My calculator answered 6.02e23 and my kid asked what the e means — how do I explain it without hand waving?

Write 6020 on the board and ask her to say it as six point zero two times something. She will get to times a thousand, and then you write 6.02 times 10 to the 3 next to it and point out the three is just how many places the point moved. Do 60200 and 602000 so she sees the number after the e is a counter, not a magic symbol. Then do 0.00602 and let her notice the point moves the other way, which is what the minus is recording, and stop there. The whole thing is ten minutes and she will finish it herself if you let her do the last two.

Anchor it in things she can picture, because the point of the notation is that ordinary numbers stop being picturable. A human hair is around a tenth of a millimetre across, an atom is around a ten billionth of a metre, and the gap between them is not something you can hold in your head as digits. That is why the notation exists, not to be tidy but because at that range the zeros stop carrying meaning and the exponent starts carrying it instead.

I would push back on the moving the decimal point framing, because it teaches a procedure that survives one term and blocks the concept underneath. The idea is how many times bigger than one, and once you say it that way, the minus is obviously how many times smaller than one, with no rule to remember. Ten to the 3 means a thousand times one, ten to the minus 3 means a thousandth of one, and the mantissa is just where you land after that. I have taught both ways and the students who learned the counter rule get stuck the first time they meet something like ten to the two point five.

My mistake with my son was opening with the words mantissa and exponent, which are the correct terms and completely useless to a child. He glazed within a minute and I had to leave it for a year. Names last, idea first, and if you never say the names at all that is fine.

The e is short for exponent and just means times ten to the. Show her the same number four ways on paper. Stop while she is still interested.