Piecewise Functions are a fun twist at the beginning of Pre-Cal. Graphing them however, has always been a bit annoying. The first year I taught Pre-Cal this topic was a disaster. Slowly I found a better way to do it. Start with evaluation. Have students evaluate a function over a wide range of points. Give them a grid and have them plot the results.
Some errors will surface. Overlapping pieces, boundaries drawn in the wrong location. This is natural. But, it gets better.
As a culminating move, I teach them how to graph with restrictions in Desmos. This year, I tried something new. You, dear child, are going to DESIGN a piecewise function. Normally, I suck at open ended stuff. But, my few weeks of bizzaro world gave me the impression that my new crop of students would be up to the challenge.
Design a function that at a minimum
- has 3 pieces
- has 3 different types of behavior: increasing, decreasing, constant, or undefined
- has 2 different types of functions: linear, constant, radical, quadratic
Present a large version of your function, the equation that generated it, and a description of the intervals that exhibit different behavior.
I set up the project with some practice on describing intervals and how to manipulate various function types (a review of transformation rules), and then set them loose.
Most went with something pretty standard, but I did get a few clever applications. Students primarily built their functions on an iPad and then made a copy. However, since I don't have a enough iPads to go around, they had the option of using their phones if they wanted. At least 3 or 4 kids in each class successful pecked out a function on the web or app version of Desmos. A few shared an iPad with a partner.
Super happy with the result. I expect this open ended thing will continue.