I set up a recent lesson by asking students to deliberately practice SMP7, look for and make use of structure.

This practice requires us to make visible what isn’t showing. In geometry, that often means drawing auxiliary lines.

We don’t always see structure in the same way or at the same rate, so once you’ve found one way to solve the problem, I want you to also deliberately work on your mathematical flexibility. Find a second way to work the problem.

I had 5 questions prepared, the last of which I learned about in Justin’s and Kate’s posts last year about a Five Triangles task. I’ve been thinking a lot this year about not only planning learning episodes but also planning ahead what instructional adjustments I’ll make based on the feedback I get from my students. In my planning, I struggled with which question to use first. Which question would you use first with your students?

Last year, I had the following results in the following order.

After this first Quick Poll, I didn’t display the correct answer, asked students to team with someone else in the room, and sent the Poll again.

After this Quick Poll, we had a student who answered 53˚ share his reasoning with the rest of the class so that we could figure out where the reasoning went wrong.

The class went fine. But I wondered what would have happened if I had started with a question that required the use of auxiliary lines (even though students struggled with the question that already had them drawn). So I tried that this year.

I could “hear” thinking and I could “see” productive struggle as students started out working the problem individually. Once they started sharing some of the ways that they made visible what wasn’t pictured, I saw evidence of SMP7. Because I had deliberately asked them to work on their math flexibility, they weren’t satisfied with only one way to solve the problem.

Many wanted to share their way with the whole class.

They tried another one, and again, you could “hear” thinking. I didn’t even have to suggest individual think time to the class, as they naturally all wanted to try it by themselves first.

I posed the folded rectangle problem, but the bell rang before students could really dig in to solving it. Maybe next year I’ll be brave enough to start with it, as the journey continues …

p.s. I’m currently reading Ilana Horn’s Strength in Numbers: Collaborative Learning in Secondary Mathematics, and before I was able to publish this post, I happened to read a section entitled “Turning Some Pet Ideas about Mathematics Teaching on Their Heads: Start with Challenging Stuff, Not Easy Stuff”. Her premise is that starting with easy stuff is inequitable, as students who get the mathematics quickly can take over the problem, and those who don’t miss out on the opportunity work with their team. Starting with challenging stuff levels the playing field for all students to contribute and learn.