Two interesting mistakes here. The first has to do with the Pythagorean Theorem, the other (more interesting) has to do with the angle of inclination.
I wonder what she’s looking at that the angle always stays the same. My guess, based on her first triangle, is that she thinks that the diagonal of a rectangle always bisects the right angle.
This might make for a nice bit of feedback for her. I could ask, “Is it possible to draw a rectangle whose diagonals don’t always make 45 degree angles? The answer matters for what you wrote here.” Or maybe the feedback I supply here should be a counterexample — a very long rectangle whose diagonals clearly don’t make 45 degrees? What’s my goal in this feedback, anyway?
I suppose my only goal is to have her know that the diagonals don’t bisect the angles, and to believe this in a way that she’ll remember and be able to reproduce on a new problem. So I want to equip her with the means to prove it to herself.
Given all this, I think I should probably be more direct in my feedback about the fact of non-bisection. I should leave the proof up to her, though. “Try to draw a rectangle whose diagonals don’t make 45 degree angles.”
One last worry. What if I’m wrong about my diagnosis of her thinking? What if she is seeing 45 degrees in these ramps in some other way? Maybe the best thing is to check in with her verbally before giving her any written feedback, to confirm that my theory is correct?
Update (4/23/15): Here’s the feedback and her post-feedback work. In conversation, I was able to confirm that my “every rectangle’s diagonals bisect a right angle” theory was right.
This is fairly representative of the class’ work. What would your next step be with this class?
Lots to notice here, including the formula that the student is using for the area of a triangle.
Let’s help Lisa out in the comments, mmk?
What interesting mistakes! Let’s make everything that’s puzzling about these explicit.
“6, 8 are equal, but 10 isn’t equal.”
- Does this mean that 6 and 8 are equal to each other? Or that 6 is equal, 8 is equal, but 10 is not equal. (To what???)
- What on Earth does it mean that 10 isn’t equal!
- What exactly does this student think “right triangle” means? Does he think it means that all the sides are equal?
“Yes, because all the sides are equal.”
- How? You drew a picture showing that the triangle has sides of length 6, 8 and 10!
- So a right triangle needs to have all equal sides?
This is mysterious to me, but what’s important is to not dismiss these students as hopelessly confused. Take the second mistake. What we’ve discovered is that you can know a lot and still think that a 6, 8, 10 triangle has all equal sides. That’s really cool!
As far as shedding light on these mistakes, I’m really having trouble coming up with anything that makes sense. I’d say that the top student is not saying that 6 and 8 are equal to each other, but then what is that student saying?
This mistake seems ripe for theorizing. Would a kid make this mistake if the hypotenuse was 4 instead of 10?
Here’s a mistake from a trig class. Would the question be easier in a Geometry class?
Here’s my theory: teachers underestimate how weak most of our students’ knowledge is, and how weakly in transfers. In particular, this problem became twice as difficult as soon as it was offered in the context of a trig class, without carefully writing the right angle in there with the lil’ square.
Am I right? Wrong?
Thanks to the Uncanny Tina Cardone for the submission.
What other mistakes would you expect to see from this problem? How do you teach so as to help students avoid these pitfalls?
Thanks to Tina Cardone for the submission.
Say something smart in the comments about why kids forget “little” things on problems. Or, alternatively, disagree with the premise of my first question. (How’s that for a lousy prompt?)
Thanks to Mark Kingsbury for the submission!
What made this question hard for the student? How come they got it wrong? Why did the student get it wrong in this particular way?
Today’s submission comes from Tina Cardone, who blogs at Drawing On Math.