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.

9th & 10th Grade Math - Properties of Quadrilaterals*Cathy Humphreys, Fremont High School, Fremont Union High School District, Sunnyvale, California*

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CATHY HUMPHREYS: It can be a square. It’s a non square rhombus, yes. So if you want to do a square then that’s different. So in other words, we’re doing the most generic one. It’s true that a square is a rhombus but what you’re going to try and do, if you’re doing a rhombus then do a rhombus that’s not a square. Alright? So think about all the things that have to be... Do you have a list somewhere? See, I think maybe yesterday you didn’t write down enough stuff as you were going along. Yeah, alright. Excuse me, which one are you going to do?

STUDENT: We just proved it.

STUDENT: Rhombus.

CATHY HUMPHREYS: You’re going to do a rhombus? And do you understand that it’s a not square rhombus? Okay. You just got it?

STUDENT: We just got it. Yay!

CATHY HUMPHREYS: You’re excited? Okay.

STUDENT: Do you guys want to do a two column proof or…

STUDENT: Write A, B, C, D.

STUDENT: Paragraph.

STUDENT: But what if she tells us that we don’t know what a parallelogram is? Whatever, just say that we know what a parallelogram is. So a rhombus is a quadrilateral that is a parallelogram with all congruent sides.

STUDENT: Yeah.

STUDENT: Exactly. There we go. So then now let’s prove it. Jerry. Wait, we need to name them first; name each side ‒ A, B, C, D and then the midpoint. A two column proof...do we have given? What’s given? That angle...that they are congruent to each other right? That’s all that’s given? That BEC is a right angle?

STUDENT: Wait, hold on. AB is congruent to BC is congruent to CD is congruent to BA.

STUDENT: No, I think we need to prove that. We don’t know that yet.

STUDENT: No, but that’s…

STUDENT: No, we are proving that it is a rhombus. If we already know that these are all congruent then it’s already proved; and all we have to do is prove that they're parallel.

STUDENT: Wait, what’s given?

STUDENT: Given, I think the only thing that’s given is these two diagonals. We said the diagonals are only…

STUDENT: Oh, oh, oh.

STUDENT: So, so you say AC is perpendicular to BB; that’s given.

STUDENT: So AC is perpendicular to BB by given.

STUDENT: And two AB is congruent to EC and that BB is congruent to...

STUDENT: And then BE is congruent to ED. So now let’s prove that it’s a triangle. Do we have to prove…yeah, let’s prove the triangle...

STUDENT: Wait, so we have side angle side right?

STUDENT: Side angle side. So we know that all triangles are congruent.

STUDENT: So they should be all like perpendicular.

STUDENT: So all the triangles are congruent?

STUDENT: All triangles should be congruent.

STUDENT: Wait, do we know that…

STUDENT: So if they are perpendicular, if they’re perpendicular then yeah. You said that AC... If they’re all perpendicular then they’re all right angles. So I say BEC, triangle BEC is congruent to triangle DEC. Yeah because of side angle side and then if triangle BDC... So then BDC is congruent to BAD by side, side, side. Right?

STUDENT: Yeah.

STUDENT: So there we go. So it would be triangle BDC is congruent to triangle DEC and that’s because of side angle side. And then five is triangle BDC – no wait, wait. Can we just jump to that conclusion; that this whole…?

STUDENT: We know that these two are congruent right? So these sides have to be congruent and then they share this side right? So how would we write that?

STUDENT: Let’s write EC bisects...no. We already know EC bisects ED.

STUDENT: Yeah, so it’d be like... How do we say that BC is congruent to DC?

STUDENT: Because we already…okay, say that BC is congruent to DC.

STUDENT: Or should we just write over here…

STUDENT: Right? CPCTC right?

STUDENT: Wait, BC is congruent to DC because of CPCTC and then…

STUDENT: Wait, first let’s say we proved these.

STUDENT: So number six is triangle BEA is congruent to triangle DEA because of side angle side. And then BA is congruent to DA because of CPCTC.

STUDENT: And now how do we prove that all the sides are congruent to each other? No. Then we’ll say that the triangles…

STUDENT: These two triangles are congruent because of side, side, side.

STUDENT: Okay, let’s prove this first. So triangle BAD is congruent to triangle BCD.

STUDENT: Wait, which are we proving first, the sides or the parallel?

STUDENT: The triangle.

STUDENT: Wait, how do we know that these sides are congruent to each other?

STUDENT: We have to prove it like this. Oh, oh wait.

STUDENT: I thought we were going to prove alternate interior angles first?

STUDENT: First or second?

STUDENT: First because it’s easier.

STUDENT: So these are not necessarily parallel?

STUDENT: Yeah.

STUDENT: So there can’t be alternate interior angles.

STUDENT: No, these are supposed to be parallel; it’s supposed to be like this. It’s supposed to be a parallelogram.

STUDENT: Not necessarily.

STUDENT: But it’s supposed to be…listen.

STUDENT: Because a rhombus could also be like this.

STUDENT: But are these parallel? A rhombus has to be...look at the definition ‒ a quadrilateral that is a parallelogram. What is a parallelogram? Two pairs of opposite parallel sides and it's all congruent. So that’s why a square can also be a rhombus because they’re all congruent sides and they're both two sets of parallel sides. That’s why! Wait, how are we going to approach it? We have to change...we have to get...okay, what do we have to do? We have to prove that all sides are congruent and we have to prove that alternate interior angles are congruent too. So how do we...?

STUDENT: Wait why do we need to prove…

STUDENT: To prove that the lines are parallel; to prove two sets of lines are parallel.

STUDENT: But we already know that it’s a parallelogram.

STUDENT: That’s the definition. We just need to prove it.

STUDENT: How do we do that?

STUDENT: Where are we at right now? Wait, BA and DA so then…BA and DA.

STUDENT: So we’ve proved that these two are congruent and these two are congruent.

STUDENT: A quadrilateral has four congruent sides then opposite sides are parallel...diagonals are perpendicular to the sides.

CATHY HUMPHREYS: Congruently angles…um, so let me look at this. You know, since you have barely started, I would really love it if someone would do a kite. Would you mind? So a kite is um…

STUDENT: Okay. It’s pretty much the same thing.

CATHY HUMPHREYS: It just doesn’t have as much – it has…it doesn’t have the parallel it has in which two pairs of adjacent sides are equal and so I think...I would love – we don’t have anyone with a kite. So a convex quadrilateral in which two pairs of adjacent sides are equal – what does that mean?

STUDENT: It means that these two sides are equal and these two sides are equal but all four sides aren’t equal.

CATHY HUMPHREYS: Okay, now do you have notes on what the diagonals look like?

STUDENT: Um...

CATHY HUMPHREYS: You guys are so disorganized what am I going to do with you?

STUDENT: Oh, here is a kite.

CATHY HUMPHREYS: Oh you got it? Okay, good Tianna. Oh that’s good! Okay, so what do we have here?

STUDENT: We have one long or two long ones…

CATHY HUMPHREYS: Okay, the diagonals are perpendicular? Okay, so why don’t you think about how those are different from that.

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.