CHROMATIC NUMBER, 2, ACTING IT OUT
(March 17, 2015) “What do you remember of our problem from last week?” I asked. The students called out details until the problem was pieced together.
Here it is in everyday language: What’s the fewest number of colors an astronaut can use to mark out territory to be sure that you are standing on a different color of land when an alien is 1 noozle (arm length) away from you? (Click here to see last week’s report, which details the reason you need to be on a different color, and other plot information.)
Here it is in math language: What’s the minimum number of colors you can use to color the plane if no 2 points that are 1 unit away from each other are the same color?
ACTING IT OUT
One astronaut, Hector, says 2 colors is enough. He opens the hatch and is about to jump out with 2 cans of paint. Astronaut Miriam says “If you go out there with only 2 colors, you’re a dead man.” Should he listen to her?
We spent the whole session today trying to find a coloring of the plane that used only 2 colors – the chalk, and the background. First we went outside with sidewalk chalk. Then we came inside to do it with crayons and paper. All attempts were child-led. So far, I have suggested nothing to the kids.
Outside, we designated the gravel as spaceship (safety!) and the asphalt as the fertile land outside. I had no clear idea or instructions on how role-playing was going to work logistically, so it evolved naturally. It worked best to let every child play either an alien or an astronaut and stand on the appropriately-colored spaces. One parent helped (Thank you! On such a beautiful spring-like day, the lure of running around with sticks was great, especially after the snowy winter we’ve had.)
The first student suggestion for coloring was a semicircle at the spot designated as the spaceship exit. (Very astute, I thought.) Everyone stepped into the semicircle. The next coloring of the plane was a square about 2 “units” away. We designated one adult human arm length as a unit. Some people moved onto this square. Then another square was created. A few kids moved into that one. At this point the group naturally divided itself into aliens and astronauts.
Unfortunately, we didn’t find this coloring plan that helpful because of a mistake I made in presenting the problem: I said that the aliens could come after you, but very very very slowly so that you could easily get away. This overcomplicated things. (Next week we will creatively get rid of this premise.)
Finally, the kids came up with the plan of stripes across the terrain - each stripe with a width a little more than one unit apart. Again, the alien ability to move made the problem a bit different, and more difficult, than the pure Hadwiger-Nelson problem. The good thing about this muddying of the problem is that the extra drama created has helped the kids to really understand the problem.
If you walked in on our class, it might look like the kids were just running around playing aliens, but the mathematical thinking going on was deep. Next time, though, we really need to reframe the problem so that we only work with distances of 1 unit if we want to explore a truly unanswered question in mathematics. (Our question today may be a truly unasked question in mathematics.)
MAPPING IT OUT
We came inside. Each child did his or her own “mapping of territory” (coloring of the plane) on paper. All students first drew the ship, an astronaut, and/or alien to anchor their thinking in the narrative thread. This makes sense, since the story I’ve told requires the astronauts to step out of the ship and immediately be safe from aliens. I wonder if we’ll be able to move to the more abstract coloring that doesn’t include such a reference point. What’s really exciting to me is the geometric thought that went into each diagram. We were very short on time at the end, so I didn’t get to adequately give each students’ work my attention. I took photos, though, of each, and promised to post them to the Talking Stick website. (So please, go to the photo gallery and show your children that their work is proudly displayed in this public online gallery.) Another piece of advice to myself for next time is to manage the time so that all of the children have time to show and explain their work to the group.
Since not everyone arrives to class at the same time, we’ve already established a custom of starting with the ever-popular function machines. At the beginning of class today, I presented the same function as last time: f(x) = ½ x. (See photo gallery for the language/symbolism we used – it’s not the “f of x” notation at this point.) The kids had a much easier time deducing the rule this week. Why? I’m not sure. I wish I had asked the kids. Sigh – another lost opportunity for further development of mathematical thinking.
Anyway, please do look at the photos – they tell the story so much better than I can in words.