## Saturday, January 19, 2008

### Conics Continued

I wanted to start my parabola lesson with paper folding to create a parabola. I'd done it before with other classes, and it went well since it was something different. This year I wanted to add an extra "twist" and experimented on my 3rd period (guinea pig) class .... (we don't LIKE being your guinea pigs).

Previously, they would draw a line on a piece of paper (directrix) and a point (focus) and then start folding the line over to lay on top of the focus many different times, and then the resulting pattern emerges as a parabola. Done. Move on to notes. This year I wanted to actually have them plot some of the points that were on the parabola, so after each fold, they were to use their ruler place it perpendicular to the directrix, through the focus, and make a dot where it intersected the fold.

Oh my goodness. Confusion city. Some kids got it, but lots didn't, so I just moved on and chalked one up to experience. But I think I heard the funniest line so far. One girl who has been struggling with math all year and is not getting the draw-the-dot-on-the-fold directions shrugs and funny-resignedly says, "even paper confuses me in math class".

1. Anonymous4:53 PM

i've done the paper-folding before... but the perpendicular line part.... it does work (when done correctly)... right?

2. Anonymous6:47 PM

It DOES. And I think it's a cool understandable proof. Since the paper is folded with the directrix over the focus, then any point on the fold is the same length from the focus and that spot on the directrix that's on the focus. You just have to find the spot on the fold that's "perpendicularly" away from the directrix.

Maybe I can guide them through it better next time.

3. I haven't seen this before (until 5 minutes ago when I came across this blog), but I plan to try it in my next period, since I was going to do the proof of "why are we dividing by 4 to get the focus?" in the same period anyway.

I'm using patty paper. I'll let you know how it goes.

4. Done: it worked wonderfully, thank you! I think it helps to have them draw the entire line to the fold, rather than just the dot (tried it with just the dot 6th period, with the line 7th); you still get the parabola picture anyway, just you have all the directrix-to-parabola lines in there (and that makes explaining what happened clearer as well).

I also ended up doing the Human Ellipse because I noticed that and it was more interesting than the other thing I had planned. And I still had time for the "4p" proof.

Hence goes my experience using lesson plans off someone's blog I just discovered the period before. Ah, the wonders of the Internet.

To return the favor, I'm uploading a slide to my blog (see name link) that might help if you're teaching rewriting general second-degree equations. I could use comments too, I'm still a little unhappy with it.

5. In my college geometry class we did something similar to this but used wax paper. If you press on the paper it draws the lines for you.

Of course you don't have an axis system or anything without pressing on the paper and then possibly messing up the pattern but for people who have never seen how a parabola is constructed it works well as a visual.

I don't know if this other approach would be a good "let's see a general one, but wouldn't it be nice to know the equation?" kind of parabola and then do one where you plot points as well.