Guest Post: A Math Class That Requires a Party Playlist For Your iPod? Yes, Please!

This guest post by Dr. Diana S.Perdue (a fulbright scholar) first appeared here on White Group Mathematics, 25 January 2013.

All too often, math is the one class that seems to be a “no fun” zone. It doesn’t have to be that way though, and today’s blog post will prove it. I’ll describe a favorite lesson that includes a rich mathematics problem and will have your students actively engaged and enjoying class. Envision for a moment walking by a classroom and, when you stop and look in, you hear music blaring and you see students moving all over the room, shaking hands, counting, and laughing. Sounds like a fun zone to me! Even better, the students are focused on rich mathematics concepts like recursive pattern recognition, finding direct formulas for the Nth term, and quadratic functions. Here’s how you can create this experience for your class.

Perhaps you’ve heard of the “classic” handshake problem. It has many variations, but goes something like this:

You and your significant other go to a Valentine’s Day party with 49 other couples. If everyone shakes hands with everyone else, how many total handshakes happen at the party?

I like to use a three-part activity plan when I do this problem with my students where Part I requires them to do a small amount of pre-class work, Part II is our in-class work, and Part III is their post-class reflection. For the handshake problem, Part I asks them to review Polya’s Problem Solving Process: (1) Understand the Problem (2) Make a Plan (3) Carry out the Plan (4) Reflect Back including the list of heuristics (e.g. strategies) like act it out, use a smaller number, look for a pattern, make a table, and find a formula.

When my students enter the room on “party day”, I present them with the problem and give them about 60 seconds to consider the problem and give an estimate of their answer. We record the estimates in a big list on the board (so we can get a visual of the range of responses). Note: it’s not uncommon for students’ estimates to be all over the place, with a range in the thousands. Next, we discuss Polya’s process and they tell me everything important about the problem. Ideally, I want them to realize several key elements in this Understand the Problem phase: that a “shake” involves 2 people, that you can’t shake hands with yourself, that the total number of people at this party is 100, and that the problem is asking us for the total number of handshakes at the party (not the total number of hands you shake). This should about 3 or 4 minutes so don’t get “bogged down” with too many details here.

Since I usually use this problem as my introduction to problem solving, I guide them in step 2, Make a Plan, by suggesting several heuristics that we will try. Specifically, we choose to combine the strategies of using a smaller number, making a table, and acting it out. Here’s where the fun begins!

I whip out my iPod and tell them I’ve created a “party playlist” just for class (which I have done, trying to choose some good party tunes in the process – email me if you’d like to know my current playlist or if you’d like a copy of the activity sheet I use in my class). I plug in my mini speaker, tell them when the music starts that they are going to “party around the room”, systematically getting into “parties” of increasing size, acting out the problem (i.e. shaking hands with everyone), carefully counting their total number of handshakes, and recording their data in an organized table of information. I turn on the music and the party starts!

As they party around the room in Step 3, Carry out the Plan, students quickly figure out that they need to systematically get into “parties” of 3 people, 4 people, etc. and also to determine a strategy so they can know when they are ‘done’ (i.e. everyone has shaken hands with everyone else). Students will find many different ways to do this. A common circular pattern occurs where one student starts, shakes hands with everyone, then steps out of the group, the next person goes, shakes hands with everyone still in the party, then steps out. The process repeats, with the entire group keeping count of the total number of handshakes. They record their answers in a table as they go. It starts something like this:

 # People in the Party Total # of Handshakes 1 0 2 1 3 3 4 6

Depending upon the class size, we usually get through about 3 songs on the playlist as they collect their data. Then, I either turn off or turn down the music (their choice), and we get to the work of figuring out the pattern and determining the direct rule to solve our problem. It’s amazing how, after just these 10 or 15 minutes of “fun”, my students are motivated to sit and work hard on figuring out the rest of the problem. It is usually only a few minutes into our whole-class discussion that they discover the recursive pattern: how to determine the number of handshakes for a party of 5, for example, if they know the “answer” to a party of 4 is 6. We discuss how this is helpful only if you know the previous answer and I explicitly teach “recursive pattern” here. Since our problem asks for 100 people, the students immediately see the downside to this type of pattern (that to get the answer for 100 people, you have to know the answer for 99 people, which requires knowing the answer for 98 people, which means you need the answer for 97 people, which requires… aaaahhh!). Consequently, they long for a “better way”. That’s when I introduce the notion of a direct formula and the “Nth term” terminology. Students agree that what we need is a type of rule or function machine where we can put “in” the number of people in the party and it spits “out” the total number of handshakes (directly, hence the name). Side Note: this can have a nice visual connection as well if you add the ‘draw a picture’ heuristic to the mix and relate the various diagrams to the parties of different numbers of people.

The next 15 minutes or so involve me guiding them in the process of determining that the relationship between n (number of people in the party) and H (total number of handshakes that occur) is not linear. They love learning this “secret” hidden in the table with the technique of looking at the consecutive differences of the outputs. Once we figure out the rule is quadratic, it is a rather simple matter of beginning with a logical (but incorrect) first guess H = n2 and “tweaking” it until we arrive at the correct formula. I love this part in the class discussion because the wrong answer is actually crucial in figuring out the right answer. I spend quite a lot of time quizzing them about our guess and whether what we get is “over shooting” or “under shooting” the target (correct) outputs. Here’s part of the table so you can see what I mean:

 # People in the Party Total # of Handshakes 1st GuessH=n2 ? 1 0 1 2 1 4 3 3 9 4 6 16

Notice that every time we are getting an answer that is greater than what we want (e.g. we get 1 but it’s supposed to be 0, we get 4 but it’s supposed to be 1, etc.) – that means our next guess needs to be a function that is less than n*n. This is the part of the classroom discussion where I suddenly stop and ask, “Now if I’m over here partying by myself (I start dancing around), how many handshakes are there?” After they tell me that no handshakes happen with a party of 1, I ask them why to make them remember that first key element of the problem: you can’t shake hands with yourself. YES! n*n means we’d shake hands with ourselves! It should be n*what? Of course my students by now are used to my antics and they figure out that this is a hint to help them determine the next guess for the direct rule. We usually go through at least one more incorrect guess before the light bulb comes on. It is always a glorious moment in the class where they “get it” and discover the correct formula that works!

During the last step in the problem solving process, Reflect Back, I make a point of illustrating how the two key elements from step 1: it takes 2 people to make 1 handshake and that you can’t shake hands with yourself, manifests in the final correct formula. The formula really makes sense to them as they understand where the parts of it come from and, of course, we use it to both check our actual data and to answer the original problem. I then quiz them with a party of 5,000 people just to illustrate the power of a direct formula and how this is not a problem any more, it’s easy!

You have noticed, of course, that I’ve purposely left out the actual formula that works – let’s say, “the proof is left to the reader” – but I will mention, for those of you who want to check yourselves, that the answer for 100 people is 4,950 total handshakes (and the answer for 5,000 people is a whopping 12,497,500 handshakes!). I will also state, for the teachers who are reading, that this activity is a beautiful one to use in Geometry classes as the pattern is so very close to that of the number of diagonals in a polygon (in fact, that’s where I first used this activity).

I’ve even done this lesson in other countries: here’s a picture of my students comparing their data after acting out parties of various sizes. It was taken at the Kigali Institute of Education (KIE) in Kigali, Rwanda. Note the students outside the classroom peering in the windows – my class drew quite the crowd as the “fun” place to be on campus.