Additive Reasoning in Ratio Problems
Ms. Rivera's sixth graders are solving ratio problems. Given 'A recipe uses 3 cups of flour for every 2 cups of milk,' 11 of 27 students solve 9 cups flour → 6 cups milk correctly, but when asked how much milk for 7 cups flour, many write 6 cups because 'you add 4.' Others set up 3/2 = 7/x and cross-multiply correctly but cannot say whether 4.67 cups is reasonable. The class has ratio tables, double number lines, and tape diagrams. Ms. Rivera has one 50-minute period before a quiz.
1. Interpret the two error patterns and name the instructional need.
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Correct answers on 9 → 6 suggest some students can scale by a whole-number multiplier (×3). The 7-cup errors show additive reasoning: students see 3 → 7 as +4 and apply +4 to 2, getting 6, or they cannot coordinate two quantities. The cross-multipliers have a procedure but not a meaning for the result; they cannot judge 4.67 cups. Both groups need work on multiplicative comparison and unit rate, not more cross-multiplying.
2. Plan the 50-minute lesson using at least two representations.
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Use a ratio table and a double number line. Build a table: 3/2, 6/4, 9/6, then ask for 7. Students see 7 is not a multiple of 3; find the unit rate: 1 cup flour → 2/3 cup milk, so 7 × 2/3 = 14/3 = 4 2/3 cups. Connect to a tape diagram: 3 equal parts for flour, 2 for milk; 7 cups is 2 1/3 groups of 3, so milk is 2 1/3 groups of 2. Have students explain why 'add 4' fails. Timing: 10 min number talk, 20 min task, 15 min share, 5 min exit.
3. Write a quiz item that tests transfer and reasonableness, with a scoring focus.
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Ask: 'A recipe uses 3 cups flour to 2 cups milk. Show how to find milk for 7 cups flour using a table or double number line, and explain why adding 4 to the milk amount is not correct.' Score for representation, correct value 4 2/3, and an explanation of the multiplicative relationship. Include a reasonableness check: 7 cups is a little more than 6, so milk should be a little more than 4.