"The Area Doubles" - Scaling in Two Dimensions
After a lesson on area, a seventh-grade teacher gives this item: "A rectangle is 6 cm by 4 cm. Each side is doubled. What is the area of the new rectangle, and how does it compare with the original area?" All 27 students correctly computed 12 × 8 = 96 cm². But asked how the areas compare, 17 wrote "the area doubles," 6 wrote "it quadruples," and 4 wrote "it increases by 4." In discussion, one student explains, "The sides went up by 2 and 2, so the area goes up by 4," and another says, "The perimeter doubled, so the area should double too." The next unit covers circles and then volume of prisms, and the state assessment includes items on how area and volume change under dilation.
1. Name the misconception(s) at work and explain why the correct factor is 4, including why the perimeter argument fails.
Read the worked answer
Correct factor is 4: 24 cm² → 96 cm², and 96 ÷ 24 = 4 = 2². Area scales by k² under a dilation of factor k because area is a product of two lengths, each of which is multiplied by k.
Two distinct errors are present:
- "Doubles" (17 students): a valid proportional intuition applied to the wrong dimension. Perimeter is one-dimensional (a sum of lengths), so it scales by k. Students transfer the perimeter result to area without noticing area depends on two lengths.
- "Increases by 4" (4 students): an additive comparison of the scaling factor itself - using difference where a factor is required. Note they land on the numeral 4, the correct factor, but express it as an addition (24 + 4 would be 28, not 96). Right number, wrong relation.
Diagnostic point: every student computed 96 correctly, so the failure is entirely in the comparison, not in area computation. The probe should ask for a picture or a sentence, never just a factor.
2. Design a representation sequence, with specific numbers, that builds the k² idea.
Read the worked answer
- Concrete tiling (6 min). Draw the 6 × 4 rectangle on grid paper, then the 12 × 8. Students physically see four 6 × 4 blocks inside it. State: "the new rectangle is 4 copies of the old one."
- Push to k = 3. Draw 18 × 12; it holds 9 copies. Perimeter factor is 3, area factor is 9. Students who guessed "area factor = 2 × scale factor" from a single data point will now be contradicted.
- Table and conjecture. k: 1, 2, 3, 4 → perimeter factor 1, 2, 3, 4; area factor 1, 4, 9, 16. Ask "what is the pattern, and why must it be that?" The why: each of the two dimensions is multiplied by k, so the product carries k twice.
- Reverse item. "A rectangle's area is multiplied by 25. What happened to the sides?" Requires square-rooting the factor - separates rule-memorizers from reasoners.
- Non-integer check. k = 1.5 → area factor 2.25. Catches students whose rule only works for whole numbers.
3. How would you probe whether a student who says "quadruples" has genuinely generalized, and how do you extend to circles and volume?
Read the worked answer
Probe with a new factor, not a new shape. Ask the student who correctly says "quadruples" for k = 2: "What if we triple each side?" A student with the structure says 9 (or reasons "3 × 3"). A student with a memorized doubling rule says 6, which is linear thinking on the exponent - they have the answer, not the idea. Follow with k = 1.5 (expect 2.25).
Circles: radius tripled → area ×9, because A = πr² and r enters twice. Emphasize that π does not change, so only the r² matters.
Volume: edge doubled → ×8, one factor of k per dimension. Sequence 6 × 4 × 2 = 48 → 12 × 8 × 4 = 384; 384 ÷ 48 = 8. Then contrast: perimeter k, area k², volume k³ - a single table across all three so the exponent matches the dimension count.
Watch for: students who conclude "area always quadruples" regardless of k, and students who confuse the surface-area factor of a solid (k²) with its volume factor (k³). Include one item of each on the unit test.