Apply the sphere volume and surface area formulas, and complete a coordinate proof using variable coordinates.
A sphere's two formulas are worth memorizing on their own, since a sphere is not a "base times height" solid the way every other shape this unit has been: Volume = (4/3) * pi * r^3, and Surface Area = 4 * pi * r^2. Both depend on nothing but the radius -- no separate height or slant height to track.
A coordinate proof places a figure on the coordinate plane using variables for its coordinates (chosen strategically -- often at the origin, or along an axis, to keep the algebra simple) and then uses the distance, midpoint, or slope formula to prove a general geometric fact, true for every triangle or quadrilateral of that type, not just one specific drawing. This connects the year's two big toolkits -- classical proof-writing and coordinate geometry -- into one method: assign convenient coordinates, compute, and let the algebra finish the proof.
Mastery looks like: They correctly apply the sphere volume and surface area formulas, and can set up and complete a basic coordinate proof using the distance, midpoint, or slope formulas to prove a general geometric fact.
Common stumbles: Forgetting the 4/3 factor in the sphere volume formula, or misapplying it to surface area; choosing specific numbers instead of variables for a coordinate proof, which only proves one example rather than every case.
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