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

[neatlynoted.livejournal.com]

(1) This is such a cool class idea! I may try to teach a 5-4-3-2-1 class at Mathcamp sometime. It's just so entertaining as a format. (Although I might set it up backwards -- use V-E+F=TWO and then when we get to FIVE, prove the FIVE color theorem.

(2) Hmm, thoughts about what to do for ONE... well, the obvious thing is the euler characteristic of the projective plane (and then you get to tell them about the projective plane, which is cool; you can use the circle identification polygon, let them figure out that v-e+f /neq 2 and then esplain why. (And you could continue on to ZERO and do the euler characteristic of the torus.)

ONE is the crossing number of K_5 and K_{3,3}...

Adjacency matrices are made up of 0s and 1s...

A graph is a tree iff it satisfies V-E=1...

Actually, that last one might be nice. You could try to get them to prove both directions.


Post again after your class -- I want to know how it goes!

Cheers,
Marisa

[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! :)

[aki-moto.livejournal.com]

I have absolutely no idea about any of this stuff you're talking about. I took Calc 3 my freshman year and that was it, and since then anything with derivatives has melted out of my brain...but I'm glad you're having fun and still teaching! Those little buggers haven't gotten you yet, I see... :D

I like embedding graphs in constellations too.

[r33na.livejournal.com]

Are you a Sagittarius or a Leo? Or maybe a Torus? ;-)

Re: I like embedding graphs in constellations too.

[ch3cooh.livejournal.com]

Torus - Gah! My journal is profained with veggies! Noooooooo!