Properties of Arithmetic Objective: To review the identity, commutative, associative, and distributive properties.

What is the identity property? The multiplicative identity is multiplying a number by 1. 5(1) = 5 The additive identity is adding 0 to a number. Example: 5 + 0 = 5

What is the Commutative Property? An operation is commutative if you can change the order of the numbers involved without changing the result. What kinds of expressions allow us to change the order of the numbers, without changing the answer? Addition and Multiplication

Commutative Addition and multiplication are both commutative. For example: Addition: 2 + 1 = 1 + 2 Multiplication: 5 x 9 = 9 x 5

Commutative Why are division and subtraction problems not commutative? Subtraction is not commutative because 4 - 3 is not equal to 3 - 4 Division is also not commutative because 6 ÷ 2 is not equal to 2 ÷ 6

Commutative = Change order

What is the Distributive Property? When you distribute something, you give pieces of it to many different people. The most common distributive property is the distribution of multiplication over addition. It says that when a number is multiplied by the sum of two other numbers, the first number can be handed out or distributed to the other two numbers and multiplied by each of them separately.

Distributive Property For example: 3(2x + 1) Can you combine these two terms? Why or why not? = 6x + 3 = 6x + 3

What is the Associative Property? An operation is associative if you can group numbers in any way without changing the answer. It doesn't matter how you combine them, the answer will always be the same. What kinds of equations will allow us to do this? Addition and multiplication

Associative Property Addition: (3 + 2) + 1 = 3 + (2 + 1) Multiplication: (4 x 5) x 9 = 4 x (5 x 9) Subtraction is not associative: (4 - 3) - 2 is not equal to 4 - (3 - 2) Division is also not associative: (12 ÷ 2) ÷ 3 is not equal to 12 ÷ (2 ÷ 3) Notice that the numbers stay in the same order, but the parentheses are moved.

Associative = Same order

EXAMPLE 1. Indicate which property is illustrated below. 8(3m)=(8·3)m A. identity B. associative C. commutative D. distributive

EXAMPLE 2. What property is shown below? 3 x (3 + 8) = (3 x 3) + (3 x 8) A. Distributive B. Commutative C. Equality D. Associative

EXAMPLE 3. What property is shown below? 10 + (5 + 3) = (10 + 5) + 3 A. Associative B. Equality C. Transitive D. Distributive What do you see taking place from the left side of the equation to the right side?

YOUR TURN… 4. What property is shown below? 4 x (4 - 2) = (4 x 4) - (4 x 2) A. Transitive B. Commutative C. Associative D. Distributive

ONE MORE… 5. What property is shown below? 2x + 0 = 2x A. Associative B. Equality C. Identity D. Distributive