(March 23, 2017)

“What’s that,” asked L, the first student to arrive for math circle. It was a balance scale. I showed him how to calibrate it, then, as the others trickled in, showed everyone how to use it to weigh things.

Once our group was assembled, people wanted to get right to work on function machines.

A went first, drawing his machine, accepting “in” numbers from students, and telling them what numbers come out. After a few pairs of numbers, I mentioned that it felt to me that the rule might be changing from one example to another. “Oh, you mean it has to be the same rule for each number?” asked A.

“Yes. Otherwise the function machine police will come.” The students wanted to draw the function machine police on the board too, but I discouraged that because we are sending photos of the machines to our two absent students, and the students in attendance agreed that this addition might be confusing.


M soon had a conjecture about what the rule was. “Don’t say the rule,” instructed A to everyone. I reminded them that they can predict the out numbers.

“Put in 30,” said M.

“You can only put in numbers less than 30,” said A. So she put in 20, and out came 21.

Then, at the same time that S said, “Put in zero,” a visiting younger sibling said “Put in circle!”

“Do you mean zero?” I asked, and he nodded in the affirmative.

“Sorry,” said A, “but you can’t put in zero.”

“Does it break the machine?” I asked him. Yep. So we put the rule on the board that 0<in<30, introducing the restriction of the domain of a function.

“So put in 25,” said M. “Then 26 would come out.”

“Is she right?” I asked A. He hesitated; it looked to me that he knew the answer but didn’t want the game to end.

“Yes, but the machine only makes out numbers that are less than 26,” he declared. So I wrote another rule on the board: out<26. I can’t even begin to tell you how excited I am that a six-year-old just invented domain and range!

After a few more numbers in and out, it was S’s turn to lead. His intricate machine took a long time to draw. It was a biologically accurate representation of the human digestive system. (I found that – as usual – just ignoring the scatological comments vanquished them immediately.) As soon as someone proposed 100 as an in number, he immediately declared a range that excluded anything above 100. This tactic was different from A’s. A wanted to exclude suggested numbers to extend the game. S wanted to include suggested numbers but cap it there to keep the calculations under control. I could really sense the hard thinking going on in the room. As soon as the domain was set, S declared a range as well, although it turned out he didn’t need it. The calculations did get a bit tricky, which is very common, so I assisted on one of them. Then we were ready to move on to something else.


Math Circle Problem Solving 3My plan for today was to continue the ogre problem that we had worked on for the past two weeks. But since two students were absent, including the student who was the most enthusiastic about this problem, I decided to save it for next week. I was surprised that no one even mentioned it once during class. Usually when a problem lasts for weeks in a math circle, some of the students ask to keep working on it. Did they forget? Were they not as interested as I thought? I didn’t mention it at all this time. It will be interesting to see if they ask for it next time.


The graph theory problem that we did last week was in the book Avoid Hard Work to illustrate the problem-solving technique of wishful thinking, which means that you can be free with changing the problem and experimenting to gain some insight. The students last week didn’t need this technique for that problem, though, so I didn’t mention the technique. This week, I gave another problem from the wishful thinking chapter:

“Three sheep, Albert, Bilbert, and Cuthbert, are standing in a field, each happily munching away on grass. Is it possible for the distance between A and B, the distance between B and C, and the distance between A and C to be the same?” (p41)

This is the set-up for the rest of the problem, which introduces a fourth sheep, Dilbert, and asks the same question. A bit trickier, you may be thinking. This is where wishful thinking might come in. (“I wish we weren’t stuck to a flat field,” say the authors.) But we didn’t get to the fourth sheep. The challenge of three was enough for this week.

The group dynamic was different this week in our smaller group. The students were not as focused, and had a bit of trouble keeping their attention on this problem. They tried lining up the “sheep” (colored papers labeled A, B, and C) in a row in different orders, and quickly saw that this didn’t meet the requirements of the problems. Then they made them overlap next to each other, but that didn’t work. Most people were antsy now. One student stacked the three in a pile and said “there!” This looked like a solution to most, and everyone wandered away. Everyone but S. He saw that only the middle sheep was adjacent to all the others; the top and bottom sheep were not adjacent to each other. “Hey guys! This really is a problem!” he called out to the rest, and they all came back to see. Math Circle Problem Solving 3People were handling the sheep roughly now and they got bent and folded. This seemed to provide an aha moment. The students together folded the “sheep” in such a way that each one was adjacent to the other two. The students were now satisfied that this was a solution that met the requirements of their redefined question (the original question was about distance; they reframed it to make it about adjacency). The students acknowledged that this solution did require a bit of wishful thinking – that you can fold sheep like you can fold paper – but I didn’t explicitly use the term “wishful thinking” or explicitly state it as a strategy yet.

We now had less than 10 minutes left, but I decided to introduce a new problem anyway.


There are 3 bags of apples and 4 single apples in one tray of a balance scale, and 1 bag of apples and 5 single apples in the other tray. The scale indicates that it is balanced. How many apples are in one bag? (paraphrased from Avoid Hard Work, p42)

I asked the students to remember the first technique of problem-solving, to acknowledge their feelings about the problem. (Before reading more, readers, ask yourself what your emotional reaction to that problem is.

My emotional reaction came in the form of some negative self-talk, something like “Are you out of your mind, giving such a hard problem to such little kids! Surely you’re setting them up for failure and a subsequent hatred of math!” But then I calmed down, and remembered that we have weeks to work on it, there are strategies, and I’ve worked on hard problems with little kids for years.)
The students initially said what has become their habit to say about their emotional reactions: “crazy,” “want to run into a wall,” etc. But then S said “No! Cross off crazy. This problem makes me feel smart!” The others nodded in agreement. They like the problem.

Math Circle Problem Solving 3“So what can we do to try to solve it?” I asked. The scale I had brought in today was too small to hold apples. I introduced the strategy of drawing a picture. The kids were surprised to hear that this is a useful strategy for mathematics. As soon as I drew it, the students formed some conjectures about possible solutions: 500, 1, 2, 8, and 250. We decided to test one conjecture. If there are 250 apples in each bag, how many apples would be in each tray?

As we were trying to calculate this, L said “Wait a minute. What does it mean for the scale to be balanced?” Good question. I wrote it down, and stressed the importance of question asking in math. We discussed this, and I gave them more info about the problem: each bag has the same number of apples, and each apple is the same size. We were then out of time, and had to go home. To be continued next week!

— Rodi

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