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 found a pattern, we found a pattern for the rectangle and the squares right? Because she told us that if we get a right angle then it will be a square and if it is not a right angle, it will be a rectangle. So that’s what we need to do for every single shape, like a trapezoid. Like for trapezoid I don’t know, I just play around with it. We need to find a way that every time we’ll get it. Like this, every time we are going to get a square or a rectangle. And then we need to do that for every single shape – like for a parallelogram, a trapezoid, ah what else, a rhombus. So do you want to split it up? Like you do rhombus and you do a trapezoid or a parallelogram. What do you want to do? Which one?
STUDENT: Um, trapezoid.
STUDENT: You’ll do trapezoid. Which one do you want to do?
STUDENT: You do the kite and I’ll do rhombus. You want to try that? Let’s try to find...let’s try to find like patterns and if you find something then share it; and then we will work from there. Okay?
STUDENT: So how should we do this?
STUDENT: So how do we know how far up the stick makes a regular and at the placement, if it makes a regular or irregular shape; and if the length matters?
STUDENT: It has to be like in the middle or something.
STUDENT: Yeah, I think they both have to be centered to make a regular. Well, in some cases. Like in the diamond case, it can’t be centered.
STUDENT: Wait, what is the diamond one?
STUDENT: The diamond, you take this short one and the long one and you make them both centered and then…
STUDENT: So you have to have an equal amount of dots on each side. Does that look right?
STUDENT: I think so.
STUDENT: So that’s a way to measure where you put it. What I want to know is does the size of the stick matter with like regular or irregular shapes?
STUDENT: I think it does matter.
STUDENT: So let’s try putting these two in the same spot and see if we get two different shapes, or if it’s the same shape but different size.
STUDENT: I have a square and a rhombus. This is a square and a rectangle.
STUDENT: No it isn’t.
STUDENT: How is it a rectangle?
STUDENT: Because it’s not.
STUDENT: I know but I made it...
STUDENT: I know I made the other two too and it’s a rectangle.
STUDENT: Oh wait, you’re right. Okay, my bad. Oh, I see why it is a rectangle because if they cross like this.
STUDENT: Oh, yeah. Like this shape.
STUDENT: Okay, so I have a rectangle and a rhombus. What do you guys have that isn’t a rectangle? This isn't a rhombus.
STUDENT: I have a rectangle and I have this. I'm not sure if it is a kite.
STUDENT: This side is kind of a kite.
STUDENT: This side and this side have to be the same for a kite.
STUDENT: This side is longer than that side so it's not a kite.
STUDENT: So it would be that too?
STUDENT: Then what would it be? But if you can’t tell what it is then that's what it is.
STUDENT: Okay, so you also have a kite.
STUDENT: And a rectangle.
STUDENT: Okay, so let’s put our rhombuses all together and then same with the rectangles.
STUDENT: So what is yours? A square, rhombus, trapezoid?
STUDENT: Do we need a trapezoid?
STUDENT: It acts the same as this.
STUDENT: So that means we need a trapezoid and we need an actual square. So we can say like, we took two pieces of paper together or something like that and... Whatever stuff you need.
STUDENT: Let’s put the paper in the middle.
STUDENT: We can go like this.
STUDENT: That’s not a square.
STUDENT: It would be a rectangle.
STUDENT: Let’s just tape two pieces of paper together.
STUDENT: This way?
STUDENT: No. So wait. How about you tape two pieces of paper together and then what you do is like, put these two things approximately like this and make a square.
STUDENT: So we’ll tape it like this?
STUDENT: Yeah, just like that.
STUDENT: Can we tape it? We should ask Ms. Humphreys.
STUDENT: Do you guys know what other shapes we need?
STUDENT: What about a rhombus?
STUDENT: We have a rhombus – wait, do we have a rhombus?
STUDENT: Yeah, we do. Trapezoid, parallelogram.
STUDENT: And a square which we are doing now.
STUDENT: Okay, so you are doing a square and trapezoid. Do you want to do a parallelogram? Okay, I’ll do something else.
STUDENT: I’ll do a trapezoid.
STUDENT: They have to be the same length.
STUDENT: Yeah, again they have to be perpendicular.
STUDENT: Would a trapezoid be the same way?
STUDENT: A trapezoid would not be perpendicular.
STUDENT: So they wouldn’t intersect on the mid-point?
STUDENT: They would.
STUDENT: They could.
STUDENT: Yeah, they do…not really.
STUDENT: You have to do them like rotating.
STUDENT: No, I think they do because it’s like this. Two sides are always parallel.
STUDENT: I think they have to be like this.
STUDENT: No. Okay, you have to imagine the sides. So on the top of the configuration I’m holding up...
STUDENT: Okay, now I see it. It doesn’t work.
STUDENT: So it’s really more like, like um, these two are congruent and these two are congruent.
STUDENT: It’s like two isosceles triangles.
STUDENT: That doesn’t, it doesn’t always necessarily have to be a true for a trapezoid right? A trapezoid doesn’t have to be an isosceles triangle; it can be something else right?
STUDENT: Well, so I suppose you could do that. That strikes me as somewhat weak; it would flex the wrong way.
STUDENT: Like in half.
STUDENT: Did we get our stuff we need to get our resources?
STUDENT: I thought we had all our resources.
STUDENT: I guess not.
STUDENT: Do you think that one of the sides has to be smaller than the other one?
STUDENT: We could try that out again.
STUDENT: Because I couldn’t get it with the large stick; I don’t know how to.
STUDENT: We have a paper that describes all the types of things like what a trapezoid is ̶ isosceles trapezoid.
STUDENT: Put in the middle it could be a rhombus every time.
STUDENT: Wait, what’s a rhombus?
STUDENT: A rhombus is a parallelogram with all congruent sides. That’s what I’m trying to do right now. So wait, I’m going to try the other way and see if we get a rhombus.
STUDENT: Do they have to be connected?
STUDENT: This is a kite too right?
STUDENT: Do they have to be connected anywhere other than the mid-point?
STUDENT: Wait – congruent, congruent. These two would be congruent and these two are congruent. Yeah, that would be a kite, I think.
STUDENT: Then we wrote that it would be a square or a rectangle. Then if it’s not, it doesn’t matter if it’s on top or bottom either. So it creates a trapezoid.
STUDENT: And then the segments must be congruent?
STUDENT: Yeah, this pair and this pair.
STUDENT: Can a trapezoid be a rectangle?
STUDENT: I don’t think so because...
STUDENT: Because isn’t a trapezoid only like the two sides are parallel?
STUDENT: Two maybe but I don’t think so. That’s a rhombus.
STUDENT: Are we supposed to find a formula or I mean write descriptive things about it?
STUDENT: I think we need to define for each shape what creates it, what diagonals.
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.