The Commutative, Associative, and Distributive Properties

Sometimes, seeing a mathematical concept demonstrated visually will help you remember and understand how it works. Here are some animated pictures whose graphics demonstrate the commutative, associative, and distributive properties.

Pick a property, any property:

Commutative Associative Distributive

Note: You need to use a modern browser to see the animations properly. Modern browsers include: the latest two major versions of Google Chrome, Mozilla Firefox, Microsoft Internet Explorer, Apple Safari, Opera, or the browsers on the iPhone/iPad or Android smartphones and tablets. (Adobe Flash Player is NOT required).

The Commutative Property

The commutative property is a property of some mathematical operations, where changing the order of the operands does not affect the result.

"Ok, but what on earth does that mean?"


3 + 4 is the same as 4 + 3.


2 X 5 is the same as 5 X 2.

So basically when adding or multiplying numbers, their order doesn't matter.

And here's a little animation demonstrating this:

blue cupcake blue cupcake blue cupcake yellow cupcake yellow cupcake

3 + 2 = 5

yellow cupcake yellow cupcake blue cupcake blue cupcake blue cupcake

2 + 3 = 5

And here's multiplication:

star 1 star 2 star 3 star 4 star 5 star 6
3 yellow right width arrow yellow left width arrow 2 red right width arrow red left width arrow 2 red up height arrow red down height arrow 3 yellow up height arrow yellow down height arrow

3 X 2 = 2 X 3

Basically for things like addition and multiplication, it doesn't matter what order the numbers that you're adding or multiplying (those are the "operands") come in.

Note: The Commutative Property applies to addition and multiplication, but not to subtraction and division, so:

4 - 2 does NOT equal 2 - 4

Explanation of Animations

The first animation shows that it doesn't matter if you start with two cupcakes and add three more, or you start with three cupcakes and add two more, the result is five either way.

The second animation is showing something similar for multiplication, whether you have two rows of three stars each or three rows of two stars each, the result is the same: six stars.

Ready to go on to the Associative Property?

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