One of the commonest mistakes made by students, appearing at every level of maths education up to about early undergraduate, is the so-called “Law of Universal Linearity”:
$$ \frac{1}{a+b} \mathrel{\text{“=”}} \frac{1}{a} + \frac{1}{b} $$
$$ 2^{-3} \mathrel{\text{“=”}} -2^3 $$
$$ \sin (5x + 3y) \mathrel{\text{“=”}} \sin 5x + \sin 3y$$
and so on. Slightly more precisely, I’d call it the tendency to commute or distribute operations through each other. They don't notice that they’re doing anything, except for operations where they’ve specifically learned not to do so.
Does anyone have a good cure for this — a particularly clear and memorable explanation that will stick with students?
I’ve tried explaining it several ways, but never found an approach that I was really happy with, from a pedagogical point of view.
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