 10 = 13 2/5 (4/10 is the same as 2/5)
134 ÷ 10 = 13 remainder 4
Understanding that this last is one way of dealing with the fact that there are not a whole number of 10s in 134 is precisely what we mean by the understanding of remainders.
So how is this skill taught?
The best way of teaching children about remainders is to allow them to do divisions in a practical way. Provide a number of interlocking bricks or cubes (Lego™ will do very well).
- Take a division which does not have a whole number answer. E.g. 32 ÷ 5 = ?
- Ask the child to create a tower of 32 bricks.
- Remind children that 32 ÷ 5 = ? can be read as ‘How many sets of 5 are there in 32?’ or in this case, as ‘How many towers of 5 bricks can we get from a tower of 32 bricks?’
- Ask the child to break their tower of 32 bricks into towers of 5 bricks. How many can they make? (6 with two left-over bricks)
- Record this answer. 32 ÷ 5 = 6 with 2 left over.
- Explain to the child that we call the left over bricks a ‘remainder’. 32 ÷ 5 = 6 r2
Children will get the hang of remainders if they perform sufficient divisions of this type in a practical context like this. Keep the numbers well within the easier times tables, e.g. divide by 2 or 3 or 5 or 10 so that the numbers are not that difficult. It is the concept that we want to get across to the child.
Once they are confident in the practical situation, move to paper and pencil calculations, still keeping the numbers fairly simple.
- 25 ÷ 3 = ? Read this as, ‘How many 3s in 25?’
- Remind children of the 3x table.
- What 3x table fact comes closest to, but not over, 25?
- 8 x 3 = 24. So we know that we can make 8 towers of 3 from 25 cubes and there will be one left over.
- 25 ÷ 3 = 8 r1.
Gradually increase the difficulty of the numbers and the times tables facts drawn upon in doing this type of division.
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.