Competition · AMC preparation · step 4 of 4
AMC 8 · 2015 · #1
Grade 5 geometry-2dPick an answer.
AMC 8 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The trap of this problem is units: the dimensions come in feet but the answer must be in square yards. Tool #8 (Analyze the Units) keeps the bookkeeping honest — convert each side from feet to yards first (12 ft = 4 yd, 9 ft = 3 yd), then multiply to get area in square yards. Tool #1 (Draw a Diagram) helps younger solvers see why the answer is 12 and not 108: sketching the 4-yd by 3-yd rectangle and tiling it with 1-yd squares shows exactly 12 tiles, making the unit conversion concrete instead of abstract.
Convert feet to yards
Divide each side by 3: 12 ft becomes 4 yd and 9 ft becomes 3 yd.
Dividing feet by (feet per yard) cancels the "feet" unit and leaves "yards" — a Grade 5 measurement conversion.
5.MD.A.1Analyze The UnitsTile the floor with unit squares
Tile the 4 yd × 3 yd floor with 1-yd squares — 4 columns by 3 rows make 12 squares.
Tiling a rectangle with unit squares and counting them is the Grade 3 visual definition of area.
3.MD.C.7Draw A DiagramMultiply the two yard lengths
Both sides are in yards, so area = 4 × 3 = 12 square yards directly. → (A)
Multiplying yards by yards produces square yards — the units track the geometry exactly.
With its two sides measured in yards, the floor is 4 yards by 3 yards, so its area is 4 × 3 = 12 square yards.
▸ Why?
The area of a rectangle is just the count of 1 yard by 1 yard squares that tile it with no gaps, and a 4 yard by 3 yard floor lays out as 3 rows of 4 such squares.
▸ Why?
Three rows that each hold four unit squares is three equal groups of four, and three groups of four is 3 × 4 = 12.
▸ Why?
The sides come out as 4 yards and 3 yards because the given feet lengths are the same lengths renamed in yards, and only yard-sides make the product land in square yards.
▸ Why?
One yard is a fixed 3 feet, so 12 feet regroups into 4 yards and 9 feet regroups into 3 yards.
This AMC 8 problem only needs Grade 5 unit conversion plus the Grade 4 rectangle area formula you already know!
- Convert feet to yards
- Tile the floor with unit squares
- Multiply the two yard lengths
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