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:

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 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?"*

Well...

3 + 4 is the same as 4 + 3.

and

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:

3 + 2 = 5

IS THE SAME AS

2 + 3 = 5

And here's multiplication:

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

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?

If a mathematical operation follows the associative property, that tells us that we can group the operands differently and the result will be the same.

*Or more simply...*

(2 + 3) + 4 = 2 + (3 + 4)

The parentheses tell us that we should perform the operations within the parentheses before applying any other operations to whatever is inside the parentheses. The associative property then tells us that for addition and multiplication, it doesn't matter where the parentheses go.

(2 + 3) + 4 means we first do 2 + 3, then add 4. So...

(2 + 3) + 4

= 5 + 4

= 9

And similarly, 2 + (3 + 4) tells us to first do 3 + 4, then add that to 2. So...

2 + (3 + 4)

= 2 + 7

= 9

As they say, a picture is worth 1000 words... (I wonder how much a moving picture is worth)?

(1 + 2) + 3

= 3 + 3

= 6

IS THE SAME AS

1 + (2 + 3)

= 1 + 5

= 6

And the same is true with multiplication:

(2 X 3) X 4 = 2 X (3 X 4)

Action!

4 trays

of

6 donuts

(2 X 3) X 4

= 6 X 4

= 24 donuts

IS THE SAME AS

2 trays

of

12 donuts

2 X (3 X 4)

= 2 X 12

= 24 donuts

With the first animation, we can see that the grouping of the carrots doesn't affect the total number of the carrots. Whether the middle set of carrots "associates" with the one on the left or the three on the right, either way the total is the same: six carrots.

The animation for multiplication is a little more complicated, but basically it looks at a three-dimensional group of donuts. You can look at it two ways: on the left it's shown as four trays, each of which contains two rows of three donuts each. On the right, it's the same set of donuts, but instead of looking at the four trays, you can instead view it as a front set and a back set of donuts, each of which has four rows of three donuts each. Either way, it's a total of 24 donuts.

If you use this property together with the commutative property, it means that for addition and multiplication, you can order and combine things any way you want, for as many numbers as you want. So:

1 + 2 + 3 + 4 + 5

= (2 + 3) + 4 + 1 + 5

= (2 + 3) + 5 + (1 + 4)

= 5 + (4 + 3 + 2) + 1, etc.

Note: the associative property doesn't apply to subtraction or division. And also it only applies to operations of the same type, so you can't mix addition and multiplication and use the associative property:

3 + (4 X 2) does **not** equal (3 + 4) X 2

In fact, using addition and multiplication together is addressed by the next property discussed here.

All right, I think we're ready to tackle the Distributive Property!

Unlike the other two properties in this section, the distributive property, or distributive law, applies to both addition and multiplication at the same time. It tells us that you can multiplying a number with a sum of two other numbers (addends) is the same as multiplying the number by each addend separately, and then adding the results.

Well gee, that wasn't confusing!

*Ok maybe this will make it a little bit clearer...*

2 X (3 + 4) = (2 X 3) + (2 X 4)

You can see how the 2 X operation has been "distributed" over both the 3 and the 4.

2 X (3 + 4)

= 2 X 7

= 14

And similarly,

(2 X 3) + (2 X 4)

= 6 + 8

= 14

But it might be even easier to understand by taking a look at this animated picture:

3 + 4 = 7

2

2 X (3 + 4)

= 2 X 7

= 14

IS THE SAME AS

3

4

2

2 X 3

2 X 4

2 X (3 + 4)

= 2 X 3 + 2 X 4

= 6 + 8

= 14

This animation basically shows that if you have two rows of seven carrots each, you can split up the big group into two smaller groups, so that you have one set with two rows of three, and another set with two rows of four. Either way you end up with 14 carrots.

Be careful with which operation is which - the operation inside the parentheses has to be addition, and the one outside has to be multiplication. So...

2 + (3 X 4) is **NOT** the same as (2 + 3) X (2 + 4).

The left one is 14 while the right one is 30.

This property doesn't quite work with subtraction and division, although if you use negative numbers and fractions, you can achieve a similar effect in some situations. If you're not ready to tackle negative numbers and/or fractions, please ignore this example:

(4 - 3) ÷ 2 is the same as (4 ÷ 2) - (3 ÷ 2)

This is because you can treat - 3 as the same as adding -3, and dividing by 2 as the same as multiplying by ½.

So (4 - 3) ÷ 2

= (4 + -3) X ½

= ½ X (4 + -3) (using the commutative property)

= (½ X 4) + (½ X -3)

= (4 ÷ 2) + (-3 ÷ 2)

= (4 ÷ 2) - (3 ÷ 2)

= 2 - (3/2)

= ½

Note this doesn't work if you switch the order around, so it doesn't work with 2 ÷ (4 - 3). That's a more complicated operation not covered here.

But when in doubt, just remember the carrots in the animated picture above, and that may help you remember how this property works!

All done! But if you'd like, you can click to go back to the Commutative Property...