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This is my (CAM) very favorite of the activities in the whole Paradigms Program. Light bulbs go on everywhere. Student leave the activity with a definition for eigenvectors that is easily generalized to eigenfunctions and eigenstates in quantum mechanics.

In 2008 (include videoclip and Emily's transcript), during the class discussion about what each of the matrices does, I (CAM)tried having the class make hypotheses about the geometry of the determinant. They loved it! Here is what one student said:

“Watching each group come up, present their case, show their transformations, and then show their determinant, and make a hypothesis, then watching the next group blow that hypothesis out of the water, the whole class come up with a new hypothesis, would never have happened in a 10-minute lecture, for sure. And I think watching that process, and watching others thinking through that process while you think through it yourself is definitely something that has never come up in any other class. Why is that? I'm not sure how it matters, but it's definitely a different way to think, and I think it makes me think about the process, more than the final answer, and I think that's pretty much what it comes down to, it's focuses on process more than focusing on the material.”

## Comments by David McIntyre (Winter 2011):

I taught the preface this year. This activity works well for teaching students basic mechanics of matrix-vector manipulations, and also gets at key concepts. There is a good mix of obvious transformations/operations with some crazy ones that are hard for students to describe. It is important for instructor to wrap up after all groups have presented their results, so that key points are made. As someone with more QM experience than geometric transformation experience, I found it useful to have my cheat sheet handy so I knew what each operation did. I made a Mathematica sheet with all the answers, which should be posted here somewhere.

## Comments by Mary Bridget Kustusch (post-doc, co-teaching Winter 2012):

I thought this activity worked wonderfully as a start to the term in re-establishing classroom norms. It also worked out perfectly that we had 12 groups, so each group got their own matrix.

I was a little surprised at how quickly they got through the actual activity part. I knew that the focus of this activity is in the wrap-up, but that came up much more quickly than I expected and I probably should have started it earlier or had each group actually do two matrices.

I found that reading through the narrative was incredibly helpful in identifying the points to cover, but also in seeing how they might be brought out naturally in the conversation. I made a list of the points that I wanted to make sure that we talked about and others that it would be good to emphasize if it came up, but then tried to let those come up naturally. For the most part, I think that it worked well. There were a few places (particularly in our discussion of projections) where I think I may have let them explore the space a little too long, but on the whole, I felt that the wrap-up was much more of a conversation, than a lecture, which was great.

Corinne also pointed out that I brought up the idea of hypotheses in a way that was pretty leading, in that there was really only one logical hypothesis at that point. This restricted the exploration probably more than I wanted, but I was able to open up the space again as we went in a way that really allowed for some good discussion.

The two lengthiest discussions with this group were about reflections and projections. The differences between reflections about a line and through a point stimulated a lot of comments, even from some students who don't often speak up in class, as did how to revise the hypothesis based on the $A_4$ matrix (rotation by $\pi$ or reflection through the origin).

We didn't talk about the matrices in “right” order. Someone suggested looking at $A_10$ after discussing $A_6$ since both are projections. Then, since we were running short on time, Corinne skipped from Group 8 to Group 12, so that we could talk about scalar multiplication. Then, as we were about to wrap it up, someone asked a question about whether projections with determinant of 0 had to be along the line $y=x$, at which point, Group 11 piped up and presented theirs. So, I think the only one we didn't actually talk about at all was $A_9$.

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