Primary Objectives. Content Standards (CCSS) Mathematical Practices (CCMP) Materials

Transcription

1 PAIR- ALYSIS How many different shoes can you design on NIKEiD? Mathalicious 2013 lesson guide NIKEiD.com allows users to customize their own pair of shoes. In this lesson, students use the fundamental counting principle to calculate how many color combinations are possible for the popular Nike Free Run running shoe. The answer? A LOT! Sounds great but is it? Students learn about the negative consequences of having too much choice, from paralysis- by- analysis to a loss of satisfaction. When we go shopping, how much is too much? Primary Objectives Calculate the number of possible color combinations for the Nike Free Run running shoe Analyze the advantages & disadvantages of having too many options, including paralysis- by- analysis Given a maximum number of possible combinations, modify the number of color options for each component Describe strategies companies use to help consumers make choices when faced with many options Content Standards (CCSS) Mathematical Practices (CCMP) Materials Grade 7 SP.8b MP.3, MP.4 Student handout LCD projector Computer speakers Colored pencils Before Beginning It will be helpful if students are already familiar with describing sample spaces using organized lists or tree diagrams, and have developed an understanding of the Fundamental Counting Principle from simpler sample spaces. This lesson is an application of the Fundamental Counting Principle with a large sample space.

2 2 Preview & Guiding Questions Students see a screenshot from the ipad version of the NIKEiD website, which shows a Nike Free Run shoe. Explain to students that NIKEiD is a service that allows users to customize their own pair of shoes, and ask whether they ve ever used a service like this. In the image, students see that there are 16 color options for the upper part of the shoe. (The last tile is black, which may be hard to see.) Of course, there are still other components to customize. Ask students to predict how many color combinations will be possible when all is said and done. If students ask how many components there are, you can refer them to the eight tabs along the right side of the image, which correspond to eight customizable components. If they ask how many color options there are for each, you might respond, I don t know, but let s not worry about that right now. Just take a guess that you think is reasonable. It will be interesting to see which student predicts the largest number, and consider pretending like it s an auction and you re the auctioneer. Can I get 100 thousand color combinations? 1 million? Can I get one million? Have you ever tried to design a shoe on NIKEiD? If so, did you end up buying it? How many color combinations do you think there are for a pair of Nike Free Run shoes? 100,000? 1 million? Act One In Act One, students watch as someone designs a pair of Nike Free Run 3.0 running shoes on NIKEiD.com. With each component, they determine how many color combinations there are in total. In the end, they find that there are more than 100 million possible color combinations for this single pair of shoes. Sounds great but is it? Act Two In Act Two, students watch a video in which Harvard psychologist and author of The Paradox of Choice Barry Schwartz explains the negative consequences of having too much choice. According to Schwartz, having too many choices makes it very difficult to choose at all ( paralysis by analysis ). Even when we do choose, though, we end up less happy with the choice we make than we would be if we d had fewer options in the first place. Students discuss their experiences with paralysis by analysis, and consider how it might affect users of NIKEiD. They then revisit the Nike Free Run and determine what they could do to limit the total number of color combinations to something more manageable. To conclude the lesson, students return to NIKEiD.com. It turns out that Nike is aware of the dangers of paralysis by analysis and has made accommodations for it. The website includes an option to start with an inspiration which allows users to select a design that s almost complete. Users can make tweaks, but most of the work has been done for them. Students discuss whether they think this is an effective solution, or whether the ability to customize at all will invariably lead to the black hole of paralysis by analysis.

3 3 Act One: My Nikes 1 Watch as someone customizes a pair of Nike Free Run 3.0 shoes on the NIKEiD website. After the first two components upper & mesh how many possible color combinations are there? Color in four of them below. Color options for the upper: 16 Color options for the mesh: 13 Total combinations (mesh & upper) = = 208 Explanation & Guiding Questions For the upper section, there are 16 color options. For the mesh section: 13 color options. If students are familiar with the fundamental counting principle namely, multiplying the color options together determining the total number of color combinations should be relatively straightforward. If students need more guidance, though, you might encourage them to start with the first upper color (yellow), and then determine how many shoes they could design with this: Upper Upper, Mesh Yellow Yellow, Yellow Yellow, Red Yellow, Light Purple Yellow, Dark Purple Yellow, Navy Yellow, Royal Yellow, Sky Blue Yellow, Aqua Yellow, Green Yellow, Lime Yellow, White Yellow, Grey Yellow, Black From this, students see that there are 13 ways to design a shoe with a yellow upper. Of course, there are also 13 ways to design a shoe with an orange upper 13 ways to design a shoe with a red upper etc. Since there are 16 upper colors, there must be = 208 ways to combine the upper and mesh colors. How many shoes can you design that have a yellow upper? What are some examples? How many shoes can you design that have an orange upper? A red upper? If there are 16 options for the upper, each of which can be combined with 13 mesh colors, how many color combinations are there in total? If you could add one more color option to either the upper or mesh, which would result in more possible color combinations? (Adding one more upper color: = 221. Adding one more mesh color: = 224. Adding an addition color to the mesh will result in more possibilities.)

