Quadratics have become my new nadir, which is cheerier news than it sounds since it means I’ve kicked linear equations into obedient submission. For the first two and a half years of my teaching career, I felt good about quadratics because if nothing else, most kids remembered how to factor, and remembered that factors had something to do with zeros on the graph. Which was a big step up compared to what they retained of linear equations. But then, last year, I cracked linear equations in a big way, which is great except now I just feel bad about quadratics, because as I develop as a teacher I realize the suckers are absurdly complicated and don’t model very easily. The kids learn a lot, but at their level of ability I’d need to do two months to have them internalize quadratics the way most of them internalize linear equations. And I don’t have two months. I just tell myself they still learn a lot. Consequently, I am relieved to see quadratics in the rear view as I move them onto the third of the models that define second year algebra (at least, as I teach it).
Exponential functions are awesome. First, they’re absurdly simple compared to both lines and quadratics. Second, they model actual, honest to god, real life situations. I’m not a big teacher for “Hey, this is something you’ll use again” but automobile depreciation or interest payments are, in fact, something they’ll use again. Third, they provide a memorable and again, useful, reason to review (or learn for the first time) percentage increase and decrease. Finally, they present a situation in which any kid who has even somewhat grasped the course essentials can see hey: Given y, I can’t solve for x. This leads beautifully and meaningfully into logarithms.
So like linear equations, I can kick off the unit with a modeling activity and get the kids moving easily into the math.
I begin with a brief lecture reminding them of the two previous models.
No. Quadratics aren’t repeated multiplication. Exponential functions involve repeated multiplication, as they’ll see in the lesson.
Then I review percentage increase and decrease. I am of two minds about this review. On the plus side, it’s immediately relevant, easy to apply, and gives them a good reason to remember it long term. The downside: the kids never remember what I taught them when they get to the percentage problems. So I explain it up front, knowing that 90% of the kids will forget everything I said just 20 minutes later, when they get to the first percentage exponential increase.
So I explain it, go round the room asking “So, if I want to increase a number by 8%, what do I multiply it by, Jose?” “1 point…..8?” “Watch that leading zero!” “Oh, 1.08.” “Right.” Do that with five or six times, think everyone gets it, and set them to working on models. This is one side of the worksheet, crunched for space so I could “snip” it.
And sure enough, the kids work through the models, making great progress, and stop cold at the third one.
“I can’t do this. How do you increase by a percentage?”
“Excuse me while I beat myself on the head with this whiteboard.”
“Nothing. Do you remember me just talking about percentages?”
“Do you see it on the board there? All the stuff about turning it from two steps into one step, and why you need to do that?”
“DO YOU SEE ANY POSSIBLE CONNECTION BETWEEN THAT CONVERSATION AND THIS PROBLEM?”
“Man, I don’t see why you’re so mean.”
“Read what it says on the board. Right there. In red.”
“Increase x by a%.”
“Yes. Can you read problem 3 and tell me what you think might possibly qualify as x?”
“Yes. And do you see the value that might possibly qualify as a%?”
“Um.” Long pause as the student stares at the problem, and finds the ONLY OTHER VALUE MENTIONED. “Twenty percent?”
I repeat that four or five times to four or five groups and then, miracle of miracles, find a student with a full table of five values for the population problem. There is a god.
“But I don’t know how to find the equation for this one like I did the first two. This one isn’t repeated multiplication. I had to take 20% of 250 and then add it….why are you hitting yourself on the head?”
“We need a function. We need an operation in which we can plug in x—do you have any thoughts on what x might be?”
“How many months?”
“How is it you know that, you smart child, and yet make me go through this torture? Yes. We need an operation that we can plug in the number of months (x) and get the population (y).”
“Right. But this is like three steps.”
“And we need only one.”
“Wouldn’t it be cool if there were a way to increase a number by a given percentage in just one step?”
“How do you do that?”
“LOOK AT THE BOARD!”
“Oh, is that what you were talking about? I was already doing the worksheet.”
And still, the lesson is largely a success. Kids are absolutely freaked out at the cell growth caused just by doubling and yes, I bring up the million dollar mission example, but at the end of the lesson, not as part of it. Most of the kids correctly graph the models, although a few end up with lines that I correct. The flip side of the handout is a blank graph, which they use to take notes on the basic exponential growth model.
Total Amount = Initial Amount * Ratetime
Initial Amount > 0
Rate > 1
One thing I mull over—the book, and the state test, go through the exponential equation (basically, Initial Amount = 1), along with the transformation model (f(x) = ax-c +- k. I haven’t focused on this in previous classes, because in my experience the kids don’t even get tranformations of lines and quadratics. But I’m going to give it a try on Monday.
“So if I take away a third of something, how much is left?”
Pause. Pause some more. Pause still more. I grab three whiteboard pens.
