Mt 322 Topics in Geometry - Fall 2002

Assignment 6: Chapter 7

November 14, 2002

Symmetry in the Plane

We will spend the class period on Thursday, November 14, working on the activities for Chapter 7, Symmetry in the Plane. This chapter builds on some of the ideas that were presented in Chapter 5. We will be investigating ways that a plane figure (such as a square or a triangle) can be congruent to itself. This leads us into a mathematical structure called a group, in particular, we will be investigating symmetry groups.

Test #2

We did postpone Test #2, and I think that we really should have one more test before the Exam. Looking at the calendar, I think that the best day to schedule this test will be Tuesday, November 26. This test will focus on Chapters 4, 5, and 7. As usual, there will be some kind of construction to do using Sketchpad. I am thinking of ways to structure this test so that there is a part that you will do with your group. I'm trying to work out how to do this -- perhaps you could have a 10-minute group meeting during the test, or perhaps there could be one problem that you do with your group. Because of including a group activity in the test, I will have to make the test a bit shorter than either Test #1 or #1.5. The Study Guide for this test will be posted by November 20.

The Proof Book

By the end of the semester, you are to have compiled a portfolio of 15 - 20 of your best proofs. Your proof book must include proofs from every chapter of the text. Each proof should be written very neatly by hand or word-processed. Your diagrams may be neatly drawn by hand, or done using GSP worksheets. Some of you are writing the steps of your proofs directly into the GSP worksheet, and you may do this for the proofs in your Proof Book.

• You should be turning in one or two proofs almost every class period. You may re-write any proofs that are weak, so that your Proof Book really is a collection of your very best proofs of the course. (Some students have already turned in nearly 10 proofs already, so you are well on your way!)
• You may find statements to prove in the Activities and the Exercises for each Chapter. You may also make up a conjecture based on your own explorations using GSP -- then develop a proof (if it is true), or a disproof (in case it turns out not to hold in general). The statements and proofs (or disproofs) that you include in your Proof Book should be statements that you have spent enough time experimenting with that they are familiar to you.
• You may include two different proofs -- using different proof strategies -- for the same theorem. Then write a short (one pararaph) reflection comparing the two proofs. -- Does one proof strategy lead you to see things differently than the other? How?
• As you organize your proofs into a portfolio, you should give some though to how to organize these proofs. You might group together several proofs that prove related conjectures or theorems. You might have a proof of one statement, and a disproof of a statement that looks very similar.

For example, the diagonals of both a parallelogram bisect each other. Since a rectangle is a special kind of parallelogram, if follows immediately that the diagonals of a rectangle also bisect each other. However, a rectangle is a cyclic quadrilateral -- but a general parallelogram is not! So include the proof for the rectangle -- followed by a counter-example showing where things break down for the parallelogram.

• The Proof Book is due on December 5, which is the last day of class.