Do you see the mistake? How would you help this student?

"Find the midpoint of the line segment connecting (2,5) and (2,396)," at the beginning of the unit. Can you predict the top three responses?

— Michael Pershan (@mpershan) April 7, 2014

What would you predict? Here are some twitter responses:

@mpershan Is (0,391) one of them? That seems like a way to get an answer from those two numbers.

— Evelyn Lamb (@evelynjlamb) April 7, 2014

@mpershan (3.5, 199) (2, 200.5) (2, 198)?

— David Wees (@davidwees) April 7, 2014

@mpershan (2,198) is probably one of them. Maybe(1,198). But I am otherwise stumped. Oh! (201.5) maybe?

— Christopher (@Trianglemancsd) April 7, 2014

@mpershan er…(2, 195.5) as a top answer.

— Christopher (@Trianglemancsd) April 7, 2014

@mpershan I’ll bet at least one of the top 3 responses includes *two* points.

— Chris Lusto (@Lustomatical) April 7, 2014

—

Here’s your answer key…

**First Place:**

**Second Place:**

**Third Place**

I keep on seeing this in my Geometry classes this year. Tasked with finding the area of a right triangle, kids move toward the hypotenuse even if two of the other sides are given. Then they end up stuck looking for a height that they can’t find.

I’m pretty convinced — based on talking to kids and looking at their work — that this is all about how they see right triangles. These kids must be seeing hypotenuses as bases, and it must feel weird for them to treat the legs as bases. Or maybe instead it’s about the height? Maybe it feels strange to them to use a leg as a height?

Decimals are *hard*.

What would we even want the student to do here if he’s working in decimal? Like, how do standard multiplication algorithms handle something like a repeating digit?

That’s what I’m getting out of this mistake right now: the deviousness of decimal representation, and the way it can obscure numerical properties.

How about you? What do you make of all this?

What is this student doing? Where is their misunderstanding? @mpershan#mathmustakespic.twitter.com/7YQNGnopwi

— Lisa Bejarano (@lisabej_manitou) January 20, 2014

Let’s help Lisa out in the comments, mmk?

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?