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, students then return to working in their small groups. The teacher illustrates how to use selected notation 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: So then let's go back here. I’m going to - the next step then is to prove what we are going to do. So we have to prove it’s a parallelogram. Now on your tests we had a little trouble with proving. Um, on the test questions I used the word “show” and a lot of people just marked the marks on the figure, but did not go through a logical sequence of steps to show beyond a shadow of a doubt that the proof was true. So before we even start our prove though – what are we going to need to prove to show that it’s a parallelogram? So what is a parallelogram? You have your definition sheets um, some of you do. What would be the definition of a parallelogram? What is? Okay, Drew?
STUDENT: A quadrilateral is two pairs of parallel lines.
CATHY HUMPHREYS: Okay, I don’t think everyone could hear you, could you say it...even though you have a loud voice, I think some people didn’t hear.
STUDENT: A quadrilateral is two pairs of parallel lines.
CATHY HUMPHREYS: Okay, that is enough for it to be a parallelogram. A quadrilateral with two pairs of parallel, oops, one "r" and then two "l’s" – and Drew I’m going to use the sides instead of the word lines, okay. Alright, okay. So that’s what we have to show. So how would we do that? Alright, so what I’m...I actually thought that was kind of a rhetorical question because what I want to do is I want to try to sketch this. So I’m going to sketch diagonals that are different lengths and they don’t intersect at right angles; and they bisect each other. Okay, let's try to do this. So there is one length, there’s another length – they’re different alright and then I am going to…so did I meet all the criteria? What’s the matter? Oh, thanks. Did I meet all the criteria? Yes, they intersect at their midpoints, so I have to show that. So this is going to be the same as that and that’s going to be the same as that; they are different lengths and they are not perpendicular. Okay, so now I’m going to draw in my parallelogram. Okay, now with your groups I want you to see…um, do you have enough scratch paper or something to work on? Um, I don’t need to collect this but what I’d like you to do right now is see in your group how would you make ̶ it could be a paragraph style or two columns, but see if you could make a proof that those three things will guarantee a parallelogram. Okay, go.
CATHY HUMPHREYS: There was nowhere on this diagram that I showed that the diagonals were not perpendicular. How might I do that? How could I mark it on a diagram so that I would know that they were not all right angles? Okay Carmel, how?
STUDENT: By marking the two large angles with one hash mark and the other two too.
CATHY HUMPHREYS: Okay, okay. Like that? And then I could mark the other ones with that? Good, that is – that’ll show that these two are congruent but they are different from those two. Excellent! And you know what, him pointing that out to me was really important because that shows you the level of detail that you have to have; you have to be careful that you don’t forget any of these things. So you have three things that you have to show. Alright, back to work now and see if you can do that.
STUDENT: And then if these two are congruent then these two are congruent because look side, wait.
STUDENT: You could also say…
STUDENT: Let’s use a little piece of paper.
STUDENT: If these two are congruent then it would be like side, side to side.
STUDENT: Yeah but these two aren’t congruent. It’s like look, it’s this, this, and this; and these are congruent and then these are congruent, you know? See, so if this is congruent to this then it's side angle ̶ side angle right? And then if you start from here it's side angle side. So these two triangles – this triangle right here and this triangle right here are congruent. So if these two triangles are congruent then this is also side angle side and then side angle side. So if this triangle is congruent to this triangle and this triangle is congruent to this triangle then it’s congruent; and the angles are congruent and if the angles are congruent…wait, did I just…we just proved that this triangle is congruent to this triangle and this triangle is congruent to this triangle. So what does that prove...how does that prove it’s a parallelogram? We just proved that the insides are congruent. How do we prove that they have two sets of opposite ̶ two sets of parallel lines/sides?
STUDENT: Well, aren’t the alternate interior angles congruent?
STUDENT: Oh! Oh Yeah! Like see, this is the transversal right here and oh, okay. Wait, let’s try writing a proof somewhere. Is it...are they...look, let’s say this is the transversal right here, this diagonal right is the transversal.
STUDENT: But we don’t know what these angles are.
STUDENT: But we know that we can prove the triangles are congruent right? So if this triangle is congruent to this triangle then all the angles inside the triangle are going to be congruent. So the alternate interior angles are congruent also. So then if the alternate interior angles are congruent then you have parallel sides – parallel lines.
STUDENT: So the same thing for this side.
STUDENT: Exactly, exactly. So there, there we go. Now we just need to write a proof.
STUDENT: How do we write it?
STUDENT: Okay, you write it. No, no. Say, let's name the triangles…triangle A, B, C and D.
STUDENT: So if triangle A is congruent to triangle B by side angle side conjecture then triangle C is congruent to triangle D through side angle side.
STUDENT: And then say if the triangles are congruent, all angles within the triangle are congruent.
STUDENT: So if all triangles are congruent then...
STUDENT: Try CPCPT.
STUDENT: What’s CPCPT?
STUDENT: Congruent triangles are congruent.
STUDENT: Okay, so if CPCPT – all angles of triangle C are congruent to triangle D, and all angles of triangle A are congruent to triangle B. And then say that…
STUDENT: Should we give this one a name or something so they know which one is…
STUDENT: Look, angle one and angle two. So since angle one and angle two are congruent to each other ̶ so say if all angles of a triangle are congruent then triangle...then angle one is congruent to angle two. If all angles of triangle C are congruent to triangle D then angle one is congruent to angle two.
STUDENT: What did you write there?
STUDENT: They have to be two different lengths because if they’re the same length then it will be a rectangle or a square. Is that what we are supposed to do?
STUDENT: Then we are done.
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.