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.

• Max Ray

I had the exact same misconception about diagonals of rectangles until I built a shed in my backyard last fall (at the age of 30)

• i think that she is completely unaware that degrees measure sizes of angles. Give her a protractor and let her measure the angles (assuming that the drawings are to scale).

• Phase Shifting

Love this one. I’m wondering along similar lines as Howard: what does the 45 mean to her? The square completed around the first example does seem to be a clue. And in the first example, she takes the time to write that she got 45 by calculating 90/2. So she seems something as being 90, and something as being in two parts. But what? I agree that it would be helpful to check in verbally along the lines of “how are you thinking it through when you give the angle?”

Ok, she thinks something about all of these examples is the same. Is there something that is the same, that she might be looking at? They *are* all right triangles. Is it possible that the 90 she’s looking at is the 90 degree angle in the triangle? Or, maybe she’s aware that in a right triangle, the other angles have to add up to 90, and she’s dividing by 2 because there are two of them.

I think it would be hard to hang on the idea that diagonals of a rectangle always bisect the angle, if you have a feel for what an angle measures, and you look at these pictures. It’s possible, like Howard suggests, that she either doesn’t have a feel for the meaning of “angle”, or possibly she’s not fully taking in the pictures.

If that’s the case, a question about the possibility of drawing a rectangle where the diagonal doesn’t bisect the angle may not help — without a feel for angles as “spread” or “width” or “pointiness” or “steepness” or some other related concept, that question doesn’t make sense… it almost doesn’t mean anything. So knowing how she’s thinking about the meaning of that “90” may help determine which questions will make sense.

Another possibility: closer to what you’re thinking about bisecting. If she really is thinking about bisecting, is there something here that does get bisected, that she might be looking at? I think there is: if you complete the square around these rectangles, you can conclude that a diagonal always bisects the *area* of the rectangle. This is a powerful idea, that she will need later on — especially when using vectors. I only wish my students (college-level electronics technicians) had fully internalized that idea. If that’s the case, it’s the correct thinking about a different problem.

When checking in with her, I’d be especially interested to know which other ideas she sees as connected to angle. If she sees angles as being connected to area,

Something I wonder: how would these problems look to her if you presented triangles with the above ratios but inside a rectangle of constant size, with the base of the triangle equal to the width of the rectangle but the height of the triangle varying, so that it hit the side of the rectangle at different points? It might help her make sense of angles while preserving her intuitive sense of connection between angle and proportion of total area. Bonus points if you can make an animated gif of the angle decreasing gradually 🙂

I’m also interested in the question formats the physics ed researchers call TIPERS. What would happen if you asked her to rank them in order of steepness? Or “choose/draw a triangle that might have properties of ….” Or, along the lines of your initial suggestion, “is it possible to have a triangle with these properties?”

• Chester Draws

Given her ability to have a triangle 5 across, 1 up and 13 on the diagonal, this is maths done without any relation to the real world. And certainly without looking at the diagrams.

The 45 degrees is some sort of learned response, and I suspect trying to source it won’t help much. I don’t imagine she thinks of degrees as being real things at all — they are some obscure thing that teachers want.

She might just be writing a random answer because some students don’t like teachers thinking that they are not trying. So they always answer questions, even when they know that they are wrong.

• daqu

If you just look at her work, she correctly writes L^2 + 1^2 = c^2 (where L is the long leg), then divides both sides by 2, but on the RHS she gets c^2 / 2 = c (!!!).

This is the source of her errors.