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

I think in a broader context, Dill becomes a confidant for Jem. As the story progresses, Jem changes from a child who, plays games with his little sister, to a more independent young man. Dill sits somewhere in the middle of that progression. Dill brings Jem into the world of boys rather than Scout who, to Jem, represents the world of children. As Dill and Jem go off together, Scout feels excluded. Fortunately Calpurnia recognizes what is happening and gives Scout the attention she needs. I think change and maturity differences are typical for siblings.