Infinity, Fall 2008, Assignments

August 28: Introduction

• Kaplan + Kaplan, "An invitation" p. 1-2
• Maor, Preface, p. vii-xiii
• Rucker, Preface to 2005 edition, p. ix - xi (up to ***)
• The backs of all three books

September 2: Counting

• Kaplan + Kaplan, Ch. 1, p. 3-17 (up to "Nameless Dread")
• Maor, 7., p. 40-43
• Dr. Syntax 1, Syntax Models

• Writing: Write one paragraph about your earliest encounter with counting.
  

  Your tutorial classmates are your audience.

• Dr. Syntax 1, Syntax Models, Exercises 1 and 3
  Note there are three sets of exercises on this one web page. You can do Exercise 2 if you want more guided practice.

September 4: Irrational numbers

• Kaplan + Kaplan, Ch. 1, p. 17-28
• Maor, 8., p. 44-53
• Maor, Sqrt[2] is irrational, p. 236-237
• Maor, How were...?, p. 239-241
• Dr. Syntax 2, Kinds of sentences and punctuation

• Exercises in Dr. Syntax 2, Kinds of sentences and punctuation

September 9: Approaches to truth

• Kaplan + Kaplan, Ch. 2, p. 29-55
  This chapter contains several sophisticated ideas. You will paraphrase one point from this chapter in the associated writing assignment due September 11.

• Bring a written outline of the points in Ch. 2, Kaplan + Kaplan, with you to class.
• Revision: Rewrite (don't just edit!) your Paragraph from September 2.
  Goals:
  • Tighter focus.
  • More descriptive language.
  • More efficient language.
  Turn in the original with the revised version.
• Oral presentation: Refer back to your Paragraph from September 2. Choose the sentence from your paragraph which you like best. Prepare an account of the sentence (follow the directions in Syntactic Analysis Exercise 1). Be prepared to present your analysis of the sentence to the class.

September 11: Prime numbers

• Kaplan + Kaplan, Ch. 3, p. 56-74
• Maor, The Prime Numbers, p. 21 - 24
• Maor, Euclid's proof, p. 235-236

• Writing: Read the section on paraphrase from the academic honesty booklet. Choose one point from your outline of Ch. 2, Kaplan + Kaplan, and write one paragraph which contains a paraphrase of what the Kaplans say about that point. Follow the MLA citation instructions for paraphrase on p. 20.
  

  Your audience is a student who has completed 4 years of high school mathematics.

  Your entire paragraph does not need to be a paraphrase, but make sure you use this technique in your paragraph.

• Oral presentation: Prepare an account of your Ch. 2 paragraph's topic sentence (follow the directions in Syntactic Analysis Exercise 1). Be prepared to present your analysis of your topic sentence to the class.

September 16: Arithmetic progressions and series

• Kaplan + Kaplan, Ch. 4, p. 77-86 (up to infinity symbol)
• Maor, 4., p. 25-28
• Dr. Syntax 3, The function of word order and parallel structure

• Exercise in Dr. Syntax 3, The function of word order and parallel structure
• Find two examples of parallel structure in Kaplan + Kaplan and/or Maor, and follow the directions for the exercise in Dr. Syntax 3. Be prepared to share with the group.

September 18: Geometric progressions and series

• Kaplan + Kaplan, Ch. 4, p. 86-99
• Maor, 5. and 6., p. 29-39
• Find the following 1914 article by Kempner from the American Mathematical Monthly:
  "A curious convergent series"

  Read the article and be prepared to explain the result and a broad outline of the method of proof to the class.

• Writing: In a one-page, double-spaced paper,
  • explain how the Kaplans use the "locked chest" metaphor
  • argue why the metaphor is or is not effective.
  Your audience is one of the students in Professor French's Numbers tutorial. Since this piece of writing has an argument, you should pay careful attention to your thesis sentence.

