[Edited for space]:
I have been teaching elastic collisions problems using an elastic equation : Vf + Vi = Vf + Vi to solve problems that are missing two of the velocities. We discuss that as long KE and momentum are conserved then we can take KE and momentum equations and divide them to get the elastic collision equation with just velocities.
I recently was tutoring a former student who is now in a local college about solving these types of problems and she told me that her professor said that the equation does not work and that its not physics! She was told that I was completely wrong!!! I immediately went to a Giancoli textbook. Giancoli does derive this equation following same reasoning that I derive for my students.
BUT... I hate to think that I have been teaching this wrong! I was hoping you might be able to offer some clarity. I went through [the professor's] problems and compared my solutions to the professor's solutions and I do get the answers he gets, just in a lot fewer steps. Any suggestions? Should I not teach elastic collisions this way?
Fascinating question. My answer is twofold -- one answer on philosophy, one answer on content:
1. You are teaching absolutely correctly. I don't know what her professor is on about. Remember, "professor" means neither "good teacher" nor "better than you at introductory physics." It's so easy for a high school physics teacher to be intimidated by folks with PhDs, or by education "experts." As long as you are carefully self-evaluating -- and you obviously are, based on paragraph 3 above -- then do things your way. I can't emphasize enough that even my well-tested methods and ideas are not for everyone. The best physics teachers, like the best chefs, are creators, not imitators.
2. On this specific issue of elastic collisions: You might consider why it's necessary to teach quantitative solutions to elastic collision problems at all. Yes, you need to be able to check whether a collision is elastic by comparing KE before and after the collision. But even with the simplified relative speed equation that you reference, solving for speed in elastic collisions is more calculation that we need for AP 1, or even for my taste in any intro course. That's not to say you're wrong to teach it, as I did for years... I just don't think it does enough to be worth the time it takes to teach and solve the problems.