## Tuesday, August 3, 2010

### Mixed Meta Four

If you haven't read about the Wolverine[1] (here and here) or Sam's Exasperating Problem, you need to get your priorities straight go read 'em now. Sam's wondering if the problem can be scaled down so a precalculus class can handle it.  I see the problem and think, "GEOGEBRA!  I CAN USE GEOGEBRA!"  Mr. H beat me to it.  (If you haven't seen his applet, I question your dedication to the cause go ahead, we'll wait.)

I love this!  We are always looking for ways to iterate problems and extend them, but there's nothing to extend with this problem.  It's all ready for the wolverine wrangler to do his stuff.  I'm looking for the guy who can make this wolverine sit and quit bearing its teeth so my 8th graders can pet it for a second.  GeoGebra does this.  Mr. H's applet makes this problem accessible to an 8 year old.  In fact, my son was so mesmerized by the animation that I swear I heard him muttering, "Heffalumps and Woozles.  Heffalumps and Woozles."  Heck, I found a strange urge to put on some Pink Floyd myself.

Can you imagine starting a problem in middle school and finishing it with calculus?  That's how beautiful (that's right I said it!) this problem is. Why can't we let these younger kids see the beauty of the wolverine without actually having to be the one to handle it?  I can see posing the problem, setting the kids up with GeoGebra (with minimal prerequisites) and turning them loose.  They'll see the pattern, make a conjecture and inductively decide the answer.  Show the applet which demonstrates the first 360 cases and inevitably, the question will be:

Why?!

Now, talk about storytelling. The table's been set for the sequel that the kid's gonna have to wait a couple of years to see.  precalculus kids can actually calculate the answer and the trilogy will be complete once they have the tools to actually prove that for n chords, the product is n+1.  This problem can span four years. At least.

[1] Apologies if I misused the metaphor.

Mr. Collins said...

So what would you say to the student who is totally convinced by the Geogebra applet, and sees no need for further investigation/exploration? (Just curious.)

CalcDave said...

This sort of thing (introducing early and working up to a finished product years later) could be done with all sorts of problems! Reminds me of this post:

http://glsr.wordpress.com/2010/07/05/the-calculus-carrot/

It would be fun to get one or two big/exciting problems that you could introduce in Algebra 1 or PreAlgebra or something and refer back to for the next few years while the students get more and more pieces of the puzzle over the next few years.

David Cox said...

Mr. Collins I'm not sure that after seeing the applet anyone would need more convincing. I do think that "why?" is the obvious next question, though. A precalculus kid should be able to grind out a few cases that the algebra kid simply trust geogebra for. However, I believe a formal proof is going to have to wait until calculus.

CalcDave I'd love to see vertical articulation get to the point where we can keep adding to each others' foundations on a single problem. The algebra kids will already know the answer, but figuring out how and why will take a while.

Jason Buell said...

Was that last comment a hint at your future plans in your new (half) job?