The students work through their understandings of congruent triangles, the triangle postulates, parallel lines, transversals, and other geometric properties to apply those to create proofs for the quadrilaterals. Making sure the proofs are rigorous (including all steps necessary) is a challenge of any geometry class. Students struggle with how thorough and precise a proof needs to be. The students move between group work and whole class interaction throughout the lesson. In small groups, the students discuss and debate proof arguments. At selected times, the teacher pulls the class together to share findings, ideas, or sample justifications. After sharing ideas or arguments with the entire class, studentsthen return to working in their small groups. The teacher illustrates how to use selected otation in a proof as students work through the logic and reasoning. Instead of a more traditional approach to teaching mathematics, where the teacher presents mathematical notation and format up front, in this class the sharing of how to communicate, using mathematical symbols, occurs when learning situation arises. Once the group has worked through the reasoning of the proof, the teacher checks in with the group and instructs them to begin designing a poster that will display the proof they had created. Groups are instructed to design a poster that contains a drawing of the figure, the conjecture of what is to be proved, a list of the given from the conjecture, and what needs to be proved. The students can use a two column or a flow chart format of the proof.
CATHY HUMPHREYS: Alright?
STUDENT: I think we should do a trapezoid. Our statement would be: if two diagonals are congruent and do not intersect or do not bisect each other and create congruent segments then the figure is a trapezoid, an isosceles trapezoid.
STUDENT: Yup, but how do you say that they don’t bisect each other but they intersect at the same spot?
STUDENT: Because they have to intersect at the same spot to be an isosceles triangle; I mean a trapezoid. Got it?
STUDENT: Oh yeah.
STUDENT: How do we prove it?
STUDENT: Let’s just do a square.
STUDENT: A square?
STUDENT: Yeah, I’m not joking.
STUDENT: Let’s do a rectangle then.
STUDENT: How about a rhombus?
STUDENT: We just did a rhombus.
STUDENT: Did we really? No, we did a parallelogram.
STUDENT: Wait. Oh, let’s do a rhombus. Okay, so if the two diagonals are not congruent but are perpendicular and bisect each other then the figure is a rhombus?
STUDENT: What’s the difference between a rhombus and a…
STUDENT: And a what?
STUDENT: All sides are congruent.
STUDENT: All sides are always congruent.
STUDENT: So what’s the difference with a...thing?
STUDENT: What do you mean?
STUDENT: What’s the difference with a diagonal?
STUDENT: Between that and a rhombus? A rhombus isn’t perpendicular.
STUDENT: It is. That one is not.
STUDENT: Oh, oh. Got it!
STUDENT: I’m recorder.
STUDENT: Facilitator go the get stuff.
STUDENT: What do I get?
STUDENT: I thought Chris was facilitator.
STUDENT: I mean resource manager.
STUDENT: What do we get – paper?
STUDENT: Can you please go get some paper and markers for us? Thank you!
STUDENT: Let’s just draw one out. I was like wait, what did I just do wrong – that’s a square.
STUDENT: There you go. We’re doing the rhombus?
STUDENT: I think we should get the...
STUDENT: Okay. This is congruent to this.
STUDENT: And you have to draw the thing that indicates ninety degrees – the ninety degree mark.
STUDENT: There are other groups doing rhombus too.
STUDENT: Who cares!
STUDENT: We can beat them up.
STUDENT: Maybe we should do an isosceles triangle, I mean trapezoid.
STUDENT: That’s what I said. No, we’re not going back now; you guys disagreed with me. Should I make it bigger?
STUDENT: That's fine?
STUDENT: This is just the, like the little…
STUDENT: Rough draft?
STUDENT: Yeah, rough draft.
STUDENT: Okay, so we write down our "if then" statement?
STUDENT: No, yeah. No, yeah.
STUDENT: So if the diagonals are not congruent and they bisect each other right? And are perpendicular ̶ wait, do we say "and all sides are congruent?"
STUDENT: Yeah, you have to.
STUDENT: Should I go get the definition?
STUDENT: Yeah. Then the figure is a rhombus right?
STUDENT: Don’t they have to bisect each other at each other’s midpoint?
STUDENT: I want to do either a kite or a rhombus.
STUDENT: Why a rhombus?
STUDENT: We have to do it in a group?
STUDENT: Yeah, we’re doing group...exactly what she did on the board. So we are defining it and then proving it.
