I've been "teaching" related rates in my BC calculus class for the last 3 days, and I did change it up from last year since I saw the struggles they had with the topic. I made some improvements, but I guess there's still more to make. Finally today after watching my mixed level class stare back at me for 2 days with a wide variety of looks, from, "yea, yea, I get it already", to "great! another topic I can struggle with to add to my ever-growing pile", to "related whats? related whos?", I had an epiphany.
As they walked into class, I moved the "yea, yeas" to the back (about 1/2 the class of 19) to quietly work on old AP exam questions on related rates. My other 1/2 I moved right up front by me, and I handed out a sheet with 4 problems. I left BIG space between the problems, and under each I had 4 categories: Picture, List of variables, Equation, Solve. I walked them through each, gradually weaning them off, and then I had to cover my eyes from all the light bulbs going off by the 3rd problem with "ohhhhhh! is THAT how you do it" looks.
As one kid said, "why didn't we do this in the first place?" Why indeed. I had to be honest with him and say, "well it's a learning process for all of us. I'm also learning what works with you all".