In teaching Calculus III geometry plays a very important role. It is crucial that students get a good sense of how to visualize curves, surfaces, coordinate axis, frames to curves, vector fields and so forth. I think my pictures do some justice to the problem, but, sometimes, I just can't get the image across. Naturally software helps, however, I don't teach in a computer lab and short of attaching points to a task it is hard to get most students to take an assignment seriously.
One thing I'd like to do more of is to give little projects in class which are little craft times where we get out some scissors, maybe some tape, a sheet of paper, perhaps some markers and make something to visualize a concept or object in Calculus III.
For example, I usually get them to take a strip of paper and twist it and tape it back. Before they tape it they draw a line on one side, then after the twisted strip is constructed the line is on the only side of the strip... of course, this is a model Möbius strip.
Can you think of any other crafty ways to bring the geometry of Calculus III alive?
(I teach standard American third semester of calculus, 4 credit hours, covers from vectors and three dimensional coordinate geometry through the basic vector calculus including Stokes' Theorem, however, I'm generally interested in any geometric craft to help ingrain principles of analytic geometry.)