(March 10, 2015) As the kids entered the room, they saw on the table were some images of life pods on Mars and…
“Mars!” exclaimed someone. Immediately their curiosity was piqued.
I told the kids about the Mars One project – a private company in the Netherlands planning to send 24 people to Mars to live. We discussed the process of selecting the people, and the technology involved. You may have heard about the people from our area who want to go to Mars and spend the rest of their lives living there. (Mars One story, website, illustrations) Then I said, “Now, let’s imagine that a similar mission already left Earth, to settle on a different planet. Imagine that you are on that mission.”
On Botso, there are blind creatures that you need, but might eat you. You need them because they breathe out oxygen and breathe in carbon dioxide. (Why is that important? Without them, you can’t breathe the air, like trees on earth.)
They can’t see you, obviously, but they can sense your presence. They can sense your presence and location if you are standing on the same color ground as them. If they can reach you with their arms, they’ll grab you and eat you. (Why do you think that’s the unit of measure?) Their arms don’t bend. So if you’re standing on the same color of ground as one of these creatures, you’re safe if you’re <1 or >1 arm-length away. And, of course, you’re safe if you’re standing on different colored ground.
Your mission is to landscape this ground so that you can plant crops outdoors and have food. How can you do it?
Immediately the questions began:
- >What technology did they use to get there? (I don’t know)
- >What are the creatures called? (IDK)
- >How long are their arms? (One noozle long.)
- >How long is a noozle? (As long as a Botso creature’s arm. This unit of measure evolved like the “foot” in England.)
- >How do they perceive color if they can’t see? (IDK)
- >Can’t the people just live in the ship forever? (Yes, but who wants to do that? Besides, their mission is to terra-form the land.)
- >Can’t the people sedate the creatures with tranquilizer darts? (No)
- >What color is the land? (Brown)
- >Can’t the people wear shoes with soles that are a different color from the land and anything on it, so they’re always standing on a different color? (No, the creatures’ perception is of the land itself, not the soles of your shoes.)
- >What if you stand on platforms that you can carry around with you, and those platforms enable you to always be on the different color from the creatures? (IDK whether the creatures would perceive the platform colors, but this, and other engineering solutions suggested – including raising cockroaches to feed to the creatures - take so much time and attention and work that you wouldn’t have enough time to sufficiently farm your land to get enough food.)
- >A fence? (I quickly shut down that line of inquiry by saying “No, creatures can bite through any material.” This comes up a lot when I attach a narrative to a math problem – the kids think of non-mathematical solutions. Great thinking, but, well, this is a math)
- >If we are going to map out our territory, can we bring along something to change the color of the land? (YES! You can make paint that you can spray on the ground. The creatures sense this paint and respect its colors. But it’s very expensive. The production facilities to produce a single color of paint take the same amount of the ship’s energy needed to produce a year’s worth of food on board the ship. So you really need to minimize the number of colors you use.)
That last student question really lassoed us into the realm of mathematics. It took a really long time to get there. Was that time investment worth it? Can attention be sustained at this level once we move deeper into the math? I guess we’ll see over the next 5 weeks.
IS THIS MATH?
We began class today playing function machines. This activity allowed people to arrive at different times without missing a big math concept. The kids had great fun. We spent about 10 minutes on function machines, then about 40 minutes on the Botso problem. Just when we were starting to discuss strategies for the coloring of the plane (in math-speak), Z asked if we could go back to function machines.
“I thought that this was going to be a math class,” he said, a bit disappointedly.
“I want to keep talking about the aliens,” said a few other kids.
“Talking about the aliens is math,” I replied.
“It is?” said a few.
“Yes, it’s a type of math called geometry.”
“I didn’t know that geometry is a type of math,” said R.
“It sure is. This specific type of geometry is called combinatorial geometry. It’s a branch of math that people have only recently starting exploring. The question we’re asking, in fact, is one that no one in the world knows the answer to for sure.” (Eyes opened wide.)
No one knows the smallest number of colors you need to make sure that no aliens get you no matter how you map out your territory. The famous wording of this unsolved question is something like this:
What’s the smallest number of colors you need so no 2 points of same color are 1 unit apart in the plane?
“It’s really called the Hadwiger-Nelson problem, posited about the chromatic number of the plane, and it was first posited in the 1950s.”
“And kids can try to answer it?!!!” asked someone in wonder. (The kids loved hearing that they were doing real work, and that the work they were doing had fancy-sounding names.)
“Yep. Kids can do real work in mathematics, can contribute to what people know. It’s happened before and can happen again.”
Everyone was impressed with that. But we only had 5 minutes left now, too little time to make progress on our big problem, so we did return to function machines. (Before making the transition, I gave them a tease for next time about the astronaut Hector, who says “2 colors are enough! I’m going out there right now!” and his colleague Miriam who says, “You will die then!”
Earlier today, we did some pretty simple functions that everyone “got” quickly – add 2, subtract 1, etc. Now we made the machine a lot more complex in order to do a higher-order task. The kids called out numbers to go “in.” When they put in 1,645, out came 822 1/2. Etc. After we had a bunch of input/output pairs on the board, conjectures began. Almost everyone thought that the machine was either doing subtraction or division because the out numbers were smaller than the in numbers. A few students saw the sense in trying smaller numbers to discern the pattern (many were having fun putting in very large “numbers” like a hundred billion thousand six million). This helped a bit, but then we were out of time without a conclusion. To be continued next week.