When 11 Cups Looks Reasonable: Additive Reasoning on a Ratio Quiz
Mr. Lindqvist’s sixth graders took a five-item ratio quiz. Item 4 read: “A recipe uses 3 cups of flour for every 2 cups of milk. If a baker uses 12 cups of flour, how many cups of milk does she need?” Of 24 students, 14 answered 8 cups, 7 answered 11 cups, and 3 answered 18 cups. In brief interviews, every student who answered 11 said some version of “3 plus 9 is 12, so 2 plus 9 is 11.” The three who answered 18 multiplied 12 by 3/2. The other four items were answered correctly by all 24 students. The adopted textbook’s next lesson teaches cross-multiplying a proportion, and the district pacing guide has the percent unit scheduled to start Monday.
1. What do the 11-cup responses reveal about student reasoning, and why is re-teaching cross-multiplication unlikely to fix it?
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The 11-cup answer is not carelessness; it is a consistent additive structure. “Add 9 to both quantities” preserves the difference between flour and milk (3 − 2 = 1; 12 − 11 = 1). The student is reasoning about an invariant, just the wrong one. The correct structure preserves the ratio: both quantities scale by the same factor, here ×4, giving 12 : 8.
Cross-multiplication will not dislodge this, because it is a procedure that executes on any equation, including a wrong one. A student holding additive reasoning may set up 3/2 = x/12, or 12/3 = 2/x, or 12/8 = 3/2, and produce a confident wrong answer with no internal signal that something is off. The procedure carries no proportional meaning, so nothing inside it contradicts the additive belief.
The 18-cup group is a second, different error - they used the flour-to-milk ratio as a scalar in the wrong direction. Two distinct misconceptions call for a task that contrasts them, not one demonstration of a correct procedure.
2. A colleague says, “Just show them 3/2 = 12/x and cross-multiply - it takes five minutes.” Evaluate that plan.
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It will probably produce 8 cups and leave the reasoning intact, which is the problem. Consider what the seven students must do: decide that flour and milk correspond, decide which quantity is the numerator, and trust the setup. Those decisions are exactly where they failed. Handing them a correct equation removes the decision, so the quiz improves and the understanding does not.
Better sequence, one period:
- Contrast cases. A 3:2 recipe versus “Ana is 3 years older than Ben; when Ana is 12, how old is Ben?” Additive reasoning is correct in the second and wrong in the first. Ask which is which and why.
- Ratio table. Build 3:2 → 6:4 → 9:6 → 12:8, with students stating the scaling factor aloud: “both multiplied by 4.”
- Then connect to the equation, as a way of recording the table, not as a replacement for it.
The comparison step is what makes the multiplicative structure visible; the equation alone hides it behind symbols the students can manipulate without understanding.
3. How should Mr. Lindqvist justify departing from the pacing guide, and what quick check would show whether the reteach worked?
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Bring item-level data and the dependency argument, not just a request for time.
Justification: 10 of 24 students (42%) left the item holding either an additive structure or an inverted ratio. Percent - the next unit - is built entirely on multiplicative comparison. A student who believes 12 : 11 preserves a 3 : 2 relationship will compare and convert percentages using the same faulty invariant, and the error will be much harder to see inside percent contexts. Two days now is cheaper than reteaching percent in the spring.
The check before moving on must require justification, not just an answer:
- “A mix uses 4 cups white paint to 1 cup blue. A painter uses 12 cups white. How much blue?” (proportional)
- “Sam is 4 years older than his sister. Sam is 12. How old is she?” (additive)
- “Explain why 11 cups is or is not correct for the recipe.”
Success looks like an explanation that names scaling both quantities by the same factor - not merely the number 8.