One problem with fully planned lessons is that it narrows the opportunities for the teacher to be surprised. This is a story about a lesson that was not fully planned.

I teach in a high school full of kids who passed a hard test. Even the students who are, for us, ‘lousy at math,’ are actually at least ok, and usually pretty good at it. But I teach a class of seniors, most of whom are not taking calculus. Self-selected. One trick we use to get some motivation is NOT to teach them the things they already have been taught, but just make it harder. Instead, I teach Matrices and Vectors – material is not too hard, but it’s all brand new, so that helps with the interest level. Plus, like I wrote, in my school they are not the ‘math stars’ but they are actually as a group pretty good at it, and some individuals are quite good.

I’m transitioning from vectors in the plane to vectors in space. That’s where I was when the Wednesday/Thursday before break was disrupted. And we are about a day and a half ahead of the other section. Yesterday we reviewed coordinates in 3 space, set up coordinates in the room, discussed midpoint, developed the extension of the “distance formula” – but we also discussed why slope was tricky, and what a vector perpendicular to a plane might look like. We were previewing work that would come later, and reviewing work that came before, but with almost half the class out, that was all we did.

I came today ready to improvise – I had a few directions to move in. What made sense depended on who came. Attendance was up enough, almost, for regular class. But almost. And we were still ahead of the other section. So I would do a little new, and review a topic from October: determinants of matrices.

More specifically, determinants of 3×3 matrices, which we had played with, but the students had not passed a quiz on that topic (we use mastery quizzes). We will need this skill next week to find cross products of vectors (and do some work with planes).

I remind the kids about the notation for determinants: absolute value bars, or double bars, or “det” + parenthesis, ie |M| or ||M|| or det(M). They calculated a couple of determinants of 2×2 matrices. And then I choose for a first 3×3 example one with variables in the first row. That’s what it will look like when they calculate cross products. Plus, less arithmetic. Make the first example easy.

I wrote in a then b then c for the first row. For the second, avoiding 1s and 0s, I wrote 2 then 3. I chose 5 next to avoid creating an arithmetic progression, but instantly saw that I had numbers from the Fibonacci Sequence.

$\begin{bmatrix} a & b & c\\ 2 & 3 & 5\\ & & \end{bmatrix}$

So I continued:

$\begin{bmatrix} a & b & c\\ 2 & 3 & 5\\ 8 & 13 & 21 \end{bmatrix}$

And listened as a kid told me the determinant was -2a – 2b + 2c. Hey, that’s weird. I ran the work step by step, to keep everyone on board:

$a \times \begin{bmatrix} 3 & 5\\ 13 & 21 \end{bmatrix} + b \times \begin{bmatrix} 5 & 2\\ 21 & 8 \end{bmatrix} + c \times \begin{bmatrix} 2 & 3\\ 8 & 13 \end{bmatrix}$

And yes, for those of you jumping up and down, those are all plus signs. Not an error. I always begin the minors below and to the right of the entry I am multiplying by.

Then I tried an “easier” one:

$\begin{bmatrix} a & b & c\\ 1 & 2 & 3 \\ 5 & 8 & 13 \end{bmatrix}$

And when I heard that the determinant was 2a + 2b – 2c I was pretty sure we had stumbled onto something at least a little interesting. That’s when I was pretty much ditched the rest of the plan. I told them that that was weird, and that I had not intended it. We talked a bit about Fibonacci numbers, how to calculate them, a couple of problems that modeled them. We calculated a few. I wrote a recursive formula, wrong, without dwelling on it (too many terms before the general term). And now the room was highly engaged. We filled in some more runs of 6 Fibonacci numbers, and sure enough, the determinant when filled starting with $\emph{f}_n$ will be 2 if n is odd, -2 if n is even.

This surprised me a little, and was new to me. But it made sense that there was something going on with Fibonacci. And I told them. Maybe it was a discovery. And then I got the most interesting questions: how would I know if it had already been discovered? Where would I check? What work would I do? Who would I ask? And if it was new, what would I do? Who would I tell? And some of the questions were coming from kids who ask fewer questions. I answered them. I took pictures of the board. And I promised I would report back.

And an hour later one of the students, one with a bit less than average enthusiasm, stopped me in the hall to ask if I’d determined if the result was original.

But I have started. I went straight for Proofs that Really Count by Arthur Benjamin. And I found straight off several identities that will help. I think I can make our result fall out of them. And then I played around. And then I googled a paper on a related problem (with answers -1 and 1 instead of -2 and 2. Cool!)

I may try to write something up. I’m not sure there is enough meat here to be worth more than a note. But I will have a class full of fans egging me on.