I spent a day on matrices and then we had winter vacation. Two weeks off. We came back and it took us two days to polish them off. In Algebra II, all we do is teach students some basics. I go over how to add, subtract, and multiply matrices. I remind students about multiplicative inverses. Then I introduce the identity matrix — so that we can talk about how . And finally we write systems of equations in matrix form, and use our calculators to solve the systems.
Early on when introducing matrices, I threw the following two slides on the board:
And then I asked, without students doing calculations, which grade took in the most money? We took a poll. Then I asked how we might figure it out. A student answered “well we take the number of sweatshirts and multiply it by the cost of each sweatshirt and add it to the…” and I said “hmmm, this sound like you’re doing a lot of multiplying and adding… we just did a lot of multiplying and adding in this funny way.” MATRICES!
So we were able to figure this out using matrices (and I showed them how to use their calculators to do this). Turns out that no student guessed the 10th grade (which was the right answer). They were so enamored by the sweatshirts that they ignored the socks! (Next year I might have them do a ranking — who made the most to who made the least.)
The next day, before we embarked on using matrices to solve systems of equations, I threw the following on the board as a do now:
FIND THE PRICES OF THE ITEMS! They just sort of sat there blankly. Well, a few said “I remember how much things cost from yesterday” but I said the school store was under a new regime of leadership and the prices have changed. I told my kids to guess and check or try anything they wanted. Most just sat there dumbfounded. We left it.
We went through class as normal, going over home enjoyment and solving systems (which is not easy to teach, btw, because you have to talk about how matrix multiplication is not commutative, how there isn’t matrix multiplication, how you need to have an inverse matrix, and how there is something called the identity matrix and how it acts like the number “1”). At the end of class I threw up the same slide.
Most kids knew what to do. They saw the system of equations, and how matrices could help them solve it.
I don’t know if I’ll keep the ordering of these problems the same — in terms of when I introduce them in class. I don’t think I gave them due deference. But for some reason, I really enjoyed them. Although it doesn’t really answer why we do matrix multiplication the way we do it, the first day slides really show them that there is some logic to wanting to multiply and add, multiply and add…. The second day’s slide really highlights how intractable some problems might be at first glance, and how powerful matrices are to get us out of a seemingly impossible quandary.