Well, that settles it then. 
No… it’s George Bush’s fault!
Well, of course. But which one of them?
Yes
For the algorithm to work, you only need a common multiple, not the least common multiple. It’s easier to teach them to get a common multiple of a and b by multiplying together. You do need to reduce to the lowest terms in the final answer, but you have to do that if you’re working with the LCM anyway.
That is a half-assed way of teaching fractions.
What I just posted? Yes, it is half-assed: it makes the algorithm a bit easier, but at the expense of the student not understanding what’s going on. It would make more sense as a computer algorithm, since a computer doesn’t understand anything.
I think, though I’m not sure, that The Second Stone is referring to the OP, not your post, Giles. I think it’s actually great to point out to students that you don’t need the least common multiple, just any common multiple, and that one easy way to get a common multiple is by taking the product. Come to think of it, I don’t remember how we were taught to get least common multiples, or even if we were taught a way to do so beyond “Just know it”.
I think it’s even worthwhile to also point out that there’s no need to reduce fractions to lowest terms, except as one may self-impose a desire to do so.
(Not quite the same topic, but coming to mind nonetheless in discussing reducing to lowest terms, it annoys me to no end when (college) students of mine run screaming away from having square roots in denominators; “But we have to move them to the numerator!” “No, you don’t.” “But… but…” “You’ve been trained to always put things into a certain standardized form, but there’s nothing to be gained by this except when you have some need for that standardization”. Showing that it’s perfectly fine to treat multiple representations of the same number on equal footing early and often would perhaps help avoid this kind of thinking.).
You don’t need common multiples in the denominator to multiply fractions, it achieves nothing, makes extra work, and obscures what’s going on. It would not make one-third of four-fifths of three-quarters of fuck-all sense as a computer algorithm either. Multiply the numerators, divide by the denominators - if you like, multiply the denominators and divide into the product of the numerators; better yet, simplify down if either of the numerators have common factors with either of the denominators.
Good luck working out 1/2 x 2/3 x 3/4 x 4/5 x 5/6 x 6/7 x 7/8 x 8/9 by the common denominator method. But by the direct method, it’s piss easy. All the numerators and all the denominators cancel except 1 and 9 respectively, leaving 1/9 as the answer.
I’m sure Giles was referring to the comment regarding the hypothetical situation where addition were being carried out instead; Giles was noting that, were it addition, there would be nothing wrong with using a common multiple other than the least common multiple, and that using the product is particularly convenient.
He’s quite right as to that, of course.
Yeah, a lot of people seem pretty retarded at very basic math . . . like even “grocery-store” math, figuring out what some % off will be or estimating totals or how to calculate tips.
In fact, most of the people I know who have a good intuitive grasp of numbers are engineers or hard (non-biological) science-types. Move even into biological sciences and people are lost. Let’s not get started on the English majors.
Yes: for multiplication there’s really no reason to find the LCM for all the fractions.
Objection: my first degree was an English major, but I did round off my education by completing the equivalent of a mathematics major four years later.
If math had been taught that way when I was in grade school, I probably would have aced it. I too had the “memorize the multiplication table” and other methods mentioned. It was painful. I couldn’t make heads or tails of anything. Fractions? Percentages? :eek::eek::eek:
Hell, I still can’t make heads or tails out of fractions unless they’re the common 1/4, 2/3, and such. Throw me a problem similar to what your students had on that test and I’d still be sitting there 
IME, the people I know who have the biggest problems with simple math are the ones who are the best at the hardest stuff (the ones who are, e.g., PhD engineering students).
I should probably also point out that my BA is in English, my minor was CompSci, and I had AP Calc in high school before getting sick of math.
I think we should blame the images on TV.
I blame Canada.
I blame the snake in the Garden of Eden – it was an adder, so it couldn’t multiply fractions.
I can’t stand all this division! 
Why do they insist on deradicalizing the denominator, anyway? I had an algebra teacher once with a big stick up her ass about that, and I never saw the point. x = 3/sqrt(5) is just as good an answer as x = 3( sqrt(5)/5 ). Easier to type, too.