We finished up our unit on Rigid Motions by taking a brief look at transformations using matrices. At the beginning of class, we made sure each student could transform a given point in the coordinate plane and generalize the transformation.
Our students do not have any experience multiplying matrices. I didn’t want to get into a huge lesson on multiplying matrices, but I also wanted my students to realize that they had the background they needed to make sense of transforming triangles using matrices.
We used TI-Nspire technology to look for regularity in repeated reasoning. We started by multiplying the matrix that represented the vertices of our triangle (1,4), (4,–3), and (–2,5) by the matrix shown to observe how matrix multiplication works.
By what matrix should we multiply if we want the output to be the vertices of the triangle after it has been reflected about the x-axis?
It took NR only two tries before she got the correct transformation matrix. It took others more than two tries. But once students had the matrix for a reflection about the x-axis, it didn’t take long to get the matrix for a reflection about the y-axis.
And then reflections about the lines y=x and y=-x.
And then rotations 90 degrees and -90 degrees about the origin.
I honestly do not care that students remember the matrix that produces a certain transformation. I am not even sure that I should have spent any class time on this topic (although I have seen questions on some state assessments about this topic). What I care about is that students know that they can make sense of problems and persevere in solving them. I care that they know that they can figure out the matrix that produces a certain transformation instead of me giving them a list to memorize.
And so the journey to create students who can figure out mathematics instead of being told mathematics continues …