4. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

The first part of this standard is refreshingly straightforward. Circles are all the rage these days, so we want students to be able to whip out the area and circumference formulas without even hesitating.

As long as they've got the radius (or diameter), students should be able to figure out the area or circumference of any circle, no questions asked. This is just your basic plug-and-chug math, as long as they know which quantity to plug in for r.

But this standard's about more than just regurgitating the area and circumference formulas, which actually sounds kind of disgusting now that we think about it. We also want students to understand how the formulas relate to each other, and be able to cook 'em up from scratch like a homemade soufflé. And speaking of delicious metaphors, the key to this skill is making sure students know their way around a crispy, flaky slice of π.

Your students have probably come across π before. Emphasize the fact that it's really just the ratio of a circle's circumference to its diameter: . That means we can rearrange this ratio to find the circumference in terms of the diameter: C = πd. And since the diameter is just double the radius, we can re-rewrite it as C = 2πr.

This also means the circumference and area of a circle have a weird but interesting relationship, like Tim Burton and Helena Bonham Carter. Students should be able to work through the derivation that since a circle's area is A = πr², a little more rearrangin' gives us , or ;. Plug that bad boy back into the circumference formula, and we've got <img src=" />. Walk your students through the steps to whittle this down to . Boom! Now they'll be able to handle any circle problem that comes their way, whether they're given the radius, area, circumference, or diameter to start with.

Knowing the nuts and bolts of how these formulas work will help your students out big time when they get into more complicated problems down the road involving sectors, spheres, and irregular geometric shapes.

It's like the difference between knowing a person's name and really knowing them as a friend. Anyone can memorize a formula; we want students to know a circle's personality, inside and out, like they do with their besties.