I regularely teach math for university students, and get favourable feedback.

When somebody asks why isn't 1/(x+y) = 1/x + 1/y, then I ask why it should be. Just because something looks intuitive or pleasing does not mean it's true. While this is only a meta-level answer for the question at hand, I consider it to be more important to learn than the special case. The students need to learn basic scientific methodology, for then they wouldn't ask such question, as one of the basic pillars of science, scepticism, easily empowers the student to answer the question on their own. Many haven't learned the scientific method in school, since in school pupils are conditioned to say what the teachers want to hear, not what is true.

When I'm convinced that the former point has been made clear, I proceed to the next step of how to approach a question like this. First, what is the statement supposed to mean? What are x and y allowed to be? Naturals? Integers? Reals? Complex Numbers? Matrices? Polynomials? All, of course, restricted to those where the expressions are defined, since worrying about the truth of an undefined statement is not sound. The context usually defines the possible range of the variables, but one needs to learn to identify that, which is akin to type interference.

Then, maybe someone simply forgot whether that equation was one of the proved ones, or not. The overwhelming majority of wrong, practically ocurring mathematical statements have small counterexamples. That's why I recommend plugging in arbitrary values for the variables, which seem interesting or make the evaluation easy. Whereas intuition and beauty don't imply truth, they remarkably successfully guide us to where to look first. To this specific problem, x = 1 and y = 1 seem like a good idea. We immediately see that they form a counterexample inside all the mentioned possible ranges for the variables. Now, we now the statement is in general wrong, and we can move on.

Depending on whether the time allows, I would show that for real x and y we have

1/(x+y) = 1/x + 1/y

<=> (multiplied with xy(x+y)

xy = y(x+y) + x(x+y)

<=> (use of law of distribution and substraction of xy)

0 = y^2 + xy + x^2

<=> (use of well-known p-q-method)

y in {-x/2-sqrt(x^2/4-x^2), -x/2+sqrt(x^2/4-x^2)}

However, the expression under the root, x^2/4-x^2, simplifies to -3/4*x^2, and is for real, nonzero x always negative, so there is no real solution in y. We just showed that the statement is not just in general wrong, we proved that it is always false, when the variables are only allowed to be real, and that there do exist true cases for complex variables.

It's a bit funny that I explain the things in that order while I despise philosophy. Learning the scientific method is not philosophy, or — to state it better — it's as much philosophy as learning mathematics as a physicist is.

Now, I don't really know anymore why exactly I wrote this down here. Probably it was just the pedagog in me wanting to make clear that questions like these do not indicate lack of mathematical talent, but lack of scientific education.

Do you know a way to teach science and math in a more interactive way than blog posts or youtube videos on the internet? I would love to do that.