I have a friend whose experience of math in school was horrible (each teacher, she says, either left or was fired the year after she had them). I've been teaching her about various things: cantor's diagonal argument, irrationality of $\sqrt{2}$, etc. and she gets very excited and interested, and she picks up on mathematical concepts and thinking easily.
I am looking for some resources people would recommend for a curious but unsophisticated mathematical mind. She is not in the target audience of most textbooks at her level; I suspect a traditional scaffolding approach would not spark her passion. How can I help deepen her mathematical thinking without falling into the mundane algorithmics typically associated with the teaching of lower-level math?