[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!]

## 7 replies on “Verticle Angles”

I’m not trying to be difficult here, or pretend that I haven’t written problems like this, but I’m not so sure anymore that we gain much from tying an obvious attempt to assess algebra skills to a geometry concept. Does this really show that the student doesn’t understand vertical angles, or that their focus was on “solving” an equation? And do problems like this have much pull with student experiences? I wish I could say I have the answers.

Ooh, Chris, I was just discussing with my almost-son-in-law (artist guy) about how great Geometry could be if we did drawing and observing, and how crappy it is as a vehicle to shovel more algebra down the craws of the disaffected. The other geometry teachers at my school have apparently decided not to do constructions because they can spend the time on algebra instead, to help the ACT scores. Maybe. Bleugh.

And my gut tells me that using geometry as a vehicle to practice more algebra is going to leave some major gaps in students’ conceptual understanding of geometric concepts.

I know that your reply is not for me (the person who submitted this mistake) directly, but Geometry teachers in general. However, I think that’s where many of the replies are going on this problem and I feel that I need to clarify. This problem was posed to a student who enrolled in Algebra 1 as I am an Algebra 1 teacher. He has never even heard of vertical angles. I threw this problem out there to see if the students could reason that the angles across from each other are congruent, as I feel that’s intuitive. So, as an instructor, I was able to see that he didn’t draw that conclusion, but was more concerned that he must have an equation even if he has to break a few rules to get one.

I do agree with your statements about Geometry, but when you see that this was asked of an Algebra 1 student, I feel that changes the perspective.

That does change the perspective. Now for the mistake…I’m going with underdeveloped conceptual understanding of equality. It’s an equation, so an equal sign must go somewhere. Students are used to seeing (and possibly more comfortable working with) one step equations where the equal sign would be where the student placed it in this expression.

I totally missed this one. I thought the question said what is the value of A, which made the whole algebra thing funny. Now it’s just a mundane (2 part) algebra mistake. I like my mistake better.

I think the idea is to force students to get used to solving multiple step problems. Of course, you could pick problems that are a little less aritificial, but it’s not a bad goal.

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