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: If you change the angle that you have it at…
STUDENT: And so that way, those two lines are parallel.
STUDENT: Uh-huh. So if you have something see, like this then you change it like this. See, like this. If you got these two, it'll be parallel.
STUDENT: And if you change one a little bit like that.
STUDENT: Just a little bit and then it wouldn’t be parallel. It would be parallel with this one but not this way. You would have a trapezium, exactly.
STUDENT: I am writing this down.
STUDENT: So you guys think that any rhombus we get or kite we get is also a rhombus, like exactly the same rhombus. So since we’ve been using...we know that using these two diagonals we get a kite or a rhombus depending on where we put them. So...but what would happen if you leave them off? Like, if it wasn’t to the mid-point of the diagonal or the mid-point of this one? So the thing that I noticed was that when we had either a rhombus or a kite, this was...we always used the mid-point of this diagonal. So what if it wasn’t at the midpoint of the diagonal? What if it was something like that? It would still be a trapezoid right? So you can also use the two diagonals you get from making a rhombus or a kite; you can also use them to make a trapezoid or a trapezium.
STUDENT: Okay, I am writing it down.
STUDENT: So any two diagonals that produce a rhombus or a kite may also be used to produce, I think is one any trapezoid or trapezium. So what do you guys…I guess that was just one of those...
STUDENT: AHA moments?
STUDENT: AHA moments! Yeah, one of those eureka moments. I don't know what that really tells us about...
STUDENT: I think it was like a small goal.
STUDENT: Yeah. So we have two diagonals that make two sets of polygons and we have two observations that pertain to these two. So I think we should start working on these two sets of diagonals more than the larger ones because with these ones we only have one set of observations; and with these two we have two whole sets of observations. So I think we should tinker with these two sets to see what we can get.
STUDENT: The only thing that we haven’t made with those two is a square and a rectangle.
STUDENT: I made a square.
STUDENT: With these? How?
STUDENT: Oh, I thought you meant in all.
STUDENT: No, I meant with these two because we had two observations that pertain to two sets.
STUDENT: But it is impossible to make a square with this because all sides are equal, making the diagonals equal.
STUDENT: So this has to be…so you can’t make a square or a rectangle.
STUDENT: Let’s see.
STUDENT: These two can’t produce a square so um, diagonals used in producing kites, rhombus, trapezoids or trapeziums cannot produce…
STUDENT: Okay, so here we go – can’t make a square without equal diagonals.
STUDENT: Can’t produce a square or a rectangle. Can it make a parallelogram?
STUDENT: Can these two make a parallelogram? I know that two large diagonals can make a parallelogram. Okay no. But we can also get a parallelogram with these two.
STUDENT: So now we have a third set of observations that have to do with the large set of diagonals.
STUDENT: I’m going to tinker some more.
STUDENT: I want to find out why the only type of parallelogram that we can’t actually make is a parallelogram. That’s confusing. So we can make a square or rectangle but not a parallelogram.
STUDENT: You did make one.
STUDENT: Oh, so we can make a parallelogram.
STUDENT: Yeah, we can make a parallelogram but not a square.
STUDENT: So out of all the shapes we’ve made, we produced everything except a square and a rectangle out of these two.
STUDENT: And trapeziums.
STUDENT: Yeah, we can make a trapezium.
STUDENT: Oh, I thought you said that you could.
STUDENT: No, with these two we can make a trapezium easy.
STUDENT: And the rhombus, and the kite and the trap and the parallelogram, except not a square or a rectangle because they’re not equal. The other observations we have that we're wondering about…
STUDENT: Did I make another trapezium or is that the...?
STUDENT: Yeah, that is another trapezium.
STUDENT: They should be pretty straight because this is like down at an angle but straighter.
STUDENT: Like I said…most common.
STUDENT: Yeah, the trapezium that we had…there are all sorts of shapes that just don’t work. You can’t define them. The only one we messed up on was the trapezium.
STUDENT: We can get the trapezium right?
STUDENT: Yeah, because we don’t know if two lines are parallel so we can't really...
STUDENT: What would this be?
STUDENT: That would be a parallelogram. See, two angles are…that is the exact same angle and those would be parallel to each other; and so are those actually. See, these two and those two would be parallel.
STUDENT: It has to be in the middle?
STUDENT: Wait, is it off the middle of both of them?
STUDENT: So you would get it pretty much the same way you would get a kite.
STUDENT: So this is a rhombus right here and if you turn it then you get a parallelogram.
STUDENT: And does it work if you turn it any other way? Even if you move the angle a little bit away, just a tiny bit away, you still get a parallelogram.
STUDENT: So from a rhombus you can also produce a parallelogram. Oh, we know that because of the observation we had with the kite. Essentially we are using the same steps. But what about through a kite, through a kite can you make a parallelogram? Like through the kite thing would this…
STUDENT: Well, because they’re not equal. I think you can. It does kind of look like it.
STUDENT: No, because this angle is not straight.
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.