You should definitely show them the "Dandelion" (actually, Dandelin!) Theorem, which I call the *ice cream cone* theorem. It proves the sum-of-the-distances theorem about ellipses w/o requiring analytic geometry. When my plane geometry teacher showed it to us, it completely blew my mind! In particular, I couldn't get over the fact that the ellipse was symmetrical, even though the spheres were of different sizes! Even if the kids haven't yet had 3D geometry, they can still follow this easy theorem. https://en.wikipedia.org/wiki/Dandelin_spheres At 05:43 AM 11/8/2018, Cris Moore wrote:
I'm going to talk to my daughter's 8th grade class about conic sections.
I want to focus on foci (ha), and how curves with beautiful geometric descriptions also have nice algebraic descriptions in Cartesian geometry.
But I found it surprisingly tricky to work out examples.
Consider an ellipse with foci at (-1,0) and (+1,0), and define the set of points where the sum of its distances from these two is 4.
Using Pythagoras' theorem produces an equation with a bunch of square roots.
Squaring both sides eventually turns this into 3x^2 + 4y^2 = 12 but this takes a bunch of steps of algebra, and mysterious cancellations of 4th-order terms.
Similarly, it takes a fair amount of work to get from the hyperbola with foci at (+2,+2) and (-2,-2), where the difference in distances is 4, to the simple equation xy = 2.
Am I doing something wrong?
Is there an easier way to get from foci and distances to these simple quadratic equations - without recourse to canonical forms, linear transformations, polar coordinates etc.?
Of course, I then want to talk about light waves bouncing from one focus to another…
I'm not sure how to justify this without a little calculus.
- Cris