The Oil-and-Vinegar Problem: 3/8 or 1 1/2?
A sixth-grade teacher poses this problem: "A marinade uses 3/4 cup of oil for every 1/2 cup of vinegar. How many cups of oil are used per cup of vinegar?" Of 30 students, 22 answered 3/8 and 6 answered 1 1/2; two left it blank. In interviews, the 3/8 group said, "You multiply the fractions - that's what we did last week," and none could say what 3/8 cups would describe. One student in the 1 1/2 group built a table: 3/4 cup oil to 1/2 cup vinegar, then 1 1/2 cups oil to 1 cup vinegar. The teacher has one 45-minute period before a unit test that includes both fraction multiplication and ratio reasoning, and the textbook's next lesson is the invert-and-multiply algorithm.
1. Explain what each answer reveals about student thinking. Which answer is mathematically correct, and how would you justify it to the class?
Read the worked answer
1 1/2 is correct; 3/8 is the right computation for a different question.
- Check: 1 1/2 × 1/2 = 3/4, so 1 1/2 cups of oil per 1 cup of vinegar is consistent with the given rate. Equivalently, 3/4 ÷ 1/2 = 3/4 × 2 = 1.5.
- 3/8 = 3/4 × 1/2 answers "how much oil is in half of a 3/4-cup batch" - a part-of-a-part question. The students executed a valid multiplication, just not the one the question asked.
- The error is structural, not careless: students attached an operation to a surface cue ("per," "of" both signal "multiply" in their experience) instead of building a quantity model.
Estimation check to make public: the answer must be greater than 3/4, because a full cup of vinegar needs more oil than half a cup does. Any answer below 3/4 is immediately suspect. Making that check routine is more durable than any rule.
2. Design a ~20-minute sequence, with the exact numbers you would use, that makes 1 1/2 the natural answer without stating the algorithm first.
Read the worked answer
- Whole-number anchor (4 min). "6 cups oil per 2 cups vinegar - how much per cup?" Students draw a double number line and see 3 per cup. This establishes the per-one structure before fractions enter.
- Same structure, fractional rate (8 min). Pose it as measurement division: "How many 1/2-cup scoops fit in 3/4 cup?" On a tape diagram of 3/4, mark one full 1/2-cup scoop and a remaining 1/4, which is half a scoop. Result: 1 1/2 scoops - so 1 1/2 cups of oil per cup of vinegar. Have students verify by multiplying back: 1 1/2 × 1/2 = 3/4.
- Contrast the two questions (8 min). Side by side: (a) "Oil per cup of vinegar" → 1 1/2; (b) "Oil in half of a 3/4-cup batch" → 3/8. Same numbers, different questions, different answers. Students sort 6 quick items into "per" or "part-of" before computing.
Deliberately withhold invert-and-multiply here; it arrives next lesson as a shortcut for a structure students already own.
3. The test covers both fraction multiplication and ratio reasoning. What is the highest-leverage move for the remaining class period, and what would you deliberately not do?
Read the worked answer
Do: spend the period on the meaning-plus-estimation routine, because 22 of 30 making the same structural error signals a shared gap in the class's model of division, not 22 individual mistakes. Highest leverage is the tape diagram / measurement-division question and the "should the answer be bigger or smaller than 3/4?" check, since those transfer to unfamiliar contexts on the test.
Don't: drill "invert and multiply" or a key-word strategy. A mnemonic would lift scores on look-alike items while collapsing on transfer items and on the multiplication items, where the same students would then over-apply it.
Trade-off to name: the multiplication items may suffer slightly because the class spends the period on meaning. That is acceptable if the division/ratio items carry more weight and if the algorithm is retaught properly in the next unit. If the test weights computation heavily, a 10-minute algorithmic exit routine after the conceptual work is a reasonable compromise - but never instead of it.