AMC 10 · 2010 · #2
Grade 6 geometry-2dFour identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?
Pick an answer.
AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Four identical small squares and one rectangle are fitted together, with no gaps or overlaps, to build one large square. Using the picture, find how many times as large the rectangle's length is compared with its width.
Givens: There are four identical (same-size) squares.; There is exactly one rectangle.; All five pieces fit together to form one large square.; From the figure, the four squares sit side by side in a single row across the top, spanning the full width of the large square.
Unknowns: The value of (rectangle's length) divided by (rectangle's width).
Understand
Restated: Four identical small squares and one rectangle are fitted together, with no gaps or overlaps, to build one large square. Using the picture, find how many times as large the rectangle's length is compared with its width.
Givens: There are four identical (same-size) squares.; There is exactly one rectangle.; All five pieces fit together to form one large square.; From the figure, the four squares sit side by side in a single row across the top, spanning the full width of the large square.
Plan
Primary tool: #4 Introduce a Variable
Secondary: #1 Draw a Diagram, #7 Identify Subproblems
The figure has no numbers, so give the small square a side length $s$. Once $s$ is named, the large square's side and the rectangle's two sides all become simple multiples of $s$, and the requested comparison turns into a ratio of those multiples.
Execute — Answer: B
6.EE.B.6 Step 1 Name the small square's side
- Let $s$ stand for the side length of one small square.
- Since the four squares are identical, they all have side $s$.
💡 With no numbers given, a single letter lets every length be measured against one common piece.
1.G.A.2 Step 2 Read the large square from the four squares
- In the picture the four squares lie in one row along the top edge, and together they reach all the way across the large square.
- Four sides of length $s$ in a line make a length of $4s$, so the large square has side $4s$.
💡 Four equal squares placed in a straight line stretch to four times one square's side.
6.EE.B.6 Step 3 Find the rectangle's length and width
- The rectangle fills everything below that top row.
- It runs the full width of the large square, so its length is $4s$.
- Its height is the large square's side minus the one-square-tall strip: $4s - s = 3s$, which is the rectangle's width.
💡 The rectangle gets what is left of the big square after the top strip of squares is removed.
6.RP.A.1 Step 4 Compare length to width
- "How many times as large" means divide the length by the width: $\dfrac{4s}{3s}=\dfrac{4}{3}$.
- The $s$ cancels, so the rectangle's length is $\dfrac{4}{3}$ times its width.
- That is choice (B).
💡 Comparing two lengths as a ratio erases the common unit and leaves just their proportion.
6.EE.B.6 Let $s$ stand for the side length of one small square. Since the four squares ar 1.G.A.2 In the picture the four squares lie in one row along the top edge, and together 6.EE.B.6 The rectangle fills everything below that top row. It runs the full width of the 6.RP.A.1 "How many times as large" means divide the length by the width: $\dfrac{4s}{3s}= Review
Reasonableness: In the picture the rectangle is a little wider than it is tall, so its length should be a bit more than its width, not double or triple it. The value $\dfrac{4}{3}\approx1.33$ matches that gentle stretch, while choices like $2$ or $3$ would make the rectangle look far shorter than it does. So (B) is sensible.
Alternative: Use concrete numbers. Take $s=1$, so each small square is $1\times1$ and the large square is $4\times4$. The top row holds the four unit squares, leaving a rectangle that is $4$ wide and $3$ tall. Then $4\div3=\dfrac{4}{3}$, confirming (B) without any letters.
CCSS standards used (min grade 6)
6.EE.B.6Use variables to represent numbers and write expressions to solve problems (Letting $s$ be the small square's side and writing the large square's side ($4s$) and the rectangle's sides ($4s$ and $3s$) as expressions in $s$.)1.G.A.2Compose two-dimensional shapes or three-dimensional shapes (Seeing that the four squares in a row compose the full top edge, so the large square's side equals four small-square sides.)6.RP.A.1Understand the concept of a ratio and use ratio language (Interpreting "how many times as large" as the ratio of length to width and simplifying $4s:3s$ to $4/3$.)
⭐ Give the little square a name like $s$, and every other length in the picture becomes an easy multiple of it.
⭐ Give the little square a name like $s$, and every other length in the picture becomes an easy multiple of it.
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