Only anecdotal evidence. I’ve taught beginning algebra courses at a community college for 23 years. Students who don’t know fractions have a very hard time in algebra. Those who can’t understand that x + 5/3 x is 8/3 x have a hard time understanding that 2xy+ay is (2x + a)y.
Understanding rational functions helps to understand what vertical asymptotes are and as such are a fundamental source of examples when learning limits. They also aid in understanding why tan(x) has vertical asymptotes where cos is 0. Every complete algebra curriculum includes rational functions. I say complete because algebra is usually broken up into 3 courses (2 at the pre-college level).
Sure - but that's just as true in reverse.
> Understanding rational functions helps to understand what vertical asymptotes are and as such are a fundamental source of examples when learning limits. They also aid in understanding why tan(x) has vertical asymptotes where cos is 0. Every complete algebra curriculum includes rational functions.
Meh. x^-1 is a good example of some things, sure, but I don't remember ever doing addition of rational functions which is what you originally talked about, and I went through an extremely reputable maths degree.
No middle school, and I very much doubt it. Searching I can see them mentioned in a further maths GCSE (which is something most schools including the one I went to don't offer, and rather suggests they're not on the regular maths GCSE, which would match my memory).
> Most students who know that x + 5/3 x is 8/3 x have trouble, initially, with understanding that 2xy+ay is (2x + a)y.
Who know that first or who have been taught it? I genuinely would like to see any actual evidence that the latter is objectively more difficult than the former.
Doubt. Do you have any evidence at all for this claim?
> It would be bizarre to teach someone how to add rational functions before they can add fractions.
Sure, rational functions obviously sit at the intersection of algebra and fractions and require both. But they're hardly some deep foundational piece of algebra; I'm not sure my classes even covered them.