(March 24, 2015) “This is the last function machine I will ever lead in this course,” I announced at the start of class. “After today, you will take turns leading them.” My function machine of the day produced 7 from 3, 17 from 8, 6259 from 3129, and so on. In other words, f(x) = 2x + 1.
We never did solve it, but got close. People realized that (1) the “out” number is always bigger than “in” number, (2) using smaller numbers helps, and (3) a few predicted some results without knowing what the rule is. (Good ballparking skills?) One student showed me how he was keeping track of the different functions in his notebook. Everyone seems excited about creating their own. I can’t lead this activity without many kids coming up to the board and trying to take the chalk out of my hand, so it’s time to give the people what they want.
CHANGING A PREMISE OF THE PROBLEM
Last week I realized that I shouldn’t have said that the aliens could move. That changed the problem. So, new development in the story:
The astronaut Isabella has come up with a way to immobilize the aliens. Now they can’t chase you when you go out to do your farming. You just have to make sure you’re not standing exactly one unit away from them, or they’ll eat you.
Cheering from the group. (Surprisingly, no one asked how she did it.)
I also reminded the kids that the aliens can’t bend their arms, so you’re safe if you’re less than one unit away. No one needed that reminder, but it gave me a chance to once again dramatize a straight-arm alien grabbing a human, lifting her up, and dropping her into my mouth. I can never get enough of doing that!
THE RIDGE ON BOTSO (CHROMATIC NUMBER OF THE NUMBER LINE)
I showed the kids a photo of Devil’s Ridge, a ridge at the top of a mountain in Scotland.
This is what the astronaut Hector had seen when he looked out the window of their ship upon landing on Botso: a ridge! Only the ridge on the top of the mountain on Botso was a perfectly straight line, and very very long. When he proposed needing only 2 colors, he meant to color the ridge.1
I drew a vertical line, representing the ridge, on the whiteboard. At the top of the board, I drew a scribble representing the space ship. “If Hector gets out and steps onto the ridge, can he walk along the ridge without an alien eating him, if he only colors the ground 2 colors?” The kids then talked me through how to color the line so that no point is the same color as another point one unit away. Well, I used the word unit. They used the word noozle. In fact, R corrected me every time I used that word unit. And of course I conveniently forget next time. That way the mathematical language will ease its way into the conversation. I also gave the students the mathematical language for what they had just done: “You’ve just determined the chromatic number of the number line!”
“It’s two!” announced R (correctly), before I even had a chance to define the word chromatic. It takes a minimum of 2 colors to color the number line to ensure that no points one unit away are the same color.
Hence the astronaut Hector was right, two colors was enough, assuming that the territory to be mapped out is a straight line.
CHROMATIC NUMBER OF THE PLANE
“Is the top of a ridge enough land to farm, so that the astronauts/settlers will have enough food to feed everyone?”
“No!” answered the class.
“Then we need to move outwards, and map territory2 to both sides also – in the plane, not just on a line.”
In the meeting room of the spaceship, the astronaut Carp made a proposal. He even had handouts. He passed out square grids like these. He told the others, “I think two colors is still enough if we use this pattern for the regions.”
I gave each child a piece of paper with a tiling of squares on it. “This is what Carp handed out. Is he right about the colors?”
Some students immediately started coloring their papers. Others just looked at me for direction. I drew a grid on the board.3 Then I asked for a suggestion of which square to color first.
I reminded the kids that we are trying to minimize the number of colors used. I also explained to L, who had been absent the last two weeks, that this is an unanswered question in mathematics. He had the same initial reaction as the other two weeks ago: first, incredulity, then wonder/excitement/joy that kids could solve real math problems. (“You mean, you don’t even know the answer?” asked someone of me.)
We spent a lot of time testing different squares to color that would guarantee no points the same color one unit away. Very very tricky. We were coloring orange marker onto the whiteboard, so the colors were orange and white (see photo gallery). It seemed that no colorings with two colors SO FAR met our goal. I took a vote: “How many of you think we’ll find a coloring that works with two colors?” (5 yes, 2 no, 2 not sure.) We kept at it for a bit more, while some kids did their own colorings. (PLEASE see photo gallery!)
Two kids got so caught up in the storytelling that they wanted to focus on the narrative – what are other possible turns of the plot, what do the aliens look like, etc. That does seem to always happen with one or two kids when the story supporting the math question is so compelling. I wonder if there is a fine line of acceptable drama that gets the kids interested/involved, but not so engaged that the narrative totally sucks them in. If that line exists, I don’t know where it is, and moreover suspect that the line would be different for each child. Let me know what you think.
Oh, one other thing about the power of the narrative: by the end of the class, we were calling the aliens “animals” instead, to be kind/sensitive/tolerant/accurate, at Z’s suggestion.
By now, everyone was experimenting on paper. “Does this work?” “Look at mine!” kids were still asking after class was dismissed. But one more thing before class was dismissed:
I passed out formal invitations to lead Function Machines:
OPTIONAL INVITATION. You may present your own function machine to the class on this date: ______________. You can design the machine, and/or set the rule.
“But I can’t read!” exclaimed someone. Not a problem. Helper J was there to save the day. I made a list of whose turn is when:
3/31 – S, Z, and N
4/7 – A and M
4/14 – R, R, and L
So we’re halfway through the course, and I’m worried. I’m worried that we won’t have enough time to sufficiently delve into the Hadwiger-Nelson Problem while at the same time letting everyone pursue function machines. But we’ll keep going, pursuing both, having fun, learning a lot, and see where we end up.
BTW I owe so much thanks for Tatiana Shubin for making the Hadwiger-Nelson problem accessible to children, and to Amanda Sereveny, for suggesting this problem for our math circle, and for suggesting more kid-friendly approaches. Also Asha for taking photos. And J for helping to manage the group.
1 If you have no idea what I’m talking about here, you need to read the reports from weeks 1 and 2 of this course. In these reports, the relevant parts of the problem are in italics.
2 I continue to use the phrase “map territory” because this is the phrase the kids used in session one. If it were just up to me, we’d probably just be calling it something more boring like “coloring the land.”
3 When I drew the line on the board earlier, and then the grid later, I wanted to have consistency in lengths – not just for accuracy, but to model careful diagramming for the students. I didn’t think ahead of time about the need for this, so had no planned object of appropriate length. I pulled my car key out of my pocket. “This key will represent 1 unit.” I used that for the rest of the class for measuring, to the amusement of the class. The key was way better than a ruler, in terms of fun and in terms of math – no preconceived notions of length and units, keeping the definition of a “unit” precise. (I’d rather call the units digit the “ones” digit, even though in base 10 they’re the same. That assumption that we’re always in base 10 just rubs me the wrong way somehow. Maybe it’s because many of my high school tutoring students are unfamiliar with the term “units digit” when they encounter it on the SAT.)