Area Scale Factor in a School Garden Model
A Grade 8 class is building a scale model of a rectangular school garden. The drawing measures 8 cm by 5 cm, and the scale is 1:50. Students must report the actual area in square meters and the length of a diagonal walkway. A student submits: actual area = 8 × 5 × 50 = 2000 cm² = 0.2 m²; diagonal = √(8² + 5²) × 50 ≈ 471.7 cm. The teacher notices that 15 of 24 students made the area error, but most got the diagonal correct. The class has rulers and calculators. The teacher wants a 20-minute intervention that uses similarity and the area scale factor, not just a formula reminder. The student's diagonal method is actually correct; the area method is not.
1. Compute the correct actual dimensions, area, and diagonal length. Identify the student's error in the area calculation.
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Scale factor k = 50. Actual dimensions: 8 × 50 = 400 cm = 4 m and 5 × 50 = 250 cm = 2.5 m. Actual area = 4 × 2.5 = 10 m². The student multiplied the drawing area 8 × 5 = 40 cm² by 50 once, getting 2000 cm², but area scales by k² = 2500: 40 × 2500 = 100,000 cm² = 10 m². The diagonal is a length, so it scales by k: drawing diagonal = √(8² + 5²) = √89 ≈ 9.434 cm; actual ≈ 9.434 × 50 ≈ 471.7 cm ≈ 4.72 m. The diagonal method is correct because it uses a linear scale factor; the area error comes from using a linear factor for a two-dimensional quantity.
2. Explain, using similarity, why linear dimensions scale by k but areas scale by k². Include a representation or proof sketch suitable for Grade 8.
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Similarity means corresponding lengths have a constant ratio k. A rectangle's area is the product of two perpendicular side lengths. If each side is multiplied by k, the area becomes (kL)(kW) = k²(LW). A Grade 8 representation: a 1 cm by 1 cm square on the drawing becomes 50 cm by 50 cm in reality, which contains 50 × 50 = 2500 of the original squares, so its area is 2500 times as large. More generally, subdivide the drawing into unit squares; each unit square maps to a k-by-k square of area k². Thus linear measures such as perimeter and diagonal scale by k, while area scales by k².
3. Design a quick check that would reveal whether students are applying the area scale factor or merely copying a formula.
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Quick check: 'A map has scale 1:200. A rectangular park measures 3 cm by 4 cm on the map. Find its actual area in square meters. Then explain what happens to the area of any similar figure if all its lengths are tripled.' Correct response: actual sides 3 × 200 = 600 cm = 6 m and 4 × 200 = 800 cm = 8 m; area = 48 m²; tripling lengths multiplies area by 9. A student using the linear factor would compute 12 × 200 = 2400 cm² = 0.24 m², or would say area triples. Evidence of understanding is using k² and explaining that two dimensions each scale by k. A student who only writes 48 m² without explanation may have copied a formula, so ask for the side lengths or the square-subdivision argument.