4 4 2 Now watch as the person designs the rest. Below, specify which components are customizable and how many color options for each. Based on this, how many color combinations are possible for the Free Run 3.0? Mesh: 13 options Tongue id: 13 Upper: 16 options options Lace: 19 options Swoosh: 16 options Midsole: 7 options Midsole Rim: 18 options There are = 103,574,016 possible color combinations. That s over 103 million! Explanation & Guiding Questions As students watch the rest of the video, they will count the number of colors that are available for each component. To help reinforce that multiplication is the operation we use to determine the total number of color combinations, you might start by focusing on just three components, e.g. the upper, the mesh, and the Swoosh. For instance, if we start with a yellow upper and yellow mesh, there are sixteen ways to include the Swoosh (as shown below.) Upper Mesh Upper, Mesh, Swoosh Yellow Yellow Yellow, Yellow, Yellow Yellow, Yellow, Orange Yellow, Yellow, Red Yellow, Yellow, Pink Yellow, Yellow, Light Purple Yellow, Yellow, Dark Purple Yellow, Yellow, Maroon Yellow, Yellow, Navy Yellow, Yellow, Royal Yellow, Yellow, Sky Blue Yellow, Yellow, Aqua Yellow, Yellow, Green Yellow, Yellow, Lime Yellow, Yellow, White Yellow, Yellow, Grey Yellow, Yellow, Black Of course, we could do the same thing for orange mesh (16 more combinations) and red mesh (another 16) and green mesh. Since there are 13 color options for the mesh, students will see that there are = 208 ways to combine a yellow upper with the mesh (13) and the Swoosh (16). And that s limiting the upper color to yellow! And if we vary the color of the upper, too? There are 16 choices for the upper, so there must be = 3,328 ways to combine the upper, mesh, and Swoosh. Once students recognize that multiplication is the operation used to determine the number of combinations i.e. that multiplication undergirds the fundamental counting principle they apply this to the rest of the components to determine the total number of color combinations. How many ways are there to combine a yellow upper, yellow mesh, and the Swoosh? How many ways are there to combine a yellow upper, orange mesh, and the Swoosh? How many ways are there to combine a yellow upper with the mesh and Swoosh? Which operation do we use to determine the total number of possible color combinations? How many options are there for each component, and how many color combinations are there in total? Which component has the greatest impact on the total number of color combinations? (The laces.) If you could add an additional color to one of the components, which would result in the greatest number of possible color combinations? (The midsole. Adding one more color option to the midsole will result in more than 118 million combinations, whereas adding another color option to the laces will result in 109 million.)

5 5 3 Imagine it takes 30 seconds to both design a pair of shoes on NIKEiD.com and also decide whether you like them or not. If you wanted to see every possible color combination, how long would it take? 103,574, seconds = 3,107,220,480 seconds 3,107,220,480 seconds = 51,787,008 minutes = 863,116.8 hours = 35,963.2 days = 98.5 years! Explanation, Guiding Questions In the previous question, students determined that there are over 100 million possible color combinations for the Nike Free Run shoe. However, they may not realize just how big this number really is. To help contextualize it, students calculate how long it would take to design them all. Mathematically, students need to know how to convert between units, e.g. from seconds to minutes, minutes to hours, etc. How many total seconds would it take to design all 103,574,106 shoes? How many minutes is this? Hours? Days? Years? Instead of 30 seconds, what if it took a full minute to design a pair of shoes? (Then it would take almost 200 years to design them all.) What are some other questions you could come up with to contextualize how big 103 million is? (Answers may vary. Options include, If Nike had to include every possible combination in a catalogue, how many pages/how thick/how heavy would it be? If you had to try on every possible shoe, how long would it take?)

