? Name 8.3 Essential Question Multiplication and Division Equations How can you model and solve multiplication and division equations? Algebraic Reasoning 5.4.B MATHEMATICAL PROCESSES 5.1.F, 5.1.G Unlock the Problem Hands On Jake is buying a movie ticket and a box of popcorn for $12. The ticket costs 2 times as much as the popcorn. How much does the popcorn cost? How much does the ticket cost? How does knowing that a ticket costs 2 times as much as the popcorn help you choose an operation to write an expression? The cost of the ticket is 2 p, where p is the price of the popcorn. STEP 1 Use a related equation. Complete the strip diagram to represent the popcorn and ticket cost. Popcorn Popcorn and ticket The popcorn costs p. Label the strip for the popcorn. It represents 1 unit. The cost of the ticket is _ times as much. Label the strips for the ticket. What is the total cost? _ How many units are shown in the model? _ Ticket Popcorn: 1 unit = _ Popcorn + ticket: 3 units = STEP 2 Use the strip diagram to write a multiplication equation. Cost of popcorn and ticket Total cost 3 p = _ Math Idea You can write 3 p as 3 p, 3p, 3(p), or 3 p. To find a missing factor, write a division equation that is related to the multiplication equation. Then divide. _ 3 = p _ = p So, 1 bag of popcorn costs _. The cost of 1 ticket is 2 times the cost of 1 bag of popcorn. So, 1 ticket costs _. Module 8 335
Example Use a related equation. Katrina buys a package of trading cards. She divides the cards into 4 equal piles. Each pile contains 12 cards. The equation that describes the number of cards in the package is c 4 = 12. How many cards were in the package when Katrina bought it? Write a related equation. MODEL SOLVE Use a strip diagram to represent the problem. c 12 12 12 12 So, the package Katrina bought had _ trading cards in it. Share and Show Use the strip diagrams to write an equation. Then solve. Write a related multiplication equation. Multiplication equation: c = _ Math Talk Mathematical Processes Explain how the strip diagrams for multiplication and division are related. 1. The cost of 3 pencils is 96 cents. Each pencil costs the same. What is the cost of 1 pencil, p? p p 96 Equation: _= 96 p p =_ One pencil costs _ cents. 2. A gardener is planting 6 rows of spring bulbs. Each row has 15 bulbs. How many bulbs is he planting? Let b equal the total number of bulbs. 15 15 15 15 Equation: He is planting _ bulbs. b 15 15 b = _ Problem Solving Practice: Copy and Solve Use a strip diagram or a related equation to solve. Check your solution. 3. 128 = 8 d 4. r 9 = 17 5. 6m = 78 6. 7 = b 17 7. Write Math How can you justify that j = 348 is the solution to j 12 = 29? Explain. 336
Name Problem Solving Use the table to complete 8 9. Use a strip diagram to write each equation. Then solve. 8. Multi-Step On Friday, the snack bar made $992 selling large buckets of popcorn. How many large buckets of popcorn did the snack bar sell on Friday? In your equation, let p represent the number of large buckets of popcorn sold. Snack Bar Menu Large Popcorn $8 Medium Drink $5 Fruit Snack $4 Yogurt Bars $3 Family Combo $18 9. Multi-Step On Thursday, the snack bar made total of $340 in medium drink sales and a total of $216 in yogurt bar sales. Which item did the snack bar sell more of on Thursday? Explain how you found your answer. Write Math Show Your Work Image Credits: Photodisc/Getty Images 10. Multi-Step Asher wants to buy a video game console. In order to save the money needed to buy it in 5 months, he divides the cost by 5. He finds he needs to save $37 a month. What is the total cost, c, of the game console? 11. Representations Mariah bought a bag of buttons. She divides the buttons equally among 15 containers. Each container has 10 buttons. How many buttons does Mariah have? In your equation, let b represent the number of buttons Mariah has. Module 8 Lesson 3 337
Daily Assessment Task Mathematical Processes Fill in the bubble completely to show your answer. 12. Mr. Tune took a vacation to Neptune. His vacation lasted for 144 hours. A day on Neptune lasts for 16 hours. Which equation and solution shows how many days Mr. Tune s vacation lasted? A 16d = 144; d = 9 B d = 16 144; d = 2,304 C 16 + d = 144; d = 128 D d = 16 + 144; d = 160 13. Which equation does the strip diagram represent? 12 12 12 12 12 12 a A 6a = 12 C 12a = 6 B a 6 = 12 D a + 6 = 12 14. Multi-Step Noah bought 5 CDs and a DVD for a total of $57. The DVD cost $12. The CDs were each the same price. How much did Noah pay for each CD? A $12 C $11 B $15 D $9 TEXAS Test Prep 15. Rick earned $133 for 7 hours of work. How much does Rick earn per hour? The equation 7p = 133 can be used to find p, the amount Rick earns per hour. 338 A $19 B $6 C $931 D $69 w
Homework and Practice Name Algebraic Reasoning 5.4.B MATHEMATICAL PROCESSES 5.1.F, 5.1.G 8.3 Multiplication and Division Equations Use a strip diagram or a related equation to solve. Check your solution. 1. c 5 = 13 2. 112 = 7 b 3. 4p = 68 4. 9 = d 21 5. 105 = 3 a 6. 9g = 99 7. m 10 = 16 8. 22 = n 7 9. 92 = f 4 10. h 9 = 14 11. 13 = j 8 12. 165 = 11r Problem Solving 13. Sierra arranges fabric squares to sew together to make a blanket for her baby sister. She divides the fabric squares equally among 12 rows. Each row has 8 squares. How many fabric squares does Sierra have? In your equation, let f represent the number of fabric squares Sierra has. 14. A school district purchased 90 new computers. The computers were divided equally among 6 classrooms. How many new computers are in each classroom? In your equation, let c represent the number of computers in each classroom. 15. Write a real-world problem for the equation 4 y = 60. Then solve. Module 8 Lesson 3 339
Lesson Check TEXAS Test Prep Fill in the bubble completely to show your answer. 16. Which equation does the strip diagram represent? 14 14 14 14 14 17. Which equation does the strip diagram represent? d d d d A b 5 = 14 B 14b = 5 C 14 5 = b D 5b = 14 b A d + 4 = 88 B d 4 = 88 C 4d = 88 D d = 4 88 88 18. Dori earned 95 points on her science test. She received 5 points for each question she answered correctly. How many questions did Dori answer correctly? The equation 5q = 95 can be used to find q, the number of questions answered correctly. A 100 B 90 C 19 19. The Adams family drove 270 miles to reach their vacation destination. If they drove 45 miles each hour, how many hours did they drive? Which equation can you use to represent the problem? A h = 270 45 B 270 = 45h C 45 h = 270 D h + 45 = 270 D 11 20. Multi-Step The movie theater has 176 seats. There are 9 rows of 18 seats plus an additional section of handicapped accessible seats. Write and solve an equation to find the number of handicapped accessible seats in the theater. 340 A 9 B 14 C 4 D 12 21. Multi-Step Sergei had $75 when he arrived at the mall. He spent $9 at the food court. Then he bought 3 posters that each cost the same amount. Sergei has $27 left. Write and solve an equation to find the cost of one poster. A $9 B $3 C $13 D $7