Mathematical Challenges for a Group Project, Study Guides, Projects, Research of Data Analysis & Statistical Methods

Download Mathematical Challenges for a Group Project and more Study Guides, Projects, Research Data Analysis & Statistical Methods in PDF only on Docsity! M316K – Foundations of Arithmetic Spring 2009 Project 1 Instructions and Guidelines 1. This project consists of several mathematical challenges; each challenge has the point value indicated. Your group should choose at least one hundred points’ worth of challenges to complete. You may do more than a hundred points’ worth of challenges if you wish, but your score will be adjusted so that it is out of a hundred points. (However, I may make some adjustments in your favor if you do a significant amount of extra work and your grade doesn’t reflect this.) 2. Your group must work on each challenge together; you are not allowed to assign challenges to various group members and have one group member complete one challenge on his/her own, another group member complete another challenge on his/her own, and so on. However, you will need to appoint a group member as a “scribe” for each challenge; this group member is responsible for any writing that needs to be done for the challenge, though other group members are allowed to review and make suggestions regarding the scribe’s work. Each group member must scribe for at least one challenge. 3. Each member of the group is responsible for everything that the group does. You should be familiar with all of the work that your group turns in, to the point that you would be able to explain and/or defend it to somebody else. 4. Please cite any books, notes, websites, or other resources you use to complete this project; do this at the end of your group’s submission for each challenge. I’m not picky about citation styles and formatting, but it is very important for you to give credit to any source you receive information from, even if this information is not quoted in your group’s final work. 5. If you anticipate that you will not be able to work with your group on this project, because of conflicts of schedule, personality, or otherwise, please let me know immediately and I will either reassign you to another group or assign you an individual project. (I will assign an individual project only as a last resort, and if I do, it won’t be any less work for you than the group project would be.) Under no circumstances should you let this matter slide without telling me and try to claim credit for work you didn’t do; it won’t work. 6. Scoring: I will grade your group’s submission for each challenge on a scale of zero to N points, where N is the number of points assigned to the challenge. However, work that goes above and beyond the call of duty may earn more than the “maximum” number of points. Everyone in your group will earn the same grade on this project, provided that I determine (and the group agrees) that everyone contributed equally to the work on all parts of the project. 7. The project is due in class on Monday, March 30; no exceptions. Make arrangements so that unexpected circumstances won’t keep you from turning in the project on this day. Tips and Suggestions 8. Start working early. If you wait until the last day (or even the last week) to do the entire project, you will not produce work of the quality I am expecting. I strongly recommend that you start working this week so that you can get a feel for the issues and difficulties that will come up. This will also help protect you in case it turns out that your group cannot work together due to scheduling difficulties. 9. Use technology to communicate; if you use e-mail and instant messaging well, you won’t have to meet in person quite as often. I do recommend that you meet in person at least once a week to touch base and discuss what’s going on, but that’s a suggestion, not a rule. 1 10. Keep in mind that Spring Break will eat up a week in the middle of your work time for this project. It’s very likely that at least one of your teammates will be out of town for that week, so don’t count on being able to get everybody together during that time. If you work efficiently and don’t procrastinate, you will be able to complete the project without doing any work over the break. 11. If you have any questions about the challenges, or want to get some feedback from me on what you’re doing, please feel free to visit me during office hours. I won’t give you much advice on how to complete the challenges, because I want you to figure that out as a group, but I am happy to clear up anything that may be confusing about the tasks. And I am definitely willing to look at your work and give you my honest (but non-binding) opinion. A Note on Academic Integrity The concept of academic integrity is very important in any educational setting, and it is especially im- portant to this project. I have chosen mathematical tasks that I hope you will find worthwhile and in- teresting to work on. It may be possible to short-cut some parts of this project by plagiarizing