Proportional or Additive? Scale Drawings in Seventh Grade
Ms. Ibrahim teaches seventh-grade mathematics. Her class of 25 students has been working on proportional reasoning and scale drawings. On a quiz, 20 students set up a correct proportion for a map-scale problem, but 9 of those gave an answer in the wrong units or inverted the ratio. Six students used cross-multiplication on a non-proportional problem, adding 3 inches to both the length and width of a 4-by-6-inch rectangle and claiming the new rectangle was 'the same shape, just bigger.' Two students who recently transferred in can solve proportions fluently but could not explain why scaling both dimensions by the same factor preserves shape. The district pacing guide says to move to percent next week. Ms. Ibrahim has one day she can spend before the unit test and is deciding whether to use it for review or push forward.
1. Interpret the error pattern. What does it reveal about students' understanding of proportional relationships, and why is correct proportion setup not sufficient evidence of understanding?
Read the worked answer
The class has split into procedural and conceptual difficulty. Twenty of 25 can set up a proportion, which shows the algorithm is available. But nine of those mishandled units or inverted the ratio, which means they are not tracking what is being compared to what; they are manipulating numbers without a referent. The use of cross-multiplication on the rectangle problem is more serious: those six students apply a proportional procedure to a relationship that is not proportional. Adding 3 inches to both dimensions increases the perimeter by a constant but changes the ratio of length to width, so the shape is not preserved. The two transfer students show the same gap in a different form: fluent execution with no explanation of why scaling works.
Correct setup is not sufficient because a proportion can be assembled by matching numbers in the order they appear. The additive error appears when students have no way to test whether the relationship is multiplicative. They need experience comparing equal ratios with equal differences and seeing that only one preserves shape.
2. Describe a task using the 4-by-6 rectangle that would confront the additive error, and explain what evidence would show that students have moved beyond procedure.
Read the worked answer
Give students grid paper and a 4-by-6 rectangle. Ask them to produce two new rectangles: one by adding 3 inches to both dimensions (7 by 9), and one by multiplying both dimensions by 2 (8 by 12). Then have them compare the ratios of length to width, measure the angles, and check whether the rectangles are similar. The 7-by-9 rectangle has a ratio of 7:9, not 4:6, and will not align with the original when scaled; the 8-by-12 rectangle is similar. Students should record the ratio as a fraction and as a decimal and explain why equal differences do not preserve shape while equal factors do.
Evidence of movement beyond procedure would be a student who, given a new pair of rectangles, checks the ratio or scale factor rather than subtracting, and who can explain that adding the same amount to both dimensions changes the relationship because the starting sizes differ. A correct answer alone would not be enough; the explanation should refer to the multiplicative comparison. Ms. Ibrahim can also ask students to write a counterexample to the claim 'adding the same number to both sides keeps the shape,' which requires them to reason about the relationship rather than execute a rule.
3. Should Ms. Ibrahim spend the review day on this topic or move to percent as the pacing guide directs? Justify your decision using the assessment evidence and describe what she should prioritize in that day.
Read the worked answer
She should spend the day, but narrowly. The pacing guide matters, yet percent is itself a proportional relationship, and the same additive error will reappear in percent change, discounts, and scale. The quiz shows that roughly 15 of 25 students are not secure: nine mishandled units or inverted ratios, and six applied a proportional procedure to an additive situation. Moving on would build new content on an unstable base and create a larger remediation problem later.
The day should prioritize two things: first, the multiplicative-versus-additive distinction with the rectangle task and a ratio table; second, unit reasoning, where students label every quantity and write 'for every 1 unit on the drawing, there are 50 units in real life' before computing. She should not reteach the whole unit or drill more proportions, because setup is already fluent. A short exit ticket at the end, containing one scale problem and one 'same shape?' judgment with justification, will tell her whether the class is ready for percent. If most students can explain the difference, she can move on and revisit percent applications as they arise.