Patty explains her use of formative assessment to gauge students’ understanding of the Pythagorean Theorem. She asks students, “what are the relationships" among the legs and hypotenuse of the right triangle? She describes how students did fine using the Pythagorean Theorem when they were provided the lengths of a triangle’s two legs. But they were more challenged when they had to figure out the length of a triangle leg when given the length of the hypotenuse and only one leg. She plans her re-engagement lesson and supports for the kind of mathematical problem she will ask her students to engage with next.

8th grade math - Pythagorean Theorem*Patricia Ferrant, Thomas R. Pollicita Middle School, Jefferson Elementary School District, Daly City, California*

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PATTY FERRANT: They had an assessment last week and I've done a few but on...like, we'll say the Pythagorean Theorem, but I don't care if they don't even know the name of that, um, but the whole relationship if you see right triangles, so that relationship of "a squared + b squared = c squared.”

They had a quiz, or a, you know, a little assessment and some of them did very well, and some of them are still struggling. So I'm going to, um, just show some student work and have a discussion about that. The assessment I gave was the right triangles and students had a story that had to do with areas of land, and we used students' names, um, in our classroom and we discovered that the areas of Damian's land was equal to the area of Nickolas's land. When I looked at the students' work we found that when the two legs...the length of the legs were given, students could use the Pythagorean Theorem, but when a hypotenuse and a leg was given there was some difficulty, um, finding out the length of the other leg.

That's my formative assessment and it's guiding me on what I'm going to ask next, or what type of problem I want to propose next. Knowing what they know and don't know just makes every decision for me, so I did a re-engagement lesson. They couldn't just say...like, I would always say, "What relationship? Like, what relationship do you know?" And they can't just say, "a squared + b squared = c squared." I'm like, “No. But what is that relationship?”

So then that's when they would say, " Nickolas's and Damian's land," or whatever class it was from and then they'd have to...they would connect it to the picture. So it's, like, what's interesting is on the MARS assessment that was just given, that screen ratio, um, and even the irrational—irrational numbers has Pythagorean Theorem and I know when I saw some of my students' work, they're drawing those areas in.

And so, to me, like, the kid who's drawing that area in is proving to me they understand that relationship of what "a squared + b squared = c squared” means. They realize what supports are around them and it's not just the teacher but it's the other students, and that talking to them, "This is building on this and this person," and then the restating, and then the adding-on, and those strategies they're getting that that's making it make sense for them.

I'm really trying to build, like, everybody else holding each other accountable — the culture where it's like, I can't expect that my partner is going to be sharing this with me, and what am I going to do if they're not. I'm going to call them out. To really be thinking deeply about what we're doing, that, I think, for all of them took a long time. It's just being asked over and over and over to think and to try and have access to something, and to know it matters what they think, and it's okay if it's wrong. I remember, like, us having conversation about, like, "So what does it mean when I'm asking you to think in your head? Red flag should be going off in your head. Like, what does this remind me of? Or when did I see this last?" And it's like all these questions.

I want students to be able to engage each other in thinking, and to develop self-efficacy and confidence related to their own work in math. I want them to think deeply about what we’re doing, and that takes time.