Piecewise Functions Scaffolding Up the Wahzoo

Out of habit (aka. a compulsive need to organize my work), my lessons are always digitized and numbered sequentially/labeled with keywords, collected into chapter folders and further grouped by classes. At the end of every school year, I zip up the entire folder tree and archive it in multiple places, labeled by year. At the beginning of the school year, I download all of the past archives onto each computer I might be lesson-planning on (one at school and one at home), so that when I need to look up an old lesson, it'd be in front of me in a matter of minutes, complete with handouts and relevant quizzes and tests.

So, anyway, the other day I started to teach my Precalc kiddies the basics of piecewise functions. After the first 1.5 lessons or so (in which I had introduced the mandatory Christmas bonus breakdown in El Salvador and had asked the kids to graph the bonus and to graph a simple progressive tax system*, and also had gone over the basics of evaluating a piecewise function), I thought of digging through my old stuff to pull up some scaffolding material for writing piecewise function equations.

I pulled some of my favorite scaffolding things together into one packet this year. Here it is (I threw a couple of word problems in there for fun, since this packet I intend on grading as a project):








I am posting it because maybe you'd find it helpful to see how I break down piecewise equations into, well, pieces! (If your kids don't need these skills broken down quite so much, you might want to check out Sam Shah's collection of piecewise worksheets, which is a little more comprehensive in the skills they need. Again, I make no claims that my worksheets are comprehensive! They just step in and break it down a little more, like the problems in the textbook don't do...)

*Speaking of which, funny thing about tax systems: I randomly wondered the other day whether there are any countries in the world that actually implement a flat tax. Found this article that talks about how Estonia and two of its neighbors implement the flat tax system, and some of its less-apparent benefits and drawbacks. (For example, did you know that when neighboring countries all implement flat tax systems, it becomes a negative competition sort of thing, where each country may keep lowering their taxes, in hopes of staying competitive?) I shared some of the interesting tidbits with my kiddies, and even they thought it was fun!

And, coincidentally, El Salvador's own tax brackets are exceedingly simple. I am going to give it to the kids sometime next week, as additional practice for making graphs and writing equations.