books or websites, but I strongly warn you not to do so. Plagiarism includes, but is not limited to, quoting part of a book or website without crediting the author, borrowing ideas or knowledge from another source without giving proper credit (even if these ideas or knowledge are stated in words other than the original author’s), or writing a report that is basically a recycled or rephrased version of somebody else’s work without adding your own ideas or analysis (even if you do credit the source). In the case of this partic- ular project, plagiarism may also consist of claiming that a solution to a problem is original when in fact you retrieved it (or a solution to a similar problem) from the internet. For more information, please see http://www.plagiarism.org/learning center/what is plagiarism.html. I don’t want to dwell on this issue very much, because I genuinely like my students and feel like I have always been blessed with very good classes of students. Sadly, even in a very good class, there may be one or two students who would rather try to cheat than make an honest effort. I’m usually a very nice guy (most of you probably know this by now), but that’s partly because I save all my mean-spiritedness for cheaters. So please don’t put yourself in a situation where you’re likely to see my mean side. 2 Challenge 3: Some Texas-Sized Numbers – 25 points In this challenge, you’ll solve several problems in which you have to make reasonable estimates and deal with large quantities. For each problem, you’ll probably need to do some “research” in order to figure out what assumptions you need to make. Feel free to use creative research methods: polling people, measuring objects and places featured in the problems, and trying things out for yourself. You may also use the internet to obtain data for this problem, but preferential treatment will be given to solutions that obtain information in direct, creative ways. If you do obtain data from the internet, please cite your sources. When writing your solutions, please explain where all of the data and assumptions in your solutions come from. If they’re blind estimates, say so. If you got them off the internet, say where you found them. If you used some other creative approach to get the data, explain what you did, and which member(s) of your group did it. In this problem, you don’t have to collect all of the data as a team; for example, if one person wants to go and measure birthday cakes for Problem 1, that’s okay. But all team members should have a substantial share in the data-gathering process. 1. Suppose that we took all of the birthday cakes that people in the United States ate in the year 2008, and combined them into one giant birthday cake. Would this giant birthday cake fit inside Darrell K. Royal-Texas Memorial Stadium? (Obviously the stadium is not an enclosed building, so by “inside the stadium,” I mean inside the stadium without rising above the highest bleachers in the stadium.) 2. What is the combined weight of all of the books in the Perry-Castaneda Library? Is this amount greater than a thousand tons? A million tons? A billion tons? (Please count only those books on shelves that are accessible to the general public; don’t interrupt librarians’ duties or students’ studying in order to complete this problem. Also try to find a way to solve this problem without checking any books out. Part of the challenge here is to find a way to get the data you need without being disruptive or abusing the library’s services.) 3. In a typical week (Monday through Friday), how many passengers do Capital Metro and UT Shuttle buses pick up at the bus stop on Guadalupe across from the Chipotle? I’m talking about the bus stop on the university side of the street, not the storefront. Be sure to account for daily fluctuations in bus ridership; you should expect the number of passengers picked up between 4 PM and 5 PM to be quite a bit greater than the number picked up between 9 AM and 10 AM. (Please feel free to make reasonable assumptions about the number of passengers picked up at night; I don’t want you to feel like you have to hang out at a bus stop by yourself at night. Also, if you happen to see your friendly instructor on one of your bus-spotting trips, be sure to say hello.) 4. Suppose the alumni of the McCombs School of Business decide to flaunt their collective wealth by papering all the walls of the business buildings (both CBA and GSB in the UT building lexicon) with dollar bills. Assuming they only paper the walls in the halls and public areas, not the classrooms, would they be able to do this using only taxpayer money if all of the money allocated by the American Recovery and Reinvestment Act were devoted to this one ridiculous project? If so, would they be able to paper the walls with hundred-dollar bills instead of dollar bills? 