Area Ratio vs. Linear Scale Factor in Similar Figures
Ms. Alvarado is grading a similarity quiz in a high school geometry class. Item 4 presents two similar trapezoids: the smaller has bases 6 cm and 10 cm, and the larger has bases 15 cm and 25 cm. Students must state the ratio of the areas. Devon writes, "The scale factor is 15 ÷ 6 = 2.5, so the ratio of the areas is 2.5 : 1." On Item 5, a blueprint problem, a rectangular patio measures 12 ft by 18 ft and the drawing uses the scale 1 in = 3 ft. Devon draws the patio as 4 in by 6 in, which is correct, but then writes that doubling the patio's dimensions doubles its area. Ms. Alvarado wants to write feedback that targets the underlying concept rather than simply marking both items wrong, and she has one class period before the unit test to reteach.
1. Identify the single conceptual error that explains both of Devon's responses, and explain why the error is consistent across the two items.
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Both errors come from treating area as if it scales the same way length does. In similar figures, lengths scale by a factor $k$, but areas scale by $k^2$ because area is a two-dimensional measure: each of the two dimensions is multiplied by $k$.
On Item 4 the linear scale factor is $15/6 = 2.5$, so the area ratio should be $2.5^2 = 6.25$, i.e. $6.25:1$ (equivalently $25:4$). Devon applied the linear factor directly to area.
On Item 5 he made the same move in the opposite direction: doubling both dimensions is $k = 2$, so the area grows by $2^2 = 4$, not 2. A 12 ft by 18 ft patio is 216 ft²; a 24 ft by 36 ft patio is 864 ft², exactly four times as large. The two items are the same misconception applied forward and backward, so one reteach can address both.
2. Write the feedback Ms. Alvarado should give Devon, including a concrete counterexample or decomposition argument that makes the squaring visible rather than asserted.
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Feedback should name the rule, show why it is true, and give Devon something to do.
"Devon, your scale factors are right, but area is measured in square units, so it responds to both dimensions at once. Take a 1 cm by 1 cm square and double each side: you get a 2 cm by 2 cm square, which holds four of the original squares, not two. Area ratio = (linear scale factor)². For the trapezoids, $2.5^2 = 6.25$, so the larger area is 6.25 times the smaller. You can check with the bases: $15/6 = 2.5$ and $25/10 = 2.5$, but the heights also grow by 2.5, and area uses base × height. For the patio, doubling gives $24 \times 36 = 864$ ft² versus $12 \times 18 = 216$ ft² - a factor of 4. Redo Item 4 and then write the area ratio for a scale factor of 3 without computing any areas.""
3. Devon now claims he understands. Describe one brief follow-up item that would reveal whether he can transfer the idea, and state what a correct response and an incorrect response would each look like.
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A good transfer item changes the representation and asks for justification, not just a number.
Item: "Two similar rectangles have areas 50 cm² and 200 cm². What is the ratio of their perimeters? Explain how you know."
Correct response: the area ratio is $200/50 = 4$, so the linear scale factor is $\sqrt{4} = 2$, and perimeters are lengths, so the perimeter ratio is $2:1$. The reasoning must go through the square root.
Incorrect response characteristic of the original misconception: "area ratio is 4, so perimeter ratio is 4" - applying the area factor directly to a length, which is the same error in a new direction. A second useful probe is to give the perimeter ratio and ask for the area ratio. If Devon handles both directions, he has the relationship, not just a memorized phrase.