3-4 Multiply Integers. Multiply. 1. 8( 12) SOLUTION: The integers have different signs. The product is negative. So, 8( 12) = 96.

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1 Multiply. 1. 8( 12) The integers have different signs. The product is negative. So, 8( 12) = ( 4) The integers have the same sign. The product is positive. So, 15( 4) = ( 6) ( 5) ( 2)( 8) 64 esolutions Manual - Powered by Cognero Page 1

2 6. 3( 2)(1) 6 Write a multiplication expression to represent the situation. Then find the product and explain its meaning. 7. Ethan burns 650 Calories when he runs for 1 hour. Suppose he runs 5 hours in one week. The expression 5( 650) represents the number of calories Ethan burns in 5 hours when he burns 650 Calories in 1 hour. 5( 650) = 3,250 So, Ethan burns 3,250 Calories each week. 5( 650) = 3,250; Ethan burns 3,250 Calories each week. 8. Wave erosion causes a certain coastline to recede at a rate of 3 centimeters each year. This occurs uninterrupted for a period of 8 years. The expression 8( 3) represents the recession of the coastline after 8 years if the coastline recedes 3 centimeters each year. 8( 3) = 24 So, the coastline s new position is 24 centimeters farther inland than it was 8 years ago. 8( 3) = 24; The coastline s new position is 24 centimeters farther inland than it was 8 years ago. esolutions Manual - Powered by Cognero Page 2

3 9. Multiple Representations Refer to the graphic novel frame below and on page 189 in the book. How many black T-shirts would Hannah and Dario need to sell to make up the loss in profit? The cost and selling price for selling 1 black T-shirt and giving 1 free black T-shirt to the mascot can be represented by the expression 1( 15) + ( 15) + 1(18). Simplify the expression. 1( 15) + ( 15) + 1(18) = 12 This means that the loss acquired from selling 1 shirt and giving away 1 free shirt is $12. Make a table to show the profit or loss when selling a certain number of black T-shirts and giving 1 black T-shirt to the mascot for free. Number of T- Profit or Loss shirts Sold 1 1( 15) + ( 15) + 1(18) = ( 15) + ( 15) + 2(18) = 9 3 3( 15) + ( 15) + 3(18) = 6 4 4( 15) + ( 15) + 4(18) = 3 5 5( 15) + ( 15) + 5(18) = 0 They will break even when 5 shirts are sold. So, Hannah and Dario need to sell 5 black T-shirts to make up the loss in profit when giving 1 black T-shirt to the mascot. 5 black T-shirts esolutions Manual - Powered by Cognero Page 3

4 10. MULTIPLE REPRESENTATIONS When a movie is rented it has a due date. If the movie is not returned on time, a late fee is assessed. Kaitlyn is charged $5 each day for a movie that is 4 days late. a. WORDS Explain why 4 ( 5) = 20 describes the situation. b. ALGEBRA Write an expression to represent the fee when the movie is x days late. a. Because the amount of money Kaitlyn had decreased by $5 each day for 4 days, 4 ( 5) = 20 describes the situation. b. Let x represent the number of days the movie is late. The expression 5x shows that the late fee is a deduction of $5 per day. a. $5 was deducted 4 times, so her balance decreased by $20. b. 5x 11. Identify Repeated Reasoning When you multiply two positive integers, the product is a positive integer. Complete the graphic organizer to help you remember the other rules for multiplying integers. Describe any patterns in the products. Sample answer: When you multiply a negative and a positive integer, the product is negative. When you multiply two negative integers the product is positive. Fill in the graphic organizer with the appropriate sign. Sample answer: When you multiply a negative and a positive integer, the product is negative. When you multiply two negative integers the product is positive. esolutions Manual - Powered by Cognero Page 4

5 12. Model with Mathematics Write a multiplication sentence with a product of 18. Student work should include a multiplication sentence with at least two factors resulting in a solution of 18. Sample answer: 6 3 = 18 Sample answer: 6 3 = Justify Conclusions Explain how to evaluate ( 9)( 6)(15)( 7 + 7) as simply as possible. Since = 0, and any number times 0 is 0, the value of the expression is 0. Sample answer: Evaluate first. Since = 0, and any number times 0 is 0, the value of the expression is Find the values of a, b, and c that make each statement true. If no values exist, write not possible. a. and b. and c. and d. and e. and a. Choose any number for a and b so that a < b (example: a = 0, b = 1). If c = 2, then and the inequality is true. b. Choose any number for a and b so that a < b (example: a = 0, b = 1). If c = 2, then and the inequality is not true. Check to see what happens if c is negative. If c = -2, then and the inequality is still not true. So, this is not possible. c. Choose any number for a and b so that a < b (example: a = 2, b = 3). If c = 4, then esolutions Manual - Powered by Cognero Page 5

