Amanda Reckonwith draws five circles with radii 1,2,3,4 and 5. Then for each circle she plots the point (C,A), where C is its circumference and A is its area. Which of the following could be her graph?
Pick an answer.
(A)
(C vs A scatter) five points with x equally spaced; y-values 2, 4, 7, 11, 16 — gaps grow 2, 3, 4, 5 (concave-up, quadratic-like)
(B)
(C vs A scatter) five points with x equally spaced; y-values 9, 6, 6, 9, 15 — dips then rises (non-monotonic)
(C)
(C vs A scatter) five points with x equally spaced; y-values 2, 6, 8, 6, 2 — rises then falls (inverted-U)
(D)
(C vs A scatter) five points with x equally spaced; y-values 2, 5, 8, 11, 14 — gaps all 3 (linear)
(E)
(C vs A scatter) five points with x equally spaced; y-values 15, 10, 6, 3, 1 — strictly decreasing
Try it yourself first — the explanation is most useful after you’ve attempted it.
View mode:
Toolkit + CCSS Solution
Understand
Restated: For each radius $r = 1, 2, 3, 4, 5$, Amanda plots the point $(C, A)$, where $C = 2\pi r$ is the circle's circumference and $A = \pi r^2$ is its area. Which of the five scatter plots could be her graph?
Givens: Five radii: $r = 1, 2, 3, 4, 5$; $x$-coordinate of each point is $C = 2\pi r$; $y$-coordinate of each point is $A = \pi r^2$; Answer choices: (A) y-values $2, 4, 7, 11, 16$ — gaps grow; (B) y-values $9, 6, 6, 9, 15$ — dips then rises; (C) y-values $2, 6, 8, 6, 2$ — inverted-U; (D) y-values $2, 5, 8, 11, 14$ — linear; (E) y-values $15, 10, 6, 3, 1$ — strictly decreasing
Unknowns: Which scatter plot matches the actual $(C, A)$ values for the five circles
Understand
Restated: For each radius $r = 1, 2, 3, 4, 5$, Amanda plots the point $(C, A)$, where $C = 2\pi r$ is the circle's circumference and $A = \pi r^2$ is its area. Which of the five scatter plots could be her graph?
Givens: Five radii: $r = 1, 2, 3, 4, 5$; $x$-coordinate of each point is $C = 2\pi r$; $y$-coordinate of each point is $A = \pi r^2$; Answer choices: (A) y-values $2, 4, 7, 11, 16$ — gaps grow; (B) y-values $9, 6, 6, 9, 15$ — dips then rises; (C) y-values $2, 6, 8, 6, 2$ — inverted-U; (D) y-values $2, 5, 8, 11, 14$ — linear; (E) y-values $15, 10, 6, 3, 1$ — strictly decreasing
Plan
Primary tool: #5 Look for a Pattern
Secondary: #2 Make an Organized List
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 ($+2\pi$ 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.
Execute — Answer: A
#2 Make an Organized List 7.G.B.4Step 1
List $C$ and $A$ for each radius using $C = 2\pi r$ and $A = \pi r^2$.
Keep $\pi$ as a common factor so the pattern is easy to read.
💡 A linear graph has equal step-ups; a quadratic graph has step-ups that keep growing. (A) shows the quadratic shape.
[1]
#2 7.G.B.4List $C$ and $A$ for each radius using $C = 2\pi r$ and $A = \pi r^2$. Keep $\pi
[2]
#5 6.RP.A.1Check the $x$-coordinates. The $C$-values $2\pi, 4\pi, 6\pi, 8\pi, 10\pi$ are eq
[3]
#5 6.EE.A.1Check the $y$-coordinates. The $A$-values $\pi, 4\pi, 9\pi, 16\pi, 25\pi$ are th
[4]
#5 8.F.A.3Match the pattern "equal $x$-spacing, growing $y$-gaps" to the choices. (B), (C)
Review
Reasonableness: Sanity check with $\pi \approx 3.14$. Exact $(C, A)$ values are approximately $(6.3, 3.1),\ (12.6, 12.6),\ (18.8, 28.3),\ (25.1, 50.3),\ (31.4, 78.5)$. The $A$-values $3.1, 12.6, 28.3, 50.3, 78.5$ accelerate upward — gaps $9.5, 15.7, 22.0, 28.3$. That is the concave-up curve in choice (A), not the straight line in (D). Also, since $A = \dfrac{C^2}{4\pi}$, plotting $A$ against $C$ traces a parabola opening upward — and (A) is the only choice whose five dots lie on such a curve.
Alternative: Tool #10 (Find a Related Problem): substitute $r = \dfrac{C}{2\pi}$ into $A = \pi r^2$. That gives $A = \pi \cdot \dfrac{C^2}{4\pi^2} = \dfrac{C^2}{4\pi}$, the equation of a parabola in $C$. A parabola through the origin opening upward is concave up and strictly increasing on the right side, instantly matching choice (A) without computing each point.
CCSS standards used (min grade 8)
7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve problems (Computing $C = 2\pi r$ and $A = \pi r^2$ for each radius $r = 1, 2, 3, 4, 5$.)
6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship (Noticing that $C$ grows in equal $2\pi$ steps, so the $x$-coordinates are evenly spaced.)
6.EE.A.1 Write and evaluate numerical expressions involving whole-number exponents (Evaluating the squares $1^2, 2^2, 3^2, 4^2, 5^2$ to get the $A$-values $\pi, 4\pi, 9\pi, 16\pi, 25\pi$.)
8.F.A.3 Interpret the equation $y = mx + b$ as defining a linear function; give examples of functions that are not linear (Distinguishing the quadratic growth pattern of $A$ (gaps $3\pi, 5\pi, 7\pi, 9\pi$) from a linear graph with constant gaps, so the correct choice (A) is the non-linear one.)
⭐ Circumference $C = 2\pi r$ grows steadily, but area $A = \pi r^2$ grows by the odd numbers $3\pi, 5\pi, 7\pi, 9\pi$ — so the graph must rise faster and faster. That concave-up shape is choice (A).
⭐ Circumference $C = 2\pi r$ grows steadily, but area $A = \pi r^2$ grows by the odd numbers $3\pi, 5\pi, 7\pi, 9\pi$ — so the graph must rise faster and faster. That concave-up shape is choice (A).
