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.
TEACHER: See what she wrote on the board? Does that work with what you know?
STUDENT: Um…yes. Let me copy that one. Okay, so this and this are the same because if the side um, I mean yeah, side, side...can we prove that this and this are the same length?
TEACHER: Does that say side, side, side or does it say something else?
STUDENT: Oh, side angle…
STUDENT: Yes, side angle because look right there.
STUDENT: Yes and you can prove for all four of them. So...
TEACHER: Are all four of them congruent triangles?
STUDENT: Yeah because side angle side and this and side angle side for this; and these two are the exact same.
TEACHER: But do all those match up?
STUDENT: No, not these ones…this one and this one. This one and that one would be.
STUDENT: Why wouldn’t this work then?
STUDENT: Ah, even if it did work, it would be different from this one because it's two different ones.
STUDENT: But what are you trying to say exactly? Are you saying that I have these marked congruent because I didn’t change it yet?
TEACHER: So when you say side angle side, what does that mean? I mean, I know that you go side angle side but what does that mean?
STUDENT: Ah, it means that if you have whatever a set of…
TEACHER: You have two triangles.
STUDENT: It proves congruency.
TEACHER: It proves what?
STUDENT: It proves congruency.
TEACHER: Of the whole triangle, how does that match up? When we say side angle side, what are we saying?
STUDENT: When you have side angle side that proves the triangle looks like this. It’s just going to be like this always if it has the same side angle side. And if this is side angle side also, that proves that…
TEACHER: It has side angle side also…I mean, every triangle - two different triangles – this triangle and this triangle. This triangle has a side, an angle, and a side; and this triangle has a side, an angle, and a side. Does that mean these two are congruent?
STUDENT: What I meant by that…I didn’t say this but what I meant was that they have the same angle and the same side and I…
TEACHER: Okay, so my question is, does this triangle here...you said that this and this are the same but do they all have the parts that are exactly the same?
STUDENT: No. This angle is bigger than this one.
STUDENT: I know. I’m not talking about these two though.
STUDENT: Oh, I thought you were.
STUDENT: No, I’m talking about this and this.
STUDENT: Oh okay, that’s what I thought. No. I thought at first...
TEACHER: I thought I heard at one time that all four were congruent.
STUDENT: Oh, no.
STUDENT: That’s what I thought you said too.
STUDENT: No, that’s not what I was trying to prove. It’s just that I think you guys are getting confused because I had a congruency mark of one for all four sides.
TEACHER: So are they all one for all four sides?
STUDENT: No, I was just doing that…I didn’t change it yet; I’ll change it right now. Okay.
TEACHER: Are all four of those sides of that parallelogram equal?
TEACHER: Then why did you mark them?
STUDENT: You marked it right there.
STUDENT: I did three, three and four, four.
CATHY HUMPHREYS: You could measure that for sure – that’s true. And if you measured, you’d have um, an example of exactly one particular parallelogram. What you’re trying to do is show that no matter what angle it is, if it fits those criteria, if it fits those three criteria then it’s a parallelogram. So what else do you know about parallel lines? What else besides…
STUDENT: They don’t intersect?
CATHY HUMPHREYS: So have you heard about...remember the transversal things and those angles?
CATHY HUMPHREYS: Okay, so would you refresh your memories about that because if you know what you need to show for parallel lines, you can kind of work backwards form there. Alright?
STUDENT: Are you sure this is right? So it would be…
STUDENT: I’m going to a get scratch paper.
STUDENT: Transversal is when the lines are parallel right?
STUDENT: So it’s like this right? So...and then you have a line that intersects right?
STUDENT: Or there.
STUDENT: It would be right here right…so let me get it that way because then we could definitely get the transversal itself– the opposite interior angles are congruent and alternate angles are congruent; and the corresponding angles are congruent and then the alternate interior angles. You know what this rough sketch doesn’t have to look great right? So does this angle equal this angle? And this angle would be equal to this angle right? From this we get that a measurement of both pairs of opposite sides are parallel and the opposite sides are parallel and the opposite angles are parallel. Yeah, so we have that the opposite angles are equal and that the opposite sides are parallel or equal. Well, they have to be. They kind of have to be because the distance between this and the distance between this would be the same. The distance can only be the same because these angles are the same. Right? These angles are the same so this is equal and this is equal. So we’ve got two and there’s one left. The measurements of both pairs are equal – both pairs are equal and the opposite sides, wait. Okay, so both pairs of opposite sides are parallel and equal and the opposite angles are equal. We've got the angle pairs down but I don’t know if we have the opposite sides right. How can we prove that…can we prove it using the transversal? The transversal is normally what you use to prove angles right? So what, how do we prove the...
STUDENT: The angles.
STUDENT: We did prove the angles with the alternating interior angle conjecture.
STUDENT: But we can’t prove the sides.
STUDENT: Wait, what if we had an isosceles triangle? Remember the SAS like this? If the angles are congruent...
STUDENT: But we have side, side, and side.
STUDENT: Oh, that makes sense. So the same thing right? And this triangle – you have angle – these angles right?
STUDENT: So two we have...
STUDENT: I know, I know. So...and then SSS...no it wouldn’t be SSS. It would be...
STUDENT: It would be AAA.
STUDENT: No. Remember when we had that one problem? It was like a large triangle and the triangle was cut into...it was like these guys – like this right and it was kind of like this – it was like this I think. Remember that? What was the conjecture of that proof? Didn’t this solve this?
STUDENT: Is it the same thing?
STUDENT: It is the same thing. What is it?
STUDENT: It was SSS because they gave you the information and you had to mark it here.
STUDENT: And the only things we have are angles. We have two set of angles and AAA doesn’t work.
STUDENT: Wait, we do have sides?
STUDENT: This one is congruent to this one and this one is congruent to this one. That’s the midpoint of this diagonal and this is the midpoint of that diagonal.
STUDENT: Oh yeah, that’s cool!
STUDENT: So now we have sides we need.
STUDENT: Oh, so now we have ASA, ASA, and ASA. So it proves that this is equal to this, which means that this is equal to this.
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.