What are interesting applications that I could work through with pre-calculus level students? I'd like to go from pop science explanations to an actual building of physical intuition and appreciation for the importance/power of math, but keep it interesting by doing this through investigations of interesting applications.
Basically, I'd like to avoid teaching in the style of "the precession of Mercury's orbit was a mystery until Einstein applied his theory of general relativity. Using his mysterious new equations, he did some magical math, and lo and behold Mercury's orbit was explained."
The Long Version:
I have the great pleasure of tutoring 3 gifted students ages 12-14 for 1.5 hours each week (individually) in exciting science topics. These students excel in the classroom, but their parents are concerned that they won't have a "passion" for STEM topics. This is where I come in.
Although in college my classes focused mostly on Newtonian physics (I'm a mechanical engineer), I personally really love learning about other kinds of physics! Especially when it comes to "the math", or any good explanation of how things "work". I think I'm particularly motivated by the idea that if we can understand enough about how things like gravity and dark matter work, we can come up with the sci-fi tech of the future! Naive? Maybe, yeah. But motivating nonetheless!
In any case, this passion has shaped how I've approached these sessions so far. For example, we've investigated the physics behind exciting sci-fi topics like Alcubierre drives, wormholes, "teleportation" via entanglement, and so on. In essence, I guess we've been doing interactive YouTube videos, with me frantically researching the material in the week between sessions and then explaining it via whiteboard when I meet with the kids in person.
I want to keep covering these kinds of topics. I and the kids both find them motivating and exciting! However, in what I think of as Phase 2, I'd like to go more than skin deep. I think I have an opportunity here to promote a passion for physics, and I don't want that passion to die for these kids when they get to class and realize that it's not all anti-particles and negative matter - that there are probabilities and field equations and experimentally derived constants involved (for example).
In other words, I'd like to get into the math a bit, and give these kids a good start at enjoying the actual nitty gritty involved with investigating the natural world. (As well as learn these topics for myself!) I also think it might be cool to investigate geometry/math in the way that I've read Einstein did as a kid, with interesting thought experiments and math challenges.
If you read this far, thanks! Any thoughts?