The biggest thing I picked up this summer is that it's better to start with the question then load kids up on math when you have a problem. It's early and with Boot Camp complete, Pre-Cal is trudging through a little Algebra II review before we jump into trig. That includes simple things like slope, point-slope equations, etc. Here's how I started with a picture of a mountain and ended with describing function behavior. This took two days.
Step 1: Picture
I pose the question "which part of the mountain is steepest?" I annotate the picture using DeskScribble to highlight the mountain profile to make it a little better. I offer kids the chance to take ownership of a particular piece. After two or three kids have claimed a slice of mountain I poll the audience "who picked the steepest part of the mountain?" and we save the result. I have clickers in my room, more low-tech methods work too.
Step 2: Add some math
I pose the question "how can we figure out who is right?" which leads us to the keyword "slope." I offer some help by leaving my profile annotation but removing the picture of the mountain. I add a grid behind the profile (you may have a software tool that does this, or simply doing a Google for "math grid" will probably get you something that works). At random we set an x/y axis. I have them identify some endpoints for the segments that were claimed. We hammer out how to find slope when the points are known and we determine who really won. In one class two students picked seemingly different segments of mountain only to find the slope was the same.
Step 3: More math
I use the slopes we found to jump into point slope equations. We hammer out the formula together then talk about what's necessary to use it. We have several points and slopes from dividing up the mountain, so I separate responsibility for the different pieces. End result is three equations in y-intercept form. I end the class by having them save the equations we came up with.
Step 1: Seemingly random video
Step 2: Recall
They recite our equations from yesterday to me. A bracket and some intervals later, instant piece-wise! My kids are in groups so I take their new piece wise function and give a turn at some f(x) values to find given the options.
A nice corollary to piece-wise functions is behavior, knowing how to use things like "increase" "decrease" "local maximum" and "local minimum." For that, another picture:
We talk about what's going on and have them feed me a description of the ball. Like yesterday, I add math layers by annotating an axis and a flightpath on top of the picture. One more step and we identify some points and tease out the word "maximum." After that I give them some plain old graphs and have them generate a description as a group making sure to cite the proper words.
Perfect lesson? No, I'm sure next year I'll have more to add. But when we worked through it it certainly felt more interesting that starting with an out of context math layer.