This site contains six research projects that investigate topics in geometry and number theory. Each provides the objectives, prerequisites, a summary, the project itself, and reference.
Type of Material:
Identify Major Learning Goals:
The successful student will learn how to apply mathematical reasoning and exploration to make conjectures and develop an ability to rigorously prove these conjectures.
Target Student Population:
Students in an upper division geometry class, number theory, an introduction to proofs class, or other related courses. These projects would be suitable for use in an honors course or independent study at the high school or college level.
Prerequisite Knowledge or Skills:
Some experience with abstract proofs would be helpful, however the emphasis is not on proofs but on developing creativity in mathematical thinking and applying that creativity to investigate problems and obtain results.
There are six activities, Pascals Simplices, Pythagorean Triples, Regular Polygons, Regular Polyhedra, Regular Polytopes, and Sums of Consecutive Powers. Each is a self-contained project that provides more depth than a standard textbook exercise. Hints are provided to answer the questions posed and a full solution is provided on the solutions page. The layout is color coded and clearly presented. Links to references are also provided. A limited number of animated GIFs accompany the static text.
The nontrivial and open-ended projects are designed to inspire students to make conjectures and research mathematics. The emphasis of each project is on obtaining results. Proofs are suggested and provided in the solutions, but students are not expected to be able to prove all the results obtained.
Potential Effectiveness as a Teaching Tool
For an instructor who wants students to solve deeper problems than typical two line exercises, these projects are useful. The projects would be particularly appropriate for honors projects or independent study. They may also be useful as guides for instructors to develop additional projects by generalizing a known mathematical result.
The projects are constructed to provide limited direction, encouraging students to follow their own lines of thought. Motivating questions and project guides are provided in each project with many of the questions in the project guides linked to hints. Very thorough solutions are also provided and are not intended for use until the project has been thoroughly investigated.
Mathematica can be used as a supplement to the projects or students may write their own programs using available software.
Very short hints are provided and very comprehensive solutions are provided. It would take a very sharp student to come up with such sophisticated solutions based on some of the hints.
Ease of Use for Both Students and Faculty
Since the learning objects are text and GIF based all students and faculty can easily access the site and read through the projects. Navigation is clearly marked, with effective use of links and bookmarks. Many of the projects have accompanying Mathematica notebooks containing supplemental programs, tables and graphs. For the student or instructor who does not have access to Mathematica, MathReader is recommended.
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