I’m mad I didn’t actually take photos or record any of today’s precalculus lesson. Apologies. But even though this is going to be a textheavy post, I think it’s pretty awesome.

TL;DR: We fistbumped in precalculus. It was awesome. Super complex math got done.

One of the things I’m working on is improving my questioning this year. One of my strengths is scaffolding, but sometimes — in my desire to be super overzealously prepared — I scaffold too much. Today we had our first “long block” (90 minutes) in Advanced Precalculus. This is how class unfolded, after our warm-up.

I asked students to fistbump everyone at their table.

How many fistbumps did you just do?

How many fistbumps just happened in class?

Then I showed them there were many fantastical ways to fistbump besides the standard “clink knuckles” method. Blow it up. Snail. Squid. Turkey. [1] That was a random impromtu aside. But now, next class, I *must* show my kids the following video:

If you do this in your class, you should definitely have this video queued up. [2]

Then: everyone had 20 seconds of *individual think time for this question:*

If you wanted to devise an efficient way for everyone in the class to fistbump everyone else in the class, what would that way be?

Kids asked what “efficient” meant. I said “it should be as quick as possible, with the least chance of someone not actually fistbumping someone else.” Now you, friend, take a guess. I have 14 kids in my class. How long do you think it would take my kids to fistbump everyone else with an efficient strategy!

Seriously… reader… take a guess! Good. I’ll reveal the answer in a bit.

After the 20 seconds of individual time, each group shared with each other, and had to converge upon their proposal to the class. We went around. The four groups had three ideas:

- Line everyone up. The first person fistbumps with everyone else, then leaves. Then the second person fistbumps with everyone else, then leaves. And so forth.
- We have four groups in our class. The first group goes around and fistbumps with the members of other three groups in order. Then the second group does that with the remaining groups. And so forth.
- Do the exact same thing as proposal #1, except as you don’t wait until the first person is done fistbumping everyone else.
*As soon as the first person is done fistbumping with the second and third person (and continues on down the line), the second person starts fistbumping down the line*. And so forth.

(I had also anticipated students talking about getting in a “circle” and having one person fistbump with everyone, then another person, then another, etc. It’s organized, but not very efficient. One thing kids asked: can we all fistbump each other at the same time, in one giant mass of fists? I nixed that. I also had kids ask if you could fistbump with both hands simultaneously — to two different people. I said yes! But I didn’t give enough time for students to devise something super efficient with that so that never got turned into a proposal.)

As a class, we decided proposal #3 was going to be the most efficient. So I had them all file into the hallway and try out their fistbump method. I got my stopwatch out. And they went at it, after organizing themselves.

You may wonder what all of this has to do with math. That’s coming. This was just the setup. I honestly think by this point in the class, some kids were wondering what the heck we were doing this for…

So how long did it take them?

Yup. Under 12 seconds! I! Was! In! Awe!

Then each group got out a giant whiteboard and markers and answered the following questions:

How many fistbumps did you just do? What was the average time per fistbump?

Once they answered that question, they called me over to discuss their findings with me. Then I had two extensions:

We have 998 students at our school. How many fistbumps would that be? How long would it take, if we used our efficient method and assumed the same average time per fistbump? [3]

Can you find a method to answer that question?

And clearly, this is where the math comes in. This — in case you hadn’t seen it — is the classic handshake problem.

And from this point on, you have to facilitate class based on what your kids are doing. Some advice?

Advice 1: If kids are struggling, have ’em start noticing patterns about the number of handshakes for smaller numbers of people. Two people? Three people? Four people? Continue working up. Make a table. Look for patterns.

Advice 2: If kids have seen the “rainbow method” or some variation (see below), have them think about the difference between an even number of things being added and an odd number of things being added.

Advice 3: Have kids work on coming up with a single formula that works for even and odd numbers of things being added. Then have them explain why that formula works.

Advice 4: Lead kids to the idea of “double counting”: if we have 4 people, then have each person fistbump with everyone else. Since each person fistbumped with 3 people, there were a total of 4*3=12 fistbumps. However we’re *double counting* in this, so there are really only 6 fistbumps. (If kids don’t see the doublecounting, have a group of four act it out.)

Advice 5: If a group needs an assist, have individual members circulate to other groups and gather ideas, and then return and share what they found.

I loved doing this activity. Kids got into it. They felt ownership and camaraderie. Kids were up and moving. Because we had a long block, kids had time to play and productively struggle with the ideas. And most importantly: I didn’t overscaffold. I built up motivation and then sprung a good open-ended question for kids to work on.

[1] If you don’t know what I’m talking about, clearly you’ve never hung out with middle school students.

[2] queue is such a strange word, right? 80% of the letters are unnecessary. “q” is the same pronunciation as “queue.”

[3] ~~The answer is around 18 hours. What I loved is that when a group got that — after we got 12 seconds for our class — I was like “come ON guys? does that make sense? it would take almost a whole day with no breaks? REALLY?” I wanted them to see the answer was kinda absurd. But it is right, because although it might seem absurd on the surface, each time we add more people, we’re making the number of handshakes grow pretty darn fast! (Follow up? How fast? Let’s make a graph! Ohhhh, quadratic? PRETTY! And grows super quickly for higher numbers, unlike linear graphs.)~~ Turns out the answer is much shorter than 18 hours. I had a misconception that someone helped me see on the betterQs blogpost! I liked admitting to my class i was wrong!

Are you in a sequences and series unit? Also I was curious how many minutes do you have per week with your precalculus class?

This is actually a precursor (some “play”) before a sequences and series unit. Usually in a 7 day cycle, I have 4 50 minute classes and 1 90 minute class. (This is a new schedule.) This was almost 2/3 of my 90 minute class. The remainder of the time was counting subsets and noticing patterns.

I can’t be the only one reminded of the (pretty NSFW) season one finale of Silicon Valley [EDITED OUT LINK]

I actually thought of that (!), but didn’t say anything (obviously)! Hahahaha. We’re of a mind.

I hope you don’t mind, but in case a student of mine is reading this, I’m gonna feel weird that this clip is linked. I’m going to take out the link. I hope that’s okay.

Here’s a similar problem I was bothered by recently and just finally sat down to figure out:

I know the odds add up to perfect squares. Could I use that face to find the sum of all integers formula?

1 + 3 + 5 + … + k = (k+1)^2/2

So:

0 + 2 + 4 + … + k-1 = (k+1)^2/2 – (k+1)/2

Thus: 0 + 1 + 2 + … + k = 2(k+1)^2/2 – (k+1)/2 = (k^2 + k)/2

Success!

I’m curious as to where the lesson after this went? Did this lead into anything specifically? This was a really cool idea and I like how you actually had the students create ways to fistbump every student in the class.

Is the final formula for the number of fistbumps?

n(n-1)/2

I would’ve maybe liked to see a little more of the math explained, but this was very neat. Seems like it was probably very fun for the students as well.

Hi David. This was an introduction to series. Your formula is correct. The interesting thing about this the various ways *kids* started thinking about this, and the various math ideas they came up with. I didn’t really have the time to outline all of that in this post!