5. If you wish, your team may craft its own UT-themed “big numbers” problem and solve it in place of one of the four problems given above. (Bonus points if the problem is especially creative and/or clever.) Notes Remember, no matter what methods you use to get information for this challenge, please be safe, stay out of trouble, and try not to get in other people’s way. If done correctly, this challenge will require a fair amount of work, but if you divide up the research work, hopefully no one team member will find it too much of a burden. (I do want you all to discuss how you will approach each problem together before you divide up the work, and also work together to answer the questions after you’ve done the research.) 5 Challenge 4: Survey Says . . . – 20 points The goal of this challenge is to further explore sets and Venn diagrams in the context of polling. Sup- pose I want to answer a question such as “Is a person more likely to go to college if he/she has an older sibling who has gone to college?” One way to try to answer this question would be to ask a lot of people the following two questions: 1. Are you, or have you ever been, in college? 2. Do you have an older sibling who is or has been in college? I might then let C be the set of people who go to college or have gone to college, and S the set of people who have older siblings who went to college. Based on the results of my poll, I could analyze the size of sets such as C, S, C ∩ S, C − S, and so forth. This analysis would then allow me to make an educated guess about my original question. Answer the following questions: 1. Suppose that I enter the results of my poll in a Venn diagram as shown below. (The percentage entered in the region C −S is the percentage of respondents in the set C −S, not the percentage in the set C.) Based on the results of this poll, what would be your answer to the original question: is a person more likely to go to college if he/she has an older sibling who has gone to college? Justify your answer. 2. Explain why the poll does not answer this question with absolute certainty. Give as many reasons as possible. 3. Suppose you haven’t seen the results of my poll, and that the results of my poll are not necessarily the ones shown in Problem 2. But you do know that I conducted the poll for this experiment on Facebook. Would this knowledge make you more confident or less confident in the scientific value of the poll? Explain. (Hint: The fact that Facebook is a social networking site and not a “scientific” or “serious” site may be relevant, but it’s not the most important thing here.) Now it’s your turn to play the pollster. As a group, come up with three sets that you think might be related in an interesting way, but are not obviously guaranteed to be related. (An example of two sets that are obviously related is “the set of people who have played team sports in high school” and “the set of people who have worn shirts with numbers on them”; people in the former set are practically guaranteed to be in the latter. The idea is that the poll should give you a chance to discover a relationship that you suspected but weren’t sure about.) My sets were “the set of people who have gone to college” and “the set of people with older siblings who have gone to college”; you should come up with three sets. The sets should be designed so that people will know with pretty good certainty whether they belong to them. A set like “the set of people who wear tennis shoes at least five days per week” is better than a set like “the set of people who wear tennis shoes regularly,” because the latter is much more open to interpretation. Similarly, a set like “the set of people who are related to someone famous” is a terrible choice, not only because it’s a matter of interpretation whether someone is famous or not, but because many people won’t have enough genealogical information to be able to determine for sure whether they’re in the set. 6 4. Define the three sets that your team came up with, and give each set a name. Try to use names that are somewhat suggestive; don’t use the names A, B, and C unless you really think these letters are the best choice. 5. As a group, brainstorm five questions related to these three sets that you’d like to try to answer. The majority of these questions should involve relationships between at least two of the sets (in other words, don’t limit yourself to one-dimensional questions like “do at least 20% of Americans smoke?”). 6. As a group, decide how you will carry out the polling for this question. Think about how many people you want to poll, how you want to select people to poll, and how exactly you will pose the questions. (At this point you may find yourselves tweaking the definitions of your sets; this is okay. You want to make sure to set up the questions so that they are precise and likely to elicit truthful responses.) Write a few paragraphs about the issues that your group anticipated, and how you decided to deal with them. 7. Conduct the poll, and represent your results in a Venn diagram. Represent your results in the Venn diagram as percentages, but also indicate how many people you polled. 