AMC 8 · 2006 · #10
Grade 5 geometry-2dProblem
Jorge's teacher asks him to plot all the ordered pairs of positive integers for which is the width and is the length of a rectangle with area 12. What should his graph look like?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: Jorge must plot every ordered pair $(w, l)$ of positive integers where $w$ is the width and $l$ is the length of a rectangle whose area is $12$. Among five candidate graphs, choose the one whose dots are exactly those pairs.
Givens: Both $w$ and $l$ are positive integers; The rectangle's area is $w \times l = 12$; The horizontal axis is $w$, the vertical axis is $l$; Five candidate graphs (A)-(E) each plot six dots
Unknowns: Which of the five graphs shows exactly the integer pairs $(w, l)$ with $wl = 12$
Understand
Restated: Jorge must plot every ordered pair $(w, l)$ of positive integers where $w$ is the width and $l$ is the length of a rectangle whose area is $12$. Among five candidate graphs, choose the one whose dots are exactly those pairs.
Givens: Both $w$ and $l$ are positive integers; The rectangle's area is $w \times l = 12$; The horizontal axis is $w$, the vertical axis is $l$; Five candidate graphs (A)-(E) each plot six dots
Plan
Primary tool: #2 Make a Systematic List
Secondary: #1 Draw a Diagram
The whole problem reduces to one question: which positive-integer pairs $(w, l)$ have $wl = 12$? Tool #2 (Systematic List) handles this cleanly — sweep $w = 1, 2, 3, \dots$ and keep only the values where $l = 12/w$ is also a positive integer. Once the list is complete, Tool #1 (Draw a Diagram) translates each pair into a dot on the $w$-$l$ plane and we match it to one of the five graphs. There is no need for algebra or special tricks; listing the factor pairs of $12$ is the whole job.
Execute — Answer: A
4.MD.A.3 Step 1 - Translate the picture into one equation.
- A rectangle with width $w$ and length $l$ has area $w \times l$.
- The problem fixes that area at $12$, so we need positive integers $w$ and $l$ with $wl = 12$.
💡 Grade 4 students already know the rectangle area formula $\text{area} = \text{width} \times \text{length}$. The problem is just asking which whole-number widths and lengths give area $12$.
4.OA.B.4 Step 2 - List every factor pair of $12$ in order.
- Try $w = 1, 2, 3, 4, 5, 6, \dots$ and keep only the cases where $l = 12 / w$ is a positive integer.
- Skip non-divisors like $w = 5, 7, 8, 9, 10, 11$.
💡 Finding all factor pairs of a whole number is exactly the Grade 4 factor-pair skill. There are $6$ pairs because $12$ has $6$ divisors.
5.G.A.2 Step 3 - Plot the six pairs on the $w$-$l$ plane.
- Each pair $(w, l)$ becomes a dot with horizontal coordinate $w$ and vertical coordinate $l$.
- The dots fall on a curve that sweeps down and to the right.
💡 Translating ordered pairs into points on the coordinate plane is the Grade 5 graphing-points standard, exactly what Jorge's teacher is testing.
5.G.A.2 Step 4 - Compare to the five candidates.
- Graph (A) plots the six dots $(1,12), (2,6), (3,4), (4,3), (6,2), (12,1)$ — a perfect match.
- Graphs (B), (C), (D), (E) plot points with $w = l$, with $w + l = 12$, with $l = 6$, and with $w = 6$ respectively, so none of those satisfy $wl = 12$.
💡 Once the list of dots is written out, reading them off each picture is just pattern-matching.
4.MD.A.3 Translate the picture into one equation. A rectangle with width $w$ and length $ 4.OA.B.4 List every factor pair of $12$ in order. Try $w = 1, 2, 3, 4, 5, 6, \dots$ and k 5.G.A.2 Plot the six pairs on the $w$-$l$ plane. Each pair $(w, l)$ becomes a dot with h 5.G.A.2 Compare to the five candidates. Graph (A) plots the six dots $(1,12), (2,6), (3, Review
Reasonableness: Quickly verify each dot in (A) by multiplying: $1 \times 12 = 12$, $2 \times 6 = 12$, $3 \times 4 = 12$, $4 \times 3 = 12$, $6 \times 2 = 12$, $12 \times 1 = 12$. All six products are $12$, so every dot is a valid $(w, l)$. The other graphs fail an immediate spot check: graph (B) includes $(3, 3)$ with product $9 \ne 12$; graph (C) includes $(1, 11)$ with product $11 \ne 12$; graph (D) includes $(1, 6)$ with product $6 \ne 12$; graph (E) includes $(6, 1)$ — that one works, but the rest of (E)'s dots like $(6, 3)$ have product $18$. Only (A) survives.
Alternative: Tool #1 (Draw a Diagram) alone: think of $12$ unit squares and rearrange them into every possible rectangle with whole-number sides. You can form a $1 \times 12$, $2 \times 6$, $3 \times 4$, $4 \times 3$, $6 \times 2$, and $12 \times 1$ rectangle — six shapes. Each shape contributes one dot, so the graph has exactly six dots matching graph (A).
CCSS standards used (min grade 5)
4.MD.A.3Apply area and perimeter formulas for rectangles in real-world problems (Reading the rectangle picture as the equation $w \times l = 12$, the Grade 4 area formula applied with a fixed area.)4.OA.B.4Find all factor pairs and recognize multiples; determine prime or composite (Listing every positive-integer factor pair of $12$ — exactly the Grade 4 factor-pair standard — to get the six ordered pairs.)5.G.A.2Represent real-world and mathematical problems by graphing points (Plotting each ordered pair $(w, l)$ as a dot on the coordinate plane and matching the set of dots to the correct candidate graph.)
⭐ Whenever a problem says "plot every $(w, l)$ with width times length $= N$," it is really asking for the factor pairs of $N$. Sweep $w = 1, 2, 3, \dots$, keep the divisors, and you have all the dots.
⭐ Whenever a problem says "plot every $(w, l)$ with width times length $= N$," it is really asking for the factor pairs of $N$. Sweep $w = 1, 2, 3, \dots$, keep the divisors, and you have all the dots.
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