September Review: Is it Actually a Necessary Evil?
It seems like math is the only subject in which teachers feel like they need to review for the first week or two of the school year. Teachers of other subjects seem to review as they go along, going back to old skills and ideas as needed, as motivated by their curriculum. To me, this makes much more sense… and I know because I say this a bit disheartened having plowed my students through a review of algebra for the first week and a half of school. After starting of my class with a bang, with some great metacognition and a good introduction to Standards Based Grading, I had a lot of trouble getting the groove, mostly because I knew that I wanted to push through the review to get to the good stuff, which means I ended up having an uninspiring week of a hugely teacher-centric classroom.
So that of course brings the question to my mind of “Why am I wasting time on not-good stuff?” I know some stuff is unavoidable (especially when a vast majority of my non-AP Calc students claim they have never seen or heard of piecewise-defined functions before), but I really believe, like a bunch of other math teachers I have talked to, that most of review could come as needed as the curriculum develops. And our book reviews all these crazy topics that won’t ever have much bearing on the future curriculum. Symmetry? Modeling** (the mathematical kind)? Animal Husbandry?
I got frustrated halfway through the week and decided that instead of just hammering out more material I would do a problem solving activity with my AP class that would remind my students of many of the things they needed to know while engaging them in deep problem solving at the same time. The sad part about teaching an AP class is that it totally felt like I was “losing” a day (my yearly schedule is nagging me), but it was totally worth it. This is something I am going to struggle with all year, as I have taught a very application based Calculus for a year and this is my first shot at the AP. Here is a mini-unit I organized about piecewise functions.
MOTIVATING PIECEWISE FUNCTIONS: I tried to get them to see why piecewise functions are necessary by giving them data of tourism arrivals and departures in Jordan and the US over the past 15 years and asking them to tell me the story that the numbers are telling them (thanks John for giving me the idea with your post about Telling the Story of a Number). Each group came up with 4-5 bullet points and wrote them on the board for the others to see. Quite unsurprisingly, there are major drops in both US arrivals in departures around 2001 and 2008, which most groups mentioned in their story, but others came up with some great explanations that I didn’t expect. For example, one group mentioned that Petra was named one of the new Seven Wonders of the World somewhere around 2008, so that might explain an uptick in arrivals (cool!). Another explained the rise in US tourism in the mid-1990’s to a Deep Purple tour… We talked about how it would be hard to fit ONE function to the data because it’s kind of all over the place, but we could fit a bunch of chopped up functions. The cool part about framing it like this is that the points where the function changes corresponds to major world events, which is because those events changed the relationship between the variables. I saw a bunch of light bulbs go off on that one.
ACTIVATING THEIR PREVIOUS KNOWLEDGE: Then, I played for them DJ Earworm’s 2009 United State of Pop mashup (I blogged about this about this on Sam Shah’s blog this summer). We made the metaphor between a mashup and a piecewise function and used that to give ourselves a quick reminder of how the notation works. This led into a few examples as a reminder, but none of the drill and kill – I just wanted them to remember that they knew how this stuff worked.
FINALLY, THE PROBLEM SOLVING ACTIVITY: With more-than-inspiration from Mimi’s Income Tax Unit, I presented them with how income tax works here in Jordan. They were really surprised – most thought it was some sort of flat tax. They were also confused. Why is it so complicated? So I presented them with the goal of the task, which was to make the Income Tax more easily understandable for the average person. I found the income tax for five different countries, and each group was tasked with graphing Tax Owed vs. Money Earned and then writing a piecewise-defined function that will give you your tax owed if you plug in your income. That way, an average person could either just find their income on the graph, or plug their income into the function. Trying to be “Less Helpful” à la Dan Meyer, I tried to provide scaffolding only where needed. This was so hard for me! I just wanted to give them little hints. I gave in to these urges every once in a while, but this was the most I have ever let my students really struggle. Most tried to start directly with the equation and had a lot of trouble abstracting the situation, but over the course of a period and half (everything takes roughly 24 times longer than I think it well), pretty much every group had a graph drawn and pretty much finished the equations.
(that’s Spain’s Income Tax)
- They realized that the keys to piecewise functions are the points on the boundaries of of the intervals. This will really help when we talk about continuity and differentiability.
- After some experimenting, most groups realized that the slope of each segment was the same as the percent of taxed money on that segment. Any sort concrete exploration of the idea of slope is alright by me.
- Many students became much more comfortable with point-slope form (or, more importantly, realized that this form of the line is much easier some of the time than slope-intercept), which will help when we talk tangent lines.
- One group made connections between all the ways they could have solved the problem. They actually determined their equations analytically – CATEGORY BASE TAX + (TOTAL INCOME – CATEGORY BASE INCOME) * TAX PERCENTAGE – which was impressive to me, as I always do things other ways first (graphically, numerically etc). But they came to the realization that this form is pretty much exactly point-slope form if you rearrange it a bit and then got the added conceptual understanding that arises from point-slope form here. I thought this was great.
- This was more genuine, engaging and thought-provoking than the rest of the week combined (and it’s not even a particularly rich problem).
(PS One of the sweet things about working here is that when I leave I will get most of the tax that I paid the Jordanian government right back. And I don’t pay US taxes because you have to make a boatload abroad to have to pay. Buhahahah.)
**I actually love love love modeling, but doing textbook problems about modeling is like that first chapter in science textbook that “teaches” the scientific method.