Transfiguration, it transpires, is Professor McGonagall's subject. She begins the class with a safety lecture:

No-one is to transfigure any part of themselves, another human, or someone's pet, until they have been taught to do it safely, which will not happen this year.

No-one is to eat, drink, inhale, or use in a potion anything transfigured, transfigure anything into something that looks like food or drink or anything that can be inhaled or a potion ingredient, or do anything that might cause someone else to eat, drink, inhale, or use in a potion something transfigured, until they have been taught to do it safely, which will not happen this year.

No-one is to transfigure anyone else's wand, broom, or any in-progress or completed potion, until they have been taught to do it safely, which will not happen this year.

No-one is to transfigure anything into gold, silver, bronze, Galleons, Sickles, or Knuts. This is not dangerous, but is extremely illegal and will be detected and punished by Gringotts.

No-one is to transfigure any part of the castle building. If it's stone or wood or glass and you can't pick it up and walk off with it without breaking something, it's part of the castle building. If anyone is curious about an edge case (and here her eyes flick to Bruce) they are to ask her about it.

No-one is to do transfiguration experiments on their own time using concepts that have not been covered in class. If anyone wants to do extra projects (and here her eyes flick to Hermione) they are to come to her office and ask her to supervise. She will be happy to supervise. This is not a burden and they should not worry about wasting her time. She would rather supervise a hundred student experiments than help Madam Pomfrey unfuse one student from their chair. Yes, that happened. Follow the rules.

If someone has a problem related to having broken one of these rules, they or anyone else aware of the problem and mobile should come to her office or her classroom immediately. There will be no punishment for reporting an incident, even if it was your fault.

Hogwarts has the lowest student fatality rate of any school of witchcraft in Europe and she intends to keep it that way. You will cooperate with this.

Now, does anyone have any questions before they begin learning the theory behind Transfiguration?

"How do I know if something I am considering Transfiguring could be a potion ingredient so I shouldn't, is there a list?"

"You will definitely avoid the problem if you stay away from parts of plants or animals, especially magic ones, and don't leave anything you've transfigured lying around where someone could mistake it for an original."

Very serious nodding. "Okay, thank you."

Onward to theory, then!

There are three main kinds of spell, not counting potions because those are not cast with a wand and draw magic from multiple sources instead of entirely from a witch or wizard's magical core. Those three kinds are charms, hexes, and transfiguration. A charm is anything that produces an ongoing though generally short-lived magical effect; a hex is anything that produces an instantaneous effect. A hex is not simply another word for a hostile or offensive spell. They will learn about the distinction between jinxes and curses in Defense Against The Dark Arts, but for now let it suffice to say that some jinxes are hexes and others are charms, and likewise for curses. Transfiguration is the third category, and it sits on the border between hexes and charms, because a transfiguration is a package of smaller spells cast in sequence, as close to simultaneously as possible.

The colloquial definition of transfiguration is turning objects into other objects; this is almost but not quite accurate. Transfiguration is the process of changing the nonmagical properties of objects. If you turn a teapot into a tortoise, it will have only those properties of a tortoise that you deliberately concentrated on giving it, because magic does not inherently know what a tortoise is. This is the key reason it is very unwise to transfigure food: "being safely edible" is actually a large number of very complicated properties, and without a great deal of careful study you will not be able to replicate all of them.

Magic treats objects (and their properties) as ontologically fundamental, that is to say, as things that magic can operate on directly. What is an object? It matches pretty well to what humans intuitively think of as an object but messing with your mental concept will not affect the reality more than very slightly. A book can be an object and so can a page in the book, but not a word or half of a page or "these six pages". A great deal of the difficulty in learning transfiguration comes from memorizing, and getting an intuitive sense of, which properties magic treats as "real" and can thus be operated on directly. Weight, density, color, shape, malleability, roughness, stickiness . . . it's quite a list and the best way to learn is through examples and practice. Properties which can be directly sensed in the moment of spellcasting (such as weight, color, texture) are much easier to manipulate than those which cannot (for example, flammability).

In sum, transfiguring one object into another requires changing its properties to the desired properties one at a time in quick succession while holding the completest possible specification of the target in your mind. She will now demonstrate this. She draws her wand, taps the desk behind her, and it flows and morphs and reshapes itself into a pig, which grunts and takes a step forwards before she taps it again and it returns just as fluidly to being a desk.

The transfigured pig, she explains, was capable of walking around and grunting, and had sufficiently recognizable organs to support that, but it wouldn't've been edible because she wasn't focusing on the meat at the level of detail necessary to make it exactly correct. It was partially animated by magic and would not have lived very long, and anyone who knew more than she does about what the inside of a pig is supposed to look like would have been very surprised had they tried to butcher it.

Hermione is alternating rapidly between gazing starry-eyed (McGonagall is SO COOL and SO SMART) and diligently taking notes (a process substantially eased by the fact that she's been furiously practicing writing with a quill pen for eleven months).

One might at this juncture wonder why someone with perfect memory would bother taking notes in class. The answer is fourfold:

1) Writing things down helps her keep her thoughts going in a moderately straight line.

2) Even though she has never yet forgotten anything, she would feel very silly if she ever did and had failed to write it down.

3) Notetaking in class is Good Student Behavior.

4) When she is stressed she likes to reread them anyway, just to be sure.

Bruce has not had nearly as much time (especially unobserved time with a flat surface) to learn how quills work, and is also sinister a lefty, so his notes look like an octopus sneezed on the parchment, but he is in fact taking them. He'll make a cleaner copy later when he's not trying to keep up.

