Algebra · Properties

Properties of Multiplication

Multiplication follows four reliable rules. See them proven with a grid of dots you can rotate, group, and even make disappear.

Part 01

The Commutative Property — Order Doesn't Matter

You can multiply two numbers in either order and get the same product. An array of dots proves it — rotate it 90° and you're just looking at the same dots from a different direction.

a × b = b × a
Try It — The Array Rotator

Rotate the grid — the total number of dots never changes

Part 02

The Associative Property — Grouping Doesn't Matter

When multiplying three or more numbers, it doesn't matter which two you multiply first — the final product is always the same.

(a × b) × c = a × (b × c)
Try It — The Grouping Switcher

Pick which pair multiplies first — the total won't change

Part 03

The Identity & Zero Properties

Two special numbers change what multiplication does: 1 leaves everything exactly the same, and 0 wipes everything out.

a × 1 = a
a × 0 = 0
Try It — Multiply by 1 or by 0

Watch what happens to a row of 6

Your Turn!

Practice

Eight questions mixing all four properties — identify which one is being used, or fill in the missing number.

Question 1 of 8 Score: 0/8