Math 325
Spring 04
Project Ideas
Here is a list of project ideas to help start your thinking. We will negotiate the expectations for each project separately; feel free to propose ideas that are not on this list. Each project needs to include the following:
1. Learning challenging new mathematics
2. Some kind of interactive activity (within the class or for other students)
3. A written component.
I will help you find
materials for your project I am more interested that you focus on the
above components than that you focus on tracking down appropriate
materials. I want to help you find a
project that you find intriguing (or at least useful).
Projects can be done individually or in groups; groups will be expected to take on more extensive projects.
1. Polyhedra Projects
As I’ve mentioned before, I’d love to see some people from class lead a “Zometool” afternoon for Wheelock and/or Emmanuel students. I have a book of activities; you can figure out some good ones, which might include building a dome or some three-dimensional projections of four-dimensional polyhedra (and you’ll learn what they all mean). There are other directions in which you could go.
I have lots more books on polyhedra
that you could look through to find another project. Chapter 21 of
2. Projects from
It is a little early to start thinking of these, but the chapters past
chapter 10 in
Isometries and Patterns (Ch 11)
Dissection Theory (Ch 12)
Square Roots, Pythagoras, Similar Triangles (Ch 13)
Circles in the Plane (Ch 14)
Projections of a Sphere onto a Plane (e.g. making maps of the earth…Ch 15)
Geometric Solutions of Quadratic and Cubic Equations (Ch 18)
Trigonometry and Duality (Ch 19)
Most of these can be augmented with Sketchpad activities and other resources.
3. Geometer’s Sketchpad-focused Projects
I have several books of sketchpad activities. One book follows the traditional high school curriculum, and would be a good starting point for future high school teachers who want to seriously review some high school content areas, while gaining a new perspective on them. I have other books that include more advanced projects and that focus on proofs.
There are also other books we could order that focus on lower grades (mostly middle school, not elementary), and we could craft a project from them, as long as it included challenging mathematics for you.
4. Shape of Space Projects
I have a wonderful, accessible, book called The Shape of Space by
Jeffrey Weeks. Some of the topics he
covers include topology, visualizing the fourth dimension, and how we might
figure out the shape of our universe (this is kind of an extension of flatland how can a two-dimensional bug figure out
whether s/he lives on a plane or a sphere?
Well, in analogy, how can we figure out what kind of universe we live
in?)
There is also some wonderful software (mostly games) that goes with the book, including things like playing tic-tac-toe on a Klein Bottle or Torus. There’s also an activity book for grades 6-10 that we could order.
5. Symmetry and Transformations
I have some great software called Kalediomania that’s used for exploring, among other things, the “wallpaper groups,” which are the 17-different mathematical classifications for two-dimensional patterns. There are a lot of good topics in here both for people who’d like to do something visual and for people who have taken Abstract Algebra and would like to make connections.
There are also plenty of projects on three-dimensional symmetries.
6. Origami Projects
I have a book on the mathematics of origami and also one on origami design, plus I have many books on folding polyhedra using modular origami. We can find many project ideas within.
7. Historical Projects
You can delve more deeply into
8. Other Areas of Geometry
You can look at an area of geometry that we aren’t going to touch on much in class. Some ideas:
Projective geometry -- starting with how do we represent three-dimensional space in two dimensions, e.g. by making perspective drawings.
Finite geometries -- start with some axioms, and satisfy them with a finite number of lines and points; prove theorems about them. Good for people who like discrete math.
Fractal geometry you can use sketchpad to investigate fractals;
I also have many other books. Cutting
edge math, with pretty pictures, much of it is accessible to high school
students too.
Inversive geometry prove theorems by changing circles into lines
and vice verse.
Copyright 2005, Debra K. Borkovitz. You may copy or edit this material for non-profit, educational use only.
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