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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. ( cute digression ) 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!
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. ( cute digression ) 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!
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