The students are in groups of four. Each group has chosen a group member to perform these roles: a team captain, resource manager, recorder, and facilitator. The groups have access to the problem (one page per group) and two packets of manipulatives for a group of four. They also have other resources that they can retrieve, including a page of definitions of quadrilaterals. The students begin working on the investigation in different manners. In some of the groups, students work individually for a while. In other groups, students work in pairs, and in still others, the entire group of four is collaborating. For those groups who initially retrieve the definition page, it seemed to influence the order they attacked the problem. The square, rectangle, and rhombus appeared to be the most straightforward for the students. Mathematically, if two of the diagonals form right angles, then at least a pair of sides of the quadrilateral will be equal in length. If the diagonals intersect at the midpoint of both diagonals, then the figure formed will be some parallelogram. In order for two diagonals to form a non-isosceles trapezoid, the following relationships must hold true: If AB is one diagonal and DE is the other diagonal, then trapezoid ADBE is formed only if the diagonals intersect at point P, which is not the midpoint, and AP/PB = DP/PE. This relationship was quite difficult for the students to investigate and conclude. The students did not choose to measure the diagonals with rulers, and therefore did not pick up on the proportional aspects of the diagonals in a non-isosceles trapezoid.
STUDENT: We have two pairs of congruent sides.
CATHY HUMPHREYS: So how do you know that a rhombus cannot be a kite? And I want you to refer to the definitions because that is a really important part of math. So look at the definitions.
STUDENT: And this side is not parallel to this side.
CATHY HUMPHREYS: Okay, all sides of what have to be parallel?
STUDENT: All sides of a rhombus have to be parallel.
CATHY HUMPHREYS: Are you saying that they are all parallel?
STUDENT: No, like the opposite sides are parallel.
CATHY HUMPHREYS: Okay. Alright.
STUDENT: And a kite…
CATHY HUMPHREYS: What does it say about the kite Chris?
STUDENT: Opposite sides are not parallel.
CATHY HUMPHREYS: Ah, opposite sides are not parallel. So there we go. What do we know from that?
STUDENT: That they cannot…
CATHY HUMPHREYS: What’s the “they”? Be careful of those pronouns.
STUDENT: A rhombus is not a special kind of kite.
CATHY HUMPHREYS: A rhombus is not a special kind of kite. Okay, good. Alright, you're doing great. So do you have a plan before now? How are you doing?
STUDENT: We just have to figure out ̶ we are still trying to find out how we can do a trapezoid, a regular trapezoid.
CATHY HUMPHREYS: Okay, so here’s what I’m going to…I don’t want you to get derailed on that because that is hard. So let’s do this, let's do all the other ones first, including an isosceles trapezoid. But don’t worry about the not isosceles trapezoid for a while and if you are ahead then you can try and work on that one. Okay?
STUDENT: We split up the shapes between us each; which ones we understand most and then we shared how we found it with each other.
CATHY HUMPHREYS: You did?
STUDENT: Yeah, like the requirements that it needs to have to make the shape.
STUDENT: To make the shape every time.
CATHY HUMPHREYS: Wow, so are you knowing all the…
STUDENT: Well, we’re trying to figure out how to make a right angle trapezoid.
STUDENT: Because we found out how to make a trapezoid but then we needed to make like…there are different trapezoids to make each time.
CATHY HUMPHREYS: Okay, so when you’re talking about how to make a trapezoid, what kind of trapezoid were you talking about?
STUDENT: She was the one who found out how to make a trapezoid.
CATHY HUMPHREYS: Okay, this. What kind of trapezoid is this?
STUDENT: A regular?
CATHY HUMPHREYS: Regular. What’s the word?
CATHY HUMPHREYS: Okay. What’s so special about this quadrilateral…trapezoid?
STUDENT: These two sides are congruent.
CATHY HUMPHREYS: Okay, that’s called an isosceles trapezoid then and so here’s what I’m going to suggest, if you’ve got a...if you’re sure that you can guarantee an isosceles trapezoid then don’t worry about the not isosceles trapezoid until you get all done with everything else because that’s really a problem on its own. Alright? So now, let’s see, have you…I guess where are you in the investigative process?
STUDENT: They‘re AHA proven, I guess.
