A couple days ago, Christopher Danielson (@Trianglemancsd) wrote a post about Domain and Range, under the header "College Algebra Teachers! Please try this and report back!" In brief, voting on numbers being in the range for the basic quadratic function.
By sheer coincidence, can you guess what my class happens to be studying at the moment? It looked interesting enough, so here's my results.
First, context regarding the class. I don't know what "College Algebra" actually refers to - I'm guessing a college course. I ran this experiment in my high school University Level Grade 11 "Functions" class. (In Ontario, Grade 11 has four streams: University, Mixed, College and Workplace. Only the first two of those even mention domain and range.)
|"Please get that off my vertex."|
(Aside, about intersections: They recognize that for a horizontal line to have one quadratic intersection, it needs to be the vertex. But they think that a line with positive slope can also have the vertex as a single point of tangency. EVERY year. I flat out do NOT know where that comes from. Do you? Anyway.)
On Wednesday, I'd defined domain and range as input and output for a relation, and from there talked about representing inequalities on a number line - which they've never seen before. I ended that day demonstrating that I could toss in a perpendicular number line, and stretch the domain up or down, so that now I have a range. That night, I read Triangleman's post.
To start on Thursday, we took up previous work. Then I flashed up a bunch of Cartesian graphs, and got them to identify domain and range from those. We saw our first "unrestricted domain" - a line - and the proposal was x>=0 AND x<0 to describe it. I challenged "why 0", got them to admit that a number choice was arbitrary, and hence x is simply an element of any real number. We also saw our first "step" graph with holes featured in 2D.
With about twenty minutes left, I said we were now going to consider domain and range if there is no graph, just an equation. And the yellow (for YES) and pink (for NO) voting sheets were brought into play. This is a class of 29 students, 27 of which were present. If you want to see the "script" in advance, read the post I linked to at the top, otherwise, here we go.
Quick test to see if they get the voting system. I ask if we're in classroom 128. We are, so mostly yellow, but a few held up both colours simultaneously because they were not sure of our room number. Rousing start. I ask about our principal's name (giving the wrong one), and all pink, so that goes better. Then I ask whether they enjoyed their lunch (which was just before my period), getting a mix. Onwards, to the basic parabola. I write on the board y = x^2.
1) Is 4 in the range of this function? I think this was totally "yes" yellow. Explanation: Because 2 times itself gives 4. (I probably should have written that down on the board, I didn't.)
2) Is -2 in the range of this function? 20 "no" pink, 7 yellow. When questioned, the first yellow guy admitted to mixing up domain and range. (My God, Chris is psychic!) Next yellow guy simply followed suit, so if they did think something else, I couldn't access it.
Explanation for no, along the lines of because you can't have a value times itself being a negative result. (I'd scribbled "As long as no (-)ve" on a sheet as they spoke... forget now what I meant.) Someone DID bring up what number system are we talking about, I said let's stick with real numbers.
3) Is 1/4 in the range of this function? Like 4, above. When I asked for the explanation, I got because 0.5 times 0.5 gives 0.25, so interesting abrupt introduction of decimals.
NOW IT'S INTERESTING
|Pi is for the oven, not the range!|
I don't remember exactly how this went down. I do remember that right off the bat, someone was saying "you can use the square root of pi", because "that root would cancel out when squared". The counter argument was along the lines of "what sort of radical is that when what's underneath it goes on forever" as well as "that's weird to look at". I wrote on the board again here: (√π)^2.
They seemed to accept this because I wrote it, thus I said, "So you're telling me that pi IS in the range, but -2 isn't?" That briefly brought up the idea of number systems again, which allowed me to reinforce why it's important to mention the "element of reals" when talking about domain and range. I also asked what sort of number √π was, but aside from "non terminating decimal", I didn't get much, even when I explicitly asked for an approximation.
After some cajoling, someone with a calculator gave me an "estimation" to put on the board (to 4 decimals). I'd forgotten to put them in groups or anything, but we were already over ten minutes by this point (I have lousy time management skills), so I had less than ten remaining in the period. I moved on.
5) Is 0 in the range of this function? Two people voted 'No'. One of the 'yes' people shouted out "it's the vertex!". A 'no' person bravely challenged this with "how do you take the square root of nothing?". Another of the 'yes' people thought that was a good point, isn't that an error on the calculator? Rebuttal that zero times zero gives the range of zero! There was a bit of calculator fiddling, ending with agreement that zero was fine.
At this point, I wrote on the board the range was "y greater than/equal to 0, y element of Reals", to link things back to the earlier graph work in the period. General acceptance of this. Then the wheels fell off again.
|There's always Pros and Cons|
"Infinity is a continuation of the curve! It's in there!"
"Infinity is a CONCEPT, not an output!"
They wanted me to arbitrate, I said there was merit both ways, but that it is true that infinity was not a number. Her rebuttal was fast: "If it's not a number, how come we can have both positive AND negative infinity?" Wuh huh, well then.
I basically said I didn't have an immediate answer for that, as I only had two minutes left to me at this point. Tossed a couple of textbook questions onto the board for everyone else to consider that night. (In retrospect, maybe I should have asked, what do you square to get infinity?) The two students continued to argue about it as they went to their next class.
On Friday, I started with some time to work since I hadn't given that to them at the end of yesterday. No one brought up the arguments of the previous day. Mostly they were just disappointed that I wasn't going to sing. (I've sung the last two Fridays.) So we moved on to the new lesson, where I finally defined what a "Function" was!
I have no idea if this resembles what might have been expected. I'm going to resist posting my own analysis, because it'd be brief, as well as to avoid prejudicing any comments. But as a reward for reading to the end, here is a post from my serial, about domain and range!