My tutoring sessions with Artemis* continue to go well. I never prep, except for fleeting thoughts about topics and problems he might enjoy. Today, I showed him the books that came a few days ago from James Tanton. (I will blog about them soon.) We started to look at Thinking Mathematics, Volume 3: Lines, Circles, Trigonometry and Conics, and before we'd looked at two pages, I asked if he'd like to figure out the equation for a circle. Sure.
I drew a circle, and asked him for the definition. He said, "All the points are the same distance from the center." I said that what we were going to do is called analytic geometry - the marriage of geometry and algebra that Rene Descartes helped found. I drew x and y-axes and asked him what we should call the center. He said (x,y). I felt bad that I'd asked the question, since I didn't want to use (x,y) for the center. So I talked about how we have a tradition of using x and y for the points that would move around (said while tracing over the circle), and so the center would traditionally get another name. One tradition would be just to use (a,b), another would be to use (h,k). I have no idea why we use h and k... He chose a and b.
I pointed to his definition and asked how we'd think about the distance. He said, "Use the Pythagorean Theorem." My algebra students tend to think the 'distance formula' is something separate, so of course it tickles me that he thinks of it in a way that feels more basic to me. I drew a triangle with a bad circle around it...
...and had him tell me what to do next. He worked it out to and I told him the tradition is to write it as .
Then we looked at and completed the squares together. He told me the center and the radius.
We decided to move on to other conics, and started with the definition which states that a parabola is all the points equal distance from a focus and a directrix (a line). But we also were talking about how you cut the cone to get the conics, and I said I had never figured out how we know that a plane cutting the cone parallel to the side gives us the same thing as this definition that uses focus and directrix.
We worked it out for an example, using x,y,z coordinates. After a bit of fiddling around, we called our cone and our plane . We used a 3D graphing calculator to check whether it was right. A bit of algebra gives us , which is a parabola opening to the left. That was nice, but has nothing to say about focus and directrix, and of course it's not a proof.
We decided to look it up. The explanation I found is for an ellipse, not a parabola, but we decided to work our way through it. It's titled Dandelin's Spheres, after the French/Belgian mathematician Germinal Dandelin (1794 – 1847) who came up with this proof - and it's dazzling! I've never seen this before, and I want to know why - it's so elegant, and pretty simple. At the end, it says the hyperbola and parabola can be thought through following almost the same steps. I'm going to do it!
This was the first time I did some stretching mathematically while tutoring him. It's going to happen more and more. I'm very curious when he'll "outgrow" me.
*Artemis (not his real name) is 8, and is very smart.
Note: The equations were done at CodeCogs. I had to redo the drawing because the website I used to put it up is gone now.