I thought I had a joke lined up for here, but I'm not sure, because
I periodically forget them. (Image taken from Wikipedia.)
My chemistry class was basically a college class, though - like, to the point that in actual college chemistry class, I didn't buy the textbook, own a calculator, or go to half the lectures (my buddy & I signed each other in because we're the fucking worst), and I got the highest grade on the first exam after being the first one done. This is not to toot my own horn, I'm saying that's how much stuck despite my slacker habits because my high school chemistry teacher was awesome. So I don't know what a "normal" high school education in chemistry is like, but I think this is interesting, so I'ma riff on it today.
I'm pretty sure that everyone learns about electron orbitals, because that was on my sister's homework, but I don't know if you get a good explanation of periodicity per se.
In a nutshell, you've got your s orbitals, your p orbitals, your d orbitals, and your f orbitals. And they go more or less in the order you see those groups expand on the table: each time you get a new number of columns in a row, you're dealing with a new orbital.
Now the crazy thing is this: while the proton count is what makes an element be what it is, it's the number of electrons that actually makes it do what it does. But the number of electrons can change due to ionization or covalent bonding, while the number of protons doesn't change unless you have a literal nuclear reaction, and the protons still influence how the electrons come & go and it's complicated and it's also been a while so we'll just leave it there.
So as the proton count goes up, so does the electron count (at least, when we're talking about a single atom, which - foooocuuuus...), and as you go through the electron orbitals you eventually get right back to where you were - and then that element behaves pretty much just like the one directly above it on the table, only bigger. And this is why the alkali metals all react so violently with water in their elemental form, and why the noble gases are all so prissy about every fuckin' thing, and the halogens are so great for light-up signs: periodicity!
Alkali metals reacting with water (from YouTube).
In fact, it's so regular that once we figured out the pattern, we were able to successfully predict the existence of a bunch of undiscovered elements, which we then went looking for and found! And that's why a bunch of the bigger ones in the lanthanide and actinide series (the rows that are hived off from the rest to make it legible on a single piece of printer paper) are named things like ununoctium, which means "one one eight ium." Other fun fact! When we actually isolate them in the lab, that's when they get proper names!
And to bring it full circle, this all started because a guy named Mendeleev figured out the patterns first, and then organized the elements according to that, and that's how the periodic table was born. Ta-da! And what's more, not only did Mendeleev predict eight undiscovered elements, he used periodicity to correct the properties of several elements that had already been known, which in chemistry circles is known as a power move.

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