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 Will this be an outdated analogy in the near future?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.
I am reasonably certain that 0's disappear because "they're nothing," but also that the student is accustomed to a simple inequality, so his answer looks right, and the right answer doesn't (to her).
ReplyDeleteI'm pretty sure it comes more from the "they're nothing" thing. I think the concept of the "implied zero" or the "implied one" is something that we don't really get at quite enough. Maybe because that's a bit more of an abstract thing. But it's important, I think!
DeleteI refer to the compound inequality at "between." We need all the values between 2 things. That helps a little. (Honestly, this is the best year I've had with Algebra 2s, and i'm wondering if it's because I had them all in Algebra I... not that i did such a great job with them, but that they know my style by now.) As for the email analogy... Our textbook uses numbers on a phone's keypad to explain domain and range. Many of the kids don't know that numbers on the phone have letters. Ah, for the old days. (Remember when # stood for numbers instead of hashtags?)
ReplyDeleteHaha, I remember those days. Now it's hard to imagine why anyone would need a LAN line anymore!
DeleteThe "in between" thing does help. I talked later this week with my students about this problem, and that one reason why we have to keep zero written down is because it's a "boundary" of the inequality.
I had an honors student ask me today if zero squared was zero. Great
ReplyDeleteI actually confiscated a calculator from a student who was using it to perform simple subtraction with integers and made her do it herself. Took her a moment (I was probably making her nervous), but she got it.
DeleteTIMELY. We're talking about Function as a Mail Carrier on Monday.
ReplyDeleteI'm going to adjust the language to 'package.' And I'm so glad that analogy in general continues to be useful.
Ha, I remember the function conversation in TE 408 (or 802 or 804, can't remember which). I recall that we were able to talk a lot about the *characteristics* of a function, but couldn't really pin down what a function actually *is* in general. I've actually been wondering lately if that's somewhere near axiom territory.
Delete(Also, isn't it nice to know that I still wonder about some of the things we talked about in class 5 years ago?)
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