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

(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.