(March 31, 2015) The first student to present his Function Machine to the group was N.* He drew the machine, then called on students whose hands were up to suggest “in” numbers. He stated the out number for each, as follows:
X means that the number going in “breaks the machine.” No one had a conjecture for a while. Then someone suggested that the numbers get bigger. “Yes,” said N. Another person pointed out the the in and out pairs each had the same left-most digit. But still, no rule. Finally, N had to tell the group: the rule is different for each number that you put in.
“Ah, a conditional function,” I said. Everyone was please that this type of function had a name.
Then it was S’s turn. He drew his machine (see photos), and ran the rule-positing the same way. Interesting to the group was that sometimes the numbers got bigger, and sometimes smaller. Z finally said “I’m positiving that this is another conditional function.” He was right. And I continue to be excited by the mathematical vocabulary that the students are starting to use in Math Circle.
Now it was Z’s turn. He whispered the rule to me and asked that I tell everyone the “out” number for their in numbers, since the calculating could be tricky. His was the first function with the same rule for each input. When 2 went in, 18 came out. When 8 went in, 78 went out. When 3 went in, 28 went out. When 1 went in, 8 came out. And so on. Lots of conjectures were posited, but none worked in all cases. We had two older kids participating in the function machines as spectators today – R (age 15) and J (age 10). They put their heads together and came up with a rule: you take the number and subtract 1, then stick an 8 on the right side of that result. It worked every time with Z’s rule, which was stated differently. They even tested it with 220: yep, you get 2198 with either rule. Do you know yet what Z’s rule was? It was 10x-2. So the food for thought from today is why to both those rules give the same result?
Then it was time to return to our ongoing problem, the Hadwiger-Nelson Problem.**
BACK TO BOTSO
Since A had been to the Kennedy Space Center two weeks ago and shown up today in his astronaut suit, I gave him the privilege of naming the next astronaut in our ongoing story. A declared that person to be a man named Blake. Then the story/problem continued.
Blake presented his idea to the group. He says that last week’s attempts were not working because we were not working systematically. We could tile the plane with only 2 colors on a square grid if we just start in the top left, and work outwards from there. Is he right?
I drew a grid on the board. Freehand – oops – each side length wasn’t the same. I hadn’t measured. But it turns out that each side length was pretty close to the length of the edge of the chalk box. So that length became our unit for the day. (This may seem a small detail, but mathematically this is huge. In week 1, our unit was the length of an imaginary alien’s arm. In week 2, it was a real live human’s arm. In week 3, it was the length of my car key. Now it was the length of the chalk box. The unit size keeps changing, but we’re still attacking the same problem, and in the problem, one unit always means the same thing. I’m pretty excited that the kids are accepting that at such a young literally-minded age. They’re thinking pretty representationally.)
Kids took turns suggesting where to color next. (see photos) We reached an impasse pretty quickly, where it seemed like it was going to be very hard, if not impossible, to color this plane with only 2 colors. Should we go on? What would a mathematician do? (Engineering solutions related to space travel came up – I declared that since this is a math circle, we’re only looking for mathematical solutions, and that I would listen to their engineering solutions after class.) Z showed the class a solution that involved 6 colors. Everyone accepted then that it could be done with 6 colors in a grid. But what is the minimum number of colors? N suggested trying it with 3 colors instead of 2.
“It looks like we have 2 options. Option A: continue what we’re doing even though it’s getting intimidating and frustrating. And option B: add in another color. Let’s vote.”
“Wait,” said R (the student R, not visitor), “there’s another idea. We could start over our coloring with 2 colors and try it a different way.” Others agreed. So we voted:
- Option A (keep going as is) – 0 votes
- Option B (continue working in 2 colors, but different coloring pattern) – 5 votes
- Option C (add in a third color) – 5 votes
What to do?
After loudly groaning, I told the kids that since we only had about 8 minutes left, that they should all do their own work, and that I would come around and insert some doubt in their conjectures, as usual. All these brave kids liked that idea.
N came with me as I looked at everyone’s work. He was faster than me at spotting solutions that didn’t actually work. It was back to the drawing board for many people.
Z showed a solution involving lengths of 4 units instead of 1. During the earlier discussion, S had proposed a solution involving lengths of 13 units instead of 1. We discussed how there could be a scale factor involved, and if a creature’s arm length was 13 times larger, then everything could just be 13 times larger without changing the problem. I wasn’t sure I understood Z’s work, and I told him that I was having trouble wrapping my mind around solutions that seemed to be slightly changing the problem, which I think his was doing. Then we all discussed briefly whether you could take creative liberties and keep the problem the same, still asking the same problem.
Finally, J took photos of everyone’s work and we were out of time. Be sure to check out the photo gallery to see everyone’s work.
*While N was drawing, getting his machine set up on the board, I gave the group last week’s challenging function, 2x+1. This time the kids got it almost immediately.
**Click here if you missed the reports on the weeks 1-3 explorations of this problem.