Exam 1

Exam rules

The exam is modeled on https://mathinthetimeofcorona.wordpress.com/2020/06/11/june-11-day-95-finals-part-ii/.

You can use all available resources to prepare, including but not limited to: each other, books, notes, the internet. During the exam, however, please use only a single 8.5x11 sheet of notes.

You will have 20 minutes to answer two questions. The first question will be your choice and the second will be mine. Since you don’t know which question I will choose, you should prepare answers to all the questions.

However, you may reject my chosen question, and I will choose a different one. So it’s okay if there are one or two questions that you don’t want to answer. If you reject my chosen question, there will be a 4 point reduction in your score. You can reject my chosen question twice (limit of 2 times).

At the end of the exam I will ask you for a self-assessment of your performance.

During the exam we will talk in a conversational style. As part of this I will ask you to turn on your microphone and camera.

Please be prepared to present your work, including responding to questions. The following formats are suggested:

Writing on a tablet screen using a stylus
Writing on paper with a document camera
Writing on a chalkboard or whiteboard

Prepared slides may be part of your presentation, but they are not well suited for responding to questions where your response might involve writing. So even if you include slides, please be prepared to also use one of the above formats. If it will not be possible for you to use any of the above, please let me know in advance.

Questions

You should prepare answers to all the questions (or, as many as you can). However, during the exam you will answer two questions.

Compare the algebra in this class and the algebra that you learned in secondary school (high school). Discuss the similarities and the differences. In particular, which strategies work the same way and which do not? Illustrate your discussion with examples.
Consider one mathematical idea from the course that you have found beautiful, and explain why it is beautiful to you. Your answer should explain the idea in a way that could be understood by a peer student who is familiar with rings but has not yet taken this class.
Discuss how systems of polynomial equations arise in other areas of mathematics or science. Discuss how varieties arise. Explain what kinds of problems can lead to systems of polynomial equations or varieties, and what kinds of systems or varieties can arise. Illustrate with examples.
Let \(R\) be a ring (commutative ring with unity; not necessarily a polynomial ring).
1. If \(I\) and \(J\) are ideals in \(R\), is \(I \cap J\) necessarily an ideal? Prove or disprove. (Prove that it is an ideal or give a counterexample.)
2. If \(I\) and \(J\) are ideals in \(R\), is \(I \cup J\) necessarily an ideal? Prove or disprove. (Prove that it is an ideal or give a counterexample.)
3. Suppose \(I_1 \subseteq I_2 \subseteq I_3 \subseteq \dotsb\) is a chain of ideals in \(R\). Prove that \(I_1 \cup I_2 \cup I_3 \cup \dotsb\) is an ideal in \(R\).
  
  (Optional: prove it for a more general chain, a collection of ideals \(\{I_t\}\) such that for any \(t_1\) and \(t_2\), either \(I_{t_1} \subseteq I_{t_2}\) or \(I_{t_1} \supseteq I_{t_2}\).)
Textbook exercise 2.3.5
Textbook exercise 2.5.18
Choose one interesting proof problem from the text that was not turned in for homework. Describe why you find it interesting. Then either solve it, or find a solution online and work through it, using your own understanding to critique that solution and improve it (be sure to cite your sources).

Grading

Questions will be graded out of 50 points (each) using the following rubric (copied from the web page linked above).

A	50	Well-executed, well-communicated, essentially correct. Minor errors quickly corrected. No nontrivial errors.
B	43	Generally well-executed. Some minor errors not recognized or corrected. Nontrivial errors corrected when identified.
C	38	Adequately executed but numerous or repeated errors (minor and nontrivial) without satisfactory resolution.
D	33	Flawed execution with nontrivial errors.
F	25	Unsatisfactory execution with fundamental errors and deep misunderstandings.
0	0	Very little of relevance. No evidence of preparation.

Since there are two questions graded out of 50 points each, the exam total score is out of 100 points.

Scheduling

You will schedule an online, one-on-one meeting with me for the exam. Meetings will be scheduled for half-hour blocks either from :00 to :30, or :30 to :00 (ie., aligned with hours). Exams are expected to take 20 minutes, not 30; the extra time is for safety, in case of delay or technical difficulties.

Available meeting times are:

Monday, October 11	9:00am-10:30am
Monday, October 11	11:00am-11:30am
Tuesday, October 12	10:30am-4:00pm
Wednesday, October 13	9:00am-10:00am
Wednesday, October 13	1:30pm-3:00pm
Thursday, October 14	1:30pm-3:00pm
Friday, October 15	9:00am-10:00am
Friday, October 15	12:30pm-2:30pm

You can reserve an exam time with a Google Calendar appointment.

If other days/times are needed, please contact me.

Time limit

Due to the number of enrolled students, it is not possible to extend exam times. It will be very important for you to have your answers prepared and rehearsed; thinking through an answer on the spot is not recommended.

There will be about 10 minutes for each question. As a guideline, plan to spend about 5 minutes giving your prepared answer, then 5 minutes on discussion and follow-up questions. Please don’t prepare answers that will take much longer than 5 minutes. For a discussion/reflection question, this means preparing a short, focused answer, with just a few carefully selected examples. For a proof question, this might mean preparing a summary or overview of the key steps of the proof, not trying to present every single detailed step.

This can be challenging (it’s hard enough trying to figure out a tricky proof, even harder to present it, and then much harder with a strict time limit on top of that). For what it’s worth, focused and to-the-point communication is an important skill. Some tips include:

Work together with other students to plan and practice your answers.
Decide in advance which question you will choose to answer, and which questions you will reject if I choose them. (Don’t use up your time deciding this during the exam!)
You can assume that I am familiar with the questions and with basic definitions. You do not need to repeat the question, or repeat the definitions of basic terms (unless there’s some detail that’s important for your answer). Once you’ve told me what question you want to answer (or, once I’ve told you what question I want you to answer), you can simply begin answering it.
Personally, I enjoy clear, well-prepared, well-executed answers, even if they are for “easy” questions. (But if you want to answer a challenging question, that’s fine too!)

That means you should choose a question that you can answer well, even if it’s “easy,” rather than trying to impress me with a “difficult” question that you can barely answer.
Think about what follow-up questions might be asked. Some possibilities include: asking you to expand on a detail of your prepared answer; a “what-if” question asking you for a different version of the main question (e.g., “what if we used a different monomial order?”); asking how you would teach the answer you just gave. As part of your preparation, think about how you might handle these or other questions.
Don’t write down or memorize answers to every possible follow-up question. It’s okay to think about the follow-up questions during the exam. In fact, that’s a good thing, since it shows your thought process. The point of thinking about possible follow-up questions ahead of time is just to be prepared and calm during the exam.

How to prepare

I encourage you to work together with other students in the class to plan and practice your answers. I want you to work together. This is the whole point of setting up the exam in this way. Solve the exercises together. Work together to think of examples.

Please choose different examples (don’t all show up with the exact same example of how a system of polynomial equations arises in science). But you should share your examples with each other, to help each other think of examples, and also to get feedback how your example can be improved.

Practice your answers, including using a timer and asking each other follow-up questions. (In a friendly way!)