If only... (updated)

What I could have done in the classroom if I had had GeoGebra, a smartboard, and a tablet/laptop of every student. I would toss the textbook, and go.

Here is a sketch easy to create, but full of mathematical questions.
I have included the option of showing some parametric info just as a "spur".

The blue point, the center of the blue circle, rotates on the green circle.
The red point rotates on the blue circle. The red and blue points complete one rotation at the same time.
Does this generate an ellipse?

Start with parametric equations as above.
\(\begin{array}{l}x = n\cos ( - \theta ) + m\cos (\theta )\\y = n\sin ( - \theta ) + m\sin (\theta )\end{array}\)

Use negative angle identities.
\(\begin{array}{l}x = n\cos (\theta ) + m\cos (\theta )\\y =  - n\sin (\theta ) + m\sin (\theta )\end{array}\)

Factor
\(\begin{array}{l}x = \left( {n + m} \right)\cos (\theta )\\y = \left( { - m + n} \right)\sin (\theta )\end{array}\)

Rewrite:
\(\begin{array}{l}\frac{x}{{n + m}} = \cos (\theta )\\\frac{y}{{m - n}} = \sin (\theta )\end{array}\)

Square and add
\({\left( {\frac{x}{{n + m}}} \right)^2} + {\left( {\frac{y}{{m - n}}} \right)^2} = {\cos ^2}(\theta ) + {\sin ^2}(\theta )\)

Use Pythagorean identity
\({\left( {\frac{x}{{n + m}}} \right)^2} + {\left( {\frac{y}{{m - n}}} \right)^2} = 1\)

Rewrite
\(\frac{{{x^2}}}{{{{\left( {n + m} \right)}^2}}} + \frac{{{y^2}}}{{{{\left( {m - n} \right)}^2}}} = 1\)

We now have a standard equation for an ellipse.