Sorting Fraction Errors by Cause, Not by Count
Mr. Alvarez reviews twelve fraction items from a fifth-grade unit test; seven are correct. The five errors are: (1) 3/4 + 5/6 = 8/10; (2) 4 ÷ 1/2 = 2; (3) 0.6 × 0.4 = 2.4; (4) asked which is greater, 3/5 or 5/8, he writes 3/5 "because fifths are bigger than eighths"; (5) 1/3 of 18 = 54. Among the correct items are 2 1/2 − 3/4 = 1 3/4, 2/3 × 3/4 = 1/2, and 0.25 + 0.5 = 0.75. He finishes in 18 minutes, well before the 40-minute period, and checks no answers.
1. Classify the five errors and state the common thread, distinguishing conceptual misunderstandings from procedural slips.
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- (1) 3/4 + 5/6 = 8/10: adds numerators and denominators separately. Not a slip-it is a coherent but wrong rule, and it treats a fraction as two independent whole numbers.
- (2) 4 ÷ 1/2 = 2: the whole-number belief that division always makes smaller. The computation itself is not the problem.
- (3) 0.6 × 0.4 = 2.4: digits are right (6 × 4 = 24), placement is wrong. Magnitude/place-value error, not a multiplication fact error.
- (4) 3/5 > 5/8 "because fifths are bigger": compares denominators as whole numbers and ignores that a larger denominator means smaller parts.
- (5) 1/3 of 18 = 54: applies "of means multiply" without checking whether the result is sensible; a magnitude-monitoring failure.
Common thread: the errors are not random. Items (1), (2), (4), and (5) all extend whole-number rules to quantities that are parts of a whole; (3) is the decimal version of the same reasoning. His correct items show he can execute procedures he has memorized, which is why counting errors alone would mislead.
2. Which single error is most diagnostic, and what question would you ask to separate two competing explanations?
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Most diagnostic: (4). Items (1) and (2) could each be a half-remembered procedure, but (4) is a verbal justification-it exposes the belief itself, not just an output. It also predicts (1) and (5), because all three follow from judging fraction size by counting parts rather than by part size.
Competing explanations to separate:
- He lacks fraction magnitude.
- He has magnitude but cannot retrieve the comparison procedure.
Ask: "Put 3/5 and 5/8 on this number line and tell me what the 5 and the 8 each mean in 3/5." A student who places them correctly but cannot state a rule has a retrieval/procedure issue; a student who places 3/5 farther right, or says the 5 means "five pieces," has a magnitude issue. The response changes the intervention, so this question is worth asking before reteaching.
3. Given the results and the 18-minute finish, what is the next instructional move, and what would you deliberately not do?
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Do: teach magnitude before more procedure-number lines partitioned into unit fractions, benchmark comparisons to 1/2 and 1, estimation before computing, and "does this answer make sense?" checks on every item. Keep the procedures he already executes correctly (subtracting mixed numbers, multiplying fractions) and attach them to the visual models so the two knowledge systems connect.
Do not: assign another packet of the same item types. He completed 12 items in 18 minutes without checking, which suggests fluency with the format but no monitoring; more of the same would reinforce the wrong rules and produce a misleadingly high completion rate. Also do not treat the decimal error as a separate unit-it is the same magnitude problem and should be taught alongside fraction size.
Monitor: a short mixed set including a comparison item and an estimation item, scored for justification, not just answers.