Wrt the example of fraction division, isn't this the standard way? Was anyone here taught to cross-multiply for division?
The failing here is that you don't understand why you get the right answer here.
Why is (1/2) / (1/4) = 2? It's because there are 2 quarters in every half, it's not because (1/2) * (4/1) = 2 that's just a 'convenient trick' to get the answer quickly. This is a great idea if your goal is to pass some maths exams which have a fixed format in the near future. It's a terrible idea if you want to be able to apply mathematics to anything.
The failing here is that you don't understand why you get the right answer here.
By 'you' you mean the putative student mislead by shortcuts, right? I understand perfectly well what's happening with reciprocal multiplication and why it works. ;)
That sounds like looking for a "trick" to me, rather than the much simpler generic rearrangement. Trying to cram that phrase into more complex ones I'd start thinking "Wait, how many five eighths are there in seventy sixteenths?"
If the question is "what is (1/2) / (1/4)" and you can't see it, then the idea of rearranging the equation and why rearranging it is OK is pretty important and general.
(5/8) / (70/16) = x
5/8 = 70x / 16
5 . 16/8 = 70x
5 . 2 = 70x
10 = 70x
1/7 = x
That seems simpler to me than "There is one seventh of seventy sixteenths in five eighths" and it requires knowing how basic multiplication and division works, and that the equals sign means that the things on both sides are equal, so if we multiply both sides by the same number or divide by the same number then that's OK. That can then lead into why you need to be careful about values being 0 or why sometimes you have to add ± to an answer.
You can also just generally explore rearrangements, try out different modifications and see if it makes things simpler, go back and try others.
How does your method explain how to solve "1/2 / 3/4" ? How does talking about how many 4/3 are in 1/2 help?
(I'm assuming that last sentence was typoed and "talking about how many 3/4 in 1/2" was intended.)
Why is (1/2) / (1/4) = 2? It's because there are 2 quarters in every half, it's not because (1/2) x (4/1) = 2 that's just a 'convenient trick' to get the answer quickly.
No, both of these are true. Your logical answer is correct: 1/4 goes into 1/2 twice because there are two quarters in every half. This is how our ancestors conceived division and fractions, and that's how physically dividing things works in the real world. But division is not an independent operation; it's the inverse of multiplication. As plonh mentions downthread, division is, quite literally, defined on a field (like Q) as multiplying by the reciprocal. (1/2) / (1/4) = 2 because (4/1) = (1/4)^-1 and (1/2) * (4/1) = 2. Any other mathematical explanation is simply justifying this fact in more palatable terms.