If you want to give the audience some sense for what mathematical argument is about, I like the topic of divisibility rules (by 2, 3, 4, 9, etc). Most people have seen these but take them completely for granted - indeed, some people take "ends in an even digit" as a suitable definition for even number. One main characteristic which separates mathematicians from the rest of the world is seeing such a rule and asking "does that always work, and if so why?" So perhaps one could first put some plausible false rules out there to create some doubt and the arguments that these rules work - both with algebra and if possible avoiding algebra. I found that emphasizing this material worked well in a class I taught for future elementary school teachers. I told them that even most/ all of their science major friends who passed AP calculus didn't really know why these rules work, so they had learned something special.
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