The Garden Area Design
A class is designing a rectangular garden with a perimeter of 24 feet using whole-number side lengths. The garden must have an area of at least 30 square feet. A student draws a 6 ft by 6 ft square and says it is the only option because squares always give the most area. Another student draws a 2 ft by 10 ft rectangle and says it has the same perimeter so it must have the same area. The teacher asks the class to list possible dimensions, compare areas, and judge both claims.
1. List whole-number length and width pairs with perimeter 24 ft, and compute each area.
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Perimeter 24 means length + width = 12. Whole-number pairs (length ≥ width): 11×1 = 11; 10×2 = 20; 9×3 = 27; 8×4 = 32; 7×5 = 35; 6×6 = 36. Areas range from 11 to 36 sq ft. Pairs with area ≥ 30 are 8×4 (32), 7×5 (35), and 6×6 (36). So there are three valid designs meeting the area goal, not one.
2. Is the claim "same perimeter means same area" correct? Use two pairs from your list to justify.
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Incorrect. The 2×10 rectangle and the 6×6 square both have perimeter 24, but areas are 20 and 36 sq ft. Perimeter measures distance around; area measures space inside, and they change independently for rectangles. The 2×10 is long and thin, wasting space; the 6×6 is compact. Same perimeter does not guarantee same area, so the second student's reasoning fails.
3. Evaluate the claim that the square is the only option giving the most area. Is the square the maximum here, and does that make it the only acceptable design?
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The 6×6 square does give the largest area (36 sq ft), and for a fixed perimeter the square maximizes rectangle area. But "most area" is not the same as "only option." The requirement is at least 30 sq ft, so 8×4 and 7×5 also qualify. A designer might prefer 7×5 for a different shape or 8×4 to fit a space. The judgement: the square is the area-maximizer, but the goal is a minimum, so multiple designs are valid.