CATHY HUMPHREYS: So do you have your AHA'S for every single shape?
STUDENT: That’s what we’re doing; we’re like writing them down.
CATHY HUMPHREYS: Excellent, excellent Jerry! So then the next stage would be to go on to the proving and explaining part; and you’re actually making so much faster progress then I thought that I haven’t even talked about what to do with that. But you’re doing…this looks really good. Do you feel good about the work you’re doing so far?
STUDENT: So then wait. So I am doing this just out of curiosity. So this is just because they are both congruent? While I said that opposite sides are parallel and then the diagonals cross at the same spot...so if you made them at the same spot and the same lengths, what if they are different lengths? Do you think it will make a different trapezoid? Like, it will make it a trapezoid with…
STUDENT: Well, when I said same lengths, I meant like at these two.
STUDENT: At the same hole.
STUDENT: Oh, at the same hole.
STUDENT: Yeah, right.
STUDENT: So if it went like that, would it be a different…?
STUDENT: That wouldn’t be a trapezoid.
STUDENT: What about...how would you…this is bugging me. How would you make it a trapezoid that is not an isosceles if these are both congruent every time? Wait, all you have to do is just two parallel sides right?
STUDENT: So what if you make it like this though? Wait, wait so…
STUDENT: Then the sides would be parallel.
STUDENT: What if you make it like this?
STUDENT: No, but that isn’t parallel to the other one.
STUDENT: Wait, it could be like this.
STUDENT: No, those two are parallel right?
STUDENT: Yeah, those two might be parallel.
STUDENT: Where is another piece of paper?
STUDENT: Did we run out? I'll get some more.
STUDENT: Don’t these look parallel?
STUDENT: No, that one looks like...
STUDENT: Or maybe if it was like this.
STUDENT: Put it right on the table.
STUDENT: They are not perpendicular.
STUDENT: Push it up a bit. That’s it. Which sides are you trying to make parallel? These, these, which ones? Let’s just try it, let’s just try it. I don’t think we have to prove this but let’s just plug it in. How do we test it? With a compass? You want to test it with a compass to see if it works?
STUDENT: Or like, we can measure it from different points. Like from here to there and from here to there.
STUDENT: Wait one second. This kind of looks like a trapezoid doesn’t it? Oh, that’s not parallel. Nope, these sides are not parallel. So this is not a trapezoid.
STUDENT: That’s a…
STUDENT: So yeah, I don’t think we even need to prove those. She said that it’s a problem within itself.
STUDENT: We made a trapezium.
STUDENT: We made a trapezium, yeah.
STUDENT: So do we want to find out how to make a right angle…
STUDENT: Trapezoid? We could. How about this, how about we prove – I'll write down each one...we get...I got square, I got rhombus and I’m going to do, I’m going to do a parallelogram, then I have to do a kite and trapezoid. After that we’ll try to prove...we'll try to find a right angle trapezoid. Okay? But first make sure you have every single one.
STUDENT: I don’t have a rhombus.
STUDENT: Here, I have a rhombus.
STUDENT: Only one side at the intersection is congruent. So this is the...not a midpoint but a point that segments and only this part and this part must be congruent. So then the other sides of these are parallel.
CATHY HUMPHREYS: Okay, stop for just a second. Dorothy, you are at a really good place. You were at an AHA and you kind of had that flash of insight and then it kind of vanished right?
CATHY HUMPHREYS: That’s okay. That’s good. So why don’t you try to talk yourselves through it or talk yourself through it and you can all try to fiddle around with Dorothy’s idea. But um, there is something about where you put those two together and what angle they are at because you are trying to create…
STUDENT: I think I got it again.
CATHY HUMPHREYS: Okay, you talk to your group about it then. I have complete confidence in you.
STUDENT: Okay. Because these aren’t congruent so just pretend that they are. Then we would have to cut some of the blue ones off because the sticks need not be congruent and I don’t want to do that. So pretend that these are congruent. This is congruent but this side and this side are not congruent; and these are parallel even though they don’t look parallel.
STUDENT: He tried them like that and he didn’t get one.
STUDENT: No, wait. So that means this triangle has to be perpendicular.
STUDENT: He made one like that and two different.
