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.

• What about www2.bc.edu/mark-reeder/1103quadparab.pdf? – Michael Hoppe Jun 8 at 15:34
• @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 $$f$$.

I never taught this subject, but if I were to I would start from the definition as section of a circular cone,
and their projective properties, invariants, and pass after to the metric properties. So I would start with
- how is it projected by the cone on a plane normal / parallel to its axis;
- shifting the sectioning plane, what the resulting parabolas and their projections have in common; - same, keeping instead firm the sectioning plane and widening the angle of the cone; - consider the normals to the cone in the plane of the parabola, where do they converge in the projection onto a plane normal to the cone axis, etc. - considering the cone double, and two symmetric parabolas on the two foils, where do their planes cross, what is the meaning of that line;
etc.

Concerning an interesting exercise, I would propose that of reproducing Archimedes steps in computing the area of a parabolic segment, centuries before calculus.