In the previous few classes, the students had already successfully verified the three most commonly used criteria of congruence for triangles. In each case the class had first engaged in constructing a triangle given those specific elements (that it, two sides and the angle between them (SAS), a side and the two angles adjacent to it (ASA), or three sides (SSS), respectively) and then proved that any triangle satisfying those conditions would be congruent to the first one. Rather than following the steps of constructions suggested by their textbook or by the teacher, the students themselves had been expected to come up with a way to construct the triangle and to show to the rest of the class how such a method would "work" (both by carrying their construction out successfully and by being able to justify it using geometrical and logical arguments). This kind of exercise provided the students with plenty of opportunities for genuine and challenging problem solving as well as mathematical reasoning.
In the episode reported below, the students engaged in verifying yet another criterion for the congruence of triangles: whether two angles and a side adjacent only to one of them are sufficient to determine a triangle uniquely (this criterion will be abbreviated as AAS hereafter). The students had been asked to work on this construction for homework, though actually only one student reported having being able to accomplish the task. The teacher then decided to engage the whole class in verifying the construction proposed by this student, along the lines of that done in previous classes. This time, however, the proposed construction turned out to be more problematic and controversial, thus engaging the whole class in mathematical discussions and problem solving that ultimately led to two alternative procedures for constructing a triangle satisfying the given conditions.
The teacher went to the blackboard and asked the student who had done the homework, Sam, to give her specific directions so that she could reproduce his construction on the board. He started by asking the teacher to draw two angles (specifying that they could not both be obtuse) and a side. In response to these directions, the teacher drew arbitrarily the two angles (labeled 1 and 2) and the side (labeled 3) showed in Figure 8.1.
To get other students directly involved in the activity, the teacher then explicitly asked the participation of the rest of the class in these first, easy steps of the construction:
Teacher: Who knows the next step? I know Sam does. Who else knows the next step? ... We're just checking it [AAS] out to see if it could possibly be that the situation created a triangle. What's the next step?
Tracy: Draw one of the angles on the side.
Teacher: Measure the first angle on this working line here? Okay.
Using a protractor and a rule, the teacher then proceeded to put in practice the step thus suggested, by accurately measuring and reporting in another part of the board the side and one of the angle, and continually checking with Sam and Tracy to make sure that she was following their instructions correctly. The figure reproduced in Figure 8.2 was thus obtained.
At this point, another student (Linda) suggested that the teacher replicate angle 2 on the other extreme of side 3, but then corrected herself once she realized that this would not correctly represent the condition given (since one of the angles should NOT be adjacent to the given side):
Teacher: Okay, there's angle 1, Tracy, what am I... Who can tell me what I'm going to do next?
Linda: Construct angle 2 on the other...
Teacher: Construct angle 2 on this endpoint? (the teacher points to the other extreme of side 3)
Linda: Yes. (Some students begin to comment among themselves and cause Linda to correct herself) On the opposite ray.
Teacher: Why are you changing your mind?
Linda: Because for angle-angle-side you need the angle at the top.
Teacher: Does everybody see why she changed her mind? If I put angle 2 over here (at the other extreme of the given side) it will be angle-side-angle (ASA) and we said that if we are writing them this way (AAS) the convention would be that they are supposed to go in this order. So if I stick it over here I'm doing it in the wrong order.
A brief pause occurred while the teacher tried to follow Linda's second direction literally, by reproducing angle 2 at the end of the extension of the other side of angle 1 (called "ray 1" hereafter), as shown in Figure 8.3.
The students were quick to realize that this result was not acceptable, and some of them started to look for some ways to "fix it".
Pat: It doesn't work.
Sam: Oh, oh no. You made the line too long or a ... Erase the line.
Teacher: Erase line 3?
Sam: Not the whole way. Okay.
Sam suggestion was to "adjust" line 3 so that the ray from angle 2 would meet it in the "right" spot (implicitly suggesting that when he did his construction at home using this procedure he must have been extremely lucky in choosing the correct positioning of angle 2 on ray 1 or, alternatively, must have used some "trial and error"). Most students in the class, however, were not willing to accept a "trial and error" procedure of this kind. Jane, in particular, was able to articulate precisely the key problem with this procedure:
Jane: You've got to know the lengths of the sides before you add another angle to the [figure]. Do you know what I mean?
Teacher: I think I might know what you mean, I am not sure if anybody else knows what you mean.
Jane: If you can't do it on that side (indicating line 3), then how are you going to do it on that side (indicating ray 1) in the right way? You don't know the length of that side.
This precise articulation of the problem may have contributed to its solution, since at this point other students were able to suggest a creative procedure to construct the desired triangle that made used of the first failed attempt:
Todd: Extend the line.
Teacher: Extend which line, Todd?
Todd: The side... the one that makes up the angle (indicating ray 1).
Teacher: This side? What, Shea?
