The Commutative Property

Arithmetic · Axiom Academy

LESSON The Commutative Property Why 3 × 4 equals 4 × 3 — the order of factors never changes a product Swapping the two factors never changes the product. Both of these name the very same number: Both products equal 12 — the order of the factors does not change the result. Watch the same array of counters rotate a quarter-turn: the dots never change, only their arrangement does. The commutative property holds for every pair of factors — not just 3 and 4. One array, read two ways, settles it: count 2 × 5 by rows, then 5 × 2 by columns — the same ten dots either way. You never have to check both orders. Once you know 6 × 7 = 42, you automatically know 7 × 6 = 42. Enter any two numbers (1–12). Watch both orders land on the same product : Whichever order you pick, the same counters just get regrouped — 3 groups of 4 hold exactly as many as 4 groups of 3. Whether you have 3 groups of 4 or 4 groups of 3, the total number of items is the same. Rotating a 3 × 4 array onto its side produces a 4 × 3 array — the arrangement changes, but the number of dots does not. Every fact above the diagonal has a mirror-image twin below it. Learn 3 × 4 = 12, and 4 × 3 = 12 comes free — so you only have to memorize half the multiplication table. You've seen why 3 × 4 and 4 × 3 are the same: rotate, re-read, or regroup the dots, and the total never moves. Scroll up to revisit any step.

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