Two weeks ago in my Algebra 2 Trig Honors class we were finishing up our unit on rational expressions, equations, and functions. In past years, students at this level had focused on graphing only simple rational functions – transformations of y=1/x. (In Precalculus, students are exposed to general rational functions.) I was feeling frustrated, though, at how limited this approach felt – I didn’t want students to be misled into ignoring the big vast ocean full of rational functions after looking at a goldfish bowl-sized sample. (Questions like: “so it will always look like this?” were driving me crazy.) But at the same time, I didn’t want to bog them down with a tedious approach to graphing general cases by hand. I just wanted my students to get a feel for all of the different crazy shapes these graphs can make, notice patterns, and begin the process of generalizing.
Last year I had included an investigation of general rational functions using Wolfram Alpha on a homework assignment. In hindsight, leaving this activity for homework wasn’t a good call – there wasn’t enough time for thoughtful follow-up, and by having students work alone they missed out on the magic of discussing their findings with others and building understanding together. So this year, I took an extra class day to allow students time to play around with more complicated rational functions on the Desmos calculator. (Which I am a little bit in love with.)
In my 2nd period class, I didn’t do a great job of setting up and explaining what I had envisioned for the hour. I showed them the Desmos calculator, told them they should be graphing a variety of rational functions, and gave them a sheet with 4 specific examples with bullet points for guided questions underneath to use as a framework for their investigation. On the back board, I had written “What do you notice/what are you wondering about…” with different categories for them to record some of their findings throughout the hour. The idea was that throughout class, I would circulate and peek in on their conversations, help answer some questions or poke at their thinking a bit, they would record their observations and questions on the back board, and at the end of class we’d save 10 minutes to discuss our findings on the board.
As I turned students loose to begin working, some problems became evident pretty quickly. While students were comfortable recognizing the one vertical and one horizontal asymptote for simple rational functions, they couldn’t “see” the asymptotes for the more complicated graphs. This to me indicated that they didn’t have a good enough understanding of what an asymptote truly was, but instead had mastered an algorithm for graphing the simple rational functions. I tried to chat with groups of students to see where the disconnect was, and I encouraged students to graph the lines that they visually predicted were asymptotes – then connect that to the algebra of each function. I also realized I had not done a good job explaining or modeling the way they should record what they noticed/wondered on the back board – I ended up with lots of random pictures and little interpretation or generalizations. I also did a horrible job keeping track of the timing of the class. It became clear we weren’t going to have the 10 minutes I wanted at the end of class to pull together our big findings – instead, I had to take pictures of the back board and wait until the next day to summarize their investigation.
There were more nuanced conversations than the sloppy work on the board indicated, but overall I wasn’t happy with the way things had gone. I didn’t feel like we had arrived at enough clarity or closure for the big ideas I wanted them to be tackling.
Lucky for me, I learned a bit from my mistakes and in my 4th period class I did a MUCH better job of setting up the activity. I graphed the first specific example on Desmos, talked through how I could visually “see” the 2 vertical asymptotes and 1 horizontal asymptote, showed them to graph those lines to confirm the asymptotic behavior, returned to the algebra of the example function to make connections between the algebra and the graph, then added a generalization to the board of noticing/wondering.
During this hour, the students did a better job of working together, asked me fewer questions, and were more able to generalize their findings. They even surprised me with some observations I was *not* expecting them to make (see the last one under “V.A.”) and a few students figured out how to use sliders with Desmos to more efficiently explore how changing different bits of the function they graphed would impact its shape. You can tell by looking at their noticings/wonderings how much more successful my second shot at this lesson was. During the follow-up conversation at the end of class, I addressed all of the points on the board, asked for students to help respond to some of the questions, and felt that the majority of students left with a better understanding of both the important features of these graphs and how the algebra of the functions controls their shape.
Since I first started this post a week ago (sorry, February is the worst), I have now given the test for this unit. General rational functions were not tested (only transformations of y=1/x), but I did decide to include a question that had students explain the shape of the parent function y=1/x. It was a great way for me to see if all of the investigation and conversation about the shapes of the graphs had paid off… were my students more “mathematically fluent” in the vocabulary and ideas associated with rational functions? I did have some very strong responses, and I’ve included my favorite below.
I’m sure there are ways for me to tweak and improve upon this lesson for future classes, but I was happy with the chance it afforded students to talk, play, investigate, and generalize.