Here's an idea for illustrating all the different shapes you can get out of conic sections: get some Play-Doh, roll it out into a cone shape (the "conic" part) -- and also a reasonably sharp knife (for the "sections" part).
First, here's our starting cone:
Note that if you cut off just the tippy-top part then you get a single point:
On the other hand, if you carefully take a shaving down the very edge then it produces a line:
But if you make a slice perpendicular to the base, then you get a perfect circle (of any size you want, depending on how far down the cone you take it):
Make a similar slice at a slight angle and the cross-section you get is now an ellipse:
Take the slice at a steeper angle and you'll produce our old quadratic friend, the parabola:
And increase the angle a bit more (greater than the edge of the cone itself), and you'll produce the parabola's angry cousin, the hyperbola (or really a half-branch of such):
Kind of neat. Full disclosure: the cone gets pretty "smooshed" on each cut (kind of like a loaf of bread with a dull knife), and I had to gently re-shape back into the proper section before each photo. Therefore, this demonstration probably works best in static photography, and would be somewhat less elegant live or in a video. But the nice thing about the Play-Doh is that you can sticky it back together pretty well after each sectional cut, and it's the only material I could think of that would work well in that way. Can you think of anything else?