8. Use the results of your poll to answer the five questions your team decided to try to answer. Keep in mind that your answers to some of these questions may be along the lines of “the poll didn’t give us enough information to determine the answer to this question.” If this happens, explain why this is the case. 9. The nature of any poll is that it cannot be perfectly accurate (some recent presidential election polls have been rather infamous examples of this). Discuss some aspects of your team’s polling process that might limit the accuracy of your poll (that is, the ability of the poll to answer questions about the entire population you’re studying). Notes Keep in mind that certain topics (especially religion, politics, and socioeconomic status) are sensitive subjects for a lot of people. My hope is that your team can come up with some interesting questions that don’t require you to question people about hot-button issues. If your team does want to try to explore questions about potentially controversial or personal subjects, I strongly recommend that you do the polling in as safe and unintrusive a manner as possible. The idea of using Facebook to poll people isn’t necessarily a bad one (and it would probably be very easy to do), but keep in mind that a Facebook poll will be a very biased source of information on certain kinds of questions. Be judicious about how you do the polling; try to get good results, but don’t put yourself or your team members in a position where you’re likely to annoy or alienate people. 7 as possible, who will win the game? Explain your answer. The key to winning at Nim is to be able to look at a configuration of stones and determine whether it is a “winning” configuration or a “losing” configuration. We say that a configuration is a winning configura- tion if a player who receives this configuration will win the game if she/he plays as well as possible. (The configuration (5) is a winning configuration because a player who receives this configuration can simply win the game by removing all of the stones. The configuration (1, 2) is also a winning configuration; if a player receives this configuration, she/he can win by removing one stone from the second pile.) We say that a configuration is a losing configuration if a player who receives this configuration will lose the game if both players play as well as possible. (The configuration (1, 1) is a losing configuration.) For problems 6 through 10, determine whether the configuration given is a winning configuration or a losing configuration. If it is a winning configuration, determine what move a player should make after receiving this configuration. 6. (1, 3, 5) 7. (3, 4, 7) 8. (4, 4, 4, 4) 9. (12, 17, 19, 26) 10. (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) 11. For the last part of this challenge, your job is to defeat your friendly (but very competitive) instructor at Nim. We’ll play a series of seven games; you’ll be required to win at least five. I’ll decide the starting configuration for each game, but you will be allowed to choose whether to go first or second (which gives you complete control over the outcome). I’ll let you take as much time as you need per turn, work together, and use scratch paper (which helps in this game); I will adhere to a strict one-minute time limit per turn and won’t use scratch paper. Don’t be intimidated by this requirement – if you really understand the strategy well, you will win every game handily. 10 Challenge 7: Math and Decision Making – 20 points In Chapter 1, you worked on some problems in which you had to make some decisions based on consid- erations such as feasibility and cost-effectiveness (e.g. the problem of whether to hire a new employee, the calling card problem). The following problems are similar to those, except that in these problems, you will have to do more extensive research and write more thorough solutions. In the following problems, feel free to use the internet (or the real world!) to do whatever information-gathering you need to do. 1. Your friendly instructor’s car is very old and needs to be retired. He plans to buy either a 2009 Toyota Corolla or a 2009 Toyota Prius (on that hefty graduate student salary, natch), and he needs your help in deciding which car to buy. Cody’s primary concern is cost – both the cost of the car, and the cost of the fuel he will have to put into that car. Cody’s work and personal life require him to drive an average of 10, 000 miles per year. About 75% of these miles are “highway” (high-speed) miles, and the other 25% of these miles are “city” (low-speed, stop-and-go) miles. He plans to use the car for ten years, then sell it and buy a new one. Help Cody decide which car to buy. Assume that Cody isn’t interested in extra (non-standard) features, isn’t worried about any possible difference in performance between the Corolla and the Prius, and isn’t worried about other aspects (such as aesthetics or maintenance costs) of the cars. If the problem requires information (i.e. future fuel costs) that you can’t pin down, discuss different scenarios that might occur, and determine whether different circumstances might warrant a different decision. 