5.2 Multiplication, Division, and Other Properties of Integers Remember to silence your cell phone and put it in your bag!

Definition of Multiplication of Integers For any integer a, and b an integer > 0, b  a = a + a + a a where a occurs b times If b = 0, 0  a = 0. Examples 3 x 2 = = 6 3 x -2 = = x 3 = 3 x -2 = = x -3 = ?

Modeling Integer Multiplication 1. Chips Model – m x n (p. 273) Start with an equal number of yellow and red chips in a bag. Put chips in a bag a certain number of times to multiply with a positive integer. Take chips out of a bag a certain number of times to multiply with a negative integer. The product is the number of chips that you have more of that are now in the bag.

Modeling Integer Multiplication Counters Model (cont.) Positive m x Positive n Put n yellow chips in a bag m number of times. Positive m x Negative n Put n red chips in a bag m number of times. Negative m x Positive n Take out n yellow chips from a bag m number of times. Negative m x Negative n Take out n red chips from a bag m number of times.

Modeling Integer Multiplication 2.Number Line Model – m x n (p. 277) 1. Begin at the origin facing the positive direction. 2. |m| represents the number of steps. If m is positive, remain facing in the positive direction. If m is negative, turn around and face in the negative direction. 3. |n| represents the length of each step. If n is positive take steps of length n moving forward. If n is negative take steps of length |n| moving backward. 4. The product is where you end up on the number line.

Integer Multiplication Using Patterns Procedures for Multiplying Integers Review the procedures on the bottom of p. 278.

Properties of Integer Multiplication a, b, c  I Closure Property a  b is a unique integer. Identity Property There exists a unique integer, 1, such that a  1 = a and 1  a = a.

Properties of Integer Multiplication (cont.) Commutative Property a  b = b  a Associative property (a  b)  c = a  (b  c) Zero Property a  0 = 0 and 0  a = 0

Properties of Integer Multiplication (cont.) Distributive property of multiplication over addition a  (b + c) = (a  b) + (a  c) (b + c)  a = (b  a) + (c  a) Distributive property of multiplication over subtraction a  (b - c) = (a  b) - (a  c) (b - c)  a = (b  a) - (c  a)

Integer Division Definition – For a, b  I, b  0, a  b = c iff c is a unique integer such that c × b = a. Missing factor interpretation of division Using patterns Review the procedures for dividing integers on p. 286.

Properties of Integer Division a, b,  I and a  0 1. a  a = 1 2. a  1 = a 3. 0  a = 0 4. ab  a = b 5. Division by 0 is undefined

Additional Properties a, b  I Properties of Opposites -(-a) = a a(-1) = -a and (-1)a = -a Distributive Property for Opposites over Addition -(a + b) = -1(a + b) = -a + (-b) = -a - b

Review – Order of Operations 1. Compute within grouping symbols first, working from the inside out. 2. Compute powers. 3. Multiply and divide in order from left to right. 4. Add and subtract in order from left to right.