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as I can see there are mainly two ways to introduce fractional terms. Two examples to demonstrate the two variants:

  1. $\frac{a^2+3}{a}; \frac{3}{2c}$

  2. $T(a) = \frac{a^2+3}{a}; T(c) = \frac{3}{2c}$.

In fact, the word "function" or "functional term" or "equation with variables" has not been learned at this point, but I think that variant two is able to demonstrate better that any term can be seen as a "number machine", in which you "throw" a number and get a certain result. I could imagine that the first variant leads to young learners being unsure what exactly to do with this experssion.

I made a quick research in some school books here (in Germany) and found those two variants. So even though this is really just a small difference, do you have any opinion or experience?

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    $\begingroup$ What age group do you teach? I use fraction for just numerical and rational function for the functions of polynomials divided by polynomials. $\endgroup$ – Chris C Apr 28 '16 at 15:38
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    $\begingroup$ It is (sometimes) helpful to distinguish "rational expressions" from "rational functions" for precisely the same reason that it is (sometimes) helpful to distinguish between "polynomials" and "polynomial functions". $\endgroup$ – mweiss Apr 28 '16 at 16:04
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    $\begingroup$ Personally, I would not write $T(a) = \frac{a^2+3}{a}$ without first having the notion of "function". $\endgroup$ – Gerald Edgar Apr 28 '16 at 16:20
  • $\begingroup$ "I could imagine that the first variant leads to young learners being unsure what exactly to do with this experssion." -- Well they shouldn't think that there's automatically something "to do" on any piece of math without a natural-language direction or question. $\endgroup$ – Daniel R. Collins Nov 2 '16 at 3:04
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    $\begingroup$ I don't see how fractions are relevant to the question; the same question could be asked about $a+1$. $\endgroup$ – user797 Nov 2 '16 at 12:20
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I think that this depends a lot on the context in which you are introducing fractional terms. Since the students haven't been introduced to the idea of a function though, I would definitely favor non-functional notation.

If you need to introduce fractional terms in the language of functions, like if you wanted to talk about the range of the term or you wanted to pull your parameters from an weird domain, then you should really introduce the idea of a function first.

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