Divination is one of the optional courses that students can start taking in their third year, along with Muggle Studies, Arithmancy, Ancient Runes, and Care of Magical Creatures.
The number of OWLs taken by Percy Weasley, 12, which is the most possible, suggests that he took Divination and got an OWL.
Academic Courses at Hogwarts (does not include Flying)
2. Muggle Studies
4. Ancient Runes
5. Care of Magical Creatures
11. History of Magic
This suggests that it is possible to at least take the OWL and presumably, NEWT-level Divination also exists.
I'm not sure where Firenze went during DH because he could not return to the Forbidden Forest either, Bane had banished him. Presumably Trewlaney kept teaching it.
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7th year Divination
Not sure if this belonged in trio era or the hogwarts section... but here I go.
Would Hannah have taken Divination in 7th year? I checked the Lexicon and found some valuable things, but nothing on what year this course started (3rd?) and how long a student might continue in the subject. If she were in the class, it would have been Trelawney, right? I assume Firenze would not have been permitted to continue under the Death Eaters rule of the school.
Thanks for any thoughts on this.
Note: I realize the masses are split on whether or not Hannah was even AT Hogwarts for her 7th year... I'm going with the premise that she was, obviously.
Like Aida said, Divination was one of the elective classes a student could take in third year, and I assume that it'd be offered at NEWT level because I can't really think of any reason why it wouldn't. Also, I've just remembered that ether Parvati or Lavender grumbles in HBP after McGonagall tells them that Professor Trelawney takes sixth year students. I can't really see Hogwarts offering the class in sixth year, but not seventh, or that students would take a subject that they're not going to get a qualification in.
Regarding Firenze, I always assumed that he stayed on teaching because he's mentioned being wounded in the Battle of Hogwarts before the other centaurs join the fray. To me this suggest that he was in the castle at the time Harry arrived there. If he was banished from the castle, I don't think he would have gone back to the forest because, like you said, Bane cast him out. I reckon he would have gone into the mountains and perhaps not necessarily returned to fight.
Why would he be banished by the Death Eaters? I don't think there's anything in the books that suggest the Death Eaters didn't like magical creatures like Firenze; I thought that was just Umbridge, but I could be wrong. In my mind, and my seventh-year fic I'm plotting at the moment, the Carrows go down to speak to Firenze and are very intimidated by his intelligence and his odd ways and aren't quite sure what to make of it. They're pretty thick people and can't work out if he'd be a thread or not. They go to Snape, who, trying to protect Firenze says something along the lines of, 'I've got more important things to be dealing with than petty staffing issues, and so have you', and the matter is left at that. The Carrows are wary or Firenze, but don't actually know how to go about throwing him out, or whether he's a thread to the regime.
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Belching up Slugs
Thanks, Aida and Sarah, for all the information, it was so helpful.
I asked about the teachers because I just wasn't sure about Firenze but remembered how Umbridge felt toward him. Thanks for clearing that up for me, Sarah.