Comparing 3/8 and 3/5: Whole-Number Reasoning Versus Unit-Size Reasoning
Ms. Okafor's fifth graders are comparing 3/8 and 3/5. During turn-and-talk, she hears three claims. Jamal says 3/8 is larger 'because 8 is bigger than 5.' Priya says 3/5 is larger 'because fifths are bigger pