Cathy Humphreys introduces the Properties of Quadrilaterals task. The lesson is introduced by explaining the students’ roles and responsibilities to carry out the investigation. The teacher models how to gather information for the tinkering stage of the investigative process. She also poses the math problem in the context of a kite making company. The kite making company uses sticks to make kites. The investigation involves how two sticks will be selected and positioned to determine the shape of a kite. She creates a purpose for exploring the key factors (length, intersection point, and angle position) that define the shape of a quadrilateral. She introduces manipulatives (different and like length strips with holes and a brad) and demonstrates how one quadrilateral might be determined by the arrangement of the diagonals. She demonstrates how the stick might be used to draw a rhombus.
CATHY HUMPHREYS: ...still work. Okay, just another minute. Would you finish whatever quadrilateral you’re working on right now? And as soon as you are done with that, would you please wait until everyone in your group is done? Okay, I see this table is done, this table is just about done and okay, this table is done, thank you. Let’s go around the table in a circle taking turns, but before you do that, I want us to review this investigative process; and we have our question and you are at this stage right now. So just remember all the different places you can go individually or as a group. I am actually going to put this arrow right here at systematic tinkering because we haven’t talked too much about that. But I want you to be thinking about how you tinker with a system rather than just arbitrary choices. So um, tell each other what you know so far and then I’ll talk to you about what you will need to turn in at the end of the period. Alright? So go ahead.
STUDENT: What did you guys find out about...?
STUDENT: That if you put these two equally and kind of at an x, this works.
STUDENT: Okay, so if you keep them the same distance from one point...
STUDENT: You see how it is a cross.
STUDENT: So it’s a cross. Yeah. (Inaudible). So that’s how you make a square. I don't think it's a perfect square because the opposites of the holes aren’t exact distance away from each other; and the points aren't exact.
STUDENT: And this still moves around. It’s not going to be a perfect square but...
STUDENT: It is not stable enough. So it represents a square.
STUDENT: This one, I did two random holes and then I got this.
STUDENT: So you got a trapezoid. So what are the diagonals for that trapezoid?
STUDENT: Here and here.
STUDENT: And can you make them all with a big one?
STUDENT: So was it something like that or...like this right?
STUDENT: Yeah, it was like that. I didn’t put it in yet.
STUDENT: So you used the holes as markers for your...
STUDENT: Yes in circles. Tell me what you did up there?
STUDENT: I used the end points. I think that made it a little sloppier. So Javier, what did you find out?
STUDENT: If you take a small one and a big one and you put them like a cross, you'll get a rhombus or a kite. They're all the same.
STUDENT: So you can see they are the same distance from each mid-point of the largest diagonal. I did the same thing and I also got a kite. You got a rhombus because it was...
STUDENT: You put one kite right?
STUDENT: Yeah, I put one higher than the midpoint of...
STUDENT: So if you were to put one lower then you'll get that.
STUDENT: It would still be a kite.
STUDENT: Yeah but...
STUDENT: Yes but it goes exactly in the middle then it would be a rhombus.
STUDENT: I found five of them and the rectangle you could make pretty much any size if you go outward with these two; if you put it right in the middle. And the square if you put...I’ll show you. So if you put it right in the middle like that then it makes a square. And if you turn it just like that then it makes a rectangle and you can change it to be however you want.
STUDENT: But did you notice this? But the small stick at ninety degrees angle, right in the middle makes a kite. So if you use...
STUDENT: Yeah, but it makes a diamond. So if you put it right in the middle...
STUDENT: So not just these two long ones but if we make... Let's say if we put the pink at the top or something... Oh, it makes a triangle. You know what I mean.
STUDENT: If you put these two right in the middle, I get a parallelogram.
STUDENT: Yeah, that’s what I got.
STUDENT: A parallelogram?
STUDENT: I got the kite but if you move it up...if you don’t put it exactly in the middle.
STUDENT: I didn’t get any regular sticks.
STUDENT: I got this parallelogram.
STUDENT: How did you get that one?
STUDENT: That was this one and like this.
STUDENT: Oh, I just put that straight. I got the trapezoid but…
STUDENT: So then what did we notice?
