Let’s take for granted that these students don’t have conceptual understanding of the Pythagorean Theorem, because if they did, then they wouldn’t make these mistakes. (I actually think that we need to be more careful with the ways that we toss around phrases like “conceptual understanding” but whatever.)
What do these mistakes reveal about how these kids think about right triangles and the Pythagorean Theorem in the absence of conceptual understanding? Why does this ever make sense to the student?
Thanks to Michael Fenton for the submission!
3 replies on “Special Right Triangles”
Two thoughts:
(1) In the lesson leading up to this exit slip, the class used the Pythagorean theorem to find an unknown third side length (when given the first two). They then looked for patterns among the 30º-60º-90º triangles, as well as the 45º-45º-90º triangles. Many students made meaningful observations/connections, some did not (as evidenced by the four images). So here’s my question: Did these students perform so terribly on this four-question exit slip because I was pushing them to be efficient before they were ready to be efficient? And if that’s the case, how do we know as teachers when our push for efficiency is actually detrimental for students? How do we know when an individual (or small group, or entire class) is ready to move from slow-and-steady-and-meaningful to generalized and more efficient? Should the push ever come from the teacher, or only from the student?
(2) Needless to say, the responses captured in these four images were more than mildly disappointing. They are signs of medium to massive misunderstanding. But they made for great classroom conversation at the start of class the next day. One approach students used in that followup conversation was the impossibility of many of the responses given the decimal approximations of the side lengths in radical form. For example, in the first image (isosceles right triangle with hypotenuse of 1), the legs (which must be smaller than the hypotenuse) cannot possible by approximately 1.41 in length. And so on.
Bonus thought: What looks like an outrageously misguided approach to someone “in the know” (whether a teacher or another student), might seem as reasonable to the student who is struggling as any other “throw-some-values-and-an-equation/formula/algorithm-into-a-blender” approach. If you don’t truly know what’s going on with the Pythagorean relationship, why not toss some angles in and press liquify? After all, using that a^2+b^2=c^2 nonsense has yielded the right answer in the past. (Further evidence that we—myself and my students included—pay too little attention to the hypotheses of our theorems, and then wonder why our (mis)use of the conclusions leads to wonky answers.)
I was tutoring a student on this topic recently & also wondering whether I pushed them to this “efficient” method too quickly. Do you think the students would be able to continue with the method used to prove these relationships? If it is 45/45 label the legs in a way that indicates their lengths are the same (values or x & x). If it is a 30/60, put two of the triangles together to make an equilateral triangle & label in a way that indicates the shorter leg is half the hyp. I guess I am having trouble seeing the point of ever introducing the “efficient” method (proportions with 2^.5 & 3^.5). It isn’t that much more efficient and the proof method is good practice with a technique and reasoning skills that can be applied in other situations.
I like the suggestion of drawing the rest of the square or the equilateral triangle to make at least some of the side relationships visible. It should also help make sure that the hypotenuse is actually the longest side! And cut down a lot on the memorization, too.
As you described in (2) above, I’m glad that the students came up with the idea of the impossibility of these numbers and showed the value of estimating in checking the work.
I call the “throw stuff into the blender” method “coping strategies” — you’re lost, you have no idea what’s going on, so you do something desperate to try to survive. Some students are so good at this that it can be difficult to distinguish from actually understanding what’s going on, until at some point in their mathematical career they get to the point that they simply can’t keep so many ad hoc rules in their minds at once and the precarious structure collapses. I think your conjecture in part (1) is correct, that a lot of this comes from rushing to “efficient” methods before students are ready for them. I think in many cases (as hodge suggests) there’s no reason ever to go to the efficient method. Often if you want the efficient method, you’d pick up a computer instead! Humans can understand, calculators can calculate. On the other hand you do want to generalize and abstract (“abstract” here used in the computer science sense of finding the commonalities in a bunch of things and rethinking them as a single thing with a variable parameter, for instance), so those types of efficiencies are vital. I’d characterize the “good” kind of efficiency more as recognizing regularity, generalizing, and identifying patterns. This kind is near the boundary: “Oh, I’ve seen triangles with those angles before, this side will be sqrt(3) times that side” is a kind of pattern to recognize, but its realm of application is narrow enough that it might not be worth looking at as hard as we do, when there are other much more heavily re-usable patterns like the Pythagorean Theorem or the idea of using symmetry to understand a diagram.