Two Things From a Tuesday

Or maybe I should title this post "Twosday Things." Because I like portmanteaus.

Thing #1:
Today, I was talking one-on-one with a student about functions. We were talking about the relationship between domain and range, and how to tell if two sets of values make up the domain and range of a function. We talked about how values in the domain are each assigned to one and only one value in the range by the function. I chimed in with the "mailbox analogy" to further explain the relationship: say you're mailing a bunch of letters. The stack of letters is like the domain, and the houses the letters are being mailed to are like the range. You can mail multiple letters to the same house, but you can't mail the same letter to multiple houses. "So you can't mail the same letter to Chicago, New York, and San Francisco simultaneously," I said to the student.

"Unless it's e-mail," the student replied.

HOLY CRAP. That was a really, really good point! I was utterly stunned that I hadn't thought of that. I guess the analogy kind of breaks down in that regard if you throw e-mail into the mix. I'm still pretty sure I got my point across, but it does have me thinking about the analogy I'm using to describe how functions work. Will this be an outdated analogy in the near future?

Either way, I was super-impressed by my student today.


Thing #2:
Some of my students are currently working on compound inequalities. Below is a piece of student work that I found interesting:

The left side of the compound inequality vanished! I've actually been seeing this happen with several students in my class; every time they get one side of a compound inequality equal to zero, they omit it in the rest of their work.

I've been wondering where this is coming from. I imagine it might have something to do with the fact that students are sometimes taught about the existence of an "implied" zero that isn't actually shown. (For example, what is the slope of the line y = 2? There's no x-term, but there's an implied "0x" in the equation; thus, y = 0x + 2, and the line has a slope of 0.)

Maybe it's coming from somewhere else. I don't think it's anything I've done, but I could be wrong.

Anyway, that's two things from a Tuesday. Maybe I'll try to do this weekly, so I'm blogging more often.

Multiple Solutions (A follow-up to "When Is the Right Answer the Right Answer?")

A couple of weeks ago, I wrote this post about how I wanted my students to determine equations of lines, given certain information. The broader point, I think, was realizing that my students had more than one option for determining answers to the problems they were working on, and being okay with that. (Why wouldn't I be?)

I had another "when is the right answer the right answer?" moment in class yesterday that I thought was really super-cool.

Two students were working together on the same problem. They came up with what they thought were different answers, so they were wondering who was correct. Their work is shown below:

So both students used point-slope form for their equations, and came up with two answers that looked different. This peculiarity made them wonder who was right and who was wrong. (Which, in turn, makes me realize that I still have a lot of work to do with teaching them about making sense versus being right.) They called me over to ask me who had the correct equation.

I must have been really busy at that moment and not really thinking, because I looked at their answers and said, "actually, you're both right." Not that I was wrong in saying so; but I regret that I didn't recognize the teachable moment that had presented itself. This would have been a great opportunity to ask each of them what they thought about their equations, how they came up with them, why they thought their answers made sense, why the other person got something different, and whether or not it made a difference which point they used for point-slope form. Still, it was a really cool moment: two students have a spirited debate over who had the "right" equation, when really they were both right. It was my favorite moment of class from yesterday.

Fortunately, the same thing happened today, on the same problem, with the same work as shown above, between a different pair of students. Grateful for a second chance, I was able to stop and facilitate an awesome math discussion between the two of them.

One student was adamant that the "first" point, (-4, 3), had to be plugged in for point-slope form instead of the "second" point, "because they're X₁ and Y₁," she reasoned. She said this because she had labeled the coordinates as such when using the slope formula to determine the slope:

And point-slope form was written on the board as Y - Y₁ = (X - X₁). So I could see where she was coming from.

I asked her, "so, how would you label these points if the order was swapped?" In other words, what if the problem listed the points "(6, 1) and (-4, 3)" instead of the order they were given? She responded that she would have labeled (6, 1) as (X₁, Y₁) and (-4, 3) as (X₂, Y₂).

My next question was, "So would that change things? Would you get a different slope, for instance?" The student initially thought that yes, she would get a different slope. The other student, who was working with her, said that the slope should be the same. I had both of them determine the slope of the line with the different designations for the coordinates; naturally, the slopes turned out to be the same as in their original work.

I asked, "how did changing the order of the points affect the slope?" The student replied that the order of the points didn't change the slope at all. "Cool," I said. "So what about the two different equations you guys came up with? What difference does choosing one point over the other [when plugging a point into point-slope form] make?" The first student still wasn't quite convinced that it didn't matter what point she chose; her partner said it didn't matter what point was chosen for the point-slope form of the equation.

We decided to have each of them solve their equations for y, so they'd both be in slope-intercept form. When they did so, they came up with the same equation, and the first student was finally convinced that it didn't matter which of the two points she chose. Both students were convinced that they'd both determined correct equations for the line described in the problem. "Why doesn't it matter which point you choose?" I asked. The first student wasn't quite sure. The second student guessed, "because both points are on the same line?" I replied, "that sounds like it makes sense."

I love when students find different (yet equally valid) solutions to problems like this. It makes for some great discussion. I need to keep myself aware that it's more important to ask my students to make sense of their work instead of telling them that they're right; I missed out on having a great conversation with two students yesterday, but I'm glad I had another chance at it today.