Additive Versus Multiplicative Scaling in a Paint Mixture
Ms. Okafor teaches a seventh-grade proportional-reasoning unit to 30 students. On Friday's exit ticket she asks: “A paint mixture uses 3 parts blue to 5 parts yellow. If you use 12 cups of blue, how many cups of yellow?” Of 30 responses, 11 answer 14 cups, 9 answer 20 cups, 6 give other answers, and 4 are blank. The 14-cup group explains, “12 is 9 more than 3, so add 9 to 5.” One student in the 20-cup group writes “3×4=12, so 5×4=20.” Another writes 1.67 cups and says “divide 5 by 3.” Ms. Okafor has a 25-minute lesson the next day and wants to use the responses rather than simply reteach the procedure. She also needs a quick check at the end to see whether students can transfer the idea to a new context.
1. Interpret the exit-ticket data. What does each response pattern reveal about students' proportional reasoning, and which pattern should Ms. Okafor prioritize?
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- 14 cups: additive reasoning. The student preserves the difference (12 − 3 = 9) instead of the ratio. This is the most common incorrect response (11/30) and the most instructionally important because it signals a whole-number additive schema that will fail in scaling, percent, and linear functions.
- 20 cups: correct multiplicative scaling: 3 × 4 = 12, so 5 × 4 = 20. These students coordinate both quantities by the same factor.
- 1.67 cups: likely computes 5 ÷ 3 and applies it incorrectly, or inverts the unit rate. This is a unit-rate error, not the same as additive reasoning.
- Blank/other: incomplete entry or guessing; do not over-interpret.
Prioritize the additive group, but do not ignore the unit-rate group. The next lesson should explicitly contrast additive and multiplicative change.
2. Design a 20-minute instructional sequence using the student responses. Include specific questions and representations.
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- Launch with contrast (5 min): Display 3:5 and the two answers 14 and 20. Ask, “Which answer keeps the same relationship between blue and yellow? How can you check without calculating a unit rate?”
- Build a ratio table (7 min): Blue 3, 6, 9, 12; yellow 5, 10, 15, 20. Ask, “What happens to yellow each time blue increases by 3? Why doesn't yellow increase by 9 when blue increases by 9?”
- Double number line (5 min): Mark 0, 3, 6, 9, 12 on the blue line and 0, 5, 10, 15, 20 on the yellow line. Connect 12 to 20 visually.
- Compare and repair (3 min): Have students test 12:14 by checking cross-products or equivalent ratios: 3/5 ≠ 12/14, but 3/5 = 12/20. Ask the 14-cup group to revise their explanation.
3. Write one transfer item and explain what evidence would show understanding versus fragile procedural skill.
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Transfer item: “A recipe uses 2 cups of rice to 5 cups of water. For 7 cups of rice, how much water? Show a ratio table or double number line and explain why your method works.”
Evidence of understanding: The student scales both quantities by the same factor (2 × 3.5 = 7, so 5 × 3.5 = 17.5) or uses a ratio table; the student explains that the rice-water relationship must stay constant, not the difference. Fragile procedural skill: The student writes 7 + 5 = 12 or subtracts 5 − 2 = 3 and adds 3 to 7, even if the final arithmetic looks neat. A correct answer with no explanation is insufficient evidence; ask the student to justify why the same multiplier applies to both quantities.