As part of our Teacher Federalism agreement, I now include the reciprocal function as one of my parent functions in Algebra 2. But time constraints don’t allow me to really dig into the function–plus, the kids are on overload by the end of the term, what with exponential functions, logarithms, and inverses. I don’t really have time to switch gears. Besides, they’ll be exploring rational expressions in depth during pre-calc.
But then I noticed, during Trig, that my students really weren’t completely understanding that four of the major trig functions are rational expressions and how they differed from sine and cosine. Meanwhile, I’m always doing a bit of algebra review…and so I decided to kick off my trig class with a rational expressions unit. A brief one, that enabled a review of quadratics and rational expression operations (aka adding and subtracting fractions with variables).
Task 1 is a straightforward linear function, so almost every kid who has made it to trig, no matter how weak, is able to quickly build the function.
Task 2, of course, is the introduction of division. In function terms, we’re dividing a line by a line, while I will eventually make clear. But practically, the big hop occurs when students realize that cost per hoodie is not constant. Usually students do this incorrectly, graphing either a constant line of 20 or confusedly dividing total cost by 20. So I’ll meander by and ask:
“According to your table for L(h), we spent $520 and got one hoodie. What was the cost of that hoodie.”
Silence. I wait. And inevitably, a student will gasp, “No! One hoodie costs $520!!”
It usually takes about 45 minutes for the kids to work through both tasks, including graphing the unfamiliar rational expression. Then I call them back up front for explanation and notes.
After putting the two equations on the board (linear and rational), I point out that our cost per hoodie equation is basically a line divided by a line. I point out the two asymptotes , vertical and horizontal. Why do they exist? Most students, by trig, know that you can’t divide by zero, but why doesn’t the vertical asymptote intersect y=20?
This usually prompts interesting discussions. I usually have a couple students to correct when they build the graph, as they make it linear. So now I redraw it, making clear that the drop is sudden and sharp, followed by a leveling. Why is it leveling?
Usually, a student will suggest the correct answer. If not, I ask, idly, “Can anyone tell me why the cost per hoodie isn’t $20? After all that’s the price.”
“Because you have to pay the $500, too.” and this almost always leads to a big “aha” as the students realize that the $500 is “spread out”, as many students call it, among the hoodies. The more hoodies purchased, the higher the total cost–but less of the $500 carried on each one.
At some point, I observe that certain forms of equations are much easier for modeling than for graphing. For example, when modeling linear functions, we use standard form and slope intercept form all the time–many real-life (or close to real life!) applications fall naturally into these formats. John has twice as much money as Jane. Tacos are $3, burritos are $5, Sam has $45. But you’d never deliberately model an application in point-slope form. You might use it, given two points, to find the equation. But it’d be an operation, not a model.
So take a look at TL(h) and what does it look like? Usually, there’s a pause until I remind them that we could have negative hoodies, and we graph that in. Then the kids recognize the reciprocal function.
“If we take a look at the graph and think of it as a transformation of a parent function, what’s the vertical shift?”
I draw the parent reciprocal function . “Remember this? Where are the original asymptotes?” and eventually the kids remember y=0 and x=0.
“Right, so the original parent function, the horizontal asymptote is y=0. Where is it in this function? y=20. So what’s the vertical shift?”
Now they get it, and I hear “20” from all corners.
“Right. Is there a horizontal shift?”
“So we know that h=0, k=20….what’s a=? What’s the vertical stretch?”
Someone always remembers that it’s the vertical distance between (1,1) and the actual output value for x=1, which is….
“Right. a=500. So TL(h) could also be written as 20+500(1⁄h). Notice that if we split the numerator into two terms and simplify, we get the same thing. But we’d never model it that way. Much more intuitive to create the linear equation for total cost and divide it by the line.”
In other words, I point out, the hoodie activity is actually the same function that they learned about last year, but instead of just graphing or solving transformed functions, they’re modeling with it.
The second part of this activity is about 20 minutes, and comes at the end of my 90-minute block. In between, I do a lecture on the meaning of rational expressions, vertical and horizontal asymptotes, usually bringing up something like this:
But while it works in the context of the lesson, it just pulls the focus of this post so I’ll write about that some other time. Suffice it here to say that yes, I discuss what the defining criteria of rational expressions are, what asymptotes are, and so on. One of the main reasons I teach this now is so the kids will understand both as they bump into them.
Part three comes quickly because the students see the pattern, and that alone is enough to please a lot of them. Suddenly, they’ve added an entire model type to their repertoire.
Part 4 is where so much gets tied together. Most students get all the way to part 3 without anything more than nudges. At that point, I usually bring it up front.
Using Desmos, we graph the same system. At this point, I’m obscuring the solution value.
This sets off discussion about the shift, how one starts out cheaper but stabilizes at a more expensive base cost. And then, look, the lines intersect? What do intersections mean, again? SOLUTION!
And in this first equation, the solution is quite simple because the equations have the same denominator. (note: I’m using x instead of h because I quickly copied these from Desmos):
So it’s a quick matter to solve the system, but again, good reminder. At 40 hoodies, they are $32.50 apiece.
