Additive Reasoning Behind a Wrong Ratio Answer
A sixth-grade math team gives a five-item exit ticket on equivalent ratios after a week of instruction. Item 3 reads: 'A recipe uses 3 cups of flour for every 2 cups of sugar. If Maya uses 9 cups of flour, how many cups of sugar does she need?' Of 26 students, 15 answered 8 cups, showing work such as 9 − 3 = 6 and 2 + 6 = 8. Those same 15 students answered every doubling item correctly, including 3:2 to 6:4. Only 9 students built a ratio table or reasoned that the flour was multiplied by 3. On item 5, which asks whether 4:6 and 6:9 are equivalent, 11 of the 15 wrote that the ratios are not equivalent 'because the numbers are different, not the same numbers.' The unit assessment is in two days, and the team has one 50-minute period before a scheduled review day.
1. Diagnose the reasoning that produces 8 cups, and explain why the doubling items failed to reveal it. Use the item data as evidence.
Read the worked answer
The 8-cup answer is not carelessness; it is a consistent additive rule. The student compares the two flour amounts (3 to 9, a change of +6) and applies the same change to the sugar (2 + 6 = 8). The relationship is being read as a constant difference instead of a constant multiplier.
The doubling items could not catch this because doubling is the one case where the additive and multiplicative rules give the same answer: 3:2 becoming 6:4 is both +3/+2 and ×2/×2. A correct answer on those items is consistent with either schema, so the team's data looked better than the understanding was. Item 5 is the clearest evidence of the additive schema: 4:6 and 6:9 are equal under a multiplier of 1.5, but an additive reasoner sees differences of 2 and 3 and calls them unequal. Eleven of the 15 students who missed item 3 gave that explanation, so the error is stable and rule-governed, not noise.
The instructional implication is that re-explaining the cross-multiplication procedure will not dislodge the rule, because the rule is not a broken procedure. It is a competing, workable model that must be directly contradicted with data.
2. The team must choose one 50-minute reteach: (a) re-model cross-multiplication, (b) ratio tables with non-doubling scale factors, or (c) unit rate. Defend your choice and sketch the period, including how you would know by the end of the hour whether it worked.
Read the worked answer
Choose (b), with unit rate appearing only as a payoff. Cross-multiplication is a set of steps students can execute while still believing the additive rule, so it leaves the misconception intact. Unit rate is valuable but presumes the multiplicative structure that is currently missing.
Sketch:
- 0–10 min, conflict: Put 8 cups and 6 cups on the board without labels. Students test both by building the 'for every 3 cups of flour' rows and see that 9 cups is three batches, so sugar is three 2-cup batches.
- 10–25 min, tables with awkward factors: Scale 3:2 by 4, 5, and one-half. Halving is the strongest dislodger, because additive reasoning cannot produce 1.5 cups from a 'difference' rule.
- 25–40 min, sort: Partners classify eight statements as additive or multiplicative and justify each with a table entry.
- 40–50 min, exit ticket: One non-doubling item plus one equivalence judgment modeled on item 5.
Success criterion: at least 20 of 26 correct on the non-doubling item with visible multiplicative work. If fewer than 15 are correct, the assessment should be pushed and the review day converted to small-group table work; if 20 or more, the review day proceeds as scheduled with a five-minute warm-up on halving.