Commutative Property
Change the order. The answer stays the same. This works for addition and multiplication.
Example 1: Addition4 + 9 = 9 + 4Example 2: Multiplication3 · 7 = 7 · 3It does not work for subtraction or division.Rules & properties reference
Properties of Operations
A property is a math rule that always works. Properties let us move, group, or rewrite numbers without changing the answer.
In simple words:Properties help make math easier and explain why our algebra steps are allowed.
Before distribution6(10 + 2)
After distribution6(10) + 6(2)
See how it moves
Distributive Property
The number outside must multiply every term inside.
5(x+3)
↘↙
5 · x5 · 3
5x + 15
Why should I care?
You can use properties to do mental math, remove parentheses, simplify expressions, and solve equations.
Quick-reference chart
The Rules
Read the rule, then study the example.
Associative Property
Change which numbers are grouped together. The answer stays the same. This works for addition and multiplication.
Example 1: Addition(2 + 5) + 8= 2 + (5 + 8)Example 2: Multiplication(2 · 3) · 4= 2 · (3 · 4)It does not work for subtraction or division.Identity Property
Add 0 or multiply by 1. The number stays the same.
Example 1: Add zero12 + 0 = 12Example 2: Multiply by one12 · 1 = 12Inverse Property
A number and its opposite add to 0. A number and its flipped fraction multiply to 1.
Example 1: Opposites8 + (-8) = 0Example 2: Flipped fractions3/4 · 4/3 = 1Distributive Property
Multiply the outside number by every term inside the parentheses.
5(x + 3)5 · x + 5 · 35x + 15Zero Property
Any number multiplied by zero equals zero.
Example 127 · 0 = 0Example 20 · x = 0Follow the thinking
Worked Examples
Use Distribution: 7(x - 4)
- Multiply 7 by x: 7 · x = 7x.
- Multiply 7 by -4: 7 · (-4) = -28.
- Put the two results together.
7(x - 4) = 7x - 28
Make Mental Math Easier
- Start with 25 + 17 + 75.
- Change the order: 25 + 75 + 17.
- Group 25 + 75 to make 100.
- Add 100 + 17.
25 + 17 + 75 = 117
Watch out
Common Mistakes
- Commutative means change the order. Associative means change the grouping.
- Do not use the commutative property with subtraction or division.
- When distributing, multiply every term inside the parentheses.
Check yourself
Quick Practice
1. 14 + 0 = 14
Identity property of addition.
2. Rewrite 3(x + 5)
3x + 15.
3. 6 · 4 = 4 · 6
Commutative property of multiplication.
Your turn
10-Question Properties Practice
Type the property name or rewrite the expression as requested.
Finished with this resource?