Comparing Fractions by Numerator and Denominator Separately
A fourth-grade team is reviewing two responses to the prompt “Compare 3/4 and 5/8.” Student A writes: “3/4 is less than 5/8 because 3 < 5 and 4 < 8.” Student B writes: “3/4 = 6/8, so 3/4 is greater than 5/8.” The team must decide which response to use as the basis for a whole-class discussion and what follow-up task will surface the misconception. The class has fraction strips and number lines available. Several students in the class consistently compare numerators and denominators separately. The teacher wants a task that requires students to justify with a benchmark and to connect the two representations. One constraint: the follow-up must be completable in 15 minutes and must include a written explanation, not just a circled answer.
1. Which student is correct, and what misconception does the other student’s reasoning reveal?
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Student B is correct. 3/4 is equivalent to 6/8, and 6/8 > 5/8, so 3/4 > 5/8.
Student A is treating the numerator and denominator as separate whole numbers. That reasoning ignores that the denominator tells the size of the parts. A larger denominator means smaller parts when the whole is the same. Student A also compares 3 with 5 and 4 with 8 independently, which is not a valid way to compare fractions. The misconception is not a careless error; it is a whole-number comparison strategy applied to fractions.
2. How could a number line or the benchmark 1/2 be used to justify the comparison in a way that fourth graders can explain?
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On a number line from 0 to 1, 3/4 is three jumps of 1/4, landing to the right of 1/2. 5/8 is five jumps of 1/8, also to the right of 1/2, but closer to 1/2 than to 1. The benchmark 1/2 supports the comparison: 3/4 is 1/4 more than 1/2, while 5/8 is only 1/8 more than 1/2. Since 1/4 = 2/8, 3/4 is 2/8 beyond 1/2 and 5/8 is 1/8 beyond 1/2. Therefore 3/4 is farther from 1/2 and is greater. Students should state that both fractions are greater than 1/2, then compare their distances from 1/2 using eighths.
3. Describe a 15-minute follow-up task that would surface and correct the misconception, including what students must write.
Read the worked answer
Give pairs a set of fraction cards with denominators 2, 3, 4, 6, and 8. Ask them to sort the cards into “less than 1/2,” “equal to 1/2,” and “greater than 1/2,” then choose two cards greater than 1/2 and compare them using a number line. Each pair writes two sentences: one explaining why a larger denominator does not always mean a larger fraction, and one using a common denominator or benchmark to justify the comparison. Circulate and ask students who compare numerators and denominators separately to place both fractions on the same number line. Close with one whole-class example such as 2/3 versus 3/5, where cross-multiplication or common denominators confirm the order. The written explanation is collected as an exit ticket.