Reading Fraction Multiplication Errors in Grade 5
Mrs. Okafor's grade 5 class (22 students) turns in an exit ticket: 3/4 × 2/5. Eleven students answer 3/10. Five answer 5/9. Four answer 15/8. Two answer 6/20. In follow-up interviews, one of the 5/9 students explains, 'I added the tops and the bottoms, like when we add fractions.' A 15/8 student says, 'I multiplied across, but the other way - 3 times 5 and 4 times 2 - because you flip the second one.' The unit test is in nine days, and the pacing guide says to begin division of fractions next week.
1. Sort the four responses by what each reveals about student thinking, and rank them by instructional urgency.
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3/10 (11 students): correct.
6/20 (2 students): also correct - equivalent to 3/10, just not simplified. This is not a misconception and should not be counted as an error when I plan the reteach.
5/9 (5 students): a conceptual problem. They are applying the addition rule to a multiplication situation, which means they have not reconceived multiplication of fractions as scaling - they are treating × as 'combine.' This will fail immediately when factors exceed 1 and cannot be repaired with a memorized chant.
15/8 (4 students): a procedural leak. They know a rule ('flip the second one') but attached it to the wrong operation; the division algorithm is bleeding into multiplication. Their procedure is a chant with no anchor, but the concept may be recoverable quickly once the model is in place.
Urgency ranking: 5/9 first - it is the larger conceptual gap and the one that will compound in division next week. Then 15/8. Then check whether the 6/20 students can explain why 6/20 and 3/10 are the same, which is a real (if minor) gap worth ten minutes.
2. Design a two-day reteach. Include the representation you would use and how you would surface the 5/9 reasoning publicly.
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Day 1 - area model first, procedure second. Students shade 2/5 of a rectangle, then take 3/4 of that shaded region, and count: 6 of 20 small parts, i.e., 3/10. The question I want in the room is 'Why does the answer get smaller when we multiply?' Follow with a number line: 2/5 scaled by a factor less than 1 must land below 2/5. Estimation becomes the habit: before computing, predict whether the product is greater or less than each factor.
Day 2 - surface 5/9 in public. Put the area model and '5/9' side by side and ask the class to adjudicate: does 5/9 of the rectangle match the shaded region? Then a contradiction test: 5/9 is larger than 2/5, and one factor was 3/4 - less than 1 - so the product cannot be larger. Have students generate their own example where adding across produces an absurd result. Close by placing the addition and multiplication rules side by side and asking students to explain why the denominators behave differently in each.
Do not simply announce the correct algorithm on Day 1; the 5/9 students already have an algorithm, and a second one delivered the same way will compete with it rather than replace it.
3. How will you know the reteach worked, and what will you do if some students still write 5/9?
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I need items that separate concept from procedure, not more of the same computation:
- Estimate first: 'Without calculating, is 3/4 × 5/6 greater or less than 5/6? How do you know?'
- Scaling context: 'A recipe uses 2/3 cup of oil. You make 3/4 of the recipe. Is the amount more or less than 2/3 cup? Draw it and compute.'
- Error analysis: show a fictional student's work (3/4 × 2/5 = 5/9) and ask the class to explain what the student was thinking and why it cannot be right.
A student who computes correctly but fails the estimate items has memorized a new rule without the model - that is not success.
If students still produce 5/9 after two days: they go to a small group with physical fraction strips and the rectangle model extended over several sessions, and they do not move on to division with the whole class. I would also slow the pacing guide for the class by one day rather than introduce division on top of a broken multiplication concept, and tell the grade-level team why, with the exit ticket data as the argument.