Commutative, Associative, Distributive, Identity, and Zero Properties 𝐴∗𝐵=𝐵∗𝐴 𝐴+𝐵=𝐵+𝐴 𝐴+ 𝐵+𝐶 = 𝐴+𝐵 +𝐶 Math Properties 𝐴∗1=𝐴 𝐴∗0=0 Commutative, Associative, Distributive, Identity, and Zero Properties 𝐴 𝐵+𝐶 =𝐴𝐵+𝐴𝐶 𝐴+0=𝐴

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.

Suddenly the cheerleader associates with someone else. 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 A + (B + C) (A + B) + C A + + C B The associative property is when a number associates with a different number. A + (B + C) (A + B) + C A + + C B

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?)

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 A + 0 = A A * 1 = A Identity Property of Addition Identity Property of Multiplication A + 0 = A 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.

Summary Property 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) =4X + 12 4 X 3 4X 12

Now you try these examples. 2(Z -3) = 5X + 15 7X + 28 2Z - 6

Summary Property 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