Why does Jem not want anything to do with Scout at school? Is his behavior typical of an older child?

As is the case for many younger siblings, Scout notices that Jem seems to be trying to distance himself from her. On the first day of school, Scout informs the reader that "Jem condescended" to take her on the first day. She's confident that "some money changed hands," in order for this to happen. Jem explains to Scout that she is not to bother him during the time they are at school. While at this point it is still acceptable to play together at home, school is to be an exception.
Jem is entering adolescence, which helps to explain why he desires more independence and space. Scout describes him as being "difficult to live with, inconsistent, moody." While it is perfectly normal for the older sibling to desire some distance from the younger sibling, Scout sometimes has difficulty coping with this change in her brother. Scout once asks Calpurnia why she calls him, "Mister Jem." Calpurnia explains to Scout that Jem is getting older and that he's "gonna want to be off to himself."