Linear Versus Area Scaling in a Garden Design
Ms. Patel’s students are redesigning a rectangular garden bed that is 6 m by 4 m. The new design increases each dimension by 50%, making it 9 m by 6 m. One student says the area increases by 50%, from 24 m² to 36 m². Another correctly notes the perimeter increases from 20 m to 30 m, then assumes the area must also increase by 50%. A third student draws the new rectangle on grid paper and counts 54 square units but cannot connect the count to the scale factor. The class has 24 students, and Ms. Patel wants to use this task to distinguish linear and area scaling before introducing surface area of similar solids. She has rulers, grid paper, and 20 minutes. Several students are still confusing ‘50% more’ with ‘50% larger area.’
1. Diagnose the reasoning behind the 36 m² answer and explain, with a representation, why 54 m² is correct. Include the scale factor and area factor.
Read the worked answer
The 36 m² answer comes from adding 50% of the original area once: 24 + 12 = 36. That would be correct only if area changed by the same additive amount as one dimension. But both dimensions are multiplied by 1.5, so the new area is 9 × 6 = 54 m². The linear scale factor is 1.5, and area scales by the square of the linear factor: 1.5² = 2.25. Indeed, 24 × 2.25 = 54. A grid representation makes this visible: the original 6-by-4 rectangle contains 24 unit squares; the new 9-by-6 rectangle contains 54. Equivalently, each row of 6 squares becomes a row of 9 squares, and the number of rows grows from 4 to 6. The perimeter, by contrast, is a linear measure and does scale by 1.5, which explains why the second student’s perimeter result is correct while the area conclusion is not.
2. Design a short sequence of questions using the grid paper that would help the third student connect the count to the scale factor. What would you listen for?
Read the worked answer
Sequence: First, draw the 6-by-4 rectangle on grid paper and count 24 squares. Second, extend each side by half its length to make the 9-by-6 rectangle and count 54 squares. Third, ask: ‘How many original rectangles fit inside the new one?’ The student can partition the new rectangle into 1.5-by-1.5 blocks or compare rows and columns. Fourth, ask: ‘By what factor did each side grow, and by what factor did the area grow?’ Listen for the student saying each side is multiplied by 1.5, so the area is multiplied by 1.5 twice, or 2.25. Listen also for a connection between 24 × 2.25 and 54. A response that only says ‘the new one has more squares’ is not yet sufficient; the key idea is that both dimensions contribute to the area factor.
3. State a two-item exit ticket and scoring criteria that would reveal whether students can transfer the idea to a similar rectangle. Avoid a task that only asks for a formula.
Read the worked answer
Exit ticket item 1: ‘A 5 cm by 8 cm rectangle is enlarged so each side is tripled. Find the new perimeter and area, and explain how each compares with the original.’ Item 2: ‘A rectangle’s sides are multiplied by 1.5. Its area becomes 45 cm². Find the original area and explain.’ Scoring criteria: for item 1, the response gives perimeter 78 cm (3 times the original 26 cm) and area 360 cm² (9 times the original 40 cm²), with an explanation that perimeter is linear and area is two-dimensional. For item 2, the response divides by 2.25 to get 20 cm² and explains that the area factor is the square of the linear factor. A strong response uses diagrams, tables, or numerical checks rather than only quoting formulas.