What is an example of how Jem and Scout's relationship changes as Jem matures in To Kill a Mockingbird?
The events of the summer effect a change in Jem Finch that produces a maturation of ideas and perspective. He, then, breaks the "code of childhood" that he holds with Scout.
In Chapter 14 Scout overhears her father and his sister Alexandra arguing about keeping Calpurnia. Jem tells Scout that their father and aunt have been "fussing." He adds that Atticus has much on his mind with worrying about the Tom Robinson case. When Scout makes a remark to the effect that Atticus never worried about anything, adding that the case doesn't bother them "except about once a week and then it didn't last," Jem counters,
"That's because you can't hold something in your mind but a little while," said Jem. "It's different with grown folks, we---"
Here Scout remarks that "[H]is maddening superiority was unbearable these days."
When Scout retorts, "Who do you think you are?" he threatens his little sister in the voice of an adult,
"Now I mean it, Scout, you antagonize Aunty and I'll--spank you."
Hearing this "parental voice" of Jem excites Scout into a fight. As they struggle, Atticus enters and separates them, telling them to go to bed immediately. However, as Scout passes her bed, she steps on something warm, smooth, and resilient. Thinking it is a snake, Scout runs to Jem. They discover that the "filthy brown package" under the bed is Dill, who fabricates a tall tale about his experiences before arriving there.
In another display of maturity, Jem looks at the door, stands and tells Dill he should let his mother know where he is; after saying this, he leaves the room:
"Then he rose and broke the remaining code of our childhood," Scout narrates. Jem calls to Atticus, asking him to come to Scout's bedroom, and he reveals Dill's presence to Atticus. Jem has separated himself from the ways of a child, and altered his relationship with Scout and Dill.
Scout and Jem’s relationship changes as Jem matures because he begins to see the world differently, and she still thinks like a child.
The first significant example of the divide between Scout and Jem is when Jem loses his pants on the Radley porch, and then decides to go back and get them. Scout thinks he will be shot, and she doesn’t understand why he does not just take his punishment.
Jem explains that Atticus has never whipped him, and he does not want Atticus to find out about the pants because he will be disappointed.
It was then, I suppose, that Jem and I first began to part company. Sometimes I did not understand him, but my periods of bewilderment were short-lived. This was beyond me. (ch 6)
What Scout does not understand is that Jem is not afraid of a spanking. He is afraid of disappointing Atticus. He wants Atticus to look at him as an adult. He wants him to think he makes good choices. Jem regrets the childish decision to sneak into the Radley yard, when Atticus told them to leave the Radleys alone.
As they get closer to the trial, Atticus follows it with the intellect of an adult but the emotions of a child. He is more and more distant from Scout, and treats her more like a child than a friend.