## Final Project

#2 and #3

## Part 1

**Click Here to examine Problem 2**

Click Here to examine Problem 3

## Part 2

### I thought that Fibonacci Numbers and the Golden Ratio was my best write-up.
In this write-up, I discussed the connection of Pascal's triangle, Fibonacci's
sequence, and the Golden Ratio. I included links to the Excel spreadsheets
used to discover the relationships. To close, using the Lucas Sequence (as
well as a discussion of Lucas and Recursive Relations) Excel made it easy
to see that the golden ratio existed in the ratios of terms

##

### Click Here to examine
Part 2.

Part 3

I chose the discussion of the centers of a triangle as the representative
of my best work. In the dicsussion, I defined the centroid, orthocenter,
circumcenter, and incenter. I also showed the location of the different
centers for various shapes of triangles. For each center, I also gave a
connecting link to the Geometer's Sketchpad Script. I felt that a discussion
of the Nine-Point Circle and the Euler Line were most appropriate for closing
out the idea of centers of a triangle.

- In investigating the centers of triangles, I learned a lot about the
interrelatedness of many aspects of mathematics. For example, the centroid
of a triangle is the common intersection of the three medians. At the same
time, it divides each median into two parts the ratio of whose lengths
is 2 to 1. The six small triangles formed by the medians have equal triangles.
Finally, the centroid plays a vital part in reconstructing the original
triangle ABC when given the medians triangle. Many other mathematical ideas
that I learned from this exploration are discussed in the paper "Centers
of a Triangle".

- I was most satisfied with this paper because I learned a great deal
about geometry and computers that I can use in the middle school classroom.

- I feel that this write-up was very thorough in that I connected the
definitions, positions of various triangles, Geometer Sketchpad Scripts,
and many other discoveries into one concise paper.

- New elements of mathematics were learned in every exploration. I felt
that the Centers of a Triangle discussion was the one that I learned the
most from because it was interesting to me. I tried to explore each problem
to its fullest extent, even though I only chose to discuss a few of them.
This assignment was a motivation to me in that it helped me to see that
new areas of mathematics can be discovered by students with just a little
bit of up-front guidance. I will be using the Geometer's Sketchpad as a
tool for my students to discover the properties of mathematics.