Lesson
8th Grade Math – Coordinate Geometry, Logical Reasoning, Justification and Proof
Clip 7/11: Coordinate Geometry Lesson Part 7
Overview
To conclude the lesson with her eighth-grade geometry students, Antoinette Villarin asks students to talk about justification: “Our goal is to justify our thinking using coordinate geometry… Justifying your work has a lot of parts to it — showing thinking, drawing diagrams.”
Antoinette challenges her students to create a representation that reveals how the elements of the problem are connected and related, with sufficient clarity that a “non-geometer can understand,” and so that the students can convince others. Antoinette asks the students to use numbers, words, and images, using the prompt: “I think the midpoints are ___________ because.”
She closes the lesson by asking students to do a partner check-in, sharing “something you noticed, something you’ll work on Monday, or something you finished doing today that you now understand.”
Teacher Commentary
Teacher Commentary
Pacing and time have always been an issue for me. I even just ran out of time today, in terms of the ending, and how I wanted students to check in. But I just knew that we hadn't talked about midpoint yet. They don't know the midpoint formula. We haven't gotten … to that in our curriculum.
But I do know that they know a lot about slope, and then they've graphed points on a line, and they know what a quadrilateral is, because we've talked about properties of quadrilaterals.
But they haven't seen midpoint. And so, knowing that, I knew that I couldn't do both at once, where one part was looking at midpoint, and then, I think the second half of the task — problem three and four — were looking at special quadrilaterals. I had to pace it separately.
I feel like I ended it at a point that I thought would be the middle of my lesson, which was talking about justification — which I had forgotten to announce at the beginning. I'm glad I was able to include it in at the end.
I might actually take their work and start to pair them up. I was thinking originally of having them be in the same groups, like, everybody that solved it the same way. But now that I've heard the two different ways that students have solved it, and it seemed almost — not half and half, but about approximately half — that if I pair them up with somebody else that has solved it differently — to switch justifications and read each other's, and see how convincing it is to somebody that solved it a different way — I feel like that’s going to be where my starting point is for next time.
I would like to list maybe all the different strategies they had on the board, and then see if maybe somebody had any extra ones that they had noticed. Because at the end of the class, there was one student that said, “Well, Miss Villarin, I found that if they were congruent, that if the quadrilateral was congruent, there was another way to find it.” And I said, “Can you save that idea for Monday?”
So, I feel like I can add to the list, and then also strategically pair them up so that they're showing each other how convincing their justifications are to somebody that did it completely differently.