© Mathalicious 2013 FLICK$ How can you decide which movie rental service to use? Between theaters, DVDs, and internet streaming, accessing your favorite movie has never been easier. With so much choice, though, finding your best option can be tricky. When you want to watch a movie, how can you figure out which service offers the most value? In this lesson, students will write and graph systems of linear equations to determine which of three movie rental services they would use. They will also explore how the value of the service varies with the number of movies rented, and will solve their systems algebraically to be sure their graphs are correct. Students Will… •
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Write equations for the cost of three rental services in terms of the number of movies rented Graph their equations to see which service is the least expensive Consider related costs to watching movies (e.g. buying a DVD player), and adjust their equations and graphs accordingly Solve systems of equations graphically and algebraically Discuss other features to consider besides cost, and how those features might affect the choice of service Common Core Standards Grade 8 EE.7, EE.8, F.3, F.4 Materials •
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Student Handout Presentation LCD projector & speakers Approximate Time 85-­‐95 minutes Before Beginning… This lesson can be used to introduce systems of linear equations and how to solve them. Because of this, some prior familiarity with writing, graphing, and evaluating linear equations will be helpful. 2 Plot S ummary & T eaching T ips Preview 5 min. To begin, students watch a video from the satirical Onion News Network about a Blockbuster Video historical tour – the underlying assumption being that brick-­‐and-­‐mortar video rental stores are a thing of the past. After the video, ask your students whether any of them ever rent movies from physical stores. If not, where do they go to rent movies? What do they like about the movie rental services their families use, and what don’t they like as much? Act One 40-­‐45 min. After learning about the renting habits of their classmates, students dive into Act One by considering three rental services: Redbox, Apple TV, and Netflix. Given the pricing information for which service, they will discuss which one seems like the best deal, and how the best deal varies with the number of movies rented. They’ll then write down cost equations and graph them, and will check that their graphs agree with what their earlier conclusions. Act Two 40-­‐45 min. In Act One students determined which service is the best deal for renting movies. But some costs, like the price of the Apple TV device, were ignored. In Act Two students try to account for these additional costs, rewriting their equations and solving them in order to come up with a better idea of when each service is the best. Finally, we’ve focused on cost when choosing a rental service, but there are other factors (like convenience, movie selection, etc.) to think about as well. The lesson ends with a discussion about what else we should consider, and how those other factors might influence our decision about which service to use. 3 Student H andout & D iscussion Redbox, Apple TV and Netflix are three of the most popular movie rental services in the United States. If you only cared about price, how would you decide which service to use? (Please be as specific as possible.) Redbox (DVD) Apple TV (streaming) Netflix (streaming) $1.20/movie $5/movie $9/month, unlimited movies Solution(s) Act One 1 Since Apple TV always costs more than Redbox ($3.80 more per movie), if I only cared about cost I’d never use Apple TV. Between Redbox and Netflix, Redbox will be cheaper if I only watch a few movies, while Netflix will be cheaper if I watch a lot of them. More specifically, if I watch 7 movies in a month, Redbox costs $8.40 < $9, but if I watch 8, Redbox costs $9.60 > 9. So if I watch seven movies or less, I should use Redbox; otherwise, Netflix is the way to go. Teaching Notes There are a number of ways students might approach this; some might use variables, while others may use guess and check to compare Redbox to Netflix (students should discover fairly early on that it doesn’t make sense to go with Apple TV based on cost alone). Encourage students to use whatever approach they’re most comfortable with; we’ll take a more focused look at these pricing schemes in the next question. Follow-­‐Up Questions •
