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 made it right now and then we used the end points of the blue and yellow sticks. Then if you connect the dots...
STUDENT: Oh, so it’s kind of like making a kite but it’s not perpendicular.
STUDENT: It’s kind of like making a rhombus but it’s not congruent. The sides are not congruent, see.
STUDENT: Does it have to be in the middle?
STUDENT: Yes, I am pretty sure it has to be in the middle.
STUDENT: Because it has to be both equal distance.
STUDENT: Oh, actually it might not.
STUDENT: I think it does. These have to be equal distance from these.
STUDENT: See, these two sides are parallel.
STUDENT: Yeah, it wouldn’t work if it was in the middle. These two sides are not the same.
STUDENT: That’s how we make a...
STUDENT: Like this one would be a parallelogram because the distances are not equal.
STUDENT: Okay, so we have rhombus. Do we have all the shapes now? Wait I have a question? Do you want to ask Ms. Humphreys if we have to show trapezium, no right?
STUDENT: Wait, what do I write for a parallelogram? It has to be in the mid-point?
STUDENT: It has to be the blue stick and the yellow; the short stick and the long stick have to be put together at the mid-point of both sticks.
STUDENT: At the mid-point and any angle except for the perpendicular or parallel.
STUDENT: No, say "use short stick" not blue because blue it could be red or yellow. Just say short stick.
STUDENT: What’s the word for moving them? If you connect the short and long sticks at their mid-points and move them or...
STUDENT: Or angle them?
STUDENT: So if you connect the short and long sticks at the mid-points, any angle you put them in except for perpendicular or parallel, will create a parallelogram. Wait, isn’t a rhombus technically a parallelogram? Because two sides of...look. Yeah, so it could be parallel too. So it could be a parallel too.
STUDENT: Yeah, except for parallel.
STUDENT: Wait, do you want to ask her if we have to do trapezium...to show a trapezium?
STUDENT: Okay, hold on. Except for parallel... Are there any other shapes we can do it with?
STUDENT: We have rhombus, we have parallelogram, we have rectangle, square; we have kite and we have trapezoid.
STUDENT: Are there any other shapes?
STUDENT: Trapezium, that’s it. That’s it.
STUDENT: So that can be like anything.
STUDENT: Exactly, that’s why…
STUDENT: So is there any other options we can do?
STUDENT: Other than trapezium, I don’t think so. I think we did...
STUDENT: I don’t think you can define trapezium because any other... So there is no specific definition.
STUDENT: It depends on what kind of trapezium you want to do.
STUDENT: So how about we don’t do…wait, do we have it…everything we have right? Is that it?
STUDENT: Wait, for the kite it has to be the short and the long sticks and they are perpendicular in the middle; and one has to cross at the mid-point or does it matter which mid-point on the other ones?
STUDENT: Except for the end points right? But wait, we have to do... Do we have to explain each one or can we...? Do we have to, wait…since we each explained our own and we put it together, do we have to write a copy of each other's work? No right?
STUDENT: No, we have to turn it in together. So did we do the rectangles?
STUDENT: I made a rectangle.
STUDENT: You noticed if you put the pin in the middle of the two sticks and… Oh, I need to make this clearer because this doesn’t make sense. If we put the pin in the mid-point of both sticks, of both of the long sticks, and make them perpendicular then the shape that will be created is a square. And if the angles are not right angles then a rectangle will be created.
STUDENT: Be sure you’re like...oh, what’s it called? Specific, be specific on your explanation.
CATHY HUMPHREYS: So you’re thinking that you can’t make a parallelogram with what you’ve got, is that right?
STUDENT: We’ve been trying.
STUDENT: We keep getting squares.
STUDENT: And trapezoids and pretty much every other thing you can think of but we just can’t seem to get the parallelogram.
CATHY HUMPHREYS: Huh! Are you keeping track of every single thing you are writing down and in order?
STUDENT: Well, I have them in the order of when I wrote them. All information I just wrote down, I just categorized them with what I thought they pertained to. So this would pertain to squares and rectangles and what I wrote down here about the kites.
CATHY HUMPHREYS: Excellent, excellent! So are you keeping track of what you’re trying as you are trying to find a parallelogram? So keep track of the things that aren’t successful as well as things that are. Alright? So why don’t you work together as a group and see if you can do it to get a parallelogram.
STUDENT: Do you think we can get a parallelogram?
CATHY HUMPHREYS: Do I think you can? Yes, I think you can.
STUDENT: Wait. Oh no, we need more rhombuses.
STUDENT: So a kite with two long?
STUDENT: One long and one short. Wait, a rectangle, I mean a square is a rhombus right?
STUDENT: I was asking if a square was a rhombus. And it’s also a kite right?
STUDENT: No, I think you can make a rectangle out of it.
STUDENT: That’s what I was saying but I wasn’t sure. I didn’t count on that. No, that would make a kite.
STUDENT: Isn’t that what I’m making?
STUDENT: Yeah, sure.
STUDENT: Okay, so this makes a kite.
STUDENT: Because a kite is shaped with two sets of parallel lines right?
STUDENT: I know, I know.
STUDENT: I can’t make a rectangle out of this.
STUDENT: Me neither.
STUDENT: Okay then…
STUDENT: I made a parallelogram.
STUDENT: Then a rectangle. Oh, it is a rectangle.
STUDENT: Wait, do we have a parallelogram with the short one? Did you write it?
STUDENT: Yeah. Okay.
STUDENT: I think a rhombus is a kite.
STUDENT: But that...what did you make that one with?
STUDENT: A long stick and a short stick.
STUDENT: That one looks kind of equal.
STUDENT: If it was equal, it would be more square because the...yeah.
STUDENT: But you can’t really tell which one. Like with this one, you can tell. This one is the short side and this one is the long side.
STUDENT: It could be like this?
STUDENT: When you go by the circles, the short one matches up right here. Yeah. Could you ask Ms. Humphreys if a rhombus is a kite?
STUDENT: I can but what do you want?
STUDENT: Could you ask if a rhombus is a kite?
STUDENT: So if that rhombus is a kite?
CATHY HUMPHREYS: You have a question?
STUDENT: Would that rhombus also be a kite?
CATHY HUMPHREYS: Would this rhombus also be a kite? That is where you need this.
STUDENT: Good idea. The convex quadrilateral in which two pairs of adjacent sides are equal, the opposite sides are not parallel. Oh! Okay, so it is not a kite.
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.