Fourth Graders Who Think 1/5 Is Bigger Than 1/3
Ms. Delgado gives her 24 fourth graders a four-item fractions pre-assessment before the comparing-fractions unit. Seventeen students say 1/5 is greater than 1/3 'because 5 is bigger than 3.' Nineteen correctly place 1/2 on a number line marked 0 to 1, but only 6 place 3/4 correctly. On a separate item, 14 students write 1/2 + 1/3 = 2/5. The district curriculum map introduces the common-denominator procedure in week one and names fraction circles as the primary model. Ms. Delgado has fraction circles, fraction bars, blank number lines, and 45-minute periods. A colleague advises her to 'just teach cross-multiplication; it is faster and they will get the answers right.'
1. Explain the thinking behind the 1/5 > 1/3 response and evaluate the colleague's advice about cross-multiplication.
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The students are applying whole-number ordering to a symbol system where the numeral does not stand for quantity. They read the denominator as a count of size rather than as the number of equal parts in one whole. Two other results point to the same root: writing 1/2 + 1/3 = 2/5 treats numerator and denominator as independent whole numbers, and the gap between placing 1/2 (19 correct) and 3/4 (6 correct) suggests many students are using a memorized half-position rather than reasoning about magnitude.
Cross-multiplication would produce correct answers on comparison items, which is exactly why it is tempting. But it leaves the underlying conception untouched. Those students will still err when adding fractions, when estimating whether a sum is reasonable, when ordering three fractions, and when judging whether an answer makes sense - none of which cross-multiplication addresses. It is also an unanchored procedure that is easy to misremember or invert.
The defensible sequence is to build magnitude first with number lines and same-whole models, then let the common-denominator procedure arrive as a shortcut that students understand. Ms. Delgado should also watch the 14 students who added across, since their error reveals the same idea and deserves the same treatment.
2. Sequence three instructional moves and describe a specific task for each, including where the common-denominator procedure belongs.
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Move 1 - benchmark placement on a number line. Each student gets a 0-to-1 line and places 1/2, then 1/3 and 1/5, folding a paper strip to check. The discussion question is which fraction lands closer to 0 and why. This directly confronts the 17 students' rule with a visual counterexample they generate themselves.
Move 2 - same numerator, different denominators. Compare 2/3 and 2/5 with fraction bars, then 3/8 and 3/4. Students explain in writing using the words 'size of the pieces.' This isolates the denominator's meaning before the procedure is named.
Move 3 - common denominators, but only after estimation. Students first predict whether 3/4 or 5/8 is larger, then find a common denominator to check. The procedure becomes a verification tool rather than the entry point.
Keep the district's fraction circles for the first two moves but pair them with number lines, because circles do not extend past one whole and encourage a 'bigger piece' intuition without a linear scale. Within 45 minutes, use a short number-line warm-up, one core task, and a written justification exit ticket. Reassess the same four items after two weeks to see whether the misconceptions are actually moving.
3. What evidence would show that the misconception is resolved rather than that students have merely memorized a new rule?
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Look for transfer to items that the procedure alone does not solve, and for reasoning language.
Specific items: (a) order 3/8, 3/4, and 5/8 without computing and justify with a number line sketch; (b) decide whether 1/2 + 1/3 is closer to 1/2 or to 1 and explain; (c) find and correct the error in a fictional student's claim that 1/6 > 1/4 'because 6 > 4'; (d) place 7/8 and 7/4 on the same number line, which requires knowing what an improper fraction means.
Proficient responses will refer to the size of the parts, the number of parts needed to make a whole, or the position relative to 1/2 or 1. Responses that resolve the misconception only procedurally will say 'I used common denominators' even on an estimation item, or will produce the right ordering with no justification.
Also track the 17 students individually on a simple pre/post grid rather than only looking at class averages, since a class mean can rise while a third of the class holds the original idea. Schedule a brief interview with four or five students, asking them to explain 1/5 versus 1/3 aloud; a child who has genuinely reorganized the idea will reach for a picture, a benchmark, or a story about sharing, not a rule.