Linear Scale Factor Versus Area Scale Factor
In an eighth-grade geometry class, students are working with a scale drawing of a rectangular classroom. The drawing is 12 cm by 8 cm, and the scale is 1:50. The actual room is 6 m by 4 m. A student calculates the drawing's area as 96 cm² and then multiplies by 50 to get 4,800 cm², which she converts to 0.48 m². She concludes that the actual room is too small for 30 desks. The teacher knows the actual area is 24 m². Another student says the answer should be 96 × 50 × 50 but cannot explain why. The class has 28 students, and the teacher wants to use this error to build understanding of how scale factors affect length, area, and volume.
1. Identify the student's error and explain the correct relationship between linear scale factor and area scale factor. Show computations.
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The student applied the linear scale factor once to an area. A scale of 1:50 means each actual length is 50 times the corresponding drawing length. Area is two-dimensional, so both dimensions are multiplied by 50; the area scale factor is 50² = 2,500.
Drawing area = 12 × 8 = 96 cm². Actual area = 96 × 2,500 = 240,000 cm². Since 1 m² = 10,000 cm², 240,000 cm² = 24 m². Checking with actual dimensions: 12 cm × 50 = 600 cm = 6 m; 8 cm × 50 = 400 cm = 4 m; 6 × 4 = 24 m².
The student's 0.48 m² is 24 ÷ 50, so the error is a factor of 50 too small. The second student's 96 × 50 × 50 is numerically correct but needs the explanation that each dimension is scaled by 50.
2. Design a contrasting-cases activity using 1:2 and 1:3 scale factors to help students generalize.
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Use a 2 cm by 3 cm rectangle on grid paper. Case 1, scale 1:2: actual dimensions are 4 cm by 6 cm; drawing area is 6 cm²; actual area is 24 cm²; the area factor is 4, which is 2². Case 2, scale 1:3: actual dimensions are 6 cm by 9 cm; drawing area is 6 cm²; actual area is 54 cm²; the area factor is 9, which is 3².
Ask: “What stays the same? What changes? How many copies of the drawing rectangle fit inside the actual rectangle?” Students can physically tile or draw grids. Then generalize: if the linear scale factor is k, the area scale factor is k². Contrast this with perimeter, which scales by k: in Case 1 the drawing perimeter is 10 cm and the actual perimeter is 20 cm; in Case 2 it is 30 cm. The contrast prevents overgeneralizing 'multiply by the scale factor' to every measurement.
3. Formulate a question that checks whether students can transfer to volume, with a model response.
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Question: “A model box is 2 cm by 3 cm by 4 cm. The scale is 1:50. Find the actual volume in cubic meters and explain your scale factor.”
Model response: Drawing volume = 2 × 3 × 4 = 24 cm³. Volume is three-dimensional, so the volume scale factor is 50³ = 125,000. Actual volume = 24 × 125,000 = 3,000,000 cm³. Since 1 m³ = 1,000,000 cm³, the actual volume is 3 m³. Checking: actual dimensions are 1 m, 1.5 m, and 2 m; 1 × 1.5 × 2 = 3 m³. Evidence of fragile understanding: The student multiplies by 50 or 2,500 instead of 125,000, or gives 3,000,000 m³ without converting units.