I find it difficult to brainstorm without laying down some specifics, so I pulled together a potential Unit 1 for introducing differential Calculus without actually introducing differentiation rules.
Please take a look, Calculus teachers, and let me know what you think! http://bit.ly/differentialCalculusIntro. I'll be trying to meet up with one of my new colleagues soon to discuss some of this material, since she is trying to pull together a projects-based Calculus class (somewhat more slow-paced than the regular Calculus class that I'll be teaching) and she's also working on the explicit incorporation of Habits of Mind this year into all of our curricula. But, in the mean time, I'd love any feedback on whether: 1. you think this is paced too slowly for a "regular" (non-AP) calculus class? 2. Are the worksheets relevant and appropriate for the level of course? Our classes are about 45 minutes each.
After this intro unit, depending on how my colleagues respond to my idea of "teaching backwards", I'll either be planning a similar "integral Calculus intro" unit that focuses on application and skips over the algebra skills, or we'll be doing manual differentiation skills as the next unit...
Addendum 7/31/2013: Thanks to Sam Shah, I will be sharing this link on the Calculus of saying I love you with my students during this first introductory unit to derivatives!