I briefly thought about doing something concerned with the Indy 500, but it's almost 10pm ET as I write this and that race is long over. I'm sure the excitement isn't for those present, but it would be forced now. So, continuing from last night's discussion on functions, let's talk, by which I mean complain, about piecewise functions.
Okay, one Indy-related question:
Q: How many ways can 33 cars be arranged at the start of the race?
A: One. That's what Time Trials are for!
That could've been a comic for today had I had time to make it. Busy weekend. Busier with the cleaning and the grading. But back to functions.
Just explaining to a student what a piecewise function is is not a simple task. Explaining how to read one takes time and patience, along with repetition of the phrase, "when you see the comma, think 'when'". Oddly, I once said, "say 'when'", and it sounded like I was pouring beverages out. I had to switch that up.
Worst of all are the examples that they give. They make no sense whatsoever. They are purely abstract creations that you wonder if they might ever come up -- even just in another math class.
This isn't to say that piecewise functions aren't useful in the real world, or even in mathematics. I could even justify them in Algebra as opposed to waiting for Trigonometry/Algebra 2. But do they have to make them so confusing off the bat. (Hell, the name itself is confusing -- and might I add that it isn't even recognized by my spell checker!)
Examples of reasonable math functions that they could have brought up? First, the absolute value function, which looks like this:
"When" x is less than 0, you want to flip the sign, (i.e., take the negative of x because a negative of a negative number is positive). Otherwise, leave x alone. Note that the last condition has to cover all other possibilities in the domain. You don't want to leave a value out(*), and you definitely don't want to repeat a value, because then it won't be a function. (*) Yeah, there are times you'll leave something out, but not here, not now and not with absolute value!
Another one that they can use which makes for an interesting graph, but doesn't have any variables, is the Sign function, not to be confused with the Sine function:
Negative numbers return -1, positive numbers return positive 1 and zero returns 0. This was useful when I was programming computers, something that I'm glad I've done and something to which I'll refer often. Why not steer the kids in that direction if it's something challenging that might interest them? By the way, the Sign function would be the basis for some kind of trinary system of anything when binary gets boring.
So we have two good examples to start. So what do the books give us? Something like this:
Okay, maybe nothing that nuts, and maybe nothing with e or i in the exponent, but it might as well have been. Non-continuous functions that mean nothing even in the abstract.
On the other hand, finding relevant, relateable uses for piecewise functions was a little crazy. I could've tried my Financial Algebra textbook which is constantly trying to get the students to create some of these (and eventually did). And then there are the old standbys, which don't mean as much any more. I used to use the example of different phone plans when talking about systems of equations. This is easily adaptable into a piecewise function. There's a minimum charge for a certain number of minutes and then you have to pay, say, $.10 per minute after you used your allotment. There are two problems with this example: first, the minimum means that there will be a constant, a line with a slope of zero and no variable. Second, what kid in my class a) pays for their phone, or b) doesn't have unlimited minutes. Minute plans are already a thing of the past.
Pay phones are right behind them, but you can tell them that they're at the mall or the airports and they'll believe that they're there somewhere and just haven't noticed.
My other example suffers similar problems: if you live in an apartment and aren't allowed to have a washing machine, then you have to go a laundromat where you can do it yourself, or you can pay to have them do it for you. It's usually done by weight. I gave an example (and I haven't dropped off laundry in a long time) of a cost of $.60/pound with a $6.00 minimum, and let them figure out that you're paying for at least ten pounds whether or not you bring ten pounds. Same problem with the flat minimum and just as unrelateable.
On the other hand, if I put enough of these problems on the board, they'll see enough commas and say "When".
Like I am now because this is too much.
30 Days. When!