Function Transformations

On the Friday before Spring Break, I gave my Algebra II class a quiz on function transformations. It only had reflections about the x- and y-axes, and vertical and horizontal shifting.

I know, I know, you can’t believe how cruel I am, doing something in class the last day before Spring Break.

Now that we’ve gotten that out of our system, back to the point at hand. Today, I finally got around to grading them. And I have to say that I was really pleased with the results. With the exception of one or two exams, all students did really well.

I told students they needed to memorize the eight standard functions and key points on them (the standard functions include $y=x^3$, $y=|x|$, and $y=floor(x)$). Key points are points I require to be correct on all graphs — after the transformations, they need to be in the right place. So, for example, I require students to know that (-2,-8), (-1,-1), (0,0), (1,1), and (2,8) on $y=x^3$, and then when  they were asked to graph $y=-(-x+1)^3+1$, they need to make sure each of those five points are in the correct place.

A few students — as expected — mixed up translating right/left. And a few performed the reflections last (when they have to perform them before they do any translations up/down/left/right. But yeah, few and far between.

My favorite part of the exam was giving students a graph like:

I asked my students to give me the equation describing the graph. Most students rocked that part, even though I only gave them one problem of the same sort as a warm up. I don’t know why I didn’t give them problems like this last year — they require students to really think hard about function transformations to work backwards.

The one question that students almost universally bombed, which made me want to turn myself into a sheet of paper and crumple myself up and throw myself in the wastebasket, was the “explain” problem. The question read something like “Explain in words why $y=-\sqrt{x}$ is a reflection of $y=\sqrt{x}$ over the x-axis. You may want to use a diagram/graph and a table of values to explain your answer.”

What’s clear to me is that, frankly, my students still have no idea how to explain their ideas in words. I have given questions like this on each assessment, but previously we had a discussion about the concept and how one would go about answering the question. This time, I threw the question on to see if they could do it themselves. Clearly not.

Next year I am going to have to come up with a good way to integrate these “explain” questions in the course. Perhaps I’ll come up with a list of possible questions for each test and hand them out — so students can try to properly prepare their answers. And after each exam, I’ll hand out a list of possible answers and having a discussion about which are good, which are bad, and which are mediocre — and why. (In addition to verbally having the discussion in class.) I think this year I’m just not being clear enough with my expectations.

Or maybe I’ll just have students write up good answers at home and hand them in, instead of having them on an exam. And if they aren’t satisfactory, I’ll give students the opportunity to rewrite their answers with my comments incorporated.

One comment

1. I think they need practice with the explaining part more often. What seems to help for my students: when a question asks for a written justification, I have them share their explanations with a group, then as a group they refine their answers together. Then the whole class comes together to share the group answers and come up with one (or two) clear justifications.

It gets everyone involved in the process and lets everyone see that writing is a process. Once they’re used to it, it really doesn’t take that long either.