Conway's Rational Tangles

I had a bonus long block period (85 minutes) with one of my Algebra II - Honors sections this week.  The other section lost their long block due to the holiday.  So, I decided to have them work with Conway's Rational Tangles.  I won't re-invent the re-explaining wheel here, but I will point you to some links.

I first saw this through a YouTube video by Tom Davis.  I adapted it to my classroom and I've done the activity four or five times with different groups of students.

After my first day at Twitter Math Camp this summer, I was back in my hotel room thinking about how I was surrounded by amazing educators, and it occurred to me that I'd be a fool not to ask questions when I had the opportunity. I remembered the Tangles activity, and how there were some pieces that hadn't ever quite worked.  I figured somebody in Cleveland with me had written something about this, so I set to Google and --- BAM! -- there's a great article by Amie Albrecht describing the activity and reflecting on her own experience with it.  (This is where you should start if you haven't seen the activity before.)

In a moment that sums up the joy of TMC, Amie was waiting for the bus the next morning when I walked outside the hotel.  I accosted her with my questions, and so as we walked to find some food, she told me about her experience with the activity. 

The most interesting thing Amie added to the activity was a Notice-and-Wonder exercise that allowed students to at least try to discover the rules of the activity themselves.

Which brings me (at last!) to my experience on this round.

I began by showing Amie's "Notice and Wonder" video.  It's a sped-up video of students doing the final Tangles exercise.  Students noticed and wondered the same sorts of things Amie's students did:  They only seem to rotate in one direction, sometimes two of the students switch places, what's the deal with the bag, etc.  The video was an addition to my usual presentation of the problem, and I liked it.  It engaged students in the exercise right away, and made them curious immediately.  I liked too that they were seeing other students in another classroom doing what they were about to do.

Next, I clarified the rules they had noticed to them -- we rotate clockwise, and can only "twist" from two positions in one direction.  I had four students demonstrate with our ropes, following my commands and creating a tangle.

Then I made a careless error in my approach.  I had divided them into two groups, and I was going to send them off to figure out how to untangle a tangle formed by a series of twists -- how would you untangle one twist?  Two twists? Three twists?  And so on.  To make myself clear, I carelessly wrote "T", "TT" and "TTT" to demonstrate what I meant.  Here's why that's a mistake:  one of the things I love about this exercise is that it starts out in a very tangible and physical way -- ropes are being tangled.  However, to keep track of what's going on as you try to solve the problems, it becomes necessary to come up with an abstract way of expressing what's being done.  In the past students have written out the words "Twist" and "Rotate", used arrows to indicate that one follows the other, and made up other sensible but cumbersome methods of recording their progress.  Eventually, they settle on the simple string of R's and T's.  I think it's valuable for them to have that experience because it mimics a lot of what mathematical notation does -- it takes a complicated situation and makes it simpler to express.

And I robbed them of this part of the exercise, because I was distracted by difficulties with the projector and my hurry to get to the project.  Dang.

So, I sent them off to solve the Problem of the Twisted Tangles.  I wandered between groups, and I was pleased that they each solved the problem in about 15 minutes.  What I love about this part of the process is that they discover that to undo a T, they do an RT.  To undo a TT, they guess it will be RTRT, only to discover that they need an extra T.  Finally, they guess that to undo a TTT, it'll be RTRTTRTTT.  When they try this, they discover that they've untangled it without that last T.  Then they feel confident that to untangle TTTT it will be RTRTTRTTRTT.  And it is.  The solution kind of finds itself.

We returned to the classroom, and summarized what they'd found.  A student wrote on the board:
T = RT
TT=RTRTT
TTT=RTRTTRTT
TTTT=RTRTTRTTRTT

I should have advised against the use of the equal sign here, as it implies something else, but I let it go.  (It was a source of confusion later, so I regret that decision not to nip it in the bud.)

I had students stand with the rope, and we talked about something one of the groups had discovered -- it never makes sense to do two rotates in a row.  I demonstrated with a couple of different tangles.