“Rhea, decrease these pens by a third.” Rhea obediently takes one pen.
“Class, how much is left after she decreased the pens by 33%, or a third?”
“Two……?” I wait. No. I sigh, and grab three more pens, getting the one back from Rhea as well.
“Paul, take away a third of these six pens.” Paul takes two pens.
“Class, he’s taken away 33% of the pens. How much is left?”
It all works out. Seriously. By the end of the exercise, most of the class is shouting back the correct answers as I ask “I take away 30%, how much is left?” 35%? 23%?” and the only mistakes they make are place errors—that is, 100-23 does not, in fact, equal 87.
The second day is always better, because it has slowly permeated their skulls that I’m serious about this percentage nonsense, that it has some relationship to the worksheet. So when they ask questions, it’s more of the “could you run this whole percentage decrease by me again? If they take away a third, I have two thirds left? But what’s two thirds as a decimal?” and trust me, this is a big step up for my blood pressure. Well, a step down. And they do the decay modeling and notes with no small degree of interest:
They have the model graph on the back, too, for exponential decay:
Total Amount = Initial Amount * Ratetime
Yes, it’s the same equation, so what’s different?
Initial Amount > 0
0 < Rate < 1
By day’s end, they have registered the import of the realization that Estefania has 95 cents left after ten days, and they’ve figured out that Jose is right, that his car is worth more than Stan’s after five years, which they managed by using an equation they built themselves, by golly, rather than decrease 25,000 by 5% 5 times.
You notice, of course, that I’ve spent most of this post talking about the percentage issue, something the kids
learned were first taught back in middle school, than the exponential growth/decay functions, the actual new material. This should not come as a shock to regular readers.
This twitter debate between reformers Mike Petrilli and Rishawn Biddle is typical of reform debates about “rigor”. Petrilli wants end of course exams to stop us teachers from pretending to teach a subject. Biddle wants more of the same, just shout louder and MANDATE instruction, particularly to those disenfranchised black and Hispanic youth who are being let down by lousy teachers with low expectations.
Both of them assume that the problem is ineffective teaching, that all us math teachers could actually teach percentages and fractions to all seventh graders if we were just smarter and better. Or maybe they just think we take the easy way out, that it’d be really really hard to teach the kids properly, and what the hell, we get paid no matter what and behind close doors it’s easier to just go through the motions. Well, sure.
Petrilli’s proposal, end-of-course exams, would trigger a bloodbath. People really don’t seem to understand how I’d be all in favor of that, if the result were a rethinking of expectations. But of course, what would actually happen is that we’d end the end-of-course exams. That’s what always happens whenever a state or district tries to enforce higher standards (cf Oklahoma and now Texas). And of course, that’s what’s going to happen with Common Core standards, assuming that anyone actually takes them seriously after the testing bloodbath this year. But I’d be all for end-of-course testing if reformers would accept responsibility for the 80% decrease in graduation rates among blacks and Hispanics who would never get past algebra I and understand, finally, that they believe in a myth.
But I digress. And I’m still going to like exponential functions, at least until I crack quadratics. Because you know what? The kids do make progress in understanding percentages, and they learn for the first time not only about exponential functions, but about asymptotes, as I explain Zeno’s Paradox. I don’t use Achilles and the tortoise as an example, but instead talk about how I could throw a stapler right at BTS’s head and know that the stapler would never draw blood because it wouldn’t reach his noggin, so I couldn’t get fired. Or that I could walk to the door and never get there. I do get to the door, of course, and alas, the stapler would eventually crack BTS’s skull. But even though we know that this is true, the tools for proving the paradox false, as opposed to demonstrating it, don’t come around until calculus. They get a kick out of that.
If all that’s not fun enough, I see genuine, honest-to-god intellectual curiosity among most students as they realize that they don’t have the tools to isolate x in the equation 8 = 3x. That for all these years they’ve been getting along fine with addition/subtraction, multiplication/division, nth power/nth root, but none of those will work here. Which sets us up beautifully for both logs and a proper discussion of inverses, leading into inverse functions. Yes, their skills are still basic, but I can see the glimmering of understanding of the underlying concepts. If the damn state tests would just ask questions about those underlying concepts instead of demanding underlying concepts and advanced operations, I might even be able to get the kids to show that understanding.
And in writing up this essay, I am struck by the obvious solution to the percentage problem on day one: I need a worksheet. They fill it out, and not until they are done with that do I give them the worksheets on growth and decay. Naturally, this solution is again a lowering of expectations, a realization that a clear explanation on a blackboard that they can refer to isn’t enough, that I need to give fifteen to seventeen year olds an activity so the information will sink in and they use the method right away without asking me to explain it all again group by group. But to hell with expectations. It will be much better for my bloodpressure.