I have decided to include a write-up from assignment # 9 on pedal triangles. In my opinion, this was one of the most enjoyable and interesting explorations of the quarter.

Why did I learn the most mathematics. . .?

Certainly, I enrolled in Euclidean Geometry in high school, and enrolled in two Geometry courses in college--one Euclidean, one Hyperbolic. Although I learned a great deal in the three courses, we never discussed topics such as pedal triangles, the centroid, the orthocenter, and the like. I am of the belief that if the teacher has a broad knowledge base, there is a greater opportunity for enhancing any course and breaking away from the "typical" curriculum.

Why did I feel most satisfied/happy?

I felt most happy with this write-up because it was clear and concise. In addition, I think that any visitor to my web page will marvel at the pictures. I have been satisfied with my other write-ups, but I feel as though my presentation of the material became better as the assignments went along. Moreover, I know that the assignment could be extended by students. Although I varied the location of the point P, there are other possibilities.

Why does this assignment show my best effort?

I believe that this assignment is most useful to me because I am able to use this for presentations in my classes. Indeed, this would be a good assignment for students to try different locations for P and add to my write-up. Another possibility would be to offer extra credit to the students for making conjectures about the locations, presenting their findings and thoughts using Geometer's Sketchpad, and attempt to prove any conjectures. Moreover, this assignment should provide the students with a sense that mathematics is not a "dead" subject, but rather a subject that is alive, growing, and contains ideas that may be extended in many directions.

I also feel as though this assignment is thorough, yet brief. It is very easy to become verbose in the write-up of an assignment in an attempt to explain your thoughts and your sketches, but this write-up lends itself well to letting the audience follow along without a great deal of written explanation; i.e., the sketches tell much of the "story."

This assignment also allowed me to be creative. First, I could place the point P anywhere in the Euclidean plane and draw any necessary conclusions. Second, I was able to construct a triangle of any size or shape that I wanted. Third, I enjoyed using various colors with Geometer's Sketchpad. I have done lots of reading about using color in the classroom. Research indicates that students remember more if teachers use various colors. I use different colors on the overhead projector; I use black for notes, green for examples; red for formulas; and blue or purple to go over homework problems. I'm not sure if students have figured out my color scheme, but I believe that it breaks the monotony. For that same reason, I feel that this assignment is aesthetically pleasing and that students would enjoy looking at and working with this assignment.

Why did this assignment open up new elements of mathematics?

As I mentioned earlier, there are many terms that I was not familiar with until I began this exploration. If the vocabulary words were the only new mathematical concepts that I gained, I would be satisfied! I am notorious for giving vocabulary tests or quizzes in my math classes. Of course, I have been questioned many times about this practice, and I tell parents and students that mathematics is a language and the "jargon" must be understood in order to develop a true mathematical maturity. In addition, I use the appropriate vocabulary in my classroom and I insist that my students do the same.

This assignment also extended my previous knowledge of Geometry. Our students always want to know a reason for learning different concepts, and I attempt to provide them with a reason. On the other hand, I enjoy mathematics just for the sake of learning mathematics--I don't necessarily care about the applications. It always interests me how one topic in mathematics relates to another. Mathematics reminds me of our highway and interstate system in that mathematics is intertwined, overlaps in various places, there are many routes for reaching one destination, and there is always a return path (an inverse operation).