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Ideas for getting high school aged students interested in math?

Im volunteering at a program at my university where we can teach free 1 day classes to students about whatever topic we like. I chose to do a class that tries to get kids interested in math for math's sake. The students are in high school so I'm hoping they are old enough to understand some of the underlying beauty of mathematics.

Some of the examples I want to include are the role of the golden ratio in the formation of sunflowers and the fact that a rope held up at two ends forms a hyperbolic cosine.

Does anyone have any ideas that get at the heart of the beauty of math while being understandable to an audience that has likely only taken algebra 1 and geometry?

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