## Trying to find math inside everything else

### Hiking and Slope

Last summer, I went on vacation out west to see some National Parks (Yellowstone, Glacier, Craters of the Moon). At Craters of the Moon, all the trails had these lovely signs talking about how steep they were – since one of us hikers had a bad knee, we needed to make sure the trails weren’t too tough. What’s interesting is that they didn’t just talk about the average grade, which many hiking books do (as we learned to our chagrin in Glacier), but also the maximum and minimum. I feel like this is a good opportunity to talk about average rate of change versus instantaneous.

Later on in the trip, we had a discussion about what it means if a trail is twice as steep as another one. If I told you the the next trail is twice as steep as this one, what would you expect? What would it feel like? Then we also talked about whether we were doubling the slope or double the angle. That distinction is tricky because, for angles less than 10°, which are the most common, the difference between doubling the slope or doubling the angle (up to 20°) is less than 1% extra grade.

Anyway, there’s a lot of data here, so I pose to you: what could you do with this?

### The Math of Nail Clipping?

To demonstrate how I’m such a nerd (or such a math teacher, or both):

I was just clipping my nails, and started thinking about the math involved. Often when I clip I’ll only do 1 or 2 clips per nail, and they can come out really jagged, pointy, and sharp. But this time I did about five clips, closely following the curve of the nail and it came out much smoother.

Which makes sense, because I’m basically approximating the shape of my nail (a curve) with the nail clipper (a tangent line), and so the more tangent lines I used, the closer the approximation is.

Now the question just is if I can turn that into a WCYDWT, or if it’s too gross for that….