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Apropos of nothing - a random HP thought.
Where does half the fandom get the impression that Harry is cheating by using a book with notes in it?
They are allowed to use their book while brewing and are supposed to have read up on the potion they are preparing in class.
And in written tests, I doubt they are allowed to use their books anyway, so the notes won't be any help to him there.
The only difference I see between Harry and a Ravenclaw (in this instance) is that the Ravenclaw would have made the notes himself - and probably not in the book, but on a spare bit of paper.

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As for whether Harry's cheating here, I certainly don't think so. It might be a touch underhanded, merely in that he has a resource not available to the others and the teacher isn't aware he has it, but it's not cheating. Most of the potion-brewing, especially before this book, seems to be purely a matter of following the instructions. Harry still has to brew all the potions himself, which he was never able to do while sitting in Snape's class. The difference is that he's got a different set of instructions; every time he uses them, he runs a risk that this time, the Prince didn't manage to improve on the potion, but that what's written was a work-in-progress and won't work. Harry's also got to write all the essays, which presumably he's doing on his own; otherwise, Slughorn would have realized that he's using different instructions almost immediately.
What I'm really curious about is *Snape's* advanced classes. I'd hope he didn't use the same textbook, for one thing. If he was using it but including some of his own modifications, someone would have probably mentioned it at some point in Harry's presence, thus alerting him to the modifications' origins. If he used a different text, it would make more sense for Slughorn to continue with that one for consistency (not that the Defense teachers have ever been concerned by that) rather than a text that's well over fifty years old.