Includes Java animations
Includes GSP-3 files
Includes GSP-4 files
These are some mathematics investigations I have pursued over the years. They may be of some interest to teachers, students, or hobbyists. I try to convey a conceptual understanding, usually without rigorous proof. Some of the lessons are accompanied by questions and suggestions for extensions.
In my pursuit of a teaching career, I was told to justify the study of each concept by establishing its relevance to the students' lives. Sorry, but I still have trouble buying into that one. Fortunately, poetry and music are rarely put to that same test. As I glance at the list below, I must concede that it would be difficult to convert any of the lessons into food, shelter, or money. These are things that interested me, and now I understand them better. If there is a reward, it is the fact that every time I do this I get better at figuring things out.
I occasionally get email from people seeking permission to use my work. Feel free to cite, quote, edit, reproduce, or distribute anything on this site. Do not worry about putting my name on it, but please do not put your own name on it.
Many of these presentations use the Geometer's Sketchpad, but only one of them actually requires it. If you do not have the software, let me suggest that you follow this link to Key Curriculum Press and look into it.
Paul Kunkel
|
May 10, 2008 |
Added Sphere Segment tool to the Solid Tools collection. See the Geometer's Sketchpad Workshop page. |
|
March 26, 2008 |
Updated the Tangent Circles page. There is now a new GSP4 document and a PDF of a journal article that I wrote last year. |
|
March 26, 2008 |
The Geometer's Sketchpad Workshop page now has a newer version of the Conics Plus custom tools. The new version includes a five-point conic construction. |
|
|
|
|
|
The BrachistochroneWe are given two fixed points in a vertical plane. A particle starts from rest at one of the points and travels to the other under its own weight. Find the path that the particle must follow in order to reach its destination in the briefest time. This is a famous problem in the calculus of variations. Nothing new is presented here, but the explanation is more thorough than most. |
 |
An old probability exercise is aided by an interactive Sketchpad file. By manipulating the sketch, an intuitive understanding may be gained, even if the student has had no introduction to calculus.
|
Chinese Handcuffs
This has no lesson, only questions. It is just an interesting animated
file, which invites student investigation. It has applications for geometry,
trigonometry, and probability. Fair warning: The discussion revolves about
a figure that appears only when the sketch is animated. Without Geometer's
Sketchpad, this one will make little sense at all.
|
 |
An investigation into the catenary, the curve formed by a chain suspended at both ends. It also extends into the curve of the main cables on a suspension bridge, which, interestingly, is not a catenary.
Inversion Geometry
|
 |
Linkages
|
 |
On the Square
|
 |
OrthocenterWhat is so significant about the orthocenter of a triangle? Why does it even have a name? This investigation was prepared for a geometry teachers' workshop. It includes a lesson handout in Word format. This lesson was written for use in a computer lab with the Geometer's Sketchpad, but with a little editing it should work just as well with Cabri or Cinderella. |
 |
The Planimeter
|
 |
Find the inradius, circumradius, dihedral angle, surface area, and volume of all of the Platonic solids. There is also some information about some of their other interesting properties.
Reuleaux Triangle
|
 |
How is that for a descriptive title? Here are investigations of two interesting visual effects seen from a moving automobile. It includes two interactive perspective drawings, and a discussion of the cylindrical and planar projections used to create them.
The Sliding Triangle
|
 |
Table PatternsExplore geometric patterns formed in a table of modular multiplication. This one requires some elementary knowledge of congruence classes and modular arithmetic. |
 |
|
Tangent Circles
|
 |
Volume of a TorusDo you already know the volume of a torus? Then humor me. I just figured it out, and I got a chance to try out this new graphics software. |
 |
The Geometer's Sketchpad WorkshopIt is plain to see that I am very fond of Sketchpad. This page was created after the release of version 4. The idea is to have a place to exchange ideas with other users, especially those who are using the advanced features. Here are some custom tools and some information on Java Sketchpad. I plan to compile links to other GSP4 sites. If you have anything to share, please contact me. |
 |
Sketchpad GalleryThese are some of my best sketches. You may find them useful or interesting by themselves. |
 |
Geometry Construction ReferenceThirteen elementary straightedge and compass constructions are described and illustrated. The original version, in Word format, can be downloaded and distributed. |
 |