Number & the laws of arithmetic
The laws of arithmetic
Three rules let you rearrange a calculation into an easier one. They are not tricks — they are why mental arithmetic works.
The commutative law
You can swap the order when adding or multiplying. 7 + 5 = 5 + 7 4 × 6 = 6 × 4 This does NOT work for subtraction or division: 7 − 5 is not 5 − 7. Use it to put the easier number first.
The associative law
You can regroup which pair you do first when multiplying (or adding). 50 × 16 × 2 Done in order that is awkward. But regroup as 50 × 2 × 16 = 100 × 16 = 1600 — much easier. 25 × 17 × 4 becomes 25 × 4 × 17 = 100 × 17 = 1700. Look for pairs that make 10, 100 or 1000.
The distributive law
You can split a number up, multiply each part, then add. 36 × 8 = (30 + 6) × 8 = 30 × 8 + 6 × 8 = 240 + 48 = 288 This is exactly what the column method does — the law is the reason it works. It is also handy for numbers near a round one: 19 × 6 = (20 − 1) × 6 = 120 − 6 = 114.
Choosing a method
There is rarely one right way. 54 × 6 might suit partitioning: 50 × 6 + 4 × 6 = 300 + 24 = 324. 41 × 5 might suit doubling and halving. 19 × 6 suits (20 − 1) × 6. Look at the numbers first and pick the method that makes them easy. Explaining your choice is part of the mathematics.
Worked examples
50 × 16 × 2
- Look for a pair making a round number.
- 50 × 2 = 100.
- 100 × 16 = 1600.
Answer: 50 × 16 × 2 = 1600
25 × 17 × 4
- 25 × 4 = 100.
- 100 × 17 = 1700.
Answer: 25 × 17 × 4 = 1700
36 × 8 by partitioning
- Split 36 into 30 + 6.
- 30 × 8 = 240 and 6 × 8 = 48.
- 240 + 48 = 288.
Answer: 36 × 8 = 288
19 × 6 using a round number
- 19 is one less than 20.
- 20 × 6 = 120.
- Take off one lot of 6.
Answer: 19 × 6 = 114
48 × 7 by partitioning
- Split 48 into 40 + 8.
- 40 × 7 = 280 and 8 × 7 = 56.
- 280 + 56 = 336.
Answer: 48 × 7 = 336
Where you see this in real life
- Adding a column of prices in whichever order makes pairs of 10.
- Working out 25 × 4 × 7 in your head by spotting the 100.
- Multiplying by 9 by doing ten lots and taking one away.
Key words
- Commutative law
- You can swap the order when adding or multiplying.
- Associative law
- You can regroup which pair you work out first.
- Distributive law
- You can split a number, multiply each part, then add.
- Partition
- To split a number into easier parts.
- Regroup
- To change which numbers are paired together.
- Mental method
- Working something out in your head.