September 23: Chalkboard Oral Presentations

• Team of 3
  Use pictures to motivate a formula for the sum of the first d integers, and then prove your answer is correct using the method of induction. You might want to describe the method of induction in your presentation.
• Team of 2
  There exist infinitely many primes, and there exist strings of consecutive integers of arbitrary length which do not contain any primes.
• Team of 2
  How does the definite integral measure the "area" under a curve, and what is the meaning of the definite integral on page 71, Kaplan + Kaplan?
• Team of 3
  What is the Newton-Raphson method, and how can you use this method to approximate the square root of 2 with rational numbers? Prove that the square root of 2 is irrational.
• Team of 3
  What is the golden ratio (numerically and geometrically)? Prove the golden ratio is irrational.

September 25: Geometry

• Maor, 11., p. 68-82
• Maor, Some geometric paradoxes, p. 83-87
• Maor, 12., p. 88-94
• Maor, 13., p. 95-101
• Dr. Syntax 4, Transitions within sentences

• Exercise in Dr. Syntax 4, Transitions within sentences
• Revision: Analyze the transitions between sentences in your paper about the "locked chest" metaphor. If you do not find transitions, or you find that your transitions imply something you did not intend, rewrite! Bring a written analysis of the transitions and the revised paper with you to class.

September 30: Biography/intellectual history library activity, Part 1

We will learn about strategies for our research projects from our Science Librarian Kevin Engel.

Today's activity will prepare you for your research project and paper. The paper you write will be 3 pages long and due after fall break. We will work systematically through the thesis sentence, outline, and writing after fall break; right now you need to concentrate on research.

For your research paper, choose one issue about the infinite and two thinkers. Compare and/or contrast the two thinkers' views about your issue.

You will conduct much more research than the material which ends up in your paper. Doing a lot of research will give you the evidence you need to make your thesis argument convincing.

October 2: Euclidean geometry and tiling

• Kaplan + Kaplan, Ch. 5, p. 100-106 (up to "...inside and out."); skim the rest of the chapter to get a sense of the authors' points
• Maor, 14., p. 102-107
• Maor, 20., p. 155-163

• Enjoy spending time looking at the pictures at the official M.C. Escher website. We will look at the Escher website together in class today.
• Oral presentation: Choose your favorite work by Escher which exhibits tiling. Using Section 14 of Maor, analyze the tiling pattern. Be prepared to explain
  • the tiling pattern
  • what you like about the piece of art
  to the class when we look at the website.

October 7: Biography/intellectual history library activity, Part 2

If you have a laptop with wireless access, please bring it to this tutorial session!

If you want to make Xerox copies of materials you find, you can pay with your P-card or with change. There is also a scanner available. Printing from computers is free.

October 9:

• Maor, 17., p. 136-138
• Maor, 18., p. 139-148
• Deutch, "A subway named Mobius" in Fantasia Mathematica, on reserve in Burling Library
• Dr. Syntax 5, The topic sentence
• Dr. Syntax 6, Transitions between sentences

• Choose three points for discussion about the readings, write them down, and bring them with you to class.
• Exercise in Dr. Syntax 5 and the related exercise in Dr. Syntax 6

October 14:

• Kaplan + Kaplan, Ch. 6, p. 131-166

• Writing: Both Kaplan + Kaplan and Maor spend significant parts of their books about infinity discussing geometry. Choose either Kaplan + Kaplan's or Maor's text. Defend or criticize that book's use of geometry to illustrate infinity.

  This assignment should be no longer than 2 pages, double-spaced.

  Some guidance:

  Your audience is your tutorial; each member of your audience has read Kaplan + Kaplan and Maor carefully.
  You should
  • convey your critical opinion about the use of geometry to illustrate infinity in one of these texts.
  • use evidence from the text(s) which supports your argument.
  • use snippets to support your argument. Cite the snippets in MLA style, and follow the instructions on p. 8 and 9 of Hunter, Exercise on Citation and Paraphrase.

October 16:

• Dr. Syntax 7, The thesis sentence

• Exercises in Dr. Syntax 7

• Revision 1: Now that we have studied Dr. Syntax 7 about thesis sentences, first rewrite your thesis statement in your geometry paper (no matter how much you like it!). Keep in mind that your entire essay should be encapsulated in the thesis sentence.

  Second, ask yourself how you would argue against your thesis, and write a counter-thesis sentence for your paper.

• Revision 2: Using Dr. Syntax 6, analyze the transitions between sentences in your introductory paragraph. Keep a written record of the transition type, and check if these transitions make sense. If you find that transitions are missing, provide them.