Math Properties

Commutative, Associative, Distributive, Identity, and Zero

Properties

𝐴+𝐵=𝐵

+𝐴

𝐴 (𝐵+𝐶 )=𝐴𝐵+𝐴𝐶

𝐴+(𝐵+𝐶 )=(𝐴+𝐵 )+𝐶

𝐴+0=𝐴

𝐴∗𝐵=𝐵∗ 𝐴

𝐴∗1=𝐴 𝐴∗0=0

What are properties?

Math Properties are rules in math. Properties are always true for every

number.

**Once you go beyond the set of Real numbers the properties may no longer hold.

Commute

• To commute means to travel from one place to another.

• For example, you commute to school in the morning.

Commutative Property

• Just like you commute from home to school, a number may commute from one spot to another.

• A + B = B + A (The numbers change places.)• This is called the commutative property of

addition.• Ex) 2 + 3 = 3 + 2• Both 2 + 3 and 3 + 2 equal 5.

The commutative property may be used with addition as seen previously and also with multiplication.• A * B = B * A• Ex) 3 * 5 = 5 * 3• Both 3 * 5 and 5 * 3 equal 15.• This is called the commutative property of

multiplication.

Associate

• An associate is a friend or someone you work with.

• For example, the head cheerleader is an associate of the school mascot.

Now imagine the football team played a late game and the cheerleader and mascot forgot to study for the math test.

Suddenly the cheerleader associates with someone else.

Associative Property

The associative property is when a number associates with a different number.

A + (B + C)

(A + B) + CA + + CB

Associative Property

• (A + B) + C = A + (B + C) is called the associative property of addition.

• Ex) (2 + 3) + 4 = 2 + (3 + 4)• The order in which you add does not change

your answer.• A * (B * C) = (A * B) * C is called the associative

property of multiplication.

Identity

• Your identity is who you are.

• Changing your clothes or getting a new haircut does not change your identity.

• Your identity remains the same.

Identity Property of Addition

• A number also has an identity• The identity of a number is the value of the

number• The additive identity is the number that when

added to another number does not change the identity of the original number

• 3 + __ = 3 (What goes in the blank?)0

Zero• The additive identity is zero.

• We can add zero to any number and the answer is the original

number.

Identity Property of Multiplication

• We also have a multiplicative identity• 3 * __ = 3 (What goes in this blank?)• We can multiply any number by one and the

answer will be the original number.

1

Identity Properties

Identity Property of Addition

A + 0 = A

Identity Property of Multiplication

A * 1 = A

Zero Property

• The zero property sounds just like what it is, a property about zero.

• A * 0 = 0

• The zero property tells us that any number multiplied by zero equals zero.

SummaryProperty Name Rule

Commutative Property of Addition A + B = B + A

Commutative Property of Multiplication A * B = B * A

Associative Property of Addition A + (B + C) = (A + B) + C

Associative Property of Multiplication A * (B * C) = (A * B) * C

Identity Property of Addition A + 0 = A

Identity Property of Multiplication A * 1 = A

Zero Property A * 0 = 0

Distribute

• Distribute means to deliver or pass out• If we distribute food to three boxes, we put

food in each of the three boxes

Distributive Property

• A(B + C) = A*B + A*C• The A is the food and the boxes are B and C.• We pass out A to each of B and C.• In this case that means that we multiply A by

both B and C separately and then add the resulting products.

Ex) 4(X + 3)

4

X 34X 12

=4X + 12

Now you try these examples.

1) 5(X + 3) =

2) 7(X + 4) =

3) 2(Z -3) =

5X + 15

7X + 28

2Z - 6

SummaryProperty Name Rule

Commutative Property of Addition A + B = B + A

Commutative Property of Multiplication A * B = B * A

Associative Property of Addition A + (B + C) = (A + B) + C

Associative Property of Multiplication A * (B * C) = (A * B) * C

Identity Property of Addition A + 0 = A

Identity Property of Multiplication A * 1 = A

Zero Property A * 0 = 0

Distributive Property A(B + C) = A*B + A*C