About 10 years ago I was invited by my district to attend a week long summer institute led by the instructors of a private school on the east coast called Phillips Exeter Academy. Each day we spent about six hours doing math problems from the curriculum that students use in these courses. Sometimes we would work on problems alone, sometimes together, with lots of presentations as well as integration of technology. What a fantastic week! I attended the same institute the following summer, choosing a different course to focus on, and found myself equally excited about this professional development.
I haven't run across the opportunity to attend another of the Exeter Math Teacher Institutes, but I have chosen to go back to their curriculum over the years. They follow an integrated approach which spirals different concepts throughout the year. They also develop concepts in ways that I am not always familiar with. I decided to take a look at their Math 4C Curriculum this week, which looks to be a trigonometry/pre-calculus type of course.
There were many topics to choose from in the first ten pages, but one that intrigued me was the introduction of conics. On page 2, problem 6 asks students to write an equation in polar form for the set of points P that are equidistant from focus F at (0,0), and directrix at y=-2. They do a much better job of explaining this in student friendly language, saying "Using the polar variables r and theta, write an equation that says that the distance from P to the directrix equals the distance from P to F." They provide a diagram similar to my diagram below.
I was excited to try this problem since it made use of a definition that I was familiar with, but in a different way. I'm not sure I've ever thought about how to derive an equation for a parabola using polar coordinates or equations. My next step was to label my diagrams with unknowns. I am not sure students would remember how to express the side lengths of a right triangle in terms of r and theta. This is another step to derive, but one that would not take too long.
After I wrote the lengths of all relevant sides on my drawing, I wrote an equation that matched the problem description. It is empowering that this equation came from me, not my textbook. Better yet that I put together information from a written description and a diagram that I filled out according to info from this written description.
The next step in the problem was to solve this equation for r. A nice review in algebraic manipulation could come from this step.
Students would then graph this equation, most likely using a graphing calculator in polar mode. I don't have a graphing calculator handy, so into Geogebra this goes! I always have to look up the directions for how to graph a function in polar form using Geogebra, so I finally wrote a blogpost so I never have to sift through online directions again. Below is what I see on my screen when I graph the polar curve above.
The next day I skimmed a few more pages and ran into a similar problem on page 5. This problem asked students to use the polar variables r and theta to write an equation that says that the distance from P to the directrix equals twice the distance from P to F. I was intrigued because I knew the work would be similar to what I had already worked on, but I didn't know what the shape would be. The text provided a drawing similar to my drawing below.
I went through the same set of steps. First, set up the equation.
Solve for r in terms of theta.
Plug into Geogebra, and it is an ellipse! Totally unexpected.
This intro comes without the barrage of details normally involved with the introduction of ellipses. I am intrigued because this is not the definition of an ellipse that I am used to, but it is a description that produced an ellipse. As a teacher, I am full of questions and curious to see how this develops. More on this soon!