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Archimedes's Principle and hydrostatic pressure are such topics that I've taught just in the last month.


Also, the observation that certain situations are stable (at least in some sense)  can be profoundly important.  In the late 19th Century, Rutherford's discovery that the atom had the positive nucleus (with nearly all the mass) in the middle and the electrons in a cloud around it.  The picture of electrons in orbits around the nucleus was immediately obvious, but it brings in a terrible problem.  To have orbits, the orbiting object is continuously accelerated (even if it's a circular orbit, the acceleration is towards the center so its direction changes even if the speed does not).  But an implication of Maxwell's equations is that when you accelerate a charge, that produces some radiation, and some of the energy is lost in that form.  (Synchrotron radiation is one label for this radiation.)  But for electrons orbiting a nucleus, there is only a fixed amount of energy there, and if the electrons really are in orbit, then they should quickly (much less than a second) radiate all that energy away and the electrons spiral into the nucleus and stay there.


Now, we observe that is not so: stable atoms exist for very long times.  Therefore something else, unknown at the time, has to be in operation to make the electrons' "orbit" outside the nucleus stable.


This led into the discovery of quantum mechanics and so on.

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I learned about the concept of imaginary numbers in Algebra I, probably my freshman year of high school.


I actually used imaginary numbers my final year as an undergraduate, in quantum mechanics.

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