Common Math Properties

The following math properties are formally introduced in algebra classes, but they are taught in many elementary schools. You probably don't even realize that you already know many of these properties. For example, the commutative property basically states you can add in any order: 6 + 5 is the same as 5 + 6.

In this page you will learn the following properties:

Associative Property

Commutative Property

Distributive Property

Identity Property

Zero Property

Not what you are looking for?


Vist our pages dedicated to the math property of equality or math clue words.

You should also be sure to understand the order of operations before attempting to understand these math properties.

Each property is listed below. On the left side of the table we show the general form – using all letters.  We know properties can be confusing when too many variables are use, so we also give an example with numbers on the right side of the table as well.  Aim to learn the general form, but use the numeric form as your "training wheels."

Associative Property

The associative property indicates that the grouping of numbers does not matter. By "grouping" we simply mean where the parentheses are placed. Take a look:

Addition (+)
(3 + 5) + 1 = 3 + (5 +1)
Try it!  Both sides = 9
Multiplication (x)
(a · b) · c = a · (b · c)
(3 · 5) · 1 = 3 · (5 ·1)
Try it!  Both sides = 15

Notice how the order of the numbers did not change.  In the examples with numbers, the order always goes 3, 5, 1. 

How can we remember the name of this math property?  One possibility is to think of the word associate – which is another word for friends.  You probably have different groups of friends and you hang out with them at different times.  The associative property deals with changing groups (parentheses).  You don’t change the order, you just change the groups. 

Commutative Property

The commutative property (like we described at the top of the math properties page) deals with the order that add or multiply numbers.

Addition (+)
4 +2 = 2 + 4
Try it!  Both sides = 6
Multiplication (x)
a · b = b · a
4 · 2 = 2 · 4
Try it!  Both sides = 8


In the commutative property you do change the order of the numbers.  In our example above, the 4 was first originally, and then it was switched to second.

How can we remember this property?  The word commute means to travel:  “A half hour commute to work.”  When you see the word commutative, think of travel – or of moving the order of the numbers.

Tip to remember: Commutative also sounds like com-move-ative.

Distributive Property

The distributive property applies when you are multiplying a number (or variable) times a quantity. You can multiply the number by each of the values inside the quantity seperately, and add them together.

Take a look at the distributive property below:

The word distribute means to give out.  In the example at the right, we are giving out the 3 to both the 4 and the 1 – see the arrows shown below?

Because you are multiplying 3 times (4+1), that means you have three (4+1)’s.  Instead of multiplying, you can add all 3 of them up.  Look at the figure with the 3 arrows.

You might be thinking:  I could just add up 4+1 to get 5, and then multiply 3 times 5 to get 15.  That is certainly true. 


The distributive property will be most useful when one of the numbers inside the parentheses is a variable.  Certain math properties are only useful in some situations.

Identify Property

he word identity means “who you are.”  You may have heard of identity theft.  In math, we want a number to keep its same identity – in other words, stay as the same number.  What number would you have to add to a number to keep it the same?  What about multiplication?

Addition (+)
a + 0  = a
6 + 0 = 6
Multiplication (x)
a · 1 = a
6 · 1 = 6


he identity operator of addition is 0 because any number plus 0 is always equal to that number – and yes, you can switch the order!

The identity operator of multiplication is 1 because any number times 1 is always equal to that number – again you can use the commutative prop!

Zero Product Property (of Multiplication)

This math property states that any number multiplied by zero will always be equal to zero.  You probably already knew this one.

Multiplication (x)
a · 0 = 0
8· 0 = 0

Return to other pre algebra math problems or visit the GradeA homepage.


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