6 6 Act Two: The Paradox of Choice 4 Barry Schwartz is a psychologist at Swarthmore College and author of the book, The Paradox of Choice. Watch the clip of his TED talk. What are some downsides to having so much choice, and how might this affect NIKEiD? Sample response: Barry Schwartz says that there are two downsides to having so much choice. The first is that it makes it very hard to choose in the first place. The second is that, if you are able to choose, you end up less happy with your choice than you would have been if you d had fewer options to choose from. If this is true, this could have a big impact on NIKEiD. There are over 103 million possible ways to design a pair of Nike Free Run shoes. That s a lot. If people have a hard time deciding which color combination they like best, they might not end up buying shoes at all, and just going to Foot Locker instead. Explanation, Guiding Questions As described in the sample response above, Barry Schwartz highlights two problems associated with having so much choice: the difficulty in making a choice at all ( paralysis- by- analysis ), and the angst that results from wondering whether you made the right choice. There are more than 103 million possible designs for the Nike Free Run. This is a lot, and students might reason that many visitors to NIKEiD become so overwhelmed that they end up going elsewhere like Foot Locker or Zappos. Not only would this be inconvenient to the consumer, it would also bad for Nike. (Nike probably makes less money on a pair of shoes sold at a shoe store than on a pair of shoes sold on its own website.) Even if people do buy shoes from NIKEiD, though, they may end up less happy with their choices than they would have been otherwise. In this case, they might be less likely to buy another pair of Nikes later. Have you ever experienced paralysis- by- analysis? If so, how did it end? If people experience paralysis by- analysis on NIKEiD, whom might this be bad for? If people experience paralysis by- analysis on NIKEiD, whom might this be good for? How many choices do you think is too many? A hundred? A thousand? (Answers may vary.) Barry Schwartz says that the secret to happiness is low expectations. What do you think he means by this, and do you agree? (Answers may vary. In the video, Schwartz says that the existence of so many options causes consumers to think that there must be a perfect answer out there a perfect pair of shoes, a perfect pair of jeans if they just look hard enough. This can drive people crazy, and instead he urges people to temper their expectations. After all, it s pretty hard to be disappointed when Ford only offers the Model T in one color: black!)

7 7 5 Imagine someone from NIKEiD said, Visitors to the site are getting overwhelmed. We need to limit the total number of color combinations to as close to 10,000 as possible without going over. Each component still needs at least two color options, but can t have any more than it currently has. How could you accomplish this? Answers may vary. This sample response yields exactly 10,000 outcomes: Upper: 5 options Mesh: 4 options Tongue id: 5 options Lace: 5 options Swoosh: 5 options Midsole Rim: 2 options Midsole: 2 options Explanation, Guiding Questions As it s worded, the question does not require students to land exactly on 10,000 possible color combinations. Some students might use guess- and- check to get as close to 10,000 color combinations as possible. If so, they should realize that each component needs at least two color options in order to be considered customizable. For students who need an extra challenge, consider encouraging them to find numbers that yield exactly 10,000 combinations. If they re savvy, they may realize that an easy way to do this is to find the prime factorization of 10,000: Since there are eight prime factors but only seven customizable components, students can combine two of the factors together 5 2 = 10, or 2 2 = 4 (but not 5 x 5 = 25, since the question specifies a limit of 20 color options) and will be done! What is the minimum number of color choices that each component can have? What are some numbers that, when multiplied together, approach 10,000? Can you find numbers whose product is exactly 10,000? Do you think 10,000 is a good number of total color combinations? If not, do you think it should be higher or lower? (Answers may vary.) If each component must be customizable, what is the smallest number of shoe combinations possible? (If each component has two color choices, there would be 2 7 possible combinations.)

8 Lesson Guide: PAIR- ALYSIS 8 6 NIKEiD includes an option to start with an inspiration. Watch the video of someone using this. Why do you think Nike offers this option and do you think it'll help customers find their perfect pair of shoes? Answers may vary. Explanation, Guiding Questions NIKEiD allows people to click a color, and the site will show them a bunch of shoes that have already been started in that theme. For instance, if a user clicks blue, the site will show him/her a bunch of blue shoes. It s possible that Nike does this to help visitors avoid paralysis by analysis, and make them more likely to buy a pair of shoes. However, even once a user chooses a theme, he or she can still choose to customize it. In this case, it s possible that the customer will quickly fall right back into the black hole of paralysis- by- analysis! What happens when a user clicks on an inspiration color? Do you think choosing a shoe that s almost done is better than designing a shoe from scratch?