Competition · AMC preparation · step 4 of 4
AMC 8 · 2019 · #2
Grade 4 geometry-2d
Pick an answer.
AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The figure is the whole story, so Tool #1 (Draw a Diagram) is the natural entry: label every short side s and every long side l on the picture, and the constraint that the two columns must reach the same height of ABCD pops out as the equation 2s = l. From there Tool #7 (Identify Subproblems) splits the work into three clean pieces — (1) deduce l from s, (2) compute the width and height of ABCD, (3) multiply for the area — instead of trying to attack the area in one move. Reaching for algebra (Tool #13) would be overkill: a single labeled drawing makes the relationship visible.
Link the short and long sides
The left column is two short sides (s + s = 2s) tall; the right column is one long side (l) tall. Same height, so 2s = l.
Recognizing that the same height of ABCD has to equal both expressions is a Grade 3 'shapes share attributes' move — same length, two names.
Each small rectangle's long side is exactly twice its short side.
▸ Why?
The big rectangle's height can be read two ways — up the left column it is two short sides stacked, and up the right column it is one long side — and since both readings measure that same height, the two short sides together must equal the one long side.
▸ Why?
In the left column the two small rectangles sit one directly on top of the other with no gap and no overlap, so the column's height is simply their two short sides added together.
▸ Why?
The left column and the right column are the two sides of the very same big rectangle, so each one's height equals the big rectangle's height, which forces the left reading and the right reading to equal each other.
Find the long side
Substitute s = 5 into 2s = l, so each small rectangle's long side is l = 10 ft.
Doubling a one-digit number is a Grade 3 multiplication word-problem step.
3.OA.A.3Identify SubproblemsFind the big rectangle's sides
ABCD is l = 10 ft tall; its width is a long side plus a short side, 10 + 5 = 15 ft. So ABCD measures 10 by 15.
Adding the widths of the two side-by-side pieces to get the whole width is the same compose-and-decompose-shapes idea from Grade 3 geometry.
3.G.A.1Identify SubproblemsMultiply for the area
Multiply ABCD's sides: area = length × width = 15 × 10 = 150 square feet → (E).
Using the rectangle area formula on a real-world measurement problem is exactly the Grade 4 standard 4.MD.A.3.
4.MD.A.3Identify SubproblemsThis AMC 8 problem only needs the Grade 4 rectangle area formula (length times width) you already know!
- Link the short and long sides
- Find the long side
- Find the big rectangle's sides
- Multiply for the area
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