Properties of operations: calculating faster — Mathematics, 11–13
Swap, regroup and split numbers with the commutative, associative and distributive properties, and judge which strategy is simplest.
What you can change
The properties of operations tell you what you can change without changing the result. In addition and multiplication you can swap the order (commutative) and change the grouping (associative): 4 × 9 × 25 = 4 × 25 × 9. You can also distribute: 6 × (10 + 3) = 6 × 10 + 6 × 3, with addition and with subtraction. With them a hard sum becomes one with round numbers.
Round numbers make it easy
Mental calculation means choosing the split that leaves easy numbers: 25 × 4 = 100, 8 × 125 = 1000, 10 × any number. The properties guarantee that the swap does not change the value. There is no single right way: two people can follow different paths and reach the same result, and comparing them shows which is simpler to explain and safer for those numbers.
25 × 36, two ways
Calculate 25 × 36 in your head. Path A: 36 = 4 × 9, and by the associative property 25 × 36 = (25 × 4) × 9 = 100 × 9 = 900. Path B: distributive, 25 × (30 + 6) = 25 × 30 + 25 × 6 = 750 + 150 = 900. Both give 900. A is shorter, but it only works because 36 has the factor 4. For 25 × 37, A fails (37 cannot be split into factors) and B solves it: 750 + 175 = 925. A good strategy is one that is simple and works with the numbers you have.
Not for subtraction and division
The reasonable mistake is to use with subtraction and division what only holds for addition and multiplication. 40 − 12 − 8 = 20, but 40 − (12 − 8) = 36: changing the grouping changed the value. Likewise 24 ÷ 6 ÷ 2 = 2, but 24 ÷ (6 ÷ 2) = 8, and 12 ÷ 3 = 4 while 3 ÷ 12 = 1/4. To regroup a subtraction, transform it first: 40 − 12 − 8 = 40 − (12 + 8) = 40 − 20 = 20.
Everyday shortcuts
You calculate like this in daily life without noticing: 7 × 98 = 7 × (100 − 2) = 700 − 14 = 686; 12 rows of 15 seats are 10 × 15 + 2 × 15 = 150 + 30 = 180 seats. To add 38 + 27 + 12 + 23, join 38 + 12 = 50 and 27 + 23 = 50, so 100. Always look for pairs that make round numbers.
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