2011 carnegie learning 5.1 multiplying and dividing integers • 253 learning goals in this lesson, you will: multiply integers. divide integers. equal groups multiplying and dividing integers p ick any positive integer. if the integer is even, divide it by 2. if it is odd, multiply it by 3 and then add 1. repeat this process with your result. no matter what number you start with, eventually you will have a result of 1. this is known as the collatz conjecture—a conjecture in mathematics that no one has yet proven or disproven. how do you think it works?254 • chapter 5 multiplication and division with rational numbers 2011 carnegie learning problem 1 multiply integers when you multiply integers, you can think of multiplication as repeated addition. consider the expression 3 3 (24). as repeated addition, it means (24) 1 (24) 1 (24) 5 212. you can think of 3 3 (24) as three groups of (24). – – – – – – – – – – – – – – – – – – – – – – – – 5 –15 –10 –12 (–4) (–4) (–4) –5 0 5 10 152011 carnegie learning 5.1 multiplying and dividing integers • 255 here is another example: 4 3 (23). you can think of this as four sets of (23), or (23) 1 (23) 1 (23) 1 (23) 5 212. – – – – – – – – – – – – – – – – – – – – – – – – 5 –15 –10 –12 (–3) (–3) (–3) (–3) –5 0 5 10 15 and here is a third example: (23) 3 (24). you know that 3 3 (24) means “three groups of (24)” and that 23 means “the opposite of 3.” so, (23) 3 (24) means “the opposite of 3 groups of (24).” opposite of (–4) opposite of (–4) opposite of (–4) 5 –15 –10 (4) (4) (4) –5 0 5 10 12 15256 • chapter 5 multiplication and division with rational numbers 2011 carnegie learning 1. draw either a number line representation or a two-color counter model to determine each product. describe the expression in words. a. 2 3 3 b. 2 3 (23) c. (22) 3 3 d. (22) 3 (23) use the examples if you need help.2011 carnegie learning 5.1 multiplying and dividing integers • 257 2. complete the table. expression description addition sentence product 3 3 5 three groups of 5 5 1 5 1 5 5 15 15 (23) 3 5 3 3 (25) (23) 3 (25) 3. analyze each number sentence. 4 3 5 5 20 4 3 4 5 16 4 3 3 5 12 4 3 2 5 88 4 3 1 5 48 4 3 0 5 08 what pattern do you notice in the products as the numbers multiplied by 4 decrease? 4. determine each product. describe the pattern. a. 4 3 (21) 5 b. 4 3 (22) 5 c. 4 3 (23) 5258 • chapter 5 multiplication and division with rational numbers 2011 carnegie learning 5. write the next three number sentences that extend this pattern. 25 3 5 5 225 25 3 4 5 220 25 3 3 5 215 25 3 2 5 210 25 3 1 5 25 25 3 0 5 0 6. how do these products change as the numbers multiplied by 25 decrease? 7. determine each product. a. 25 3 (21) 5 b. 25 3 (22) 5 c. 25 3 (23) 5 d. 25 3 (24) 5 e. write the next three number sentences that extend this pattern. 8. what is the sign of the product of two integers when: a. they are both positive? b. they are both negative? c. one is positive and one is negative? d. one is zero? when you multiply by the opposite, you go in the opposite direction!2011 carnegie learning 5.1 multiplying and dividing integers • 259 9. if you know that the product of two integers is negative, what can you say about the two integers? give examples. 10. describe an algorithm that will help you multiply any two integers. 11. use your algorithm to simplify these expressions. a. 6 3 5 b. 28 3 7 6 3 (25) 28 3 (27) 26 3 5 8 3 (27) 26 3 (25) 8 3 7 c. 23 3 2 3 (24) 23 3 (22) 3 (24) 3 3 (22) 3 4 23 3 (22) 3 4 3 3 2 3 (24) 23 3 2 3 4 12. determine the single-digit integers that make each number sentence true. a. 3 5 242 b. 3 5 56 c. 3 (29) 5 63 d. 3 5 248260 • chapter 5 multiplication and division with rational numbers 2011 carnegie learning 13. describe the sign of each product and how you know. a. the product of three negative integers b. the product of four negative integers c. the product of seven negative integers d. the product of ten negative integers 14. what is the sign of the product of any odd number of negative integers? explain your reasoning. 15. what is the sign of the product of three positive integers and fve negative integers? explain your reasoning.2011 carnegie learning 5.1 multiplying and dividing integers • 261 problem 2 division of integers when you studied division in elementary school, you learned that multiplication and division were inverse operations. for every multiplication fact, you can write a corresponding division fact. the example shown is a fact family for 4, 5, and 20. fact family 5 3 4 5 20 4 3 5 5 20 20 4 4 5 5 20 4 5 5 4 similarly, you can write fact families for integer multiplication and division. examples: 27 3 3 5 221 28 3 (24) 5 32 3 3 (27) 5 221 24 3 (28) 5 32 221 4 (27) 5 3 32 4 (28) 5 24 221 4 3 5 27 32 4 (24) 5 28 1. what pattern(s) do you notice in each fact family? 2. write a fact family for 26, 8, and 248.2011 carnegie learning 262 • chapter 5 multiplication and division with rational numbers 3. fill in the unknown numbers to make each number sentence true. a. 56 4 (28) 5 b. 28 4 (24) 5 c. 263 4 5 27 d. 24 4 5 28 e. 4 (28) 5 24 f. 2105 4 5 25 g. 4 (28) 5 0 h. 226 4 5 21 talk the talk 1. what is the sign of the quotient of two integers when a. both integers are positive? b. one integer is positive and one integer is negative? c. both integers are negative? d. the dividend is zero? 2. how do the answers to question 1 compare to the answers to the same questions about the multiplication of two integers? explain your reasoning. be prepared to share your solutions and methods.