© Mathalicious 2014 lesson guide TRICKS OF THE TRAY’D What’s the best way to design a food tray? It’s not always easy to get students to make healthy choices in the lunchroom, but one possible solution might be right under their noses: the cafeteria tray. By giving more visual real estate – but not more actual space – to the most enticing items, students might feel like they’re getting more of the foods they love, without going overboard on calories. In this lesson, students calculate volumes of rectangular prisms and use that information to design an appealing and well-‐balanced tray. Primary Objectives Find the volume of a rectangular prism with whole-‐number or fractional edge lengths Given the dimensions of its base, calculate the height/depth necessary to form a rectangular prism with a particular volume Design a lunch tray that holds the appropriate volumes of different food groups, while maximizing the visibility of the most desirable foods Discuss the effects of food presentation and placement on people’s food choices •
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Content Standards (CCSS) Grade 6 G.2 Mathematical Practices (CCMP) Materials MP.3, MP.7 •
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Student handout LCD projector Computer speakers Before Beginning… Students should be able to calculate the area of a rectangle with known dimensions. The lesson focuses on calculating the volume of a rectangular prism, but that’s not a prerequisite. As long as students are able to calculate rectangular area, they can make arguments about packing unit cubes in order to develop a general rule for calculating the volume of a rectangular prism. Lesson Guide: TRICKS OF THE TRAY’D 2 Preview & Guiding Questions Begin by showing the video clip about Mississippi public schools’ attempts to create a healthier student population. Students learn that part of the state’s plan includes replacing some of the unhealthy, high-‐calorie lunchroom staples with foods like fresh fruits and vegetables. There are some obstacles, though. For one, people have a strong attachment to the foods they like, and it’s not always easy to overcome that bond. For another thing, unhealthy food is readily available, and fresh options are not. At one point in the video, Dr. Al Rausa of the Mississippi State Department of Health mentions that the average person has 3,500 calories available to him or her on a given day. That’s almost triple what many people need. This lesson will focus on how the way in which food is positioned and presented affects the kinds of choices people make, so it’s important that the preview discussion include something about why people tend to make the choices they do. For instance, unhealthy food options are generally extremely convenient, and the average school cafeteria is no different. •
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What kinds of things is Mississippi trying to do in order to encourage students to be healthier? How does this situation compare to your own experience? What do you think Dr. Rausa means when he says the average person in America “has 3,500 calories available to them every day?” How do you think convenience affects people’s food decisions? Can you give any examples from your own life? Do you think that the way in which food is presented affects the kinds of choices you make? Act One In Act One, students begin by examining a standard cafeteria tray that has compartments dedicated to different food groups. They calculate the volume of each compartment and use that information as a benchmark for a balanced school lunch. Next, students are presented an alternative to the standard tray. They discover that it holds roughly the same amount of each food group, even though it looks very different to someone sitting over it. For instance, the space for dessert looks twice as big, even though it holds the same volume. Students discuss which version of the tray they would prefer and why that might be. Act Two In Act Two, students are introduced to the idea that how food is presented can affect how people perceive their meal. It can even affect the kinds of eating choices that they make. Based on this line of reasoning, students design their own tray layouts and try to make them seem as enticing as possible while still offering a balanced meal. Finally, students watch a video about some researchers who rearranged a cafeteria in order to persuade students to make healthier decisions, and they discuss what kinds of changes they would (or wouldn’t) like to see in their own school. Lesson Guide: TRICKS OF THE TRAY’D 3 Act One: You Got Served 1 Nutritionists recommend that everyone eat a balanced diet, and school lunch trays are designed to help students do this. The standard tray below has the recommended amount of room for the following food groups: protein (e.g. meat, beans), starch (e.g. rice, potatoes), fruit, vegetable, and dessert. Top View Side