2. Suppose you are the head of a household containing four people – say, the members of your group. You want to feed your group for a week so that each group member gets 100% of his/her FDA-recommended daily calories and nutrients each day. You want to do this as cost-effectively as possible, while abiding by the following requirements: 1. Number: Your group should have three meals per day. 2. Balance: Your group should get all the required calories and nutrients without getting a highly excessive amount of the bad stuff: sugars, fats, sodium. (In other words, be wary of an all-Ramen diet.) 3. Variety: The meals should not be the same each day. What is the cheapest menu you can come up with? Determine the cost of this menu as accurately as possible. 3. As a group, come up with your own problem in which you must make a real-life(ish) decision using a combination of research and basic math. Then solve this problem and report your results. Notes In this challenge, the most important thing I’m looking for is a well-written report for each problem that convinces me that (1) you’ve researched the options thoroughly and (2) the option you’re recommending really is the best one. Even though you have to designate a scribe for this challenge, I expect you to work as a team when organizing your information and deciding what to say. One more note: this challenge isn’t just a silly exercise that I cooked up to put you to work. The sort of work you’ll be doing here is the kind of work that consultants and actuaries are paid (often very well) to do on a regular basis! 11 Challenge 8: Fact Finder – 15–25 points (see note) In this challenge, you’ll solve a few logic problems, and invent two of your own. When you write your solutions to these problems, please be as organized as possible, detailing your thought process from start to finish. This will take a while if you do it properly, so don’t leave the process of writing the explanation until the last minute! 1. Cody shows up for an exam review session to discover that the five students already there have de- cided to play a prank on him. Each of these students has been designated either as a “truth-teller”, who makes only true statements, or a “liar”, who makes only false statements. After chatting with his students for a few minutes, Cody realizes that not all of his students are being honest with him. Frustrated, he asks, “Isn’t there anyone in here who won’t lie to me?” The five students reply as follows: Alicia: None of us are truth-tellers. Brandin: Exactly one of us is a truth-teller. Candyce: Exactly two of us are truth-tellers. Denice: Exactly three of us are truth-tellers. Emily: Exactly four of us are truth-tellers. Based on these statements, help Cody figure out which students are telling the truth and which students are lying. Explain your reasoning. 2. After figuring out which of his students are telling the truth and which students are lying, Cody decides to join the fun, along with some other students who have arrived at the review. Cody’s youngest student Kennan arrives at the review a few minutes late to find Cody and eight of his M316K students lined up on the stage. Kennan asks what is going on, and one of the students not on the stage (a truth-teller) explains to him that everyone on the stage is either a truth-teller or a liar. Kennan asks the people on stage, “Which of you are liars?” Rather than answering the question, the people on stage decide to tell Kennan a bit about themselves (or, as the case may be, not about themselves). They make their statements in order, from one end of the line to the other: Brian: I learned Hapkido so that I could protect myself from gangs in El Paso. Freddy: I play the accordion. Sakeenah: I have a triplet who was adopted by another family. Lesli: The first three toes on my feet are the same length. Allison: I can play seven different musical instruments. Ellen: I’ve been on television five times. Jackie: I started dating a boyfriend because I prank-called him by dialing a random number. Angelina: I am legally blind without contacts or glasses. Cody: I once took – and passed – the Texas Bar Examination on a dare. At this point, Kennan has a confused look on his face, and Marjorie (his mother) complains that the people on stage haven’t given her and Kennan any way to figure out which people are truth-tellers and which are liars. So the people on stage decide to be a bit more helpful: Brian: At least three of us are telling the truth. Freddy: There are two liars standing next to each other. Sakeenah: Angelina is a truth-teller. Lesli: There are two truth-tellers standing next to each other. Allison: The majority of the women in this line are liars. Ellen: There are more female liars than male liars in this line. Jackie: If we took Ellen out of the line, exactly half of the people in the line would be liars. Angelina: One of the two guys at the front of the line is a truth-teller, and the other is a liar. Cody: Lesli and I are both liars. 12