I tried something new this year when teaching function transformations: I had students come up with the “rules” in groups with little help from me.

Day 1

Students created a DesMan (person using Desmos) and built up an intuition for how functions move[1]. We spent a block day (1.5 hours) doing this. Students were free to share what they discovered with one another and that helped tremendously. Here are some of their creations:

And here’s one that has moving parts:

https://www.desmos.com/calculator/hco3gb5vju

Day 2

No students finished the DesMan on the first day, but I told them that they could get credit for “Modeling with Math” if they finish it.

We split into random groups of three and I had them move to their vertical whiteboards with these instructions:

- Split your whiteboard into 4 parts.
- On each part you should explain to a Freshman (these are Juniors and Seniors in Precalculus) how to transform a function in these four ways: Translate up (they suggested the word “translate”, I was going to use “shift”), translate down, translate right and translate left.
- Please include a graph, use function notation, and an explanation using the ideas “input” and “output”.
- You should use Desmos to confirm your explanations with some examples.

Here are some of the whiteboards they created.

As groups finished[2], I approached them and made sure everyone in the group understood each idea, especially the input/output language. I asked them to do the same thing on a second whiteboard, but this time with stretching vertically, shrinking vertically, stretching horizontally, and shrinking horizontally. Here are some of those whiteboards:

Even though their whiteboards are great, that wasn’t the best part of the activity. The best part was the *discussions* that I overheard. *Every* student was involved because every student (a) had some experience with all of this through their work on the DesMan and (b) felt comfortable asking “why?”. I felt as though we had reached a classroom culture of safe inquiry and curiosity, where “why does that work?” questions are empowering rather than embarrassing.

It also helped tremendously having Desmos as an outlet

Some groups finished this part before the end of class, so I had them examine vertical and horizontal reflections. Here’s their work:

Or maybe I drew those pictures in red, I forget.

Day 3

We did a short game of “guess the type of function transformation from the graph”. It helps having 2 screens: one that I only I can see, which I use to edit Desmos and the other that the class sees, when the function is hidden.[3]

Summary

In three short days, I feel confident that students will be able to use these transformations the rest of the year[4]. It’s important to note that I’m not introducing this to students the first time (it is Precalculus and they should have seen this *at least once* in Alg II), though they often act as though they haven’t seen it before.

[1] I didn’t use this Desmos page, but it’s another way to do this: https://teacher.desmos.com/desman. I just had them graph a parabola, restrict the domain {-3<x<3}, then move it and they were off!

[2] This is where the vertical whiteboards was tremendous: I could catch groups right as they got to finishing their work.

[3] I only have this setup in one of my classrooms. In the other I have to use the “freeze” button on the projector to work out the next graph.

[4] And they will use them them since I like 3-act lessons where students have to figure out what the function is. It’s important that they practice this throughout the year. Maybe one day I’ll show them how to approximate data using the tilde (~) in Desmos.