May 26, 2013

Matrix Mess

One of the challenges of PreCalculus is making it seem like a cohesive course rather than a list of skills to review and master. My theme lately has been "solving problems efficiently." Students have learned a variety of methods to solve, for example, systems of equations. My goal is for them to master each method and know how to choose amongst them. Since we had just finished the unit on conics, we solved systems of second degree equations (smoother transition plus a more challenging set of problems than the ones they'd seen in both Algebra 1 and 2). That all went fine and students worked on elimination, substitution and graphing. However, I also wanted to review matrices and so I upped the ante to three variables (but returned to first degree equations). The problem was, I didn't want to take the time to properly review matrices, I just wanted to remind students that they're useful when solving systems. This obviously backfired.

After a rough week of allergy haze, it was already after school on the day before I needed to give this lesson and I still didn't have a plan. I had a vague idea of what I wanted to happen but couldn't even put it into words when explaining my thoughts to a coworker. Despite my incoherence she helped me out by putting together this worksheet (which she fully expected me to edit but I was too exhausted to do much):

I also put this problem on the board:

And had this diagram ready to show students who needed help remembering how to multiply:

You already know this is going to go badly because the material is split into three places and is dependent on me showing up with a diagram at the right moment. Students were totally baffled why they were multiplying matrices. They didn't understand that the calculator would be taking the inverse (some tried to multiply the matrix by -1). They didn't read the instructions all the way through and were calling me over to help with the calculator. 

My attempt at a rewrite:

Thoughts and further suggestions would be much appreciated.


  1. I'm at a parade, so can't see your documents. In a pre-calc course I think the way to show this, so that the kids see the reason for doing it, is to start off with the equations, put into matrix form and to do Jordan-Gaussian eimination by hand to show what is occurring. (I ignore the multiplying by the inverse matrix explanation). Then when the kids get capable, ask how many would like an easier way. (The 84 and nspire can both simply simplify rows and/or jump directly to the solution.... share whichever you wish with the students, though I think not just jumping to the solution is a better petagogical way of doing this)

    I've not taught pre-calc but this is how I show it in Alg2 when I cover matrices. (Now if I can just get he Alg1 teachers to stop showing how the calculator can just short-cut to the solution of a system of equations with matrices. The Alg1 students NEED the practice with the algebraic manipulation and graphically understanding a solution of a system of equations.)


    1. My original plan was to do the lesson you described, but since my kids did this method in Algebra 2 and my goal was to review I went with the inverse method. I almost wonder if this lesson was even worth doing if I'm not willing to do it justice though.