Mental maths: the strategies, year by year, and how to practise

What is mental maths?

Mental maths is calculating in your head, using number facts known by heart, like 7 + 3 = 10 or 6 × 4 = 24, and a few strategies that turn a harder calculation into easier steps. Quick jottings to keep track are fine, government guidance says. What matters is reasoning with numbers, not counting in ones.

Mental and written methods go together: children keep practising mental maths right through primary school, even after they learn column addition.

Mental maths strategies children learn

These are the main mental maths strategies in the curriculum and government guidance. Schools may use slightly different names.

Counting on and counting back

In Year 1, children add by counting forwards and subtract by counting back. Later, counting on is the quick way to find the difference between two close numbers. For 402 − 398, count on from 398: 2 to get to 400, and 2 more to 402, so the difference is 4.

By Year 2, the aim is to stop counting in ones for small sums and use facts instead.

Bridging 10

Bridging 10 means going through 10 on the way. For 7 + 5, split the 5 into 3 and 2. 7 + 3 = 10, and 10 + 2 = 12. This needs the number bonds to 10.

Subtracting works the same way. For 15 − 9, take 5 to get to 10, then 4 more: 15 − 5 = 10, and 10 − 4 = 6. The same idea works across any multiple of 10: 38 + 5 goes through 40.

Partitioning

Partitioning splits a number into tens and ones so each part is easy to add. For 45 + 23, add the 20, then the 3: 45 + 20 = 65, and 65 + 3 = 68. On a number line that's two jumps.

Doubles and near doubles

Children learn doubles, then use them for near doubles: double 3 is 6, so 4 + 3 is 7. By Year 2 they know double 6, 7, 8 and 9 and their halves, so 7 + 8 is double 7 and one more: 15.

Using known facts

A fact your child knows unlocks many more. 3 + 7 = 10, so 30 + 70 = 100. 8 + 6 = 14, so 80 + 60 = 140 and 800 + 600 = 1,400. 3 × 4 = 12, so 30 × 4 = 120. This is why place value and the times tables matter so much for mental maths.

The inverse

Addition and subtraction undo each other, as do multiplication and division. Children use the inverse to check answers and solve missing number problems. If 50 = 20 + ?, then 50 − 20 = 30.

Compensation

In Year 6, children learn the compensation rule: add some to one number and take the same from the other, and the total stays the same. So 39 + 26 is the same as 40 + 25, which is 65.

Mental maths in Year 1 and Year 2

In Year 1, children learn number bonds and the related subtraction facts within 20, and add and subtract numbers to 20, including zero, using objects and pictures alongside.

In Year 2, the curriculum lists the mental calculations by name: a two-digit number and ones (34 + 5), a two-digit number and tens (34 + 20), two two-digit numbers (34 + 25) and three one-digit numbers (4 + 6 + 3). Children recall addition and subtraction facts to 20 fluently, learn the 2, 5 and 10 times tables, and check subtraction by adding. See the Year 1 maths and Year 2 maths guides for the rest of each year.

Mental maths in Year 3

In Year 3, children add and subtract mentally with three-digit numbers: a three-digit number and ones (346 + 7), a three-digit number and tens (346 + 40), and a three-digit number and hundreds (346 + 300). Mental sums with two-digit numbers can now go over 100, like 76 + 48.

They learn the 3, 4 and 8 times tables and use them to derive bigger facts: 3 × 2 = 6, so 30 × 2 = 60 and 60 ÷ 3 = 20. The curriculum also gives a neat example of reordering: for 4 × 12 × 5, do 4 × 5 = 20 first, then 20 × 12 = 240.

Column methods start in Year 3, and government guidance says quick recall of the facts within and across 10 comes first, because every column depends on them. Children also estimate and use the inverse to check. More in the Year 3 maths guide.

Mental maths in Year 4

In Year 4, mental maths moves towards multiplication and division. By the end of the year children should know all the times tables up to 12 × 12, which is what the multiplication tables check looks at.

They multiply and divide mentally using place value and known facts, including multiplying by 0 and 1, dividing by 1, and multiplying three numbers. 2 × 6 × 5 is easier as 2 × 5 = 10, then 10 × 6 = 60. They use factor pairs and the turnaround, and derive facts with hundreds: 2 × 3 = 6, so 600 ÷ 3 = 200.

They also split a number to multiply it, which the curriculum calls the distributive law: 39 × 7 is 30 × 7 and 9 × 7, which is 210 + 63 = 273. More in the Year 4 maths guide.

Mental maths in Year 5

In Year 5, children add and subtract mentally with increasingly large numbers. The curriculum's example is 12,462 − 2,300 = 10,162: only the thousands and hundreds change, so there's no need for a written method.

They multiply and divide mentally using known facts, and multiply and divide by 10, 100 and 1,000. Decimals come in too: adding tenths in their head, and pairs that make 1, like 0.83 + 0.17 = 1. Known facts still do the work: 8 + 6 = 14, so 0.8 + 0.6 = 1.4.

A written method isn't always best, government guidance says: 201 × 4 is 200 × 4 = 800, plus 4 more, so 804. Children round to check answers. More in the Year 5 maths guide.

Mental maths in Year 6

In Year 6, children do mental calculations with large numbers and mixed operations, and estimate to check answers. Mental arithmetic also helps in the KS2 maths SATs. See the Year 6 maths guide, and the KS1 maths and KS2 maths hubs for the bigger picture.

Quick mental maths practice at home

Mental maths practice works best little and often: five minutes most days beats a long session at the weekend.

• Number bond ping-pong. Say a number, your child says the bond to 10 (or 100). 7, 3. 65, 35.
• Doubles and halves. On a walk or in the car, double and halve numbers in turn.
• Ask "how did you do it?" There's usually more than one good way, and talking it through builds the strategies.
• Use real numbers. Prices, change from £5, the time until tea.
• Keep the times tables going. A short mixed round on one or two tables, like the 7 times table, keeps the facts quick.

Common mental maths mistakes

• Counting in ones. Counting on fingers for 8 + 5 works but is slow and error-prone. Steer towards bridging 10: 8 + 2 + 3.
• Losing place value. Adding 40 to 346 and getting 350. Ask "which digit changes?"
• Turning subtraction round. For 63 − 17, working out 7 − 3 instead of 3 − 7. Take away 10, then 7: 63 − 10 = 53, and 53 − 7 = 46.
• Forgetting to check. A quick estimate or the inverse catches most slips.

Mental maths questions to try

1. 8 + 7
2. 16 − 9
3. 57 + 30
4. 63 + 29
5. 482 − 300
6. 40 × 6
7. 0.7 + 0.5
8. 25 × 7 × 4

Answers

1. 15 (double 7 is 14, and one more)
2. 7 (16 take away 6 is 10, take away 3 more is 7)
3. 87 (only the tens change)
4. 92 (63 + 20 is 83, and 9 more is 92)
5. 182 (only the hundreds change)
6. 240 (4 × 6 is 24, so 40 × 6 is 240)
7. 1.2 (7 + 5 is 12, so 7 tenths and 5 tenths make 12 tenths)
8. 700 (25 × 4 is 100, and 100 × 7 is 700)