I spent about an hour yesterday playing with geometry tiles. I was:

- Fully engaged and enjoying myself.
- Experiencing OMG, WTF, and AHA moments. (AKA actively
**learning**.)

The inspiration for my mathematical play time was this photo from Sara Vanderwerf’s recent post on **Stand and Talks**, a bold routine she uses to increase the number of students talking about math. Really great stuff and worth your time to read it through.

But back to the photo, which is her example of using a Stand and Talk to introduce a new idea, in this case, the Pythagorean Theroem. My eyes perk up (is that a thing?) because I have been pondering about this very topic. Specifically, wondering about **what activities/tasks would place students in the active role of sense-making**.

The first thing that popped into my head was….wait, *trapezoids*? (Thank you, Sara!) Then I started some noticin’ and wonderin’ from my teacher-y perspective and also imagining/guessing what students might say. The more I noticed and wondered, the more I realized how FULL OF MATH IDEAS this *one* photo is.

I find this **exciting**. Really, *really* **exciting**. Can you imagine starting here, with this one little image, asking students what they notice and wonder, and then letting them run with their curiosity? Isn’t it joyful to know in your heart of hearts that this is **so** much BETTER than, “Hey kids, here’s a little something called the PT, copy it in your journal/onto your pre-made “doodle” notes, plus copy these three examples of how to get the right answer, and do this worksheet”? How *superior* this single photo and all its potential are to “I do, we do, you do” and /or anything else that keeps students in the passive role?

It didn’t take me long to drag out my tub of tiles and start messing around with them. TOTALLY what I would want my students to ~~ask~~ beg to do.

The questions I decided to guide my play were:

First, can a proportional, scaled-up version of each polygon tile (that I have) be built with only that particular tile?

If so, why? Is there a pattern?

If not, why not?

Second, can any three similar polygons be used to make any type of triangular “hole”? Why or why not? Which ones make right triangles? Why? *Is there a pattern??*

I started with squares. Don’t know why, maybe because that’s what is already familiar to me. No surprises, the areas are the square numbers, *1, 4, 9, 16….*.

So I moved on to the equilateral triangles, wondering/kinda expecting that I will make triangular numbers. Imagine my surprise (my OMG moment) to find SQUARE NUMBERS AGAIN! Holy shit! NOT what I was expecting! Now I feel compelled to keep exploring. COMPELLED. (Imagine your students feeling compelled!!)

I quickly confirmed that the next in the series takes 16 triangles, and moved on to the rhombus. *YES*, the areas are square numbers *AGAIN*! Which gets me reconsidering my understanding of square numbers and forming conjectureish questions: Can square numbers be made out of any polygon, not just squares? These are all regular; what if they are irregular?

I’ve been reading Tracy Zager’s fabulous book __Becoming the Math Teacher You Wish You Had. __ She starts Chapter 7,* Mathematicians Ask Questions*, with a quote from Peter Hilton: ** “Computation involves going from a question to an answer. Mathematics involves going from an answer to a question.” ** Brilliant distinction. Looks like I’m living Chapter 7. Know I want this for my students!

### Remember, one photo.

I finally get to the trapezoids, which are more challenging to tile and require some analysis. I managed to construct one, but the number of tiles it took (7) was a red flag for me, so I examined it closely to verify if it was proportional or not. It wasn’t, so I kept playing, keeping these two beliefs in my mind: a proportional trapezoid was possible, and it could be made out of 4 tiles. And lo and behold….

TA DA! “Square” Numbers! (Whew!)

By now, I have even more questions. By now, I have reviewed *and* applied my prior understanding of proportionality and similar figures, without ever touching a worksheet. By now, I’ve taken risks, persevered, reasoned, looked for patterns, used vocabulary, formed conjectures, and justified. All from looking at one picture and allowing myself both time and pleasure to play and be curious….and I haven’t even begun exploring that thing with the triangular “hole”. Yet.

Finally, the hexagons. And….What?!? You can’t even *make* a similar hex without using some non-hex friends! WTF!

New question! **WHY NOT?**

One.

Little.

Picture.