So I've got no idea how to offer a scientific context for functions in general, which is what I'm working on now in my Algebra 2 class. So I gave myself a bit of a challenge. I spent today trying to get my class excited and confused about the idea that besides for the countable infinity there is a larger, uncountable infinity. Then I told them that once we learn functions they have everything that they need to understand the proof.
I might have dug myself into a hole here. My plan was that I could lay the groundwork for the proof as I introduce them to domain, range, one-to-one, onto and bijection. Then I figured at the end I'd devote a bunch of class time to trying to help them grasp Cantor's diagonalization proof. This is problematic, though, because I'm going to need to devote almost a full period to help them grasp the diagonalization argument, and the confusing parts aren't the Algebra 2 parts, and I'm already crunched for time with this curriculum.
But I'm a desperate guy. Almost all my students think that what we're learning is worthless. I need to do something!
UPDATE: This might help.