Cathy Humphreys closes day one of the Properties of Quadrilaterals lesson. The second part of the investigation is getting students to justify and prove their findings about the diagonals of the kites. The students use definitions, postulates, and theorems to develop a proof about the diagonals of a quadrilateral and how they constrain the type of figure that is formed. The teacher moves between groups, checking in on the progress students are making in developing their justifications. At the close of the first period, the teacher employs the resource manager to make sure all the manipulatives and materials are collected and stored.
CATHY HUMPHREYS: So um, wow that took all period so far. What...any reflections about what that was like for you, working in a group like that?
STUDENT: I’m never going to make kites, never, never, never.
CATHY HUMPHREYS: You are kind of busy cleaning up. What was that like? Anyone want to make a comment? You don’t have to say anything you don’t mean – I mean this is not...I am just curious what that was like for you.
STUDENT: It wasn’t that bad.
CATHY HUMPHREYS: And not that bad because?
STUDENT: Because it feels good when you understand it.
STUDENT: And its fun working together.
STUDENT: I wouldn’t want to be a kite builder though.
STUDENT: Well, not all kites are made with quadrilaterals, so there’s more than just that.
CATHY HUMPHREYS: Okay, any other comments that people are willing to share?
STUDENT: We should have made a dragon.
CATHY HUMPHREYS: Dorothy? Can we all hear Dorothy?
STUDENT: It was fun to explain.
CATHY HUMPHREYS: How do you mean it was fun to explain?
STUDENT: Like, um…
CATHY HUMPHREYS: And Dorothy, could you talk so everyone could hear you.
STUDENT: Sharing your ideas with other people in as many ways as possible so that they can understand you.
CATHY HUMPHREYS: It’s not easy to explain always is it?
CATHY HUMPHREYS: Yeah, it’s not. Okay. So on this card that’s on your table um, please put your... when I’m done talking put your name on it and then I’d like you to write down please, what question you’re thinking about individually right now. What’s the most important question in your mind right now about this project? Not about the project but about the diagonal, the math of it. What question in the...I’m not saying this very well. So here’s what I mean, like right now everyone...nobody is finished completely so could you please do your best job at writing down what you’re wondering about, or what question you are thinking about right now um, right now. I'm done. Yay!
COMMENTARY BY COACH DAVID FOSTER: Cathy uses the phrase “convince yourself, convince a friend, convince a skeptic” to describe for students the level of precision necessary to justify findings.
The students were very successful at proving that if the diagonals are congruent and are perpendicular bisectors of each other, then the quadrilateral formed is a square.
Cathy checks in with each group, assessing where the students are in their thinking and how they are progressing in generating proofs. She finds a question to pose or re-directs the work around a specific task to move the group along.
Cathy makes group use of the group roles and the students respond positively by carrying out the duties assigned.