When I was student teaching, I had to be observed four times by someone from the ed school in charge of me. He and my supervising teacher would confer with me afterwards and tell me what things ‘worked’ and what things ‘needed improvement’. He was a very nice man, a former math teacher, now retired and doing his part for the ed school. As nice as he was, however, I felt that sometimes he and my teacher found things to complain about.
During a lesson on ratios for my seventh grade math class, I introduced the concept by saying how when we compare things we sometimes use subtraction, like comparing heights. That led into how ratios is a comparison by division. There are times when subtraction is appropriate and times when ratios are a better measure of comparison. This is a standard introduction in most math books–it certainly was in mine, and I’ve seen it in many.
My teacher criticized me for talking about subtraction in my lesson, feeling it detracted from (rather than setting up the discussion for) ratios. The retired math teacher agreed. I said nothing, but I felt and still feel that they were dead wrong.
I’ve taught several seventh grade classes at this point, and I never hesitate to start off with talking about subtraction as the springboard into a discussion on ratios. As a way of illustrating how the difference between what subtraction and ratios measure, (and the appropriateness of each) I show a video of the old comedy team Abbott and Costello which no student has ever heard of.
The video is very versatile. It can be used as an intro to asymptotes for rational functions in algebra 2. And can also be used as an intro to limits for calculus. Feel free to use it yourselves and let me know how it goes!