Next, I had them make a TTTRT tangle to see if they could undo it.  They struggled a bit until someone realized that it was the TTT pattern starting to untangle.  Then they could just read the solution of the board.

I made up a tangle that they couldn't solve the same way -- something like TTTTRTTTRT.  We'd need another method.

At some point, a student had referred to the starting position as "a nothing tangle."  I suggested that we follow this and give it a number.  The consensus was that 0 made sense for the starting state.  Then we needed to decide what a Twist does to the number -- several suggested that it should add 1 to the value of the tangle.

At this point, I paused to give them a moment to see that since we were defining both the starting point's value (zero) and the role of a twist (+1), there was no "wrong" way to choose a value or an operation.  We could choose the starting tangle to be a 7, and a twist to be a "multiply by 5."  Of course, this would likely make the overall solution of the problem unduly complicated, so it makes sense to keep things simple.  (I'm pretty sure that it would be solvable with these first conditions, but not certain.)

Now the tricky part:  What does a Rotation do?

We looked at tangles that we knew how to solve:

TRT would be 0, then 1, then x, then 0.  But the final zero gives us x+1=0, so x must be -1.

Which means that R gets us from 1 to -1.

Perhaps R is "subtract 2"?  But then TTR would be a zero tangle, and it clearly isn't.  Hmmm.

Perhaps R is "multiply by -1".  Then TTRTT would be a zero tangle.  And it isn't.
Double hmmm.

After some quiet struggle, I had the students start with a zero tangle, and do a rotate.  The question is: What number do we have here?

I tell them to do a twist.  What happens?  Nothing changes.  Which means the number must be a solution to the equation x + 1 = x.  Hmm.  At this point in the past, I've had a student recognize that x might be infinity.  No such luck this year.

What number plus 1 equals itself?  I gave students the hint:  "Think of a playground taunt."   Sometimes this is enough -- not this year.  I give them enough coaxing to get: "I hate you infinity," and the retort:  "I hate you infinity plus one."  They see what I'm getting at, and they also wonder aloud if I had a difficult childhood.  (Not so much, but I probably need a more positive prompt.)

Anyway, once they see that infinity fits the bill, the question becomes: What operation can we perform on the number 1 that will bring us from 0 to infinity and then back to 0?  They puzzle and guess some things, but eventually someone suggests dividing by 0.  This leads to someone else suggesting that taking the reciprocal will do it.  They piece together that it's the negative reciprocal, and we're off to the races.

As you can tell, I haven't figured out a slick way of having students discover the rotation operation.  They'll look at some tangles and makes some algebraic equations, but it's different when the operation is the variable instead of the number.  It seems like relying on insight and instinct (and hints from the teacher) is the only way to go.

We tested our hypotheses (T is "add one", R is "take the negative reciprocal") on some of the tangles we'd solved.  Then, for homework, I gave them a more difficult tangle to undo.  We calculated that the value of the knot was 14/8.

They returned the next day with some success, and enough agreement on a solution for us to try it out.  We made the original tangle.  Then I inserted each of the four ropes through a plastic bag.  We then followed the "untangle" steps which created a tangle of rope and plastic.

The big payoff:  as students stood in their spots holding the ropes, I tore away the plastic bag and - TADA! - an untangled rope.

My (continuing to develop) conclusion:  This is a good exercise for this type of situation, and a class of about 12.  My class of 16 meant I should have probably had a third group, but I was lacking both ropes and space to do so.  With 8 in a group, it was possible for students to be disengaged, or to have their ideas lost in the ruckus of a larger group. 

I also find that it's easy to think that the four students actually holding the ropes are engaged, but sometimes they are just following directions and not really thinking.

I continue to have trouble getting someone to discover the rule for rotations.  The project is always more satisfying (for me and for the students) when a student has the "negative reciprocal" insight.

Still:  I like this activity a lot.  It's different, it's accessible, and it has a nice concluding moment, worthy of cheering.  It also has some sophisticated extension questions (Can you make any rational number into a knot?  Can any rational number be untangled?), although I must admit I haven't had students pursue these in the past.

For sure, I'll do it again.