Thanks for your advice on the parabola paper folding activity. I see how that would be an improvement, so I’ll make a note in my plans to try it next time.
I like your completing the square slide. The underlining and arrows and boxes of explanation with different colors make it clear what the flow is. I have 2 suggestions and 1 comment on what I do differently (and I don’t know that it’s better).
1. I would have the problem end up where one of the x or y has a “+” and one a “-“, so the students wouldn’t think it’s always minus. For example ending up with (x + 3)^2 and (y – 2)^2.
2. I would also change the numbers so it’s not 4 that ends up being squared then added for both the x and y parts.
My difference: When I complete the square, I automatically adjust the other side of the = sign instead of doing it in place (+16 -16) as you do. I don’t know if that’s better or worse or just different.
Comment 1 & 2: Both things I’m unhappy about. I realized it after I already had the layout, but I pretty much have to start from scratch to fix this. (Maybe next year!)
Last year when I taught this I automatically adjusted the other side of the = sign. The students were so befuddled I decided to slow the process down. The main issue is the distribution, where doing a “partial distribution” when pulling -16 out of the parenthesis is one extra math concept tossed in the mix.
For his own purposes Apollonius of Perga defined an ellipse as a conic section. This definition excludes the perfect ellipse which is a cylindric section which cannot be a conic section. A cylindric section unlike a conic section has locations for foci and can be represented by the equation (x/a)^2+(y/b)^2=1. This equation does not represent a conic section.
You can if you like follow Apollonius’s example and call a conic section an ellipse. It was Kepler who recognised after much study that planetary orbits needed foci which do not exist in conic sections. Kepler’s works are not read as much as they should be largely because of religious prohibition.