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1 Hot Shirts Using Tables, Graphs, and Equations, Part 2 Learning Goals In this lesson, you will: Use different methods to represent a problem situation. Estimate values of expressions that involve decimals. Determine an initial value when given a final result. Key Terms estimation point of intersection You might be surprised to know that the word T-shirt wasn t really used until the 1920s. And, until the 1950s, people thought of T-shirts as underwear. Popular actors like John Wayne and James Dean surprised audiences in the mid-1950s by wearing this underwear on screen! Since then, T-shirts have become one of the most popular items of clothing in the world. 2.6 Using Tables, Graphs, and Equations, Part Using Tables, Graphs, and Equations, Part 2 99

2 Problem 1 Students calculate the cost for orders of T-shirts for various given values from a competitor with a different cost value. The number of shirts that can be purchased for various amounts of money is estimated. Students will create a table and graph to represent the problem situation. Problem 1 Analyzing the Competition Previously, you explored a job at U.S. Shirts. One of U.S. Shirts competitors, Hot Shirts, advertises that it makes custom T-shirts for $5.50 each with a one-time setup fee of $ Your boss brings you the advertisement from Hot Shirts and asks you to figure out how the competition might affect business. 1. Describe the problem situation and how it will affect business in your own words. I should determine the competitor s total cost of orders. The total cost of an order is the cost of each shirt ordered plus a setup fee. The cost of one shirt is $5.50, and the setup fee is $ Grouping Have students complete Questions 1 through 3 with a partner. Then share the responses as a class. Questions 1 through 3 What information is given in this problem? What is a competitor? What does the number 5.50 represent in this situation? What does the number represent in this situation? How does the set- up fee for Hit Shirts compare to the setup fee for U.S. Shirts from the previous lesson? How does the per shirt rate for Hot Shirts compare to the per shirt rate for U.S. Shirts from the previous lesson? How can you calculate the cost of an order of shirts from Hot Shirts? 2. What is the total customer cost of an order for: a. 3 shirts from Hot Shirts? Total customer cost in dollars: 3(5.50) An order of 3 shirts will cost a customer $66.45 from Hot Shirts. b. 10 shirts from Hot Shirts? Total customer cost in dollars: 10(5.50) An order of 10 shirts will cost a customer $ from Hot Shirts. c. 50 shirts from Hot Shirts? Total customer cost in dollars: 50(5.50) An order of 50 shirts will cost a customer $ from Hot Shirts. d. 100 shirts from Hot Shirts? Total customer cost in dollars: 100(5.50) An order of 100 shirts will cost a customer $ from Hot Shirts. 100 Chapter 2 Linear Functions What is your initial prediction? Is Hot Shirts a strong competitor for US Shirts? 3. Explain how you calculated the total customer costs. I determined the total customer cost by first multiplying the number of shirts by the cost of one shirt and then adding the setup fee to the product. 100 Chapter 2 Linear Functions

3 Grouping Ask a student to read the information prior to Question 4 aloud. Discuss the worked example and complete Question 4 as a class. Have students complete Questions 5 and 6 with a partner. Then share the responses as a class. Remember, you can use estimation to determine the approximate values before you do actual calculations to get a sense of the answer. For example, you can estimate the difference of and So, you could round down to 125, and round up to 85. Then, calculate the difference of 125 and 85, to You can write this as The symbol means is approximately equal to. Discuss Phase, Question 4 What does it mean to estimate a value? What symbol can be used to show that the answer is an estimate? Have you used any other symbols to show that the answer is an estimate? Why does it make sense to estimate the answers to these questions? Questions 5 and 6 If someone could pay for 10.2 shirts, what is the approximate total cost? Can someone buy more shirts than they can pay for? When you estimate the number of shirts, do you round in the same way as when you estimate the other values? If someone could pay for shirts, approximately how many shirts could they buy? 4. Estimate the value of each expression. a < 810 b < 140 c (20.44) 45(20) (20.44) < Estimate the number of shirts that a customer can purchase from Hot Shirts for: a. $50. Number of shirts: < The customer can buy no shirts from Hot Shirts for $50. b. $60. Number of shirts: < The customer can buy approximately 2 shirts from Hot Shirts for $60. c. $220. Number of shirts: < The customer can buy approximately 28 shirts from Hot Shirts for $ Using Tables, Graphs, and Equations, Part Using Tables, Graphs, and Equations, Part 2 101

4 6. Explain how you used estimation to efficiently determine the number of shirts that can be purchased. I determined the number of shirts by first subtracting the rounded-up setup fee from the amount of money available, and then dividing the difference by the rounded-up cost for one shirt. Grouping Have students complete Questions 7 through 9 with a partner. Then share the responses as a class. Question 7 What is the smallest number of shirts possible for an order? What is the largest reasonable number of shirts for an order? What are the smallest possible and largest reasonable total costs? 7. Complete the table of values for the problem situation. Number of Shirts Ordered Total Cost (dollars) Round to the nearest penny. 102 Chapter 2 Linear Functions 102 Chapter 2 Linear Functions

5 Questions 8 and 9 What bounds are appropriate for the number of shirts? What bounds are appropriate for the total cost of the order? Explain. What interval is appropriate for the number of shirts? What interval is appropriate for the total cost? Explain. Was this problem easier or more difficult to complete than the U.S. Shirts Problem? What representations did you use to model the situation in this lesson? How did the advantages and disadvantages for each representation used in this lesson compare to the advantages and disadvantages you found in the U.S. Shirts Problem? 8. Create a graph of the data from the table on the grid shown. First, choose your bounds and intervals by completing the table shown. Remember to label your graph clearly and name your graph. Variable Quantity Lower Bound Upper Bound Interval Number of shirts Total cost Total Cost (dollars) y Hot Shirts C = 5.5s Number of Shirts 9. Define the variables and write an algebraic equation for this problem situation. The variable C represents the total cost in dollars, and s represents the number of shirts ordered. The equation is C 5 5.5s x How did you define the variables in the U.S. Shirt problem? 2.6 Using Tables, Graphs, and Equations, Part Using Tables, Graphs, and Equations, Part 2 103

