Dan cried:
Do not treat me that way
OR
Do not treat me that way.

That complete sentence (with a period) is what is contained within the single quotes.

Frankly, I find it very funny that you say, "t just looks wrong…", referring to correct punctuation, when your own is so bad:

I was looking on Grammarbook.com(NO SPACES) and came across this example,(COMMA) which confuses me. It(CAPITALIZE) doesn't explain why the period goes before both the single and double quotation marks. It just looks wrong,(COMMA) as I thoughT the period would be after the single quotation. Can someone explain this to me please?(QUESTION MARK)