# Math 1319 Mathematics in the Modern World Homework

## Dr. Duval

For class on Tuesday, December 2, read section 7.6.

### Homework assignments

1.4: 9. Be sure to write your solution (including your reasoning) clearly! Don't use guessing, probability, or psychology; just logic.

2.1: 6 (do your own measurements!), 12.
2.2: 8, 10.

2.3: 8, 11.
2.4: 6, 11.

2.6: 8, 12.

2.7: 7, 22.
3.1: 8, 19.

3.2: 9, 14.
3.3: 10, 19.

4.3: 8, 9.

4.1: 15.
4.2: 7, 10, 12.

4.5: 8, 12.
5.3: 7.

4.7: 6, 8, 13.

7.1: None.
7.2: 7, 14, 19, 21.

7.3: 16, 22, 30, due Tuesday, Nov. 25.

### Writing assignments

2.3 Infinitely Many Primes.
Explain the proof that there are infinitely many primes to someone who has not studied as much math as you have. (For simplicity's sake, I will refer to this person as "your friend", even though it may instead be a relative.) Before you do this, you will probably want to study the proof on your own, outline it, or summarize it in your own words, and anticipate points that may be hard to explain. You may use any techniques you feel are appropriate in order to help your friend understand the proof.

After you do this, write a report that summarizes what you said, what your friend said, and what your friend had difficulty with. How did you get past the difficult parts? Are there any particular parts of the proof that you understand better (or worse) than before you explained it to your friend?

Format: Use correct grammar, which includes writing in complete sentences, and breaking material into paragraphs. Please type your paper, double-spaced (or 1 1/2 spacing) to allow room for my comments. Please do not use binders, folders, plastic sleeves, etc.; simply staple the pages together. You do not need a cover sheet, though you can use one if you like. Make sure your name is on the first sheet. I anticipate most papers being 2-4 pages, but you should feel free to go above or below that range.

Rubric:

5 Engaging recounting of activity, including: how you explained the proof, which should be completely correct; and an analysis of what your friend found difficult, why it was difficult, and how you got around it. Description of parts of the proof that you now understand better (or worse) as a result of the activity. Writing is inspiring or clever, very clear, and well-organized.

4 Detailed recounting of activity, including: how you explained the proof, which should be completely correct; and a description of what your friend found difficult, and how you got around it. Writing is clear and organized.

3 Detailed recounting of activity, including how you explained the proof, which should be correct, and some indication of your friend's response. Writing is readable and somewhat organized.

2 Complete recounting of activity, including how you explained the proof, which may have some minor flaws or gaps. Writing is readable enough, not very detailed.

1 Incomplete recounting of activity, or incorrect version of proof, or writing that is confusing or hard to read.

0 Minimal work, or extreme confusion about the proof.

4.3: Problems 12, 13.
Part 1, the activity. Engage with these problems in the following way:

Try these problems first by yourself, gathering enough data to make a hypothesis about what happens in each case. (Be sure to try problem 13 on a variety of rectangles.) Then check with one or two other students to see how your hypotheses compare, and try to combine them. Finally test your hypothesis on several more examples. Note that problem 12 is really just a warmup for problem 13, which should be your main focus.

Part 2, the writeup. Write a report of your experiments and conclusions in two parts as follows:

Part A. Describe the experiments you did, including pictures. Write about the process by which you arrived at, and then tested, your hypotheses. This written description of your process should be clear and detailed enough for someone else to recreate your thinking.

Part B. Clearly state your conclusions of what you think happens in each of problems 12 and 13. Show a wide enough variety of examples (especially for problem 13) to convince a skeptic that your conclusions are correct.

Format: Use correct grammar, which includes writing in complete sentences, and breaking material into paragraphs. Please type your paper, double-spaced (or 1 1/2 spacing) to allow room for my comments. Please do not use binders, folders, plastic sleeves, etc.; simply staple the pages together. You do not need a cover sheet, though you can use one if you like. Make sure your name is on the first sheet. I anticipate most papers being 1-3 pages, not including diagrams, but you should feel free to go above or below that range.

Rubric:

5 Clearly-written narrative showing your motivation for each step you did. Correct, thorough conclusions, stated clearly, for both problems, with a good variety of examples.

4 Clearly-written narrative showing some of your thought process. Correct conclusions for both problems, with many examples, including some variety.

