Day 1 - Generalizing the Yin Yang problem

This is a well known problem. I got it from the Mathematics Teacher magazine poster of100 great problems,  but it's so meaty for rising 9th graders that I like to start the year with it.
It allows me to start the year with discussions of:

• fractions vs. decimals, 
• algebra & factoring, 
• brute force vs. elegance, and 
• algebraic proofs. 

But no postulates or theorems.

The FIRST DAY'S HOMEWORK: Have them do the simple version. "Find the ratio of the shaded area to the unshaded area."

IN CLASS THE NEXT DAY:
Common issues:

1. use of decimals leads to calculation errors
2. the brute force desire to multiply everything out before dividing leads to calculation errors
3. When asked if the result would hold for different numbers, they have no idea.  

Class QUESTION: "what would allow us to prove this result would hold for ALL possible cases?"
. . . Someone will say "use variables", so then have them do this version:

Tomorrow, we can talk about how to make this discussion interesting. To end today though, perhaps you noticed that the geometry class did NOT start out with postulates? :-) 

--RJ