STUDENT: I say rhombus because I said square rhombus and you said kite rhombus, so…
STUDENT: So what do you guys think? What do you guys want to choose?
STUDENT: Rhombus. We have barely any information on the rhombus? We have more information...
STUDENT: On the rhombus?
STUDENT: We were working on the parallelogram but if we ignore the green one then this would make a parallelogram. Wait no…a rhombus.
STUDENT: That’s a rhombus. Okay, so let’s define what a rhombus is.
STUDENT: Do you need one?
STUDENT: I’ll get one for you.
STUDENT: So "if then" statement for a rhombus.
STUDENT: No, write square. Not square. Generic means it’s like a regular rhombus. A square can technically be a rhombus but we're not proving that a square can be a rhombus, you know? We are just proving a regular old rhombus. That’s it. If the diagonals are different lengths and they have to be perpendicular to each other though. No and they have to be perpendicular ‒ yeah, right and they are perpendicular.
STUDENT: First it has to be at their midpoint.
STUDENT: Yes, it has to be at their midpoints and they have to be perpendicular; they bisect each other and... Okay, wait, wait. Don’t write anything yet, so we organize our thoughts before we write it down. So we know that they have to be different lengths right? We know that they have to be perpendicular to each other and we know that they bisect each other. What else?
STUDENT: Is that it?
STUDENT: That’s it right? They bisect each other. That’s it. So now we’ve proven it. See, bisect each other, are perpendicular to each other and then the quadrilateral formed is a rhombus. I’m just checking the diagonals to see. So in a rhombus this would be congruent to this and this would be congruent to this.
STUDENT: So should we label it with ABCDE?
STUDENT: Yeah, but we can only do it on this thing though. So then...
STUDENT: Should I draw a rhombus?
STUDENT: Yeah, draw a rhombus. There you go. You want to write what a rhombus is?
COMMENTARY BY CATHY HUMPHREYS: Although a wonderful conversation is happening here between Omer and Jerry, I was struck by Julie’s absence in this conversation. Even though she was the one who had the idea to use alternate interior angles as well as the idea of proving triangles congruent, once Omer and Jerry understood her idea they were off and running on their own again. Their eye contact and body language was for a group of two, not three. This raises many issues for me –questions, for example, about the gender equity in this class (even though it is my class!). I know that the boys were not intentionally selfish or mean-spirited, but the result was that Julie was excluded from most of the group work. This clip highlights the necessity of building better group skills. It also made me want to teach interrupting as a skill to be used when necessary. I heard Madeleine Albright talk about how difficult it was at first for her as Secretary of State because women generally do not interrupt, while men do so freely. I think not only Julie, but Omer and Jerry as well would have benefited from a little interrupting so that she would become an intrinsic part of the group.
This clip also makes me wonder: would assigning the group roles and structuring the group work around those roles have mitigated what happened here?
Another thought I have is that perhaps this task would have been better as a partner task; when people are trying to wrap their heads around a logical path, four people’s inputs can make it harder. With partners, there would be more for each person to do and less ideas to form into one coherent argument; this would ensure more equal participation and hence more learning.
COMMENTARY BY COACH DAVID FOSTER: Students developed a deeper understanding of congruent triangles and parallel lines in applying that prior knowledge to this new situation of proving certain quadrilaterals are formed from the formation of diagonals. Students would work through their fleeting understanding of triangular postulates (SAS, ASA, etc.) to use those geometric building blocks to make their proofs. Cathy used students’ partial proofs to improve the students’ understanding and skills with making rigorous arguments. She stopped the group work and had the class hear from a student as she explained how to make the proof complete and valid. Students sharing their work and ideas with the entire class are an effective instructional technique to deepen students’ knowledge and improve their work products. Cathy introduces conventions and notation as the students explore mathematical ideas. This technique of waiting for “just in time” information is more conducive to learning. Too often teachers give students a lot of up front information that students often forget or never use. This “just in time” methodology is more aligned to how students learn. As she moved from group to group, Cathy checks to make sure the students’ reasoning were sound and that there were no holes in their logic. Occasionally she presses on students to explain their thinking or check out an error or faulty assumption. She then would advise the group to start to design a poster that would contain the proof of the quadrilaterals they selected. Most of the groups selected a two-column proof instead of the flow chart format.