The instinct of many students to cancel these fractions kicked in and there was some uncertainty as to whether the 2's in the first fraction or the g's in the second would cancel. No, I dutifully informed them, you can only cancel factors that are common to all terms of the fraction. Done and dusted, I hoped, and we could get back on with the main thrust of the lesson - rearranging those formulae.
But students were not convinced by this. With the silent sound of 'I didn't really get that' hanging heavy in the air, I realised my errors: (i) I had whisked quickly over an important learning point in my efforts to drive the lesson towards the intended goal, refusing to be led down this tangential avenue, and (ii) I had presented this necessary must-be maths fact to students as though it were an arbitrary fact, instead of designing a more powerful experience where they could see it for themselves.
Following these realisations, I wrote the following on the board:
...and an impromptu point of inquiry was born. My students seem to really enjoy getting to the bottom of problems posed this way and certainly came away with a clear understanding that it must be that cancelling the b's in this fraction was not a 'safe' algebraic operation. Their findings? The two fractions are only equivalent sometimes - when one (or both) of two particular conditions is met. (I'll leave finding those two conditions to the reader.)
Ideally, I believe the spirit of mathematical inquiry should permeate every maths lesson - certainly whenever students are encountering new must-be maths facts. As to what constitutes a 'mathematical inquiry', well I believe the answer to this is very broad indeed - from points of inquiry like these that may last just a few minutes to those that last an hour lesson or beyond (for example, those that can be found at www.inquirymaths.co.uk.)
I think it is also important when manipulating algebra that students do not lose sight of the fact that the letters represent numbers and operations on numbers - that it is generalised arithmetic. If forgotten, algebra can come to seem like an arbitrary language with strict and potentially confusing grammatical laws. If students have a good awareness that algebra always represents the way numbers work, they can fall back on 'substituting a few different numbers' as a check for whether manipulating an algebraic expression in a certain way is valid. (This is not foolproof though, of course)
Finally, an important distinction emerges when students are told the rule compared to when they 'see it' for themselves - the question of on whose authority is this fact true? If simply told the fact, the students have no choice but to accept the authority of the teacher. If students see the fact for themselves, the fact clearly stands as true on its own - or indeed, on the students' own authority. Must-be maths facts are not true simply because I, or any other teacher, says they are, they are true because they simply must-be! That is the nature of mathematics, and that is what students in our classes should experience.