after spending six weeks pulling teeth with my precalculus students trying to get them to actually think about functions, rather than just parotting phrases like “domain is input, range is output,” we top everything off with this project. The whole idea of this project is to get students to analyze the graphs of functions, discussing their domain and range, intervals where they are positive and negative, where their combinations are positive and negative, etc. Students are encouraged to work from the graph of the function rather than the equation, and to that end I try to use screwy equations they aren’t very comfortable with. They can use whatever method they want to come up with the initial answers, they may graph the composition on their calculator to answer questions about it’s domain and range, for example, but then they have to justify that answer based on the two original functions.
We spend a lot of time analyzing graphs of functions in this chapter, we cover most of the aspects of functions from their graphs, only (my kids are already pretty good at most of the plug and chug algebra answers) to try to break them of this idea that graphs are somehow subordinate to equations. Every year, though, the answers to this project are absolutely abhorrent. I try to remind myself that it is more about the process than the final project. One of these years I’ll get around to assigning something similar towards the end of the semester to see if they do any better.