My school bought new math textbooks for the 2009-2010 school year.  If you're a teacher, I'm sure you know how much work it can be to get new books - it's like teaching a brand new class!  I was lucky in that only 2 of my 3 classes had new ones, and the one was just a new edition (but same material).  So really I was down to one class to prep for. 

In my Honors Precalculus class we mostly cover functions in the first semester and trig in the second.  I love teaching trig and have been pretty successful in finding fun stuff to do with it, but the first semester material is usually a mishmash of topics.  The kids have seen most of the material before, and with the new Honors Algebra 2 books, it sounds like they've really done a nice job of covering a lot more of logs and conics and rational functions and such much more rigorously.  Which left me thinking last night (at 11:00 PM while I was lying in bed trying to get to sleep)...

What am I going to do first semester next year?

It would be a major waste of time for me (and the kids) to cover the topics I normally cover in the way I normally cover them.  Why spend all those weeks factoring and graphing and solving equations that they already know how to solve?  I want to be able to at least touch on the different types of functions (because we all know how well kids remember things from year to year) so they're not totally unforgotten, but I was thinking that maybe I could do that for a few days and then come up with projects for the students to do to help them cement things.

Here's what I typically cover:
1.  Functions and their Graphs (transformations, combinations, inverses, linear models)
2.  Polynomial and Rational Functions (zeros, graphing, complex numbers, quadratic models)
3.  Exponential and Log Functions (with nonlinear models)
4.  Vectors (we throw that in there first semester to help the kids out with physics)
5.  Conics
6.  Sequences and Series (with Pascal's triangle, the Binomial Theorem, mathematical induction)

Aside from Vectors and Sequences and Series, all of these topics have been done before.

Anyone have any ideas on how to gloss over them quickly and then apply knowledge with a cool project?