I've taken workshops in pedagogy of physical science teaching and I've had OJT for engineering teaching, but while I am a competent math and English tutor for some areas, I am not qualified to be in front of those classrooms except as a dire emergency when the other choices are just warm bodies to keep order.
And playing the part of the warm body- me!
Your denominator example (that I snipped out) is one reason why I've resorted to doing calculations like the gas law with R using "just" dimensional analysis instead of using PV=nRT and then manipulating it to get the desired answer. Putting R in the denominator continually resulted in a few exploding heads each class until my coworker showed me how he did it. Start with R and the "work the units" until you get what you want.
In order for this to make sense, I do have to go over the relationships between pressure, volume, temperature, and moles. I have zero data to support it, but my impression is that they have been performing better on these calculations since I stopped manipulating the ideal gas law. Well... I didn't stop completely. I show it and show how to manipulate it once.
Ah, whereas my path was to get a "teaching arithmetic" book and a "teaching algebra" book to see what I am supposed to show and set aside two whole class periods where all we do is the arithmetic and algebra in exquisite detail, showing ALL the steps to manipulate simple equations. Many of my students' math is so weak and they have such a poor grasp on the idea of units themselves (to the point of being unable to tell me what is wrong with saying that a volume is 35 degrees Celsius) that we have to do the math review before I can hope to do dimensional analysis. After all, if you think that you can add 12 feet to 35 degrees Celsius to get 2.9166666667 milliliters, then focusing on the dimensional analysis doesn't work.
I see fewer instances of T=nR/PV this way. Yeah, checking one's units should minimize that error, but the weaker students have a difficult time seeing the difference between K and 1/K.
Units? Nah, we don't got to check no units through the calculation. Just slap the one you think ought to go on the end and call it good. In the early days of the class, I consider it a victory if I can get students to pick a relevant equation from the book (whether or not they can correctly manipulate it) and plug in the right numbers with units instead of just doing some random operations on the given numbers and writing down a unit of some type next to the answer. Maybe my views of what is required to teach developmental math are skewed because I encounter such a large percentage of students who appear to think that math is just a big shell game that has no rules or procedures, just guesses that you hope get somewhere close to some random number the instructor draws from a box.
