One criticism I have is, why not include circumference too? That way students can see that one pattern is linear, while the other is not.

Here's one way I thought about the connection between diameter and circumference:

The diameter is 9 pennies; the circumference is 25 pennies. The key to the pattern is looking at the circle of diameter

**pennies.**

*8*So the circumference turns out to be pi*(d-1) pennies, which is about 25 pennies. Seeing the circumference in this way actually gives students a real insight into the area:

Area = previous area + circumference. (This is why area is quadratic: the derivative -- the differences, i.e., circumference function -- is a linear function.)

One more extension question that I love: Imagine a ring of pennies around a tennis ball. If you want the pennies to be one penny farther away from the tennis ball, how many pennies do you have to add to the ring? Imagine a ring of pennies around the moon. If you want the pennies to be one penny farther away from the moon, how many pennies do you have to add to the ring? Because circumference is linear, the answer is counter-intuitively the same for both!

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