I have been itching to try an Open Middle math problem.

As a tutor, I am always looking for ways to enrich and deepen the understanding of the students I work with, as opposed to just band-aiding their current math struggles. I finally got my chance last week. I chose this 3rd grade question: *Draw three rectangles with a perimeter of 20 units.*

First, I asked my student to draw one rectangle with a perimeter of 20 units. Here is his initial attempt.

Rectangle…check. Perimeter-is-the-edge…check. Perimeter = 20…not so much.

We add the side lengths and he sees the total perimeter is much more than 20.

**Aha! #1** “Oh, I get it!” Immediately, he draws a square and proudly labels each side 5 units long. *Yay! *

Wonderful! Now: *Draw another rectangle with a perimeter of 20.*

Complete puzzlement.

So, we talk about what perimeter means and I ask *Is there any other way to have the sides add up to 20? * **Aha! #2**, and he draws the following “rectangle:”

Hmm. Okay. Perimeter = 20…check. Rectangle…not so much. So I draw his “rectangle” to scale.

*Hmmm. What’s going on here? *We talk about the properties of rectangles and his next attempt is a rectangle, but again the perimeter is not 20.

So, I ask him to try again. He thinks for a bit and then draws:

*Great! We have two rectangles with a perimeter of 20. **Now draw another one.*

Much less thinking time this time:

And we get to **Aha! #3**, at which point he stops drawing, and starts talking: “Or, the sides can be 2-8-2-8, or 1-9-1-9, or …!)” Pretty soon he’s listed all of the different configurations/rotations of rectangles (with integer-length sides) and HE GETS IT!

This was more than an “aha!” moment; it was a groundswell of understanding – like a wave of comprehension crashing on the beach: **I get it!** **I get it! I get it!**

**And it didn’t happen after creating one rectangle, or even after two. It required him to come up with three different rectangles before he understood.**

SO fun to watch. It literally made my day. Can’t wait to try more Open Middle problems!