3 Readable narrative showing the steps you took. Some correct conclusions for one or both problems, with many examples.

2 Narrative is readable enough, not very detailed. At least one correct conclusion about one problem, with some examples.

1 Some confusion as to the procedure of the problems, or writing that is confusing or hard to read.

0 Minimal work, or extreme confusion about the problem.

7.1, Let's Make a Deal, due Thursday, December 4.
1. Find a friend or relative to help you with this. You can do this several times, with several differnt people, but you don't have to. (For simplicity's sake, I will refer to this person as "your friend", even though it may instead be a relative.)
2. Explain the "Let's Make a Deal" game to your friend.
3. Ask them whether they think one should stick or switch, and why. Get them to explain their reasoning.
4. Conduct the experiment, with at least 25 trials of sticking, and at least 25 trials of switching. You should serve as the "host", and your friend as the "contestant". Make sure you are not giving away, consciously or unconsciously, which door has the prize.
5. Explain to your friend the theory of why one should switch. Is your friend convinced?
6. Interview your friend to find out what she or he was thinking and feeling at each stage of the experiment. Was she or he ever confused? angry? happy? interested? bored? Etc.
Write an analysis of your friend's thinking. To do this, you will have to recount what happened, but most of your attention should be on what your friend was thinking or feeling. This will require you to do a careful interview both before and after the experiment. Really try to get inside this person's head!

Rubric:

5 Engaging recounting of the activity, including a careful interview before and after. An analysis of your friend's feelings, including how they changed (or did not change) as a result of the activity. Writing is inspiring or clever, very clear, and well-organized.

4 Detailed recounting of the activity, including a good interview before and after. A good idea of your friend's feelings, before, during and after. Writing is clear and organized.

3 Detailed recounting of the activity, including some sort of interview before and after, and some hint of your friend's feelings. Writing is readable and somewhat organized.

2 Complete recounting of the activity, including some of your friend's response during the experiment. Writing is readable enough, not very detailed.

1 Incomplete recounting of the activity, or conducting the experiment incorrectly, or writing that is confusing or hard to read.

0 Minimal work, or extreme confusion about the experiment.

Proof, due Thursday, December 4.
This is an optional assignment. If you do turn it in, it will be averaged with the other writing assignments (and if you do not turn it in, it will not affect your grade at all). Therefore, it only makes sense to do this assignment if you had at least one bad grade on a writing assignment (and, even then, it will not "undo" the damage of the previous bad grade, only lessen its effect).

Watch the 2005 movie "Proof", starring Gwyneth Paltrow and Anthony Hopkins.

Write an essay concerning "What is proof?", using incidents, ideas, or themes from the movie as at least some of your key examples. What does "proof" mean in math class, and in "real life"? What does the movie have to say about this? Your essay should attempt to answer the question: What does it mean to prove something?

Note: The movie is rated PG-13 for "some adult language and one discreet bedroom scene." ["Discreet" here means we know what happened in the bedroom, but we don't see much of it happening on screen.] If this is truly a problem for you (e.g., you don't see PG-13 movies as a rule because you are offended by the possible content), you may replace the movie by an important event in your life where you or someone you know had to "prove" something important to someone else, if you talk to me about it first.

Format: Use correct grammar, which includes writing in complete sentences, and breaking material into paragraphs. Please type your paper, double-spaced (or 1 1/2 spacing) to allow room for my comments. Please do not use binders, folders, plastic sleeves, etc.; simply staple the pages together. You do not need a cover sheet, though you can use one if you like. Make sure your name is on the first sheet. I anticipate most papers being 2-5 pages, but you should feel free to go above or below that range.

Rubric:

5 A well-argued essay that clearly establishes your answer to the question, closely integrates that argument with the movie, and highlights the differences and similarities between proof in math class and in real life. Writing is inspiring or clever, very clear, and well-organized

4 A well-argued essay that makes its case for your answer to the question, and integrates that argument with the movie. Writing is clear and organized.

3 Essay answers the question with some detail, and makes some good connections to the movie. Writing is readable and somewhat organized.

2 Essay addresses the question, with at least one connection to the movie, but response is not detailed. Writing is readable enough

1 Essay that does not address the question, or writing that is confusing or hard to read

0 Minimal work