Tag Archives: Discovery Activity

Function Transformations

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:


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.

IMG_20151002_150615906 IMG_20151002_150229253 IMG_20151002_150601495

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:

IMG_20151002_154405792 IMG_20151002_152324286 IMG_20151002_151618290

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]


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.


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Understanding Radicals

Vertical non-permanent surfaces (aka whiteboards hanging from command strips on the wall) gives every activity a “new look”. This activity, done during the early “let’s review our algebra” period of Precalculus gave me a new appreciation for how little students understand the procedures we teach them through the years.

Here’s the website that gave me the idea:

Nrich’s Nested Surds

Here’s the Google presentation I made from it:

(Link to the document in case the above embedding doesn’t work.)

I did visible random grouping (I’ve started to use this website because how straightforward it is. Make sure you select the “new window” output format.) in groups of 3. With 21-22 students, I had 7 or 8 groups. I showed students the first slide and had them “test out” numbers. Once they did a handful of perfect squares, I let them try it out on their calculators, for non-perfect squares.

\sqrt{a} \times \sqrt{b} = \sqrt{ab}


Here’s what shocked me: these precalculus students worked on this first problem for about 15 minutes[1]. Nobody had gotten even remotely close (okay, one group came up with a rule “if a is even and b is even, it works!”). I was within a few minutes of pulling the entire class in and having them sit down while I explain it all (as if they’d learn & understand any better, but I was panicking!). Then, within a minute or so, every group started “getting it”. It helped that they could look around (since all the whiteboards are vertical), but that didn’t help as much as I thought it would. Students must still have some inhibition along the lines of “seeing other groups’ work is cheating.”

The second problem (\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{frac{a}{b}} ) went a little faster as did the third (a\sqrt{b}=\sqrt{ab} ). I especially liked the third because it is a common mistake, so this activity gives us some ground for discussion when that type of mistake occurs in the future. Unfortunately that was as far as we got before we ran out of time.

It surprised me how difficult this activity was for these students, who should have seen radicals 2 or 3 times before in their math classes. The Alg II teacher commented that perhaps it was the open-endedness of the activity that tripped them up, but to me that simply reveals a weak understanding of the underlying math. Regardless of how open-ended an activity is, students should know and be able to apply how radicals work.

Using the vertical whiteboards, I appreciated (A) how quickly students got to work, (B) how long they worked, and (C) how I was able to quickly assess where groups were and get them back on task. Unfortunately this particular group seemed especially prone to getting off-task, distracting other groups, having conversations that were not related to math. I’m sure that standing contributed a little to this, if only because they aren’t used to standing during most classes. However, we had a good “chat” and I’m happy to report more on-task-ness more recently.

Thanks to John Golden for sharing about the Nrich site and thanks to Alex Overwijk for sharing about vertical whiteboards & random grouping!


[1] Actually, I have no idea how long it took–it might have been 30+ minutes. I just know it felt like a long time and it was at least 15 minutes.

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Dilation and Translation of Functions in Geogebra

So I created a handful of activities because I love using Geogebra.  I can’t believe I didn’t know that there are spreadsheets in Geogebra (I’ve never read the manual :/), and I’m definitely going to look into using Kevin’s lesson when we get to trig functions and graphs.  But as a review for my Precalculus students, I created a discovery activity where students get to see functions dilating and translating (I call them “stretching” and “shifting” or “sliding” because I think it’s more intuitive for the students… and I often forget the official names).

Here are two of the activities that we’ve used so far, and I like the activities because it starts assuming that the student has never used Geogebra.  Later activites, I get to assume that the student is experienced with Geogebra and I don’t have to write each step explicitly down, but I was shocked at how few of my students were able to follow such clear step-by-step instructions, even though they are all juniors or seniors in high school!

EDIT: Even after doing this for one year, students found some wrong problems, so I’ve edited the second packet to include these corrections:

Let me know if you have suggestions for these activities!

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