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7/12/2010 Section 5.2 Multiplication, Division, and Other Properties of Integers ` ` ` ` ` ` ` ` ` ` ` ` Use black counters for positive integers, red counters for negative integers. Start with a bag containing an _________ number of red and black counters. Use putting counters in a bag a certain number of times as _________ integers. Use taking counters out of a bag a certain number of times as _________ integers. The first factor tells you how many _________ you are putting in or taking out. The second factor tells you the _________ of the group that you are actually putting in or taking out. Remove zero _________. Remaining counters is answer to problem. Start with a zero-charged field. (equal number of positive and negative counters) Take out or put in # of groups according to _______ and absolute value of first integer. The h second d integer is the h amount/size off each h group and type (positive or negative) of counter to be taken out or put in. The resulting “charge” is your answer. ` Using a Counters Model ` Using a Charged Field Model ` Using the Number Line Model ` Examples 1) 4 x 2 Put in 4 groups of 2 positive counters 2) - 5 x 3 groups p of 3 positive p counters Take out 5 g 3) 2 x (- 1) Put in 2 groups of 1 negative counter 4) - 2 x (- 3) Take out 2 groups of 3 negative counters ` Examples 1) - 2 x 3 2) - 4 x – 1 3) 3 x - 2 1 7/12/2010 ` ` Start at 0. First factor: ` ◦ If positive: Face right, take that # of steps. (Walk east) ◦ If negative: Face left, take that # of steps. (Walk west) ` Examples 1) - 2 x 4 2) - 3 x - 2 Second factor: ◦ If positive, walk forward that # of units. ◦ If negative, walk backwards that # of units. ` Product: ◦ Positive integers are modeled by miles east of 0. ◦ Negative integers are modeled by miles west of 0. ` ` Multiplying integers with the _________ sign: Multiply the absolute values. The product is _________. Multiplying integers with _________ signs: Multiply the absolute values. The product is _________. (When multiplying more than two integers, count the number of negative signs. If the number of negative signs is odd, the product will be negative. If the number of negative signs is even, the product will be positive.) ` For all integers a, b and c, b ≠ 0, a ÷ b = c iff c x b = a. Procedures for Dividing Integers ` Dividing two integers with the same sign: Divide values. Di id the h absolute b l l Quotient is positive. ` Dividing two integers with different signs: Divide the absolute values. Quotient is negative. ` ` ` ` ` ` ` Closure Property For all integers a and b, ab is a unique integer. Multiplicative Identity Property 1 is the unique integer such that for each integer a, a x 1 = 1 x a = a. Commutative Property For all integers a and b, ab = ba. Associative Property For all integers a, b and c, (ab)c = a(bc). Distributive Property For all integers a, b and c, a(b +c) = ab + ac and (b +c)a = ba + ca. Zero Property for Multiplication For all integers a, a(0) = 0(a) = 0. Properties of Integer Division For all integers a and b, a ≠ 0, a ÷ a = 1, a ÷ 1 = a, 0 ÷ a = 0, 0 and d ab ÷ a = b. Remember: We do not divide an integer by 0, because no unique quotient exists. 2 7/12/2010 ` Properties of Opposites For all integers a and b, -(-a) = a, -a(-b) = ab, (( a)b = a(( b) = -((ab b), ) and d a(-1) = (-1)a = -a. ` ` Distributive Property for Multiplication over Subtraction: For all integers a, b and c, a(b – c) = ab – ac. Distributive Property for Opposites over Addition: For all integers a, b and c, -(a + b) = -1(a + b) = -a + (-b) = -a + -b. 3