Good vs. Evil

Dudes, this post has layers. Like an onion. Or an ogre.

shrek onions

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.

This picture is so freaking cool.

This picture is so freaking cool.

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.

First Week Reflection: Pinch Me

So we’re two days in to the new year, and I can’t quite believe it. Not just the time (although the first days always feel significantly longer than they are), but also the place where I work. The kids are crazy, in the best way possible. They come in, they get seated, they’re polite and pretty considerate. I ask them to stop talking and they do. I give them a fun math puzzle and they all work together to complete it, and they’re all respectful about it. This week I’ve had a general idea for how I want things to run, but I didn’t script it out or have a hard-core lesson plan or anything, and the classes went off without a hitch. The veteran teachers keep assuring me that this isn’t going to last, but even if they slip a bit this is still so much better than what I dealt with last year. 

A couple things of note:

* My school doesn’t do posters. Like, at all. I asked “Oh, do we put up like procedure posters?” and they were like “no, we just decorate around our desks.” What is this insanity? On the one hand, I’m glad I don’t have to stress about bulletin boards, but on the other I’d like to put up some posters.  All the classes are so spread out I’m not sure how it will work just yet, but I may sneak some in. 

* We also only have one bell at the end of the period, so it’s kind of teacher discretion about lateness. I try to roughly gauge 3 minutes after the bell and then work from there. My goal this year is to be a real hard-ass about lateness, so I have to start guilting these kids into just being on time. 

* I’m also intent on being a hardass about uniforms and pencils. Here’s a question for my readers: How do y’all feel about the pencil/pen debate? Personally I think all math work should be done in pencil, but does that also apply to notes? I always took notes in pencil but I could see how it would be annoying for some kids and get all smudgy. But if I let them use pen for notes, won’t that just lead to homework being done in pen?

* I’m really punting on this Notebook thing this year. I really, really liked the idea of interactive notebooks. While I will never be a foldable kind of guy, I was really drawn to the idea of having kids treat their notebooks with a degree of reverence. And I think if I did it right, I could really create that sense for the kids. The tricky part is that it’s not something our school has in place at all, so I’d have to take a stand on my own. I think for this year I’m going to pick my battles in other places and just strongly suggest taking care of their notebook. 

* Having dedicated co-planners is amazing. I was more or less completely planned for Monday by the time I left on Friday. I have stuff to work on tonight, but if I get super frustrated with it I can stop and come back to it tomorrow and still be mostly ahead of the game. And I get to bounce ideas off of them and have them talk through things with me. It’s so cool and awesome and most importantly it helps me keep some sense of self. 

* I still hate the summer. I feel like being a hot sweaty mess kind of undermines my authority around the kids. But all the other teachers are just as gross. My home-base has AC so all the teachers have taken to hanging out in there when we don’t have class.  The only downside being that the AC tends to spit and leak. But I teach in the front of the room, so it doesn’t hit me. 

* I’m starting Friday Letters this year, and even though I presented it very poorly I already got one! Now I’m trying to figure out what kind of mailbox to get. Do I just use a paper box, or do I spend the $10 on a plastic mailbox? I kind of like the idea of a real mailbox that I can then decorate and that won’t get messed up so easily by kids moving it around. 

So those are the big things. I should get back to planning a bit. Our plans for this week are as follows:

Algebra 2/Trig: We’re doing an “Intro to Functions” unit, where basically we go over terms like function, relation, domain, and range,  The idea is to set-up what we’re doing in Algebra II ahead of time, similar to what Glenn does. I’m really excited to set-up the whole value of functions and the whole point of domain and range, especially since I didn’t think I really understood it myself until recently.  This unit will only last this week, and we’ll jump right into quadratics next week.

Algebra/Geometry: We’re starting with a problem solving unit, focused on how we approach new and confusing problems. At first my co-planner was hesitant to start with this, but after seeing how the kids approach some of the problems I think it will be really necessary. Tomorrow is a general introduction to confusing problems, and then we’ll spend three days each focused on a different strategy, and a final wrap-up on Friday. Then we jump into Statistics next week. This course is going to have a weird structure but I think it has potential to be really awesome