Competition · AMC preparation · step 4 of 4
AMC 8 · 2011 · #19
Grade 4 counting
Pick an answer.
AMC 8 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Counting rectangles in a tangled picture is risky if done by eye, so Tool #2 (Make a Systematic List) is the safest plan: sort the rectangles by size or by which big rectangle they live inside, and tally each group. Tool #1 (Draw a Diagram) helps to label the three original rectangles S (central square), H (horizontal), V (vertical) and mark where their edges cross. Tool #7 (Identify Subproblems) splits the count into clean subproblems: (a) the three big rectangles, (b) extra rectangles made when H crosses the square, (c) extra rectangles made when V crosses the square, (d) the small rectangle where H and V overlap each other.
Count the three big rectangles
The three original rectangles — S (square), H (horizontal), V (vertical) — are themselves rectangles, giving 3 right away.
Recognizing axis-aligned rectangles by their four right-angle corners is a Grade 3 geometry skill.
3.G.A.1Draw A DiagramCount rectangles inside the square
Inside S the two cuts make a 2×2 grid: 4 small quarter-pieces plus 4 half-square rectangles give 8 new rectangles.
Splitting a square into equal-area rectangles and counting the combined pieces is the Grade 3 "partition shapes into parts with equal areas" idea.
3.G.A.2Make A Systematic ListRecount with combinations
The grid formula C(3,2)×C(3,2) = 9 confirms the count; subtracting the whole square S, already counted, leaves 8 new.
Listing the 9 rectangles in a 2 × 2 grid is a classic systematic-list exercise and matches the picture exactly.
Inside the square, the horizontal cut and the vertical cut make a 2 × 2 grid of regions that holds exactly 9 rectangles in all.
▸ Why?
Every rectangle in the grid is pinned down by choosing which two of the three vertical lines are its left and right sides and which two of the three horizontal lines are its top and bottom, and there are 3 ways to pick each pair, so 3 × 3=9.
▸ Why?
Each rectangle matches exactly one pair of vertical lines together with one pair of horizontal lines, and each such pairing traces back to exactly one rectangle, so counting the rectangles is the same as counting these pairings.
▸ Why?
There are 3 ways to choose the two vertical lines (list them: left-and-middle, left-and-right, middle-and-right), and for each of those the same 3 choices of two horizontal lines are open, making 3 equal groups of 3.
Check outside the square
The parts of H and V that stick outside S add no new rectangle: their outer edges are just H's and V's own sides, already counted.
Breaking the figure into "inside the square" and "outside the square" regions keeps the counting honest.
3.G.A.1Identify SubproblemsTotal every rectangle
Add it up: 3 big rectangles + 8 new ones inside the square = 11, matching choice (D).
Adding subtotals from a systematic list to get a final count is a Grade 4 multi-step word-problem skill.
4.OA.A.3Make A Systematic ListThis AMC 8 problem only needs Grade 4 careful counting — split the picture into clean pieces and add them up — that you already know!
- Count the three big rectangles
- Count rectangles inside the square
- Recount with combinations
- Check outside the square
- Total every rectangle
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