Shea: If you copy the angle that's right above [line] 3, that doesn't have a number... (she points to angle CC'A in figure 8.4) If you copy that [angle] to the end of line 3, then just make the top of line 1 be the top of the other angle. Do you know what I mean?}
The idea behind Shea's construction can be better understood by looking at her final product, reproduced in Figure 8.4 (the letters identifying specific points have been added by me so as to make the narrative that follows easier to understand).
Shea's procedure is indeed a very clever way to resolve the problem of knowing exactly how much ray-1 needs to be extended, taking explicit advantage of the information provided by a prior incorrect result. Most interestingly, Shea herself was able to realize and articulate it:
Teacher: [This] is probably about what Todd wanted me to do. He wanted me to make this line longer. ... Todd suggested: If this line were longer, it might work...
Shea:Yeah, but you don't know how much longer to make it.
[...]
Shea: It works, but you can't do it unless you do it wrong first.
The rest of the class seemed now convinced that this construction would work and were able to repeat individually such a construction on another example. Some students were also able to explain why the procedure worked by observing that given that the angles ACC' and ADB (see Figure 8.4) are equal by construction, then the lines CC' and DB are parallel and, hence, the triangles ACC' and ADB are similar.
After this activity was accomplished, the teacher also tried to come back explicitly to Shea's comment, in order to point out that making an error first was a necessary part of the procedure, something that could not be avoided if we wanted to eventually reach the correct result. Not every student seemed ready to agree with this statement. While the teacher attempted to move on to the question of whether Shea's construction would yield only congruent triangles, some students continued to argue among themselves about the appropriateness of the procedure itself. Overhearing them, the teacher decided to bring their controversy to the attention of the whole class:
Teacher: Jim ... [you said to] Jane "No, you don't have to do it wrong the first time to make it work." And Jane said "So, how do you do it?"
Jim was invited to the board to prove his point. After some unsuccessful attempts, he had to give up. At this point, however, another student (Todd) was ready to take his place at the board and show an alternative procedure to construct a triangle given AAS which would not require "to do it wrong first". His construction relied on the known property that the sum of the angles in a triangle is 1800, as he used this property to construct the third angle of the triangle, which could then be used to construct the triangle using the ASA procedure created in the previous class (as illustrated by Todd's final product, reproduced in Figure 8.5).
Todd's procedure took the teacher by surprise, since she herself had never thought of this possibility before. This reaction is reflected in the following dialogue, which occurred after Todd had completed the construction of the third angle on the left side of the blackboard and was starting to construct the triangle next to it:
Teacher: Are you starting over?
Todd: No, I'm starting the triangle now.
Teacher: Oh, wait a minute, I think we need to hear what he's doing here. You are just starting your triangle now?
Todd: Yeah.
Teacher: So what you were doing was preliminary work? And what was that?
Todd: You see the trouble was we couldn't get an angle at this end. And since all three angles of a triangle equal 1800 (He points to the picture he has just completed on the left) angle 1, angle 2 and you just... the rest is that angle.
Pat: Wow.
Shea: That's really neat.
Both the teacher and the other students were quite impressed by the ingenuity and the novelty of this procedure, though some students spontaneously questioned the validity of his procedure. This, in turn, initiated a discussion on why Todd's construction would work:
Student: It does look like hers (Shea's). Hers just leaned a little more.
Mary: Proof. Proof.
Teacher: You've got to prove what, Mary? You've got to prove that the angle 2 that we were supposed to have really does end up there.
Student: (interrupting) He did [prove it].
(As this conversation is taking place, Todd finishes his drawing and the class applauds.)
Teacher: Who is the other person in this room who obviously thought about this picture and can explain it?
Jim: He constructed angle 1 and then added angle 2 to it. Also three angles... all three angle equal 1800. The portion left from angle 1 and angle 2 is ... hum, angle 4... So then you go angle 1, line segment 3, [and then angle 4] goes from the end of line segment 3.
Teacher: Was angle 4 part of the given?
Students: No.
This observation allowed the teacher to point out that often in mathematics one has to use information and results that are not immediately given in the text of the problem:
Teacher: Before this one (Todd's procedure) could work, what had to happen?
Linda: You had to think about it.
Teacher: (With emphasis) Yeah you had to think about it! That's true.
Jim: And you had to remember that the angles equalled up to be 1800.
Teacher: So you had to bring in some other information? ... What [else] did you use to make it work? What did he make really and truly to construct that triangle?
Don: He used two angles and a side.
[...]
Teacher: Look very carefully for a minute at the triangle that he used. I got there, but I got there in rather an odd-ball way. Look very carefully at the triangle he made, remember how he made it.
Mary: He used angle-side-angle first.
Jane: Yeah, he did.
Teacher: Angle-side-angle. Which was something that we already proved. Right?
This observation led to a final discussion of the difference between AAS and ASA, as well as of the cumulative nature of mathematical results, based on the realization that previously proven results can be used in new proofs.