@Asaf: I teach a special course to (some of) the best public high school math students in New York City, so I'm often at pains to anticipate questions and get details right. Having taught this class to such students for many years, I can say without hesitation that when I walk into the room each day, I'm never the smartest person there. But the enthusiasm and discussions make it a real joy.

I think your first suggestion above is excellent: show them how vastly one can increase A without increasing its cardinality. However, I have difficulty with your other suggestion in your paragraph 3: note that x was very specially constructed so that it was easy to show that the bijection misses it. Once you form the union between A and x, you have to look for another very special element that the new bijection misses, which may not be so easily recognized...

@Pete: Thanks for the link to your notes -- very useful.