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.
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.