View Use the diagram to calculate how many cubic centimeters (cm3) of each food group the tray can hold. Food Group Protein Starch Vegetable Fruit Dessert Length 16.5 cm 8 cm 8 cm 8 cm 8 cm Width 10.5 cm 11.5 cm 9 cm 9 cm 8 cm Depth 2 cm 2 cm 2 cm 2 cm 2 cm Volume 346.5 cm3 184 cm3 144 cm3 144 cm3 128 cm3 Explanation & Guiding Questions If students are unsure how to find the volume of a rectangular prism, encourage them to think about what a compartment looks like in two dimensions from the top. Each one is just a rectangle. Begin by asking students to calculate the area of one of the rectangles. Then, students can imagine packing the rectangle with unit cubes. Since the number of cubic units that can fit into a rectangle is the same as its area, students have already calculated the volume of a 1-‐cm deep compartment! Then, since each compartment is actually 2 cm deep, have students think about what would happen to its volume if they were to pack in a second layer of unit cubes. The compartment would contain two identical layers of cubes, so its volume would simply be two times its area. In other words, the volume of a rectangular prism is just the area of its base times its depth. Students can use that relationship to calculate the remaining compartments’ volumes. •
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What kind of three-‐dimensional shape is each compartment? How can you find the volume of a rectangular prism? From above, what two-‐dimensional shape is each compartment? How could you find the area of that rectangle? If you were to pack the rectangle with 1-‐cm3 cubes, how many would fit? How much volume would that represent? What if you packed it with two layers of cubes? What would the volume be then? Can you come up with a general rule for finding the volume of a rectangular prism? Deeper Understanding •
What’s the volume of each compartment in cups? (One cup is about 236.6 cm3, so the number of cups is equal to the volume in cm3 divided by 236.6.) Lesson Guide: TRICKS OF THE TRAY’D 4 2 Imagine your school is thinking of buying new lunch trays, and is deciding between the standard tray and the one Side View 2 Top View Side View 1 below. Which would you rather use during lunch, and which do you think most students would prefer? Explain. Food Group Protein Starch Vegetable Fruit Dessert Length 8.5 cm 8 cm 7.5 cm 7.5 cm 16 cm Width 20 cm 11.5 cm 5.5 cm 5.5 cm 8 cm Depth 2 cm 2 cm 3.5 cm Volume 3
340 cm 3
184 cm 3.5 cm 3
144.4 cm 1 cm 3
144.4 cm 128 cm3 Answers will vary. Sample response: This second tray seems better because it gives more visual space to more appealing foods. For instance, the desert compartment occupies 16 cm × 8 cm = 128 cm2 of area, which is twice that of the standard tray. Explanation & Guiding Questions The goal of this question is to get students to consider that the way in which food is presented can affect the experience of eating it. If a school cafeteria hopes to get kids to eat a more balanced meal, then it might help to present that meal in the most appealing way possible, even if everything else about the food stays the same. Have students compare this proposed tray to the standard one from the first question, particularly the visual area each compartment occupies. Why might that matter? It may help to compare, for instance, the two trays’ dessert compartments. From the top, the dessert compartment looks much bigger than before because it has twice the base area, but it only has half the depth, so the volume is identical. In other words, it appears to the diner that he or she is getting more dessert, but is actually eating the same amount. That might be enticing for a student who (probably like most) really enjoys dessert. Some students may point out that in reality, foods probably don’t take up the exact volume provided. Though that’s true, the designers don’t have any control about what foods actually go into the tray; they can only provide the relative volume to allow for a balanced meal. •
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How is this tray different from the standard one? How is it similar? How does the visual area occupied by each compartment compare to its counterpart in the standard tray? Why might that matter to someone? What about someone who really likes dessert? Do you think how food is presented affects how much people enjoy eating it? What kinds of assumptions are we making about the compartment sizes and the foods that fill them? Does that affect your answers? Deeper Understanding •
