So we’re two weeks into the Rational Tangles math circle.

Rational Tangles is an activity within the mathematical realm of knot theory where the students tangle and untangle ropes to uncover mathematical properties. With students this age (12-13), topics such as negative numbers, geometry (rotations, reflections, transformations), strategies to test conjectures, order of operations, mathematical operations, adding and subtracting fractions, reducing, infinity, and fractional reciprocals have/are going to come up. I’m generally following mathematician Tom Davis’ approach to this activity.* My intention has been to give the students the basic premises of the activity and then lead it in a totally inquiry-based way where the students’ questions dictate in what direction we take it.

### The Introduction to “Rational Tangles”

At the beginning of our first week, I introduced the activity, and had four kids come to the front of the class and hold the ropes. Once we established where everyone was supposed to stand, I introduced the different moves, or ways that the kids were allowed to move to tangle the ropes. First we started with the command “rotate,” where all of the students holding the ropes were supposed to rotate 90 degrees. “This is geometry, I think!” exclaimed J when she heard the phrase “90 degrees.”

It took a few minutes to establish the fact that all of the people holding the rope were “identical,” as I explained, or that even if different people were standing in different positions, all that mattered was the position the rope was in. The people were supposed to be completely interchangeable. We did two rotates, and the ropes were back in the same positions they were when we started, but the people were in different positions. “How do we get back to where we started? Is this the same position/tangle as when we started?” I asked. “No, we have to do more rotates to get back to where we started, because we are in different positions now,” posited M. It took a little explaining for the group to understand that it didn’t matter.

“What’s this have to do with math?” J asked. I explained that there is a whole field of math called knot theory that this is part of.

“With rotate, does everyone have to rotate, or only some people?” R asked. I told her that everyone has to.

I then explained the move “twist,” where one person lifts their end of the rope high over their head and the person next to them walks under the lifted rope, pulling their end of the rope along with them.

“Why can only L go under?” asked M. I explained that only the person in one of the four positions, or spots on the floor, can lift the rope and that only one person can go under it.

After we practiced twist and rotate a few times, and switched out the students holding the ropes (I tried to switch them out as often as possible, which definitely proved to be necessary – after sitting or standing for a while, they got tired and needed a change of pace).

### Time For A Challenge

Then it was time for a challenge. “Do a twist,” I instructed. “Now, here’s a challenge for you. Try to untangle the knot in as few moves as possible.” The students were immediately intrigued, and all started positing conjectures at once. This activity is a great “accessible mystery;” everyone is interested in forming and untangling knots, and what’s great about this particular activity is that once the rules/premises are explained, the students can completely take over. And take over they did.

“Let’s rotate, then twist.”

“No maybe two twists, and then rotate.”

“Okay, we need to do two rotates.”

“Wait, no, that won’t work!”

“Okay good, the knot looks smaller now.”

They worked on the knot for a while and finally got it untangled.

“Let’s tangle again!” all the kids enthusiastically exclaimed. They were all really excited to do more.

Here’s the knot they made (T=tangle and R=rotate): TTRRR.

A lot of debate ensued about how to untangle this much bigger knot.

J suggested a move. “I think this helps.”

“No, it doesn’t do anything,” replied L.

The kids started getting frustrated, so I intervened with a guiding question.

“Does anyone have any idea about how to untangle this? Is there a certain pattern of twist and rotate that you think would work?”

At this point, the kids completely took over.

“No, no, that’s going to make it even worse!” exclaimed A. “And it’s really ugly now.”

After a lot of debate, failed attempts, and collaboration, they finally untangled the knot.

### One Last Challenge

They were looking a little tired, and class was almost over, so I introduced a new challenge.

“Rodi thinks that she can untangle any knot you make without looking,” I explained. The kids looked at her incredulously.

They wanted to make a super big knot to really challenge Rodi, but I made them stop after 15 moves. Here’s the sequence, for anyone who’s interested: TTRTTTRTRRTRTTT.

Rodi and I covered the knot with a plastic bag with four holes cut in it for each of the rope ends to go through, and then she started scribbling in her notebook. The students tried to peer over her shoulder to see what she was writing, but she kept it secret.

As she scribbled, she called out moves. “Twist! Rotate! Twist again!” Once she was done, everyone waited in suspense as she cut the bag off the knot.

“Ohhhhhh.” A collective sigh went through the room (or, more accurately, the garden, as we were outside). The knot was still as big as before. The method hadn’t worked.

“That’s so weird, I must have made a mistake. This should have worked,” said Rodi, while she consulted her notebook. “Oh yes! Here’s my mistake. Maybe you’ll give me a chance to try again next time.”

“Sure,” the students agreed.

“What was her method?” the students asked.

“I’m not telling!” I responded. “But I’ll give you a hint: she used math.”

By that point our time was up, and I dismissed class with the promise that “Next time we’ll try to figure out what Rodi was trying to do.”

As I was packing up, R came up to me. “How did Rodi work it out mathematically?” she asked. “Hmmmm, we’ll have to see,” I replied.

And with that, our first class concluded.

## Week Two

We started our next session with letting Rodi try again to untangle a knot without looking. This time, I double checked her work, and when we cut the plastic bag off the knot, there was no knot left. The kids were all shocked that she was able to do this, and clamored to know how.

“Do any of you have conjectures about what she did?” I asked.

“Maybe she did our moves in reverse,” posited L.

The other students started nodding.

“How would you test that?” I asked.

“Do a smaller example,” suggested M.

They tested out making a knot, and doing the opposite of their previous operations, TTRT, but it didn’t work, as A had previously guessed (“It won’t work,” she warned, as they were trying to untangle the knot).

J tried out a conjecture of hers (another pattern), but it didn’t untangle the knot. The students were silent, not sure what to do next. “Do you want a hint?” I asked the room. They nodded.

“Let’s try starting at our original position, where the ropes aren’t tangled at all.” This gave the group some direction, and they proceeded with more testing and conjectures. Then we started correlating the knots to numbers. The students had various conjectures about what numbers and what mathematical operations corresponded to the positions and moves. “Does the position of the ropes influence the math?” the students wondered.

We got to the point where everyone accepted that twist is plus one. They had numerous conjectures about what rotate does (subtracts 2, subtracts 3, or switches the number to its opposite). “How would you test your conjectures?” I asked. The students tested the conjectures with different knots, and ended up rejecting all of their conjectures. We were out of time, I asked what they wanted to do next time, and they said “Let’s find rotate,” so that’s the plan for our third session together.

— Rachel

* For details about the specifics of Rational Tangles, click here (PDF).

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