I had great ambitions to do a lot of schoolwork this summer. Instead I started, abandoned, and restarted a unit for a course that I’m teaching next year. That’s about it. It’s a new course for me: Advanced Precalculus. The other teacher and I have decided to totally mess around with the ordering of topics, and we put sequences and series as the second unit. Our department is also trying to integrate more problem solving in the curriculum, and so I tried to make this unit involve as much problem solving as possible . I like that we’ll be doing it early in the year, because I want them to see immediately that we are not going to be focusing on plug-and-chug but real thinking.
Those of you who know me know that I am a pretty traditional teacher, and I have gotten in the habit of creating guided worksheets as a structural backbone for a lot of my classes. This is the first time I’m creating an entire guided unit. It isn’t flashy or have a good hook. It’s simply a scaffolded way to help kids think in an increasingly abstract way. It also gets at almost all the standard things in a sequences and series unit (except for recursive equations, which I threw out). To put it out there: I would never say that what I do is inherently engaging for my kids. But it does get kids talking. I guess what I mean to say is: these packets/worksheets that I tend to create don’t make kids like/love math, but it does get them to think about math. I’m not great at the former, but I’m definitely getting better at the latter.
The last thing I have to say is that although it may look pretty traditional (the questions), try to think about the packet if you were a student and you were in a class going through it. It builds up elegantly, in my opinion. The motivation for sequences comes out of a series of IQ-test-ish puzzles, and the motivation for series comes out of a lottery problem. No formula is given to students. There are connections drawn to graphs, and to a few geometric visualizations of sequences and series. Students are asked to conjecture and defend their conjecture at various times.
I’m including two copies below. The packet with my teacher notes, and the packet without my teacher notes.
With Teacher Notes
Without Teacher Notes (Blank)
[Word version of this to download: .docx… to see my teacher notes, go to “Review” and go to “Final Showing Markup”]
Huge thanks in the creation of this goes to @JackieB, who went through a lot of it page by page and gave excellent suggestions! Precalculus guru! Also I included a few blogposts at the end of the document which I stole wholesale from or adapted in my own way.
Lastly, yes, I know this is a long packet. Usually I think classes do this whole unit in single week, and there’s no way we’ll be done with it in that time. It’s an experiment. From what I’ve heard from teachers everywhere, sequences and series always get short shrift in precalculus classes because they come at the end of the year. But I think there is so much depth and abstract thinking that can be brought out of a unit properly done. I’m super nervous, but we’ll see if this is an experiment that fails or not.
 I’m liberally defining problem solving as having students deal with situations they have never dealt with before, and generalizing from those situations. But I understand I am giving them A LOT of scaffolding with which to do it.