Fraction Multiplication and the “Add Tops and Bottoms” Error
Mr. Boone’s fifth graders are solving a recipe problem: A batch of granola uses 3/4 cup of honey. If he makes 2/3 of a batch, how much honey is needed? Priya writes 2/3 × 3/4 = 5/7 and explains, “I added the tops and the bottoms.” Several students get 6/12 and simplify to 1/2, but when Mr. Boone asks whether the answer should be more or less than 3/4 cup, the class is divided. The standard for this unit asks students to multiply fractions and interpret multiplication as scaling. Mr. Boone has fraction strips, grid paper, a document camera, and a number line on the wall. He wants to use Priya’s error as a discussion point rather than simply correcting it, but he has only 15 minutes before specials.
1. Analyze Priya’s error. What does it reveal about her understanding of fractions and multiplication, and why is 5/7 not a valid result?
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Priya is extending an additive whole-number pattern to fractions: she treats numerators and denominators as separate whole numbers and adds them. This suggests she does not yet see a fraction as a single quantity with a numerator and denominator in a part-whole relationship. It also suggests she is not interpreting “2/3 of 3/4” as scaling a quantity by a factor less than 1. Mathematically, adding numerators and denominators is not a valid operation here because the denominators name different-sized parts; 2/3 and 3/4 do not have a common unit, and multiplication of fractions is not repeated addition of the same addend in this context. The correct product is (2 × 3)/(3 × 4) = 6/12 = 1/2. A quick reasonableness check also exposes the problem: 2/3 is less than 1, so 2/3 of 3/4 must be less than 3/4. The result 5/7 is about 0.71, which is close to 3/4, and Priya has no way to judge whether that is plausible. Her error is therefore both procedural and conceptual.
2. Design an area-model sequence Mr. Boone could use in the 15 minutes to justify why 2/3 × 3/4 = 1/2. Include the questions he should ask.
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Draw a rectangle to represent one whole batch. Partition it vertically into 4 equal columns and shade 3 columns to show 3/4 cup. Ask: “What does the shaded region represent? What does the whole rectangle represent?” Then partition the same rectangle horizontally into 3 equal rows and shade 2 rows to show 2/3 of the batch. The overlap of the two shadings is the answer. Ask: “What does the overlap represent in the story? Into how many equal small rectangles have we divided the whole? How many small rectangles are in the overlap?” Students should see 12 equal parts and 6 shaded twice, so the overlap is 6/12 = 1/2 cup. Connect the drawing to the symbols: “The 2 rows and 3 columns give 6 overlapping parts; the 3 rows and 4 columns give 12 total parts. Where do the 2, 3, 3, and 4 appear in 2/3 × 3/4?” Do not lead with the rule. End with a number-line check: mark 3/4, divide that interval into thirds, and take two of those thirds; the endpoint is 1/2, which is less than 3/4. This connects the area model to scaling.
3. How should Mr. Boone respond to Priya’s 5/7 in the moment so that the error becomes a class discussion without embarrassing her?
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He can publicly treat 5/7 as a hypothesis worth testing, not as a wrong answer to be erased. For example: “Priya has given us 5/7. Let’s keep that on the board and check it against the story. If I make 2/3 of a batch, should I need more than 3/4 cup or less? Why?” This invites the class to reason about the size of the factor before computing. Then he can ask Priya to test her result with a fraction strip: “Show me 3/4. Now show me 2/3 of that 3/4. Does the result look like 5/7?” If she is unsure, the area model gives her a way to revise her thinking without being told she was wrong. Mr. Boone should also name the mathematical idea, not the mistake: “We are finding a part of a part, so the answer should be smaller than 3/4.” Later, in a private or small-group moment, he can ask Priya to compare the addition pattern with the multiplication model. The goal is to preserve her participation while making the reasoning public and revisable.