|Image: WikiMedia Commons
When I learned about this, it struck me as amazing. Who knew that math actually happened in real life? That there was a way to describe the pattern of seeds on the top of a sunflower. That leaves grow in a pattern that optimizes the efficiency of catching the Sun's energy. That math is actually interesting!
A few years ago I wrote a lesson about the Fibonacci sequence. It's really an introductory lesson, meant to give students the opportunity to explore and create a pattern based on Fibonacci. You don't need to be a mathematician to teach it (not even a mathemagician). It's a fun way to get students thinking about patterns, and that they might just run into some math outside of the classroom.
If you're interested, the plan is available here. Take it as a starting point, and enjoy the journey. Don't be surprised if your students start to find Fibonacci's numbers everywhere. Believe them!
If you didn't already, click on the math is interesting link. You'll probably enjoy Vi Hart's math doodles series - it's an interesting perspective on topics like this one.