Competition · AMC preparation · step 4 of 4
AMC 8 · 2007 · #16
Grade 8 geometry-2dPick an answer.
AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #5 (Look for a Pattern) is the right fit because the question is really asking how the area pattern behaves as the circumference grows steadily. Circumference grows by the same amount each time (+2π per unit of radius), but area grows by the differences of consecutive squares — 3, 5, 7, 9 — so the gaps between y-values must keep getting larger. Tool #2 (Make an Organized List) lets us tabulate the actual C and A values side by side so the growth pattern is visible. Once we see "equal x-steps, growing y-steps," only one choice can fit.
List circumference and area
Tabulate (C, A) for each radius using C = 2π r and A = π r².
The Grade 7 circle formulas plug in directly: C is linear in r, A is the square of r (times π).
7.G.B.4Make A Systematic ListCheck the horizontal spacing
The C-values step up by 2π each time, so the five x-positions are evenly spaced.
Doubling, tripling, ... the radius doubles, triples, ... the circumference. Steady step.
6.RP.A.1Look For A PatternCheck the vertical gaps
The A-values π, 4π, 9π, 16π, 25π rise by 3π, 5π, 7π, 9π — the gaps keep growing.
Consecutive square differences are the odd numbers 3, 5, 7, 9 — a classic growing pattern.
The area values climb by larger and larger amounts — the jumps are 3π, 5π, 7π, 9π — so the dots must rise faster and faster instead of by one fixed step.
▸ Why?
Each area is π times a square number (1, 4, 9, 16, 25), and the jump from one square to the next is (r+1)² - r² = 2r+1.
▸ Why?
Multiplying out (r+1)² gives r² + 2r + 1, so taking away r² leaves exactly 2r + 1.
▸ Why?
That jump 2r + 1 gets bigger whenever r gets bigger, so each next jump 3π, 5π, 7π, 9π is larger than the one before.
▸ Why?
2r is two equal groups of r, so a bigger r makes a bigger 2r, and adding one keeps it bigger.
Match the pattern to a graph
Only (A) fits: equal x-steps with y-gaps 2, 3, 4, 5 that keep growing — the quadratic shape.
A linear graph has equal step-ups; a quadratic graph has step-ups that keep growing. (A) shows the quadratic shape.
8.F.A.3Look For A PatternCircumference C = 2π r grows steadily, but area A = π r² grows by the odd numbers 3π, 5π, 7π, 9π — so the graph must rise faster and faster. That concave-up shape is choice (A).
- List circumference and area
- Check the horizontal spacing
- Check the vertical gaps
- Match the pattern to a graph
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