He raises a hand, waits to be called on, and asks, "Does that mean that knowing a lot of chemistry and biology can help make transfiguration work better?"

"It does! Though we won't get to much of that in class this year."

If Potter has ended up with his father's investigative spirit and knack for Transfiguration and his mother's rule-abiding conscientiousness, she's going to have a lovely time pretending not to be overworked while he experiments in her office. She almost never uses her time-turner--it doesn't actually give you extra hours, just lets you spend them twice as fast--and she probably won't have to this year either, but some things are worth it.

But they will get to it in class eventually, it sounds like this means. And she and Bruce are going to be so, so ready for it, it's going to be awesome.

They are going to be super ready for it, and it's going to be the best. 

The rest of the class is mostly taking notes half-heartedly at best, but where one might easily brush off one enthusiastic nerd as an outlier, two of them in a class of eight* is enough to gently nudge the ambient peer pressure situation a little bit Ravenclaw-wards, so everybody's at least quarter-assing it. 

Lavender and Parvati are both doodling more than they're writing, with the unthinking ease of reasonably coordinated people who learned to write as toddlers with the crayon equivalent of quills. Seamus and Neville have both clearly met quill pens before, but Neville (despite having had an entire dedicated penmanship tutor) has the fine manual dexterity of a drunk seal and Seamus (one of whose parents is a Muggle) keeps instinctively trying to hold his like a pencil. Dean, who like Bruce met a quill for the first time extremely recently, is struggling, but he has pretty good manual dexterity and is managing, slowly.

*(As they may have noticed from their schedules, many of their classes are joint - double Potions and Flying classes with the Slytherins, Herbology and Charms with the Hufflepuffs, and Astronomy and History of Magic with the Ravenclaws - but Defense Against the Dark Arts and Transfiguration below the NEWT level are always given in single house groups, for safety reasons.)

Once the theory section is complete, she hands out matchsticks to everyone in the class and instructs them to try to turn them into needles. Magic has a certain momentum to it, such that as they get more fluid it will be easier to do lots of properties in quick succession than to pause in between, but for now they may find it easier to focus on one or two at once. Focusing on the detailed sensory experiences they want the needle to produce--what it should look and feel like--will work better than concentrating on words like "sharp" or "metal" or "with a hole in the end". And no-one is to try to transfigure anything other than their own match or aim for any target other than a needle.

Bruce starts out working on getting the match as smooth as a needle, planning to move on to giving it a round cross-section instead of a square one if he finishes making it smooth. It turns out to be tricky to get the magic to go through his wand and out into the world instead of just circulating around, especially while focusing on something else, and then he gets it for a split second and the match instantly turns perfectly smooth and then breaks in half.

Intense concentration time!

Where to start? Well, if a match is to be a needle it's got to change color, right? Hermione focuses carefully on what it would look and feel like for her little matchstick to be silver wood. Not metal, silver wood - one step at a time, that's the rule.

There we go, silver wood. Next, it's got to be pointy -

No one else has managed to make their matches do anything yet, except for Seamus, whose match is on fire.

Dean looks sympathetically between Seamus and Bruce, on either side of him. "You almost had something for a second there, how'd you do that?" he wants to know.

"It was like--you know how sometimes you end up with your knee in a weird position and you move it just so and it goes pop and then feels okay again? It was like that. But I might've done it too fast or something, since the match broke." Maybe his next thing should be trying to transfigure one of the halves back to the right length; length is a property after all.

"Huh. Neat." He prods his match thoughtfully; this doesn't do anything, but he seems cheerful about figuring it out eventually. 

Seamus was about to raise his hand to ask for a new match, but now he's watching Bruce try to fix his, fascinated. It would be less embarrassing to do that instead if he can figure out how! 

Length is indeed a property! A very simple and easy to imagine one, even, for an object that strongly resembles a line segment.

The result is a bit wonkily proportioned, a little skinnier and with the sulfur end longer than it should be, but it is convincingly A Match rather than Half A Match.

Excellent. He spends the rest of the time trying to get it more cylindrical but doesn't manage to get his magic to do the thing again before time runs out.

McGonagall comes around to inspect everyone's work and encourage the people who didn't get anywhere to take their matchsticks and keep practicing outside of class. 

"Very good," she says when she gets to Bruce. "Just a little too much power a little too quickly. Try visualizing a smooth transition from the starting state to the desired state rather than simply the desired state. And take a point for Gryffindor for your progress." Then she sweeps off to look at Hermione's work.

Yay! He carefully tucks the wonky matchstick into the box his quills live in so it won't get crushed by books.

Hermione has not managed to turn her matchstick entirely into a needle yet, even though she has not really been paying attention to anything else, and she is somewhat disappointed in herself. It is, however, silver and pointy; she's stuck on the last step where she's not quite managing to visualize the nature of metal vividly enough.

"Well done, miss Granger; a point to Gryffindor for your efforts. If you have trouble with the material it may help to imagine the sound and visual of dropping it on the table, or even on an imaginary metallic surface. It is of course possible to transfigure properties one cannot sense directly, but in the early stages of learning to control one's magic every little bit helps."

Yesss, teacherly validation!!!

Hermione writes this down, beaming.

Ron hasn't gotten anywhere and it has not even slightly occurred to him that having a hand-me-down wand is making this harder than it needs to be, especially since nobody else has gotten anywhere either.