Right around here, I point out that it would be convenient if one equation could show us the information we needed. How could we show the difference between the two functions?
Fortunately, a number of the kids have lived through my algebra 2 class, and call out “subtract!”. I briefly explain that functions have operations and can be combined, for the rest. So we can subtract one equation from the other. Since ultimately the Hawk’s function will be more, we perform TW(h) – LW(h) and graph it as D(h).
Note first that it’s still a rational expression, although not the same type we’re working with. See how nice and clean the break even point is displayed!
So what we want to do is move this break even point further to the right. Luvs is ahead and has no reason to bargain. Obviously, we need to talk to Hawk’s Hoodies.
The administrators want to buy the better hoodie, but 40 isn’t enough to have a fundraiser–they want over 100. How can we get a better deal?
Suggestions? The ideas come fast.
First up is always “Hawk’s should sell the hoodies for cheaper.”
“Right. Hawk’s could lower its asymptote and slow the rate of increase in total cost. What would be a good price?”
We try $23/hoodie:
$22 is even better, putting the break even point at 100 hoodies.
“But here’s the thing–Hawk’s has real pride in their hoodies. They know they’re charging more, but their hoodies are worth it. That’s why we want those hoodies to begin with! They’re softer, better colors, hold up to wear and tear, whatever. So cutting their price by 12% sets a precedent. There’s a whole bunch of marketing research showing that customers don’t value luxury items if they sense the vendor will cut prices at a later date. So while we might like the price of the hoodies, Hawks could be hurting its brand if it cuts the retail price per hoodie. Take this as a given, for the moment. Is there any other way we could cut the price per hoodie that still maintains the same retail price?”
This always leads to good feedback: give a school discount, cut the price of the logo conversion, and so on.
At some point, I break in (unless the solution I’m looking for has been mentioned):
“All of you are coming up with great suggestions that involve reducing the value of the numerator. How about the denominator?”
“If I have a division problem stated as a fraction, reducing the numerator (the dividend, if you must) will reduce the result, or the quotient. But is that the only way I can reduce the quotient?”
Pause. New teachers, let the pause hang. If it still gets no response, say “What else can I change?” because that will lead to someone saying…
“The denominator. But why would reduce the denominator?”
“Why indeed. 6 divided by 3 is 2. If I reduce the numerator to 3, my answer is 1. But….”
“Oh, I get it! Increase the denominator?”
“What would that do? Or put it this way: what would Hawks have to do to increase the denominator?”
And eventually, everyone figures out that Hawks could throw in some hoodies for free, which would also let them maintain their higher prices while still getting the sale.
“So go figure out the equation if Hawks includes 10 hoodies for free.”
Someone will always realize that this means we could get negative hoodies. So I tell them to test some negative values and remind them to think about what this might do to the asymptotes.
When they’re done, we put the whole thing on Desmos, showing that the vertical asymptote has changed, but not the horizontal.
“See, this way, Hawks is decreasing the time it takes for our purchase to get to the lower prices, getting us to just a little over $25 per hoodie with far fewer purchased, because we’re getting $10 for free.”
Now, take a look at the new equation to find the breakeven point:
“So how many of you remember being assigned these ridiculous equations with variables and fractions and thinking oh my god, none of us will ever use this? Who would ever have to add or multiple or subtract fractions? And yet, here we are. This one has them set equal to each other, but as we said above, function D(h) is the difference between the two :
…look at that! Your math homework in real life!!!
So we discuss what d(h) is doing. I point out that “solving the system” of TL(h) and TW(h) is nothing more than “finding the zeros” for D(h).
From a curriculum standpoint, I transition pretty quickly from rational expressions to a review of binomial multiplication and factoring. So the D(h) subtraction equation gives me a great opportunity to review the procedures before I set them on their way. I work the problem–which requires the quadratic formula at the last step, ironically, but still gives us a chance to review the steps to determine whether or not a quadratic can factor. Then I show again how Desmos takes the equation and shows us how far we’ve “moved to the right”:
Then they all work out the comparison between a $3 reduction in price and ten hoodies thrown in for free. I take a moment to point out that math drives business analysis. Today, we have technology to do the work for us, but the best analysts have an understanding of the rational expressions driving the graphs.
If I had time, I’d do this in Algebra 2, but from a time perspective, I have a choice between introducing exponential equations and logs or go deep on rational expressions. That’s a nobrainer. They need to at least be introduced to logs, and there’s no opportunity in trig to bring that topic up. Rational expressions, on the other hand, forge a connection that makes sense when we get to the graphs.
And yes, it’s made a difference. I’ve been using this activity for two years, and have seen a noticeable improvement in their understanding of the four rational expression graphs. Remember, I’m not just teaching my kids, so even those who got a full dose of the rationals with other teachers are showing increased understanding. I would like to do this and more in Precalc, and will report back.
Sorry I’ve been so long without writing. We had a ridiculous heat wave and I responded by sticking to Twitter and playing Fallout Shelter, which is kind of cool.