One of my goals for this summer has been to rewrite a bit of my non-AP Calculus course. After so many years of teaching AP and focusing on making sure I was being challenging enough, I found that my tendency was to give tasks that were a bit too far out of my students' ZPD. I adjusted quickly and got better as I got to know their strengths and weaknesses, but any time it happened I knew it would hurt some of their trust in themselves...and maybe in me.
Many of these students were not enrolled in an honors level pre-calculus course last year, so algebraic concepts are fuzzier than you might expect at the very beginning of the year. They need more explicit practice with those background skills that AP students have mastered. One of those, I knew, would be dealing with composite functions- both their composition and decomposition. This is a key skills for the remainder of the course, with special importance to limit definition of the derivative, chain rule, optimization, and integration by u substitution.
As both a pre-assessment and a task to get them thinking in groups, I drafted this sorting activity. It gives students notational representations of composite function and expressions to match. There are a few that don't quite match, and students must work to answer questions about those.
I also am going to be using Index Card Questions (as described here in my Necessary Conditions blog) with my students for the first time this day, so I'm having students reflect on the classroom norms we've used to emphasize them as we move into more content-centric lessons.
This is very much in draft form, so if you have any feedback, please let me know!