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Can you think of any other ways to answer this question? How do you think Netflix can afford to charge a flat fee for unlimited movies? For each plan, write an equation to calculate the cost of renting x movies and graph these equations below. How could you use the graphs to determine which service to use, and is this what you expected from before? Solution(s) Act One 2 Redbox (DVD) Apple TV (streaming) Netflix (streaming) $1.20/movie $5/movie $9/month, unlimited movies C = 1.2x C = 5x C = 9 4 $20 $18 $16 $14 $12 $10 $8 $6 $4 $2 $0 0 1 2 3 4 5 6 7 8 9 10 movies Teaching Notes Now students get some practice writing and graphing linear equations. If students have trouble coming up with equations for Redbox or Apple TV, you can prompt them by asking about renting a small number of movies; for example, “How much does it cost to rent one movie from Redbox? Two? Three? Okay, now how about in general?” Students may also struggle with the fact that the Netflix equation doesn’t involve x at all. Looking at a small number of movies can again be useful in this case; the difference, of course, is that this time is that the amount you pay won’t change as you increase the number of movies watched. Follow-­‐Up Questions •
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What do the line intersections on your graph represent? Are there any other costs we’ve ignored? In addition to the advertised prices, let’s consider other practical costs. For example, each service requires special equipment. Also, Redbox’s price is $1.20 per movie per day, and Redbox reports that on average, customers keep a movie for two days. Based on this, write a new equation for each rental service to calculate the total cost to watch movies over the course of the first year. We’ll assume you only rent one movie at a time, and that you already have a TV and internet service. Act Two 3 Solution(s) Redbox (DVD) Apple TV (streaming) Netflix (streaming) $1.20/movie per day Gas: $0.50 per movie DVD Player: $40 $5/movie AppleTV: $100 $9/month, unlimited movies Wi-­‐Fi Adaptor for TV: $30 C = 2.9x + 40 C = 5x + 100 C = 138 5 Teaching Notes The process here is similar to the previous question, although now there are some slight modifications. As before, looking at a table with small values will be helpful in coming up with the general forms of the equations. Be sure to emphasize that we are now looking over an entire year rather than just a month. The reason we do this is because otherwise Netflix is clearly the best deal, no matter how many movies we rent! Follow-­‐Up Questions •
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What is the slope of each line, and what does each slope represent? What about the y-­‐intercept? How would your equations change if you looked over two years instead of one? Use the graph below to sketch the total cost for each service, and estimate when the different plans cost the same. Then find the exact number of movies where the plans cost the same. Solution(s) $200 Redbox vs. Netflix $180 $160 2.9x = 98 x = 98 ÷ 2.9 ≈ 33.8 $140 $120 $100 Apple TV vs. Netflix $80 5x + 100 = 138 5x = 38 x = 38 ÷ 5 = 7.6 $60 $40 Act Two 4 2.9 + 40 = 138 $20 Redbox vs. Apple TV $0 0 10 20 30 40 50 60 movies 2.9x + 40 = 5x + 100 2.1x = -­‐60 x = -­‐60 ÷ 2.1 ≈ -­‐28.6 Teaching Notes If you’re using this lesson as an introduction to solving systems of equations, one approach you may want to consider is the “Downside” method. It works like this: have students choose a pair of rental plans and articulate the downside of each one. In order for the plans to be equivalent (from a cost standpoint), the downsides of each plan have to balance out. For example: Downside of Redbox $2.90 more per movie Downside of Netflix $98 more initial cost # of movies for plans to be equal 2.90x = 98 x = 98 ÷ 2.90 ≈ 33.8. The Downside approach works just as well for comparing Apple TV to Netflix. Notice, though, that when you try 6 to compare Redbox to Apple TV, Redbox has no downside! Both the initial cost and the cost per movie are higher for Apple TV. In other words, just as in Act One, we see that Apple TV is never the most economical option. If your students already have some experience with solving systems, they may try to find the intersection of the Redbox and Apple TV lines algebraically; if so, they’ll come up with a negative answer. This is a good opportunity to talk about whether or not a negative solution makes sense within the context of the problem. Similarly, you may want to discuss the domain of these equations with your students: since it doesn’t make sense to rent a negative number of movies, these linear equations are really only defined over the whole numbers. Follow-­‐Up Questions •
What if you kept your Redbox movies for a week? Two weeks? In reality, people don’t just choose movie rental services based on cost. Why might someone choose a more expensive option, and which movie service would you choose? Explain. Solution(s) Answers will vary. Act Two 5 Teaching Notes Some things students may want to consider: convenience (you don’t have to leave your house to use Apple TV or Netflix), movie selection (Netflix has a flat fee for streaming, but fairly limited selection compared to the other services), and cost per day (Redbox is the only service that charges by the day – with Apple TV rentals, you have the movie for 30 days, though you only have 24 hours to finish the movie once you start it). Can your students come up with other things to consider? Follow-­‐Up Questions •
Can you think of any other movie rental services? How might they compare to these three?