Activities
Commutative, Associative & Distributive
Three big laws of arithmetic — try to break them with examples.
Three laws govern arithmetic. Boring names, beautiful consequences — they're why algebra works.
Quick check
Which calculation uses the **distributive** law?
Try it
Use the laws to do 8 × 25 in your head.
8 × 25 = 8 × (20 + 5) = 160 + 40 = 200 — or regroup: 4 × (2 × 25) = 4 × 50 = 200. Same answer, no calculator.
Subtraction and division are not commutative or associative: 5 − 3 ≠ 3 − 5, and (12 − 4) − 2 ≠ 12 − (4 − 2). Order and grouping matter there.
Recap
- Commutative: order doesn't matter for + and ×.
- Associative: grouping doesn't matter for + and ×.
- Distributive: a × (b + c) = a×b + a×c.
- These three laws are *why* algebra rearranging is allowed.
The three laws
- Commutative: order doesn't matter. 3 + 5 = 5 + 3. 2 × 7 = 7 × 2.
- Associative: grouping doesn't matter. (2 + 3) + 4 = 2 + (3 + 4).
- Distributive: 3 × (4 + 5) = 3×4 + 3×5. The 3 spreads across.
Try it
Quick: 7 × 23
Distribute: 7 × 23 = 7 × 20 + 7 × 3 = 140 + 21 = 161. Mental math, no calculator.
Subtraction and division are NOT commutative. 5 − 3 ≠ 3 − 5. Order matters there.