HELP on a Graph Theory Lecture for middle school students

[ch3cooh]

 On Friday, I taught for Proveit! (an ESP program where a MIT student teaches math at a local middle school for an hour once a week) and it was such a beautiful day outside that, 20 minutes before the class needed to start, I decided to scratch the planned topic and make a lecture that I could teach outside using sidewalk chalk (and where the students also get to play with sidewalk chalk a lot).  After a couple minutes, I decided that Graph Theory was excellent - and, in the 15 minutes walking over, this lecture sprang into my head as an awesome introduction with a very simple overlaying theme - 5, 4, 3, 2, 1, Graph Theory!. 

I started the lecture with a very basic introduction to what a graph is - verticies, edges, faces, and then the idea of embedding a graph in the plane. Students get to use their chalk a bit here - example: draw one possible embedding of K4 - and it's clear that there are distinctly different ones.  Then we start on the theme: counting down to GRAPH THEORY! == FIVE is a very simple 'proof' that K5 is non-planar.  It's really just intended as an example to show that non-planar graphs are possible.  (online explanation).  Then I introduce the FOUR color theorem and explain that no 'elegant' proof is known - the students get to test it of course, drawing and coloring graphs for each other.  Then, I ask them to prove that K3,3 is non-planar, although the problem is explained as the THREE houses THREE utilities problem. Actually, this is the first proof I /ever/ did - when I was in highschool, I did a summer program my freshman year, and this problem was introduced as a challenge problem at the end of a graph theory intro.  Actually, the teacher said something to the effect of - "I'm not going to show that this (the 3 houses 3 utilities problem) is impossible, but if you can do it, I'll buy you a pizza.  I proceeded to try to claim the pizza by showing a rough proof that it's not possible in the plane, and then embedding it on a taurus. :)  but he didn't get me a pizza. :(... ;-) anway, I tried to make a much more rigourous proof later and it was a very interesting experience - I wound up circling a bunch of assumptions that I didnt' know how to prove - like that every cycle/circle in the plane has an 'inside' and an 'outside' and I tried to define these terms as best as I could. I wound up with a 2 page long paragraph-form 'proof' with a lot of funny things circled. :)  I wish I had kept it, but I think I gave it to a teacher at some point... anyway- And this sets up Euler's formula V-E+F=TWO and a simple proof with some hints that expanding this idea to graphs embedded on a Taurus is interesting.  (so, that's the FIVE, FOUR, THREE, and TWO)

And now... now I need an awesome conclusion with a surprising ONE, and I can't think of anything.  The ideal would be where the one comes from nowhere - like, if each face of a polyhedron could be labeled with a positive or negative fraction somehow so that they all add up to 1... *sigh* I don't know - I just need something EXCITING (and preferably that can be demonstrated or tested with sidewalk chalk) to end the class!  All ideas welcome and appreciated! I have till this upcoming Friday to figure out how to end the class!

Comments

[ch3cooh.livejournal.com]

Yeah - Tree iff V-E=1 sounds close to perfect! Especially given that the class is becoming as much of an introduction to proofs as a graph theory introduction. :D

This is a young/inexperienced enough group that even the concept of an iff proof will be new. And it definitely presents them with another opportunity to play with chalk.

I'd like to teach a topology class for them at some point and maybe do the projective plane when I've introduced some of the terminology and intuition for alternate planes (the standard paper lab - taping the different shapes together, cutting up the mobius strip, etc.)

K, will post again after the class - here's hoping it's sunny ;-)

Thanks Marisa! :)