Coherence & Canoe Trips: Common Language for Understanding Vectors & Polar Coordinates

First, let me begin with this:

Okay, now that we've gotten that out of the way...

One challenge that I find with teaching Pre-Calc is that it's the culmination of all the content the students have learned up to this point. Very few of the topics are completely new to the students, so often there is a "we already know this" attitude when we intro a topic. However, the course requires more than just the knowledge of each specific skill; it demands an integrated understanding of the concepts. Disconnected thinking just isn't going to cut it here....coherence among pedagogical approaches needs to be a priority if we want kids to have an integrated conceptual framework in their minds. Kate Nowak mentioned it in her talk at NCTM Nashville- you don't want to piece together a string of "short stories" in your class; you want to reveal to students the "whole novel." 

My goal years ago was to develop some sort of common experience to use with my kids to talk about the concepts of vectors and polar coordinates since they are so closely related but often taught separately. I wrote and have been tweaking these explorations for the past few semesters.

Canoe Trip Part I- Finding Vector Components


Canoe Trip Part II- Converting Between Rectangular and Polar Coordinates

Photo from http://smsta.msta.org/wp-content/uploads/2014/09/460507657.jpg

Both of these are introductory level activities and I've found that some kids latch on to the "rule" that they develop quite easily, while others cling to the "drawing a triangle will always work" approach. Both methods are 100% valid to me, so I don't make much a distinction at this point in the concept (although the "rule" method is much faster!). When I see those right triangles being drawn out, I get the "teacher stock photo" feeling. My heart feels like this picture- a student choosing to use the concept instead of just blindly memorize the formula! I have a "moment", for sure. 

I like the way these work right now, but I'd like to make them more open ended in the future. Maybe give different groups different scenarios and have them critique each other. Maybe ask them to create their own scenario to model a situation. Unfortunately, these topics always wind up getting smushed at the end of the course and we don't get to have the "play time" I'd like. Anyone have any tweaks or any ways they love to teach these topics? I've love anything to improve or enhance these! Comment away!