Matrices, Social Networking, and Algebra II

At the tail end of the fourth quarter, my students and I grew tired, weak, and weary from trigonometry overload, so we did a short one week lesson on matrices and systems of equations. I taught them how to add, subtract, and multiply matrices — by hand, and on their calculators. Then, I decided I wanted to bring some “real world” stuff to them.

So I decided to do a lesson on matrices and food webs [click here to view the assignment]. I pretty much stole it wholesale from some website or another (my motto: beg, borrow, and steal!), made a few changes, and let them go at it. And even though I don’t know how interested all of them were with the assignment, I was actually extraordinarily pleased at how well they did on it and how engaged they were in the classroom [1]. They talked, debated, and came to some pretty solid conclusions. My role in the classroom was relegated to going around and asking them questions like “so you answered the fourth question… can you tell me what the 2 in that matrix represents?”

You know, just to make sure they were getting it.

And they were.

One of my favorite moments was when a group asked me “do you add or multiply the matrices?” and I asked them “what do you think?” and then they got to arguing about it for 3 minutes before they came to the right conclusion.

Literally five minutes after finishing this activity in my first class, I realized that all the social networking sites (MySpace, Facebook, and the like) can be analyzed in the same way as food webs. Hello six degrees of separation!

So at the beginning of my next class where we were going to do food webs, I first drew a bi-directional network on the whiteboard with three teachers and one student. The student I chose is one who I felt I could poke fun at because he pokes fun at me. Of course I made up funny relationships between all my characters. So, for example, I said that the student liked teacher A, but teacher A didn’t like the student one bit — she told me that she thinks he is too rambunctious. And so forth. It was a tiny, fun little network, with all these fun little stories behind each relationship, and we made a tiny, fun little matrix from it. Then we moved on to the food web activity.

After class, I thought: why not do this whole social network thing next year? So last night I made up a fake set of relationships among teachers at my school and then created a network:

It’s pretty funny actually. I have one husband who likes his wife, but the wife doesn’t like her husband, and other strange relationships. And to accompany it, I made a draft of a worksheet to use next year [click here for draft]. And you know what: I think it’s pretty good. [2]

Besides food networks, and friend networks, I had two more ideas:

  1. Actually make a small celebrity network using IMDB, connecting them only if they’ve been in the same movie. A la Kevin Bacon. Then using that matrix to calculate the degrees of separation.
  2. Have students pick an airline and a bunch of cities it serves. Look at all the flights of an airline on a particular day — and make a matrix representing the number of flights that are made between all cities that one day. Some cities won’t have direct flights between each other — but that’s when you use the square of the matrix, to find which cities are accessible to one-another via one stop over. And you can take the cube of the matrix to find out which cities are accessible via two stop overs. And so forth.
Actually, I really like the second idea for some sort of take-home student project, where we also learn and use some basic Excel. Hmmmm….
And you were wondering what my last post was all about! Ah, gentle reader, I would not leave you hanging for too long.
[1] What was interesting to me about this assignment was although I saw them all working and thinking and grappling, showing true engagement unlike other times when I’ve failed, they didn’t show a true *interest* in the topic. Which makes me question the whole equality that teachers and administrators often believe in implicitly: student interest = student engagement.
[2] Although I might make two changes: (a) not make the network bi-directional (if person A is friends with person B, then person B is friends with person A), and (b) focus more on how to figure out how many degrees of separation someone is from someone else.
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