In chapters 30 and 31 of To Kill a Mockingbird, what were Mr. Tate and Atticus arguing over?
They are arguing over Bob Ewell's death. Tate wants to concoct a story about it being an accident. But Atticus knows what happened and he will not go along with the falsehood. Here is an exerpt of their exchange (my edition Ch 30, pg. 315-318).
"Mr Finch." Mr. Tate was still planted to the floorboards. "Bob Ewell fell on his knife. I can prove it."
Atticus wheeled around. His hands dug into his pockets. "Heck, can't you try to see it my way? You've got children of your own, but I'm older than you. When mine are gorwn I'll be an old man if I'm still around, but right now I'm -- if they don't trust me they won't trust anybody. if they hear me saying downtown something different happened -- Heck, I won't have them any more. I can't live one way in town and another in my home."
As Atticus has already painfully learned, right doesn't equal justice. Tate replies, "I may not be much, Mr. Finch, but I'm still sherrif of Maycomb County and Bob Ewell fell on his knife. Good night, sir."
Atticus is forced to ask Scout, "Can you possibly understand?" Scout does. She runs to him, hugs and reassures him.
The children have not lost their respect for Atticus. Having witnessed the injustice of the trial, they now know the injustice of the world. And though they can no longer view him as a god, they can view him as a man who consistently acts morally.
Heck Tate reveals that Mr Bob Ewell is killed and found on the laying under the tree with a knife stuck under his ribs. Heck Tate calls the death and accident but Atticus, thinks that Jem his son killed Mr Bob. He does not was his son to be protected by the law because he is fair and his a man of the law. Heck Tate tells Atticus that Tom Robinson dies for no reason now let Mr Bob be responsible for his dead: "Let the dead bury the dead" -Mr Tate