Come up with some other dimensions that would make the starch compartment exactly the same volume as in the standard tray? (e.g., 4 cm long × 11.5 cm wide × 4 cm deep.) Lesson Guide: TRICKS OF THE TRAY’D 5 Act Two: Tray-‐ding Spaces 3 Nutritionists say that the amount of food people get is often less important than the amount of food people see. If it looks like people have more of the foods they like and less of the foods they don’t like, they’ll be happier. Using the diagram below, design a new tray that you think would encourage students to eat a balanced diet, and would make sense for your school’s cafeteria. Be sure to specify the dimensions of each compartment. Answers will vary. Sample response: For instance, if the desired dimensions for the fruit compartment are 8 cm × 6 cm, then in order for it to have the same volume as the standard tray (144 cm3), it would need to have a depth of 144 ÷ (8 × 6) ≈ 4.5 cm. The depths for the remaining compartments can be calculated the same way. 20 cm 25 cm Length Width Depth Protein 15 cm 10 cm 2.5 cm Starch 10 cm 10 cm 2 cm Veggie 8 cm 4 cm 4.5 cm Fruit 8 cm 6 cm 3 cm Dessert 12 cm 10 cm 1 cm Explanation & Guiding Questions Based on their conversation from the previous question, students can design their tray to give the largest-‐seeming and most prominent spots to the foods people want to eat most. It can be overwhelming to try and calculate all the volumes at the beginning. Encourage students to complete the layout first, and then – using their rectangle areas – solve for the required depth so the volume for each food is close to the standard tray. Of course there are some practical limitations; a vegetable compartment that’s 10 cm deep is probably not feasible. A compartment can’t have zero height. So if students run into those problems, have them adjust their rectangle dimensions slightly so the resulting depths are more reasonable. •
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Which foods are you giving the most area to? The least? How do you think your layout will encourage a balanced meal? If you know a rectangle’s area, and the volume you’d like for the compartment, how can you find what depth you’d need for it to have? What are some realistic limitations on a cafeteria tray? Lesson Guide: TRICKS OF THE TRAY’D 6 4 Watch the MTV video about Cornell professor Brian Wansick. In addition to redesigning lunch trays, what else are nutritionists doing to “trick” students into eating healthier, and do you think the changes will be effective? Answers will vary. Sample response: Nutritionists did several things. They moved healthier beverages to the front of the refrigerator and healthier food options to the front of the cafeteria line. They also put side items like fruits in front of less healthy options like cookies, and placed them in more attractive containers. In some cases, students would actually have to ask a food service worker to get the unhealthy snacks for them, making it even more difficult. It’s hard to tell definitively whether these strategies can help students eat better in the long term, but the video suggests that students in the study ate fewer calories. Explanation & Guiding Questions Leading up to this question, some students might be skeptical that simply changing a tray’s appearance can have a meaningful impact on someone’s impression of the meal. But the video demonstrates how psychology seems to have an important effect on the food choices we make. It details several ways that the researchers manipulated the cafeteria environment to try and get students to choose better foods. Some of the techniques just make it physically easier for students to choose healthier items, but others appeal directly to their brains (for instance putting the oranges in a more attractive bowl). Have students discuss how they would feel about these sorts of changes being instituted in their own cafeteria. Do they think it would make a difference? Encourage them to think about why they make the choices they do. Sometimes, for instance, it’s just really easy and convenient to grab a bag of chips. Finally, invite them to brainstorm some other ways that the cafeteria could make it easier or more enticing for students to choose healthier lunchtime options…or make it harder for them to choose less healthy ones! •
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What sorts of things did the researchers do in order to make it easier to choose healthy foods? Did their changes seem to affect students’ eating behavior? How do you know? How would you feel if the school did things like this in your cafeteria? Do you think you would eat less unhealthy food if it were less convenient? Would you be willing to go out of your way to ask a cafeteria worker for a bag of chips? Can you think of any other things the cafeteria could do to encourage healthy eating? Deeper Understanding •
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The researchers said that fruit consumption increased by 102%. Do you think that’s significant? (Answers will vary, but that seems like a large increase.) How many students do you think were eating fruit before? Would it be more impressive if there were lots of students eating fruit before, or only a few? (The larger the number of students who were eating fruit before, the more impressive any percent increase would be.)