I wonder if the student would’ve made the same mistake with “scaling by 2.” In any event, this isn’t necessarily a nasty misconception to fix, but I’m always interested by the circumstances when addition and multiplication get tangled up.

What is this student thinking? @mpershan #mathmistakes pic.twitter.com/ytTFS4sDV6

— Lisa Bejarano (@lisabej_manitou) January 12, 2014

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?

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.

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.

My sense is that this mistake isn’t as interesting as the rest, but it’s a pretty common one that I see in Trigonometry. The question is, what sort of activity would help this student out?