The other day in class, I took time to look at Kepler's first law and ellipses. I included a bit of history of how at one point, people believed all planets had a perfect circular orbit but that was disproved in the 17th century.
I found a short article at Khan Academy which gave a great description of this. Besides providing a bit of history, it also includes some animations which help illustrate it.
What is so cool about teaching ellipses with Kepler's law is the sun is the origin of the ellipses. A coordinate plane could be placed over the orbits so students calculate the formula for the ellipse. This article has great information on eccentricities. I teach it but this is a great topic for for showing its application so its not something taught in isolation.
Just think, Neptune and Pluto have interesting paths because at certain points, Pluto is closer to the sun than Neptune. This has to do with differences of eccentricity of orbit and is a great way of showing students that not all orbits are the same.
This site has the distances between each planet and the sun for its closest point and its furthest point. Using this information, students can create an elliptical equation for the planet's orbit. Yes, I am aware there are a lot of factors involved in the orbit but I'm looking at students creating the equation from the data.
Once they've created the equations, they can use the information to calculate the eccentricities for each orbit and compare their answers to the actual answers. This leads to a great line of questioning on why they might be different.
Yes I'm going to be doing this today in my advanced math class. I'm interested in seeing how well it goes. I'll report back on Monday and let you know how it goes.