Group them any way

The idea

For any three integers a, b and c, a × (b × c) = (a × b) × c, so three or more integers give the same product whatever order they are multiplied in.

Which two first?

What is 5 × (−3) × 4? We can only multiply two numbers at a time. We can do (5 × (−3)) × 4, or 5 × ((−3) × 4).

Will the two ways give the same product?

• Yes, the same
• No, different products
• It depends on the signs

Order and grouping

Tap two numbers to swap them. Choose which two to multiply first. Find all the ways. Does the product ever change?

The product stays the same

(5 × (−3)) × 4 = (−15) × 4 = −60. And 5 × ((−3) × 4) = 5 × (−12) = −60. Multiplying 5 and 4 first also gives (20) × (−3) = −60.

For any three integers a, b and c, a × (b × c) = (a × b) × c. Integer multiplication is associative, so three or more integers give the same product whatever order they are multiplied in.

• Way | Steps | Product
• (5 × (−3)) × 4 | (−15) × 4 | −60
• 5 × ((−3) × 4) | 5 × (−12) | −60
• (5 × 4) × (−3) | 20 × (−3) | −60

Notes

For any three integers a, b and c, a × (b × c) = (a × b) × c. Integer multiplication is associative: three or more integers give the same product in any order.

Check yourself

Find 25 × (−6) × 12. Multiply 25 and 12 first.

Answer: -1800

(25 × 12) × (−6) = 300 × (−6) = −1800, the same as 25 × (−6) × 12.

Find (−2) × (−5) × (−3).

Answer: -30

(−2) × (−5) = 10, and 10 × (−3) = −30. Grouping the other way, (−2) × ((−5) × (−3)) = (−2) × 15 = −30.

Meena works out 8 × (−5) × 2 as (8 × 2) × (−5) and gets −80. Is that allowed?

Which is always equal to a × (b × c) for integers a, b and c?

• No: we must go from left to right. Integer multiplication is associative: any grouping and any order gives the same product.
• Yes: the order of multiplication does not change the product — correct. Yes! 8 × (−5) × 2 = −80 in any order or grouping.
• Yes, but only when all the numbers are positive. It works for negative integers too: (8 × 2) × (−5) = −80.
• (a × b) + c. That adds c instead of multiplying by it.
• (a + b) × c. That adds a and b first. The product changes.
• (a × b) × c — correct. Yes! Grouping the first two instead of the last two gives the same product.