Mental calculation strategies
Perform mental calculations including with mixed operations and large numbers, using strategies such as partitioning, compensation, and derived facts
What a learner can do afterwards
- Mentally compute 45 × 8 by partitioning into 40 × 8 + 5 × 8
- Use known facts to derive 6.5 × 4 mentally
- Solve a multi-operation mental calculation such as 250 × 3 + 500 and explain the strategy used
1 · Read
At the till with no calculator, you still need to know what six packs at 52p each will cost before you pay. Mental maths asks you to do that job in your head, by breaking one hard problem into smaller, friendlier ones.
To work out 52 × 6 in your head: split 52 into 50 + 2. Work out 50 × 6 = 300 and 2 × 6 = 12. Add them together: 300 + 12 = 312. When one problem has both multiply and add or take away, do the multiply part first.
Derived facts save you work. If you already know 6 × 4 = 24, you don't need to start over for 6.5 × 4. Just add half of 4, which is 2, so 6.5 × 4 = 26.
When a problem has more than one step, like 250 × 3 + 500, do the multiplication first, then the addition. Say each step out loud so you don't lose track: 750, then 750 + 500 = 1250.
Split a hard problem into easy pieces, use facts you already know, and take one step at a time.
The rewritten reading was written for this dataset.
More to read
Easier numbers for mental maths
Use easier numbers to speed up mental maths. Partitioning splits a number while multiplying each piece: 45 times 8 becomes (40 times 8) plus (5 times 8). Add 320 and 40 to get 360. For 99 times 7, calculate 100 times 7, then subtract 7. Doubling one factor while halving the other also helps. For factors such as 29 or 199, round up and adjust the result to remove the extra. Practise aloud until useful splits come to you automatically.
Multiply by using place value strategies and the distributive property
Place value lets you break a multiplication into a tens part and a ones part. For 38 times 6, use 30 times 6 plus 8 times 6, which is 180 plus 48. Adding the parts gives 228, and every step fits in your head. The rule behind splitting is the distributive property: multiply each part, then add. It also works by compensation: 38 is 2 short of 40, so 40 times 6 minus 2 times 6 gives the same 228. Pick whichever split has the friendliest numbers.
Multiply multiples of 10 by one-digit numbers by using the associative property of multiplication
Multiples of 10 are mental maths gold because extra zeros only shift places. For 250 times 3, first do 25 times 3 to get 75, then put the zero back for 750. Adding the final 500 gives 1,250 without any column working. The associative property blesses this regrouping: 250 times 3 is 25 times 10 times 3, done in any order. Strip the zeros, multiply the friendly core, then restore the zeros at the end. If an answer looks ten times off, a zero was lost or doubled.
8.3: Mental Arithmetic,Using the Distributive Property
The distributive property makes mental arithmetic work. It says a times (b plus c) equals a times b plus a times c, so 7 times 43 is 7 times 40 plus 7 times 3. Choosing splits with round numbers keeps every intermediate answer small and memorable. Subtraction splits work the same way: 7 times 39 is 7 times 40 minus 7 times 1. The page shows how known facts (times tables you trust) grow into unknown ones. Rebuild each product from a fact you already own instead of memorising each new product.
Explore more
2 · Watch
En anglaisTake it off screen
Where it sits
Optional connections
This opens up
18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.