STUDENT: Is this perpendicular? Yes right? No?
STUDENT: No, it doesn’t need to be perpendicular.
STUDENT: No, once it's perpendicular then these two lines can be parallel. If it‘s not perpendicular then it’s not...
STUDENT: But if you shave off this, this side isn’t congruent to that side.
STUDENT: No, I am saying that the diagonals have to be congruent. Okay wait, let’s draw this out.
STUDENT: Well, I made a trapezium!
COMMENTARY BY CATHY HUMPHREYS: Reviewing the videos of group work was truly fascinating for me; it is rare for a teacher to be able to watch students’ conversations so closely.
Throughout the block I saw difficulties with the use of mathematical language and definitions. The frequent use of pronouns (“it”, “they”) obscured what students were trying to communicate. Other examples of difficulties in communicating mathematically include imprecise or incorrect uses of terms (regular, diamond) and definitions.
In Part B an important question was asked but was not taken up with interest by the other group members: “Can a trapezoid be a rectangle?” His team member answered, “I don’t think so,” but did not justify his answer using a definition. This makes me think that he had an intuitive notion that a trapezoid was never a rectangle, but he did not articulate why. I am afraid that unless students are allowed more opportunities to develop the notion of the importance of precision in mathematical language, they will not be able to use definitions to support their ideas.
In Part C as well as in the closing of the lesson, I refer to “mathematical friends.” This notion came from Thinking Mathematically by Burton and Mason, a book about mathematical problem solving in which the authors talk about a hierarchy of certainty when trying to write a convincing argument. Convince yourself (the easiest), convince a [mathematical] friend, and finally, convince a skeptic. Developing a skeptical mindset and not jumping to conclusions too quickly is another hallmark of good mathematical thinking.
It was fascinating to me that Carmel and her group thought it was sufficient to measure the alternate interior angles of one figure in order to prove that the opposite sides parallel for all like figures.
Our class had studied the triangle congruence theorems in December and had been able to show that corresponding parts of congruent triangles are congruent. They also had studied theorems (and their converses) about the angles formed by parallel lines and a transversal. And all year we had been working with the idea of a general argument and the difference between showing something is true for a particular case and showing it is true for all cases. Carmel and her group were able use the knowledge that if alternate interior angles are congruent, then the two lines are parallel. But they were unable to access the need for a general proof and to use triangle congruence to do so.
It keeps hitting me that when students are confronted with a new problem that they have not seen before, it is very often difficult to access skills they have learned in an isolated context or setting. Practicing CPCTC over and over again with different types of problems did not help these students recognize that proving triangles congruent and using this to show that alternate interior angles were congruent, rather than measuring, would accomplish the general proof they needed. Knowing how to use a tool is very different from knowing when to use it, and students need a lot more practice with problems that require them to dig deep to find, from all that they know, the tools and ideas that will help them solve a new and unexpected problem.
COMMENTARY BY COACH DAVID FOSTER: Cathy makes use of these roles throughout the class to get groups started, making sure everyone is participating, retrieving materials and cleaning up and recording findings and communicating their conclusions. The issuing of the materials was quite purposeful. Cathy knows that if the students have one problem they will need to begin by sharing the instructions together. She also handed out two sets of manipulatives so that students could have hands-on experiences, but not so many as that each team member would be off working only by his or her self. Cathy wants to promote individual think time, so how students actually reconvene as a team is dependent on their own learning needs. Most groups were able to use the manipulatives and their understanding of quadrilaterals to determine how the arrangement and size of the diagonals determined the square, rectangle, rhombus and parallelogram. The students found how to form a “geometric kite” rather easily, but struggled with how to described all the different positions and size of the sticks that would make a kite. Students used the holes in the manipulatives as a measuring tool, but very few connected how the holes of the two sets of sticks were proportional. The trapezoids (isosceles and non-isosceles) were the most challenging quadrilateral to find and define for the students. This was especially true of the non-isosceles trapezoid. Although, they were able to construct the trapezoid with the manipulatives, they were unable to ascertain the exact relationship of the diagonals necessary to determine a non-isosceles trapezoid. Realizing the proportional relationship between the intersections was not very obvious to the students. The positioning of the holes (although proportional) did not seem to help students see this important relationship.