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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.

no subject
He didn't write the notes - he didn't work on the amelioration of the formulas in the book, and yet, he takes credit for it.
I mean, if you were, say, in german class, and there were questions in your german textbook with the answers written underneath, wouldn't you be cheating if you answered the teacher with those answers ?
I understand why Hermione's angry at Harry - she works almost 24/7 to be the best, but all he has to do to is open his textbook and voilĂ ! (even though he clearly doesn't understand half of what Slughorn says during class (cf. chapter where they have to make the antidotes)). He manages to make perfect potions with the HBP's textbook, but not without it.
Which makes me wonder, aren't the official formulas written in textbooks also chosen to help distinguish between good and bad students ? I mean, if all the students had the HBP versions of the formulas they'd obviously all achieve EE on their exams, which would pretty much make them useless. So maybe the official textbooks have purposely difficult versions of the potions, so as to make it easier for the teachers to see which students truly understand potions and which don't.
(... wow, that was long)
no subject
It's not as if there were whole essays written on the margins of the book. More improvements on the potions the like of which could probably be found in more up to date potions texts.
In the chapter where actual understanding of the concepts is required, *no one* in the whole class but Hermione has any idea what to do, so I don't think the actual underlying principles are something that's often mentioned in class.
So all the advantage Harry has in most lessons is an improved recipe. Which he could have come by himself if he actually did the work.
Of course, he is lazy, but that doesn't mean he's cheating.
(And I certainly used the scribbled in answers in my English textbooks to answer the teacher's questions. As long as it wasn't during an exam, I fail to see how that's cheating.
I might regret it now, because it didn't do me much good for actually learning the language, but it's not cheating.)