# Presentation on Parabola

All right, I'm presenting a class in front of some $$20$$ or so $$12^\text{th}$$ grade students in a couple of days on the topic of Parabola. I was wondering what would be the best way to teach it to them, and also if anyone had some really amazing problems as I have to give them $$2$$ insanely hard problems\ or just problems with a really great solution or analysis to be done in the class. Also if anyone had their hands on any great lecture notes from college I would be very grateful since we have been instructed to not restrict our self to any sort of level of difficulty.
Any ideas would be welcome.

• @MichaelHoppe So from the little bit I have read, it sort of seems that the paper explores all the properties of the parabola to prove the statement given at the beginning by Archimedes. If so I think that is a great idea for the way I could structure the lesson. Thank you so much – Prakhar Nagpal Jun 8 at 15:39
• I once uses to teach my students "constructing tangent and perpendicular to a parabola" as well "constructing focus of a given parabola" and so on. Most of them enjoyed it. – Qurultay Jun 8 at 15:44
• That seems like a good idea, but since the class is virtually based that might be a little bit difficult logistically – Prakhar Nagpal Jun 8 at 15:51

Too long for a comment ... One special Archimedean Lemma I've used for a lesson: Let $$f\colon x\mapsto ax^2$$ for positive $$a$$ and let $$P(p_1,p_2)$$ a point with $$p_2<0$$. The two tangents from $$P$$ to $$f$$ touch $$f$$ in $$A_1$$ and $$A_2$$, resp., midpoint of $$A_1A_2$$ is $$M$$. Now show that the midpoint $$S$$ of $$MP$$ is a point of $$f$$.
Use this fact to cleverly construct the tangent in any point $$A$$ of $$f$$.