STUDENT: Anything like that is a trapezoid.
STUDENT: So what do we notice about this…what patterns do we get?
STUDENT: If you keep both of the points centered then you can get parallel lines.
CATHY HUMPHREYS: Because this is what everyone is going to turn in, their individual investigative work. So all the quadrilaterals that you did and I would like you put them in order. So that I...when class is over today I am going to go through your work and see what order you were doing. It’s going to give me an idea of what you’re testing and how you’re testing it. And then as you work today, keep evidence along the way of what you observed. And remember I showed you that piece of work from our class about how someone had actually written the comments um, about what they noticed, what they observed, a question they had. Those things I would actually like you to write out in words. If you have a conjecture like I think that this is true then please write that down. If you get stuck, like something is not working please write that down. Ah, if you get an AHA, a mathematical AHA then write that down. And any system of tinkering that you are doing, if you think it won’t be obvious to me by reading it, if you can write down what your system is; what are you testing for? So those are the things that I’m looking for because that’s going to help me see your process. Remember how we talked after the polygon investigations about how it's not just the end product that I want? I am really interested in your process.
Alright, so now the question is how will the manufacturer know what particular arrangement of sticks will guarantee, guarantee the quadrilateral that he or she needs to make? Um, the facilitators, you are going to coordinate your group, make sure everyone has something important to do…a part of the group task. Resource managers you are going to make sure your group has the materials you need and that the materials are also put away at the end of the period. That does not mean you have to do it but it means that you have to coordinate that. Team leaders make sure your group members are on task because sometimes it gets discouraging and people kind of drift off, so make sure that doesn’t happen. Oh, resource managers also call me over if you’ve got a question for your team. And then recorder, reporters your job is going to be more on Thursday instead of today; I think depending on how the investigation goes. Okay. So your first job – you have done some individual work so now as a team I would like you to talk to each other as a group. I would like you to talk to each other about how you are going to organize yourselves for the investigation as a whole group because you are going to have a group product as well. So would you talk that over for a few minutes? I am going to walk around eavesdropping and then when your group knows how you are going to proceed and everyone knows how you are going to proceed then you can get your materials and get started. Are there any questions? Alright, I am not going to interrupt you for awhile. Yay!
COMMENTARY BY CATHY HUMPHREYS: I have posed this problem as a “real-world” situation, which it clearly is not. The idea of the kite factory motivates a context for the investigation but it is a fanciful context, indeed. There are many truly real-world applications of mathematics (e.g., the mathematics of global warming). These applications are especially important for students to experience in order to understand the relevance and importance of mathematics in their lives.
I also want to emphasize the importance of the process of private individual thinking prior to the group addressing a task. This, as Ori said, ensures individual ideas and different ways of thinking that enrich the group’s approach to the problem.
As I closely watched video of the small-group discussions (an opportunity which a teacher rarely has), I noticed that although the students had individually made some conjectures, there was no emphasis made on recording these; and this also obstructed examination of exactly what the person had said. When I do this lesson again I think I will have one student at each group record for the group: 1) “What have you found out so far?” and 2) “What are you wondering about?” This way they will have a better starting point for their group work.
Finally, I would like to comment about the structure of the group work during this task. The idea of these roles comes from Elizabeth Cohen’s work on Complex Instruction. I have, however, not employed these roles in the true spirit and intent of Complex Instruction, in which the roles are assigned so that all students have the opportunity to perform each of the roles. Assigning the roles can help negate status issues that may arise from certain students always selecting a specific or preferred role.
COMMENTARY BY COACH DAVID FOSTER: Cathy states explicitly how students are responsible for investigating, working with others, making sense of the mathematics and sharing their findings. She describes the thinking the students will experience in the investigation, reminding them of the process they have both experienced in the past and how it is depicted in the process on their poster. She uses the notion of tinkering to help the students understand that they will have to work at making sense of the situation before they come to understanding it completely. Cathy poses the math problem, with a purpose that student relate to and can understand. She introduces the manipulative before having the students begin their investigation. She gives two sets of manipulatives to each group of four to foster collaborative exploration.