This is part two in a six-part series.
Read part one (chapters 1-5)
here.
Chapter 6
DRACO'S DETOUR
Harry receives a certain badge giving him some authority at Hogwarts.
While shopping in Diagon Alley, Harry, Ron, & Hermione spot Draco Malfoy sneaking around and decide to follow him while they hide under the invisibility cloak (which Harry carries with him at all times now per Dumbledores council).
Draco goes into a shoppe in Knockturn Alley and the trio discover that he is trying to get something fixed and reserve something in the shoppe.
Harry suspects that Draco has replaced his father (who is in Azkaban) as a Death Eater. But nobody else believes him.
Chapter 7
THE SLUG CLUB
While onboard the Hogwarts Express, the new professor, Slughorn, invites Harry and Neville to his dining table. Others are there whom Slughorn is trying to influence or impress, including a slytherin boy, who Harry follows (hiding under the invisibility cloak) to Draco Malfoy hoping to spy on him and get some more concrete evidence of his suspected 'Death Eater' status.
Chapter 8
SNAPE VICTORIOUS
As most people have guessed by the title, Snape is revealed here as the new DADA Professor, while Slughorn replaces him as Potions professor.
Harry wasn't expecting to be able to continue with potions since Snape told him last year that Harry would need to get an 'O' (Outstanding = top mark) to continue to NEWT, but now Harry and Ron are both able to take the class.
Chapter 9
THE HALF-BLOOD PRINCE
Since neither Harry nor Ron were prepared to take the potions class, they needed to borrow used books (Advanced Potion Making, pictured on UK adult cover) that had been left over from previous years. Harry's second-hand book has some notes and scribbles in the margins; some corrections or adjustments to the potion recipes.
Slughorn challenges the class to a contest for who can complete a certain potion in the most correct fashion. Using the notes in his new book, Harry wins the contest. The prize is a bottle of a certain "luck potion" which promises to give Harry 12 hours of pure luck where nothing will go wrong for him.
Harry discovers that the book used to belong to a person called 'the Half-Blood Prince'.
Chapter 10
THE HOUSE OF GAUNT
Harry meets Dumbledore for his first lesson.
Using the pensieve (pictured on the American book cover), they visit one of Dumbledore's memories and follow a man to a shack in the woods (pictured on
our current front page and Deluxe Edition cover). This is the first in a series of revelations regarding Voldemort's history/background.
Continue to part three (chapters 11-15) 
HBP Chapter-By-Chapter: 6-10 (spoilers) 
Dan
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