Math Playground

Algebra › Properties of operations

Commutative, associative, distributive

Three laws that let you reshape any sum or product. Drag, regroup, expand — and watch the total never budge.

Three laws of arithmetic

Pick a law and play with it — drag, regroup, expand.

Drag the tiles to rearrange — the total never changes.

  • 3+
  • 5+
  • 8
=16

Try subtraction or division mentally: 5 − 3 ≠ 3 − 5. Those operations don't commute — only + and × do.

Commutative law

Order doesn't matter (for + and ×).

3 + 5 = 5 + 3 · 3 × 5 = 5 × 3

1 · Commutative law — order doesn't matter

For addition and multiplication, the order of the numbers makes no difference:

  • 3 + 5 = 5 + 3 — both are 8.
  • 3 × 5 = 5 × 3 — both are 15.

Subtraction and division don't commute. 5 − 3 = 2 but 3 − 5 = −2. 6 ÷ 2 = 3 but 2 ÷ 6 = 0.333….

2 · Associative law — grouping doesn't matter

For addition and multiplication, where you put the brackets has no effect on the answer:

  • (2 + 3) + 4 = 2 + (3 + 4) — both are 9.
  • (2 × 3) × 4 = 2 × (3 × 4) — both are 24.

That's why we can write 2 + 3 + 4 with no brackets at all. The associative law guarantees there's no ambiguity.

Associative ≠ commutative

They're different! Associative is about grouping; commutative is about order. (2 + 3) + 4 = 2 + (3 + 4) doesn't rearrange any numbers — it just shifts the brackets. Subtraction is NOT associative either: (10 − 5) − 2 = 3 but 10 − (5 − 2) = 7.

3 · Distributive law — × spreads over +

For any numbers a, b, c:

a × (b + c) = a × b + a × c

This is the bridge between multiplication and addition. It's also the rule you use every time you "expand brackets" in algebra:

  • 3 × (4 + 5) = 3 × 4 + 3 × 5 = 12 + 15 = 27. (Check: 3 × 9 = 27. ✓)
  • x(a + b) = xa + xb. The same law, with letters.
  • (a + b)(c + d) = ac + ad + bc + bd. Apply the law twice — that's "FOIL".

Worked example: shopping

You buy 4 t-shirts at $12 each plus 4 pairs of socks at $3 each. Total cost?

Long way: 4 × 12 + 4 × 3 = 48 + 12 = $60.
Distributive way: 4 × (12 + 3) = 4 × 15 = $60.

Same answer, less arithmetic.

Worked example: mental math

What's 7 × 102?

Rewrite 102 as 100 + 2: 7 × (100 + 2) = 700 + 14 = 714. Much easier than long multiplication.

Common mistakes

  • Forgetting to distribute to every term. 3(x + 4) ≠ 3x + 4. The 3 multiplies both pieces: 3x + 12.
  • Distributing × over ×. 3 × (4 × 5) ≠ 12 × 15. You only distribute multiplication over addition (or subtraction).
  • Sign errors. 3 − (x + 2) = 3 − x − 2 = 1 − x, not 3 − x + 2. The − distributes too.

Try it

  1. Use the commutative law to rewrite 17 + 25 + 3 in a form that's easy in your head. (Hint: 17 + 3 = 20.)
  2. Use the associative law: which is easier, (47 + 13) + 25 or 47 + (13 + 25)?
  3. Expand 5(x + 6). Then expand 5(x − 6).
  4. Compute 8 × 99 the smart way using the distributive law.

Answers: 1) (17 + 3) + 25 = 45. 2) Both 85; the second is easier because 13 + 25 = 38 then + 47. 3) 5x + 30; 5x − 30. 4) 8 × (100 − 1) = 800 − 8 = 792.