Dudes, this post has layers. Like an onion. Or an ogre.
Also apologies in advance for formatting, I’m too lazy to figure out math type.
First layer. When we got back from winter break we started our Rational Functions unit. We talked about how to simplify rational expressions and we talked about restrictions on the domain, and then we started to talk about asymptotes. When we first introduced the idea, we showed it through transformations with simple reciprocal functions. 1/(x+4) – 3 has a vertical asymptote at x = -4 and a horizontal asymptote at y = -3. If you’re doing (h, k) form (a la Glenn) then it turns out the asymptote is usually y = k.
Our mistake here was that we weren’t talking about WHY the horizontal asymptote was y = -3, we just asked students to look for patterns. I think the patterns are useful as shortcuts, but students didn’t understand why any of that was actually true. Which I realized the second we showed them real rational functions and they all wanted to tell me the horizontal asymptote of (x – 3)/(x – 5) was y = 5.
We followed up a lesson with a sort-of discovery lesson about finding the horizontal asymptotes of different rational functions and students generating those stupid “Degree rules”. You know the ones, if the degree of the thing is greater than the thing then the asymptote is this other thing. The only reason I remember them is because I just taught them. And I know when I was in school I could never keep the rules straight and really didn’t understand it (which is just one of the many reasons I wrote off Algebra 2 as pointless when I was 14).
What should come as a surprise to no one who actually thinks about how students understand math, my kids all bombed the test. Over half of my students failed (although in their defense, not just because of the one question on asymptotes). So in a panic, I decided to offer test corrections. It was a huge mistake that I will blog about later, but long story short, I wound up having to explain asymptotes one-on-one to about 30 different students. They all knew that HAs tell us end-behavior, but they’re not clear what that means other than “When x is really big.” When I first taught it I had described “fuzzy math” where we replace x with infinity, but that seems too abstract in hindsight. This time around I started to say x = 1,000,000. It’s a sufficiently big number that kids are in awe of it, but also fairly easy to work with. I go through some examples and they kind of get the idea I think.
My big concern, though, is that it won’t stick. They have a week to their midterm and 5 months until the State Test, so I was trying to figure out a way to make it memorable, and I came up with the analogy of Good vs. Evil.
The numerator is the forces of good, the denominator is the forces of evil, and they’re fighting for control. Kids are pretty good at understanding if the numerator is bigger than the denominator, but they lose track of what that entails. So I phrase it like this: If the numerator is more powerful than the denominator, then good beats evil, and we all live happily ever after without any restrictions on our lives (No Horizontal Asymptote). If the denominator is more powerful than the denominator, then evil wins and burns everything to the ground and we’re left with nothing (HA: y = 0).
It gets a little trickier when the numerator and denominator have the same degree. If I have something like 20x/5x, and you ask “Which is bigger, the numerator or the denominator?” I always get “Numerator” but then I have to say “But is it much much stronger, or only a little bit stronger?” Because they’re fairly well balanced, I always say that they fight to a standstill. The place of the standstill is decided by the leading coefficients. I think next year I’ll have little chibi images of Yoda and Darth Vader in the numerator/denominator to make it more visual. It could also work with Harry Potter or anything else where there’s good and evil.
I hang a lantern on the whole thing at the beginning, noting how cheesy it is. But I say that if it works then that’s awesome. And some kids tell me it has. I’ll let you know after I grade their midterms this week.
But here comes the second layer, which is more pedagogical: I’ve been conflicted the last week trying to figure out if this counts as a trick or not. It’s been my goal this year to nix as many tricks as possible. I’ve trained my kids to the point that when I said “FOIL” this week they all said “NO MR. B”. So now I’m trying to figure out if this counts. My gut tells me if I still present the topic with large numbers then my method becomes more of a mnemonic. Yet here I am stressing out about it. Because obviously as a second year teacher I need more to stress out about.
Being reflective on teaching practice is great, usually. It helps me to improve my method and better reach my kids. But sometimes it’s so exhausting. I tell myself you can’t fix everything in a day. Bleh.
Anyway, tomorrow is another day, and I have to go grade some quizzes.