During the break between classes, [Harry] learns his first lesson with [Dumbledore] will be on Saturday; the trio discusses what the lesson will entail and [Harry] and [Ron] do their [Defence Against the Dark Arts] homework.
Snape's Defence Against the Dark Arts Class
On the way out of [Snape]'s first [Defence Against the Dark Arts] class, [Harry], [Ron], and [Hermione] discuss Harry's reaction when Snape tried to jinx him, and [Hermione] compares [Snape]'s love of the [Dark Arts] to [Harry]'s from the previous year.
Harry Receives Help From a Textbook
As the first [N.E.W.T.]-level [Potions] class prepares to start, [Harry] notices the [potions] sitting on the desks, and [Professor Slughorn] gives [Harry] and [Ron] secondhand books to use until their new ones arrive.
[Horace Slughorn] begins the first [N.E.W.T.]-level [Potions] class of the year by introducing the potions he's brewed - [Veritaserum], [Polyjuice Potion], [Amortentia], and [Felix Felicis] - promising a vial of the [Felix] to the student who brews the best [Draught of Living Death].
[Harry] starts brewing his [Draught of Living Death] - he has a little trouble at first, but soon starts following directions scrawled in the margins of his book, and produces the best potion in the class.
[Slughorn] surveys the finished potions, concludes that [Harry]'s is the best, and awards him a bottle of [Felix Felicis].
On the way out of [Potions] class, [Harry], [Ron], and [Hermione] - soon joined by [Ginny] - discuss the tips written in the margins of [Harry]'s book. [Harry] then looks for the name of the book's previous owner, and discovers the notes were written by the [Half-Blood Prince].
Chapter 10 - "The House of Gaunt" - In which Harry attends his first lesson with Dumbledore, who explains that they will be delving into Voldemort's past and takes Harry into the Pensieve. In Bob Ogden's memory they go to Marvolo Gaunt's house to see about his son, Morfin, who has committed a crime; there they see a mean, impoverished, racist Marvolo abusing his daughter Merope, who is watching muggle Tom Riddle. Dumbledore explains that Merope later used a love potion on Tom, and they became Voldemort's parents.
As the week progresses, [Harry] continues to use the [Half-Blood Prince]'s [Potions] book, and he, [Ron], and [Hermione] wonder about who the previous owner was.
As he heads to [Dumbledore's office] for his first private lesson, [Harry] runs into [Professor Trelawney], who is reading [Tarot cards].