Voting on How to Proceed with Conjectures:
R wasn’t here last week, so the students did a brief recap for her of what we did last time, which was testing their conjectures for which mathematical operation the move “Rotate” represented.
“I’ve been thinking about the rotate thing, and maybe it’s multiple operations, not just one thing,” M mused.
“If you want, I can tell you what Rotate does, because it’s very hard to figure out,” I informed them. “Or, you can keep testing your conjectures. Who wants to do what?” The students voted. The room was pretty split, so we decided to keep on testing conjectures.
“Rotate has to be taking away something, or we could never get back to zero, right?” M asked. The other students nodded. Since some combination of Twist and Rotate untangles a knot, or “gets it back to zero,” her conjecture made sense.
The students were pretty stuck, so I asked them if they wanted to ask me for hints. I doubted that they could figure out what Rotate does with their amount of math knowledge, so I thought it wouldn’t hurt for me to help them out a little. Here are some of the questions they asked me:
“Are fractions and decimals part of this?”
“Is it using adding somehow?”
“Does it divide and go into negatives?”
“Is it multiplying by a negative?”
“What does multiplying by a negative mean?”
We took a little break from guessing what Rotate does for the students to explain to each other what it meant to multiply by a negative, with a little help from me, and an example from Rodi.
The students asked for more hints, and made more guesses. I admitted to them that they might not know the math to figure it out. After spending forty minutes on trying to figure out what Rotate does, I paused to assess their feelings about moving forward.
“So, you’ve been working on this for forty minutes,” I informed them. They were shocked. “We can either spend more time trying to figure it out, I can tell you the answer, or we can do some other fun activity for the rest of class today, and I’ll tell you what Rotate does at the beginning of next week.”
Two students wanted to keep on figuring it out, two wanted me to tell them, and two didn’t say anything, so I decided to do a fun logic problem to finish off class.
A Bit of Logic:
We did an activity with matches, arranging them into wine glasses holding “Shirley Temples” with cherries in the middle. The challenge was to rearrange the matches into a certain configuration in only two moves. The students solved it in a matter of minutes, so I presented another logic problem, which proved to be much more challenging. The problem described a scenario with two professors and a secretary eating lunch together, and then challenged the readers to figure out which person in the story had which color of hair (giving additional information, of course). The students started working together, and writing information on the board. R was working on her own, and, after a few minutes, wanted to check her answer in the back of the book.* I warned her that the book had a lot of sexist language, and, reading the explanation, she agreed. For example, in the explanation it said that you would naturally assume that the woman in the group was the secretary, and that, instead of having white hair, the woman must be a platinum blonde, therefore assuming that women can’t/don’t have white hair.
We were out of time, so some other week we’ll finish this logic problem, and discuss the sexism in the book in terms of its historical perspective. I promised the students that at the beginning of the next class, I would tell them what Rotate was.
“You said you were going to tell us what Rotate was today!” A exclaimed as soon as she got into the classroom. “So why are the ropes here? Aren’t we done?”
“Nope,” I replied. “But you’re totally correct that I’m going to tell you what Rotate is. But once you know what Rotate is, then you can tangle and untangle the ropes using math.”
The Horse Horseshoe Boots Problem:
We started off the class with an interesting math problem that had been circulating the internet.** I showed it to the students and asked them if they could solve it. They worked together, and some debate ensued: is multiplication the same as division? How are they different?
M taught the class the order of operations, and J made up her own acronym for PEMDAS: “Please Eat My Dairy At Sundown.”
“But why is the order of operations in that order?” J asked. We discussed it.
After coming to a consensus about their answer to the problem, I surprised them by saying that different people have gotten different answers, and that the answer may depend upon interpretation of the problem.
However, their answer was the correct one, and, when I told them, they were very excited.
Reciprocals and More:
Once all the students had arrived, we got back to the ropes. “So, your conjectures were definitely on the right track,” I informed them. “You guessed that Rotate was a two-step operation, that it involved changing a number to its opposite, and that it somehow involved division, and you were right about all of those.”
“Just like we said!” They exclaimed.
I explained that Rotate was a two-step operation. First, it changes a number to it’s opposite, like 2 to -2. Then, it changes that number into its reciprocal, like -2 to -½. After briefly clarifying what “opposite” means and what “reciprocal” means, the students grasped the concept quickly.
“Is it (-1)/5 or - (⅕)?” M asked. “Like, is it turning the whole number negative, or just the top part?” I explained that it was turning the whole number negative.
“I think you’re ready for a challenge now. It’s time for the ropes,” I said. “Your challenge is to tangle a knot and then untangle it using math. I suggest that you make a small knot with only a few moves to start, because it can take a while to untangle it if you do a big one.”
The students took over, with J writing down their moves on the board. They decided together which moves and how many to do. Here was their knot: TTRTR.
“What number are we at now?” M asked.
“Should we make a number line to figure it out?” J asked.
“I think it would get too complicated,” M replied.
Adding a Positive to a Negative:
The students worked out that after two Twists, they would be at 2. Then after a Rotate they would be at -½. The challenge came when they had to add 1 to -½ (a Twist). They had lots of conjectures about what number they would end up with:
Eventually J used the number line to explain why it would be positive ½, and the class agreed. We had two minutes left in class, but R realized that, after another Rotate, we would be at -2, and we just had to do two Twists to untangle our knot.
The students did two Twists, and were thrilled that the knot was untangled. We’ll keep on working on challenges like this with the ropes in the next few weeks.
*Raymond Smullyan, “What is the Name of This Book”