Argument #1
The Court should rule in favor of Fraser. He was given a political nominating speech. Although some might find his speech offensive, they do not have the right to punish him for it. Students have First Amendment rights as well as adults.

Argument #2
the Court should rule in favor of the school district. Fraser gave a lewd and vulgar speech in school which was heard by children. His speech led to several students becoming unruly in the audience. He also violated the school's own rule against disruptive conduct. The school authorities have a right to prevent such speeches.