I feel I learned the most mathematics when I did this construction
(assignment 6). This may seem very elementary, and I've seen the construction
a couple of times. But constructing it yourself is much more enlightening.
I was really forced to think about the meaning of the focus and directrix.
It's very easy to say the definition, even to wonderfully explain, the definition
and meaning. Still, I know more now after going through the construction.
This is my best effort during the quarter. Due to the new grasp I have on the information, this knowledge will be most useful to me when teaching. Not only will my discussions of this be crisper, they also will be more thorough. And presenting it as a GSP sketch and discussing the way it is constructed (not just showing the pretty picture) will benefit my students in the same manner.
You said in class (on more than one occasion) that we should all be able to make this construction in 3 - 4 minutes. And you always proceeded to make the construction...in about 2 minutes. When I sat down to construct the parabola, you voice echoed continuously "...3 - 4 minutes." I worked and worked and thought and "...3 - 4 minutes." I could not remember anything about the construction you did, except that you used a triangle (and "...3 - 4 minutes). Suffice it to say that when I completed the task (about5 hours) later I felt very satisfied and incredibly happy, elated even.
Still, you may think this too elementary and , after finding out I spent so much time on it, that I need more help than you originally may have thought. But what means so much to me is that I struggled to figure out the construction and I finally did, all alone on a Sunday afternoon. I asked no one anything. It was wonderful! I am not convinced my construction is the most efficient yet, yours is a few steps shorter maybe. But as with many paths, longer and shorter constructions accomplish different things. However, the problem opened up new elements of mathematics for me because I awoke that ability in me to not only search but accomplish. And I did it alone.
Click here to see the GSP construction.