Category: Geometry

Pythagorean Theorem Similarity, Right Triangles and Trigonometry

“All the sides in the 6, 8, 10 triangle are equal.”

Post author By mpershan
Post date January 12, 2014
8 Comments on “All the sides in the 6, 8, 10 triangle are equal.”

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

— Lisa Bejarano (@lisa_bej) 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?

Geometric Measurement and Dimension Geometry

Sides and angles

Tina asks:

Why do kids have such a hard time distinguishing between sides and angles? They are so different in my mind, I don’t even know how to explain the difference.

Thoughts?

Algebra in Geometry Expressing Geometric Properties with Equations Vertical Angles

Verticle Angles

[Re: the title, I know I’m not the only one who has seen “vertical” misspelled in every possible way.]

It’s easy to say that this is sloppiness on the part of the student. And maybe it is. But it’s the sort of sloppy mistake that I would rarely make, and that beginners often make, which leads me to think that there’s something else going on here as well.

We’ve talked about reading on this site before, and it’s something that I don’t know a ton about. But it seems to me that part of “looking for and making use of structure” is something like what I’m trying to get at. If you’re really experienced at math, then you start seeing a problem like this as rigidly structured into two separate and equal expressions. I’d bet that for a student that doesn’t have a lot of experience with these sorts of problems that sort of structure is less apparent, and this sort of mistake is less apparent.

Did that make any sense?

[Thanks to Nora for the submission!]

Congruence Proofs

If ABCD is a parallelogram…Prove that angle A is congruent to angle E

Post author By mpershan
Post date October 23, 2013
6 Comments on If ABCD is a parallelogram…Prove that angle A is congruent to angle E

These are from my classroom, and a little bit of context might be helpful:

This is from a very strong Honors class.
Proof, right now, means something less formal in our Geometry class than it might mean in yours’. Proof doesn’t mean “Statements/Reasons.” Proof means offering an explanation for why something is true.
We do this because it’s just as rigorous without crushing the souls of anyone in the classroom. Look at the array and variety of reasoning going on in these proofs. By keeping things less formal, we’ve got enough breathing room to actually do some Geometric thinking.
As the year has been going on, though, we’re getting more and more rigorous. These exercises help reveal some sloppiness in the kids reasoning. These proofs fail by their own standards of explanation. I’m thinking that I’ll be printing these out and handing them back for discussion. This is exactly what my English teachers friend does with essays.

Feel free to comment wildly here, either on my standards, some of my bulletted statements, or about any of the student work.

Equations of Parallel and Perpendicular Lines Linear, Quadratic, and Exponential Models* slope

Slope of two parallel lines

Post author By mpershan
Post date October 22, 2013
5 Comments on Slope of two parallel lines

Tina says: “Two students have done this so far. Not a mistake, but still curious what these kids are thinking:”

She’s talking about the 4/6 thingy. Any ideas, people?

Congruence High School: Geometry Proofs

Inferring Congruence

Post author By mpershan
Post date October 15, 2013

https://twitter.com/MTChirps/status/390185033631137793/

https://twitter.com/MTChirps/status/390201190085971968

https://twitter.com/daveinstpaul/status/390198645884088321

@mpershan @MTChirps @MrPoliquin But rules of implication may be at heart of the error rather than knowledge of angles. And maybe guessing.

— Christopher Danielson (@Trianglemancsd) October 15, 2013

@MTChirps Does she know that the converse often doesn't follow from the conditional? Maybe she thinks the converse is always trouble.

— Michael Pershan (@mpershan) October 15, 2013

Your thoughts?

Geometry Quadrilaterals

Parallelogram Problems

This is a straight up student-empathy question: what was this kid’s thought process like?

Thanks to Tina for the submission!

Editor’s Note: I categorized this as Grade 7 – Geometry in the CCSS, but I’m not really sure if that’s right. Where does this belong?

Geometry Right Triangles

Special Right Triangles

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!