6 Problem 2 Students compare the cost for orders of T-shirts from U.S. Shirts and from Hot Shirts. They will graph the total cost of an order from both U.S. Shirts and Hot Shirts on the same grid. Finally, students write a report that compares and analyzes the two different shirt companies. Grouping Have students complete Questions 1 through 9 with a partner. Then share the responses as a class. Problem 2 Which Is the Better Buy? You have explored the costs of ordering T-shirts from two companies, U.S. Shirts and Hot Shirts. Your boss asked you to determine which company has the better price for T-shirts in different situations. 1. Would you recommend U.S. Shirts or Hot Shirts as the better buy for an order of five or fewer T-shirts? What would each company charge for exactly five shirts? Describe how you calculated your answer. Total cost from U.S. Shirts in dollars: 5(8) Total cost from Hot Shirts in dollars: 5(5.50) U.S. Shirts would charge $55 for five shirts, and Hot Shirts would charge $77.45 for five shirts. Because of the setup fee, U.S. Shirts gives a better price for five shirts or fewer. To calculate the total cost of ordering, I multiplied the number of shirts by the cost of one shirt, and then I added the setup fee to the product. Questions 1 through 3 What information is given in this problem? What is being asked in this problem situation? What was the equation for C, the total cost of an order of s shirts from U.S. Shirts? What was the equation for C, the total cost of an order of s shorts from Hot Shirts? How are the equations different? How are the equations similar? 2. For an order of 18 shirts, which company s price is the better buy? How much better is the price? Explain your reasoning. Total cost from U.S. Shirts in dollars: 18(8) Total cost from Hot Shirts in dollars: 18(5.50) Difference in cost in dollars: For an order of 18 shirts, the Hot Shirts price is the better buy by $ To determine the better buy, I first calculated the total cost of ordering from each company. Then, I subtracted the total costs to calculate the difference between them. 104 Chapter 2 Linear Functions 104 Chapter 2 Linear Functions

7 Question 4 What two variable quantities are important from the U.S. Shirts Problem? What two variable quantities are important from the Hot Shirts Problem? What did you use as the label for the horizontal axis on your graph for the U.S. Shirts Problem? What did you use as the label for the horizontal axis on your graph for the Hot Shirts Problem? What did you use as the label for the vertical axis on your graph for the U.S. Shirts Problem? What did you use as the label for the vertical axis on your graph for the Hot Shirts Problem? What is the smallest possible number of shirts that each company can sell? What is the largest reasonable size of order for each company? What bounds should you use for your combined graph for the number of shirts? What is the smallest possible total cost and largest reasonable total cost of an order from each of the companies? What bounds should you use for your combined graph for the total cost? How can you graph two different lines on the same graph? What do you think the graph will look like? Make sure you label each graph. 3. For an order of 80 shirts, which company s price is better? How much better is the price? Explain your reasoning. Total cost from U.S. Shirts in dollars: 80(8) Total cost from Hot Shirts in dollars: 80(5.50) Difference in cost in dollars: For an order of 80 shirts, the Hot Shirts price is the better buy by $ I first calculated the total cost of ordering from each company. Then, I subtracted the total costs to calculate the difference between them. 4. Create the graphs for the total cost for U.S. Shirts and Hot Shirts on the grid shown. First, determine the bounds and intervals for the grid by completing the table shown. Variable Quantity Lower Bound Upper Bound Interval Number of shirts Total cost Total Cost (in dollars) y Comparing U.S. Shirts and Hot Shirts C = 8s C = 5.5s x Number of Shirts 2.6 Using Tables, Graphs, and Equations, Part Using Tables, Graphs, and Equations, Part 2 105

8 5. Estimate the number of T-shirts for which the total costs are the same. Explain how you determined the number of T-shirts. The total costs are the same when about 14 shirts are ordered. The total costs are the same where the graphs cross each other, so I started at this point and moved down and read the number of shirts for this total cost. 6. For how many T-shirts is it more expensive to order from U.S. Shirts? U.S. Shirts is more expensive to order from when the order is more than 14 T-shirts. 7. For how many T-shirts is it more expensive to order from Hot Shirts? Hot Shirts is more expensive to order from when the order is fewer than 14 T-shirts. 8. Look at your graph. Describe the graphs of the lines in your own words. The graphs are straight lines that start at the bottom left of the graph and move to the upper right of the graph. The graph for U.S. Shirts is steeper than the graph for Hot Shirts. The graphs cross one another at about (14, 127). Notice that the graphs intersect at about (14, 127). This point of intersection indicates where the total cost for each company is the same. So, when U.S. Shirts sells 14 shirts, the total cost is $127, and when Hot Shirts sells 14 shirts, the total cost is $ Write a response to your boss that compares the costs of ordering from each company. Try to answer your boss s question, Will Hot Shirts prices affect the business at U.S. Shirts? Students should write a report that includes the following key points. Hot Shirts will affect the business at U.S. Shirts for those customers who order large numbers of T-shirts. The main customers for U.S. Shirts should continue to be customers who order small numbers of T-shirts. Be prepared to share your solutions and methods. 106 Chapter 2 Linear Functions 106 Chapter 2 Linear Functions

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