I'd like to hear some ideas for problems or projects related to this summer's solar eclipse, in particular for the high school or undergrad level (algebra through calculus).
I would myself be interested to know what is the algebraic form of the equation of the curve the shadow follows. Perhaps in a spherical coordinate system centered at the Earth's core? Or in some convenient coordinate system. How far does the path deviate from a geodesic?
To more directly address your question: What is the great-circle path between the shadow's West-coast and East-coast endpoints, Portland to Charleston?
I highly recommend watching Terry Tao's lecture:
It contains several ideas which could be used to combine astronomical observations with basic trig and geometry.
The first experiment (finding the radius of the Earth) would be a bit hard, but you could certainly have them time how long it takes the moon to set, how long it takes the sun to rise, how long a lunar eclipse lasts (if you are lucky enough to have one), how long a solar eclipse lasts, observing that (by pure coincidence!) the moon almost perfectly covers the sun in a solar eclipse, etc. Then you could use these facts to determine the distance to the moon, the radius of the moon, etc.
This could be on the difficult side, but the story of Eddington's 1919 confirmation of Einstein's general theory of relativity comes to mind. This was done on the basis of measurements made during a solar eclipse, using the distorted apparent positions of stars near the sun.
There are relatively elementary accounts of this, and there are multiple angles. (For example, there is a story in the literature that the measurement error was too large for the experiment to yield any meaningful conclusions as to the validity of GR; it would be fun to run a check on this -- if this is at all feasible in a student project of course.)