Year 6NumberOperationsYear 6 Maths

Division and what to do with the remainder

What this lesson covers

Standard written algorithms (e.g. column addition, column subtraction) rely on place value, regrouping, and renaming.
Remainders from division can be represented as whole numbers, fractions, or decimals, depending on the context.

Key idea

Thirty-one children, vans that hold six

How many vans do you need to book?
31 ÷ 6 is 5, with 1 left over.
But five vans leaves a child standing on the footpath.

Key idea

The division itself is the easy part

How many sixes fit into 31? Five of them make 30.
That uses up 30 and leaves 1 behind.
So 31 ÷ 6 = 5 remainder 1.

Key idea

One remainder, three ways to write it

As a whole number with a remainder: 5 remainder 1.
As a mixed number: 5 1/6.
As a rounded decimal: 5.17.
All three are correct. They are not interchangeable.

Written as	Answer
Whole number and remainder	5 remainder 1
Mixed number	5 1/6
Rounded decimal	5.17

Key idea

For the vans, the answer is six

You cannot book 5.17 vans. There is no such thing.
Five vans means one child does not go.
So you round up: six vans, and the last one is nearly empty.
Six is the answer, and the division never printed it.

Option	What happens
5 vans	1 child left behind
5.17 vans	does not exist
6 vans	everyone goes

Worked example

Sharing sweets: keep the remainder

31 sweets shared between 6 children.
Five each, and one sweet left in the bag.
You cannot cut a sweet, so the remainder stays a remainder.
5 each, 1 over. That is the useful answer here.

Worked example

Sharing cake: split the remainder

31 slices of cake between 6 people.
Cake can be cut, so the leftover slice gets shared too.
That last slice splits six ways: one sixth each.
So each person gets 5 1/6 slices.

Worked example

Measuring rope: use a decimal

31 metres of rope cut into 6 equal lengths.
A tape measure shows decimals, not sixths.
31 ÷ 6 is 5.1666… so round it: 5.17 metres.
Same sum. A decimal is the useful shape this time.

Key idea

The sum never changed

Four situations. One division. Four different right answers.
The arithmetic cannot tell you which one to give.
You have to look at what the leftover 1 actually is.

Situation	Answer	Because
Vans that hold 6	6 vans	round up: nobody is left behind
Sweets shared out	5 each, 1 over	a whole number and a remainder
Cake shared out	5 1/6 each	the leftover gets split
Metres of rope	5.17 m	a decimal, to two places

Key idea

Bigger numbers, same idea

Now 1283 ÷ 5, which is too big to do in your head.
Short division works one digit at a time, left to right.
Each step carries what is left over into the next digit.

Worked example

Working through 1283 ÷ 5

1 ÷ 5 is 0, and 1 carries over to make 12.
12 ÷ 5 is 2, and 2 carries over to make 28.
28 ÷ 5 is 5, and 3 carries over to make 33.
33 ÷ 5 is 6, with 3 genuinely left over.

Step	Result	So far
1 ÷ 5	0, carry 1	0
12 ÷ 5	2, carry 2	02
28 ÷ 5	5, carry 3	025
33 ÷ 5	6, remainder 3	256 r 3

Key idea

1283 ÷ 5 = 256 remainder 3

Reading the digits off the top: 256.
And 3 that would not divide.
Check it: 256 × 5 is 1280, plus 3 is 1283.

Check	Value
256 × 5	1,280
plus the remainder	1,283

Worked example

And that remainder, three ways

As a remainder: 256 remainder 3.
As a mixed number: 256 3/5, because 3 of 5 parts are left.
3/5 is 0.6, so as a decimal it is 256.6 exactly.
No rounding needed this time: fifths land on decimals neatly.

Written as	Answer
Remainder	256 r 3
Mixed number	256 3/5
Decimal	256.6

Key idea

A sixth of something IS dividing by six

Finding 1/6 of 31 and working out 31 ÷ 6 are the same job.
Not similar. The same.
Sharing into 6 equal parts is what both of them mean.
So 1/6 of 31 is 5 1/6.

Key idea

Which is why the answer can be a fraction

One sixth of 31 obviously is not a whole number.
31 does not split into 6 whole parts.
The mixed number is the honest answer, not a rounded one.

Worked example

A quarter of 90

1/4 of 90 means 90 ÷ 4.
4 twenty-twos make 88, so 22 with 2 left over.
Those 2 split four ways: 2/4, which is 1/2.
So a quarter of 90 is 22 1/2, or 22.5.

Step	Value
90 ÷ 4	22 remainder 2
Remainder as a fraction	2/4 = 1/2
Answer	22 1/2 = 22.5

Worked example

Estimating first, as always

1283 ÷ 5. Round 1283 to 1300.
1300 ÷ 5 is 260, because 13 ÷ 5 is 2.6.
Our answer, 256.6, sits just under 260.
If the answer had come out 51 or 2,566, you would know.

Step	Value
Estimate	1,300 ÷ 5 = 260
Actual	256.6

Your turn

Four to try

1. Work out 947 ÷ 4, with a remainder.
2. Write that remainder as a fraction, then as a decimal.
3. What is 1/8 of 60?
4. 50 people, minibuses that hold 8. How many buses?

Show answer

Answers: 1. 236 remainder 3. 2. 236 3/4, and 3/4 is 0.75, so 236.75 exactly. 3. 60 ÷ 8 = 7 remainder 4, and 4/8 is 1/2, so 7 1/2 or 7.5. 4. 7 buses: 50 ÷ 8 is 6 remainder 2, and those 2 people still need a bus. Question 4 is the van problem again; if they answer 6, or 6.25, that is the beat to go back to.

Summary

The remainder is a decision

Three ways to write one, and the situation picks.
Ask what the leftover actually is before you answer.

Written as	Use it when
5 remainder 1	the leftover cannot be split
5 1/6	the leftover can be split
5.17	you are measuring
6	everyone has to fit

Is your child on track?

Take the free placement test