Partitioning numbers: what it means and how children use it

What is partitioning in maths?

Partitioning is splitting a number into smaller parts that add up to the whole. The parts can be anything that adds up, but the most useful way is by place value: tens and ones, or hundreds, tens and ones.

Schools often show partitioning with a part-whole model. The whole number goes in the top circle and its parts go in the circles below.

The parts can be put back together in any order. 20 + 8 = 28 and 8 + 20 = 28, and 28 − 8 = 20 shows the same thing the other way round.

Partitioning in Year 1

Partitioning starts with small numbers. In Year 1, children split numbers up to 10 into two parts: 6 is 5 and 1, or 4 and 2, or 3 and 3. Learning every way to split a number to 10 is the same as learning the number bonds, and it's the first step towards taking away.

Partitioning two-digit numbers into tens and ones

To partition a two-digit number, split it into its tens and its ones. 47 is 4 tens and 7 ones, so it partitions into 40 and 7. This is a Year 2 skill, and it depends on knowing that the 4 in 47 is worth 40, not 4. Our guide to place value explains this in more detail.

47 in a place value chart

Tens	Ones
4	7

Partitioning three-digit numbers

In Year 3, children partition numbers up to 1,000 into hundreds, tens and ones. 346 is 300, 40 and 6.

346 in a place value chart

Hundreds	Tens	Ones
3	4	6

Watch the order. Government guidance asks teachers to give the parts in a mixed-up order, so children learn that 40 + 300 + 2 = 342, not 432.

Partitioning numbers in different ways

Numbers can be partitioned in more than one way, and the curriculum asks children to do this from Year 2. 23 is 20 and 3, but it's also 10 and 13. In Year 3, the curriculum's own example is 146, which is 100, 40 and 6, or 130 and 16.

This is called flexible partitioning, or non-standard partitioning. It's not a trick for its own sake: it's what makes subtraction with exchanging make sense, as the examples below show.

Using partitioning to add

To add two two-digit numbers in their head, children partition both numbers, add the tens, then add the ones. For 45 + 23:

1. Partition: 45 is 40 and 5, and 23 is 20 and 3.
2. Add the tens: 40 + 20 = 60.
3. Add the ones: 5 + 3 = 8.
4. Put them together: 60 + 8 = 68.

With more practice, children partition just one of the numbers. 45 + 20 = 65, then 65 + 3 = 68. A number line shows this as a jump of 20 and then a jump of 3.

Using partitioning to subtract

For subtraction, children partition only the number being taken away. For 45 − 23, they take away 20, then take away 3: 45 − 20 = 25, and 25 − 3 = 22.

Flexible partitioning helps when there aren't enough ones. For 146 − 28, there are only 6 ones, so think of 146 as 130 and 16:

1. Take the ones from the 16: 16 − 8 = 8.
2. Take the tens from the 130: 130 − 20 = 110.
3. Put them together: 110 + 8 = 118.

How partitioning leads to column methods

Column addition and column subtraction, which children learn in Year 3, are partitioning written down. Each column is one part: ones, tens, hundreds.

When the ones add up to more than 9, the extra ten moves into the tens column. That's partitioning 13 ones into 1 ten and 3 ones.

1. Add the ones: 6 + 7 = 13. That's 1 ten and 3 ones: write 3 in the ones column and carry 1 into the tens column.
2. Add the tens: 4 + 3 + 1 = 8 (including the 1 ten carried). Write 8 in the tens column.
3. Hundreds: there's only the 1, so write 1 in the hundreds column.

Exchanging in column subtraction is flexible partitioning. In 146 − 28, one ten is exchanged for 10 ones: 146 becomes 130 and 16, just as in the mental method above.

1. Ones: you can't take 8 from 6, so exchange 1 ten for 10 ones (the tens digit becomes 3). Now there are 16 ones: 16 − 8 = 8.
2. Tens: 3 − 2 = 1.
3. Hundreds: nothing to take away, so write 1.

Multiplication by partitioning and the grid method

Partitioning also helps with multiplying. To work out 34 × 6, partition 34 into 30 and 4, multiply each part by 6, then add: 30 × 6 = 180, 4 × 6 = 24, and 180 + 24 = 204. In Year 4 the curriculum names this the distributive law, with the example 39 × 7 = 30 × 7 + 9 × 7.

Set out in a table, this is the grid method. It leads on to short multiplication, the formal written method.

34 × 6 with the grid method

×	30	4
6	180	24

1. Partition 34 into 30 and 4.
2. 30 × 6 = 180.
3. 4 × 6 = 24.
4. Add the answers in the grid: 180 + 24 = 204.

Children need their times tables for this: the 6 times table gives 4 × 6, and 3 × 6 gives 30 × 6.

Common mistakes to watch for

• Reading the digits, not their value. Partitioning 47 into 4 and 7 instead of 40 and 7. Ask "what is the 4 worth?"
• Muddling the order. Writing 40 + 300 + 2 as 432. Put each part in its column on a place value chart.
• Partitioning both numbers when subtracting. For 63 − 17, splitting both gives 60 − 10 and 3 − 7, and a common slip is to turn 3 − 7 into 7 − 3 and get 54. Government guidance warns against this. Take away 10, then 7: 63 − 10 = 53, and 53 − 7 = 46.
• Missing zero. 305 is 300 and 5, with no tens. A place value chart shows the empty tens place.

How to help at home

• Use money. 10p and 1p coins are a ready-made way to show tens and ones. Make 47p and ask how many 10p coins it takes.
• Ask "what else?" When your child says 36 is 30 and 6, ask for another way: 20 and 16, or 35 and 1.
• Talk through mental sums. Ask how they'd work out 52 + 30 or 75 − 20 without writing anything down.
• Follow the school's method. Ask your child to show you how they partition in class, and stick to it.

Partitioning questions to try

1. Partition 58 into tens and ones.
2. Partition 274 into hundreds, tens and ones.
3. 64 is 50 and what?
4. 36 + 42, by partitioning
5. 73 − 25, by partitioning 25
6. 23 × 4, by partitioning 23

Answers

1. 50 and 8
2. 200, 70 and 4
3. 14
4. 78 (30 + 40 is 70, and 6 + 2 is 8)
5. 48 (73 take away 20 is 53, take away 5 is 48)
6. 92 (20 × 4 is 80, and 3 × 4 is 12)

For everything else in these years, see the KS1 maths guide and the Year 2 maths and Year 3 maths guides.