Mental arithmetic means calculating in your head, without a written method. Developing a bank of strategies — such as doubling and halving, near-number adjustments, and partitioning — lets you check answers quickly and solve routine questions in seconds, making it easier to spot errors in calculator work too.
Why does mental arithmetic matter at KS3?
Written methods and calculators are powerful, but mental arithmetic catches errors before they happen. If your calculator gives you 1430 when you expected something close to 1400, your mental estimate tells you whether that answer is plausible.
At KS3 you are expected to choose efficiently between mental, written and calculator methods depending on the numbers involved. A question involving 50 × 24 is much faster done mentally than with long multiplication.
What is the partitioning strategy?
Partitioning means splitting a number into its hundreds, tens and ones, then working with each part separately.
Example: 47 + 35
- Split: (40 + 7) + (30 + 5)
- Add the tens: 40 + 30 = 70
- Add the ones: 7 + 5 = 12
- Combine: 70 + 12 = 82
Partitioning also works for multiplication:
Example: 6 × 47 = 6 × 40 + 6 × 7 = 240 + 42 = 282
This is the mental version of the grid method — no paper required once you practise it.
How does the near-number (compensation) strategy work?
Round one of the numbers to a convenient value, calculate, then adjust by the amount you rounded.
Example: 57 + 38
- Round 38 up to 40: 57 + 40 = 97
- Adjust: you added 2 too many, so subtract 2: 97 − 2 = 95
Example: 83 − 29
- Round 29 up to 30: 83 − 30 = 53
- Adjust: you subtracted 1 too many, so add 1: 53 + 1 = 54
The key question to ask yourself is: did I round up or down, and do I need to add or subtract to compensate?
How do doubling and halving help with multiplication?
Doubling and halving exploits the fact that multiplying by 2 is easy — and therefore so is multiplying by any power of 2.
| Calculation | Strategy | Answer |
|---|---|---|
| 15 × 4 | Double 15 twice: 15 → 30 → 60 | 60 |
| 36 × 5 | Multiply by 10, halve: 360 ÷ 2 | 180 |
| 24 × 25 | Multiply by 100, divide by 4: 2400 ÷ 4 | 600 |
| 48 × 0.5 | Halve 48 | 24 |
"Multiply by 5" is most efficiently done as "multiply by 10 then halve" — this always works and is faster than trying to count up in fives.
How do you use the "bridge through ten" strategy for subtraction?
To subtract mentally across a boundary (e.g. crossing through a multiple of 10 or 100), count forward from the smaller number to the larger number in two steps.
Example: 84 − 57
- Step 1: from 57 to 60 = 3
- Step 2: from 60 to 84 = 24
- Total: 3 + 24 = 27
This avoids the tricky "borrowing" that causes errors in mental arithmetic, replacing it with easier addition. Think of it as a number-line hop.
How do you apply these strategies to multiplication by 11, 12 and 15?
These multipliers appear often and reward a specific mental trick.
Multiply by 11: multiply by 10 and add the original.
- 23 × 11 = 230 + 23 = 253
Multiply by 12: multiply by 10, then add double the original.
- 14 × 12 = 140 + 28 = 168
Multiply by 15: multiply by 10, add half the result.
- 18 × 15 = 180 + 90 = 270
Each of these decomposes the multiplier: 11 = 10 + 1, 12 = 10 + 2, 15 = 10 + 5. Once the pattern clicks, these become instant.
What mistakes should you avoid?
- Forgetting the compensation. Rounding 29 to 30 and forgetting to add 1 back is the single most common slip with the near-number strategy. Always ask: did I take too much or too little? Which direction do I adjust?
- Partitioning incorrectly. 47 is 40 and 7, not 4 and 7. Check you have the correct place values before splitting.
- Trying to use mental methods for unsuitable numbers. 347 × 29 is better done in writing. Reserve mental methods for calculations where the numbers are "friendly" (multiples of 10, 25 or 0.5, or numbers close to a round number).
Frequently asked questions
How can I improve my mental arithmetic speed?
Regular short practice beats long sessions. Spend five minutes daily on times tables, then layer in the strategies one at a time: master near-numbers first, then partitioning, then doubling/halving. Use a timer and aim to cut your response time each week. The goal is not memorising answers — it is internalising the strategy so it becomes automatic.
Is mental arithmetic still important if I am allowed a calculator in the exam?
Yes. Most GCSE papers have a non-calculator section, and even on the calculator paper a mental estimate tells you whether a calculator answer is sensible. Many marks are lost not because students cannot operate a calculator, but because they press the wrong key and cannot recognise the error. Mental arithmetic is your error-detection system.
Can I use partitioning for division?
Yes, though it is slightly trickier. 96 ÷ 4 = (80 ÷ 4) + (16 ÷ 4) = 20 + 4 = 24. The key is to split the dividend into parts that are each divisible by the divisor. If the split is not obvious, try the doubling/halving approach instead: 96 ÷ 4 = 48 ÷ 2 = 24.
What should I do if none of the strategies feel natural yet?
Pick one strategy and practise it exclusively for a week. Mental arithmetic fluency builds in layers — trying to juggle all strategies at once before any is automatic slows progress. Start with near-numbers (rounding and compensating), since it applies to the widest range of calculations. Once you can do it without thinking, add partitioning, then doubling and halving.
For guided KS3 number practice with Professor Pi, visit AI Tutors.