6 and the inequality is true. d. Choose any number for a and b so that a < b (example: a = -1, b = 1). If c = -2, then and the inequality is true. e. Choose any number for a and b so that a < b (example: a = -4, b = -2). If c = 0, then and the inequality is true. a. Sample answer: a = 0, b = 1, c = 2 b. Not possible c. Sample answer: a = 2, b = 3, c = 4 d. Sample answer: a = -1, b = 1, c = -2 e. Sample answer: a = -4, b = -2, c = 0 esolutions Manual - Powered by Cognero Page 6

7 15. Reason Inductively The product of two integers is -21. The difference between the integers is -10. The sum of the two integers is 4. What are the two integers? Since the product of the two integers is -21, the possible products could be -1 x 21, 1 x -21, 3 x -7, or -3 x 7. From there, you need to check to see which one has a difference of -10 and a sum of and 21 Neither have a difference of -10, so they cannot be -1 and and -21 Neither have a difference of -10, so they cannot be 1 and and -7 Since -7-3 is -10, you need to check the addition. Since the addition does not work, it cannot be 3 and and 7 Since they work with subtraction, check the addition. Since they also work with addition, the two integers must be -3 and and 7 Multiply (11) The integers have different signs. The product is negative. So, 7(11) = esolutions Manual - Powered by Cognero Page 7

8 17. 20( 8) The integers have the same sign. The product is positive. So, 20( 8) = ( 2) The integers have different signs. The product is negative. So, 25( 2) = ( 4) ( 9) ( 1)( 5) 45 esolutions Manual - Powered by Cognero Page 8

9 Write a multiplication expression to represent the situation. Then find the product and explain its meaning. 22. The average person loses 50 to 80 hairs per day to make way for new growth. Suppose you lose 65 hairs per day for 15 days without growing any. The expression 15( 65) represents the number of hairs a person loses when they lose 65 hairs a day for 15 days. 15( 65) = 975 So, the number of hairs on a person's head is 975 fewer than it was 15 days ago. 15(-65); -975; The number of hairs on a person's head is 975 fewer than it was 15 days ago. 23. Financial Literacy Lily has a $100 gift card to her favorite pastry shop. She spends $4 a day at the shop for the next 12 days. The expression 12( 4) represents Lily spending $4 a day for 12 days. 12( 4) = 48 So, Lily s gift card has $48 less than its starting amount. 12(-4); -48; Lily s gift card has $48 less than its starting amount. Copy and Solve Evaluate the expression if a = 6, b = 4, c = 3, and d = c = Replace c with esolutions Manual - Powered by Cognero Page 9

10 25. Replace b with a = Replace a with bc = Replace b with 4 and c with esolutions Manual - Powered by Cognero Page 10

11 28. abc = Replace a with 6, b with 4 and c with Replace a with 6, b with 4 and c with a 2 Replace a with esolutions Manual - Powered by Cognero Page 11

12 31. cd 2 Replace c with 3 and d with a + b Replace a with 6 and b with Find the Error Jamar is finding (-2)(-3)(-4). Find his mistake and correct it. Explain your answer. Sample answer: The answer should be -24. A negative multiplied by a negative will be positive. Then, if it is multiplied by a negative it will be negative. esolutions Manual - Powered by Cognero Page 12

13 34. J.J. needed to withdraw money from his savings account to buy tickets to the zoo for himself and 7 of his friends. Each ticket costs $5. What multiplication expression represents this situation? Use the symbols to model the situation on the number line. Write an integer to represent the total withdrawal. Since the problem says that that J.J. and his 7 friends are going to the zoo, there are a total of 8 people going. Also, since each ticket costs $5, it can be represented by -5. The multiplication expression is To model the situation on the number line, use 8 curved arrows and dots moving to the left. The integer to represent the total withdrawal is -40. $ Morgan drove from Los Angeles, at an elevation of 330 feet, to Death Valley, at an elevation of -282 feet. What is the difference in elevation between Los Angeles and Death Valley? To find the difference, you need to subtract the lower elevation from the higher elevation So, the difference in elevation is 612 feet. 612 feet esolutions Manual - Powered by Cognero Page 13

14 Fill in each with < or > to make a true sentence. 0 is further to the right than 1 on a number line. So, 0 is greater than 1. > is further to the left than 9 on a number line. So, 9 is less than 9. < is further to the left than 48 on a number line. So, 84 is less than 48. < is further to the right than 27 on a number line. So, 32 is greater than 27. > esolutions Manual - Powered by Cognero Page 14

15 40. Laura s allowance balances over the last three months are shown in the table. Positive values indicate the number of dollars she had left over, and negative values indicate the number of dollars she overspent. Order the allowance balances from least to greatest. Order the integers on a number line. -$10, $2, $5 41. Graph 1, 4, 3, 2, 0, and 2 on the number line below. Place a point on the number line to represent each integer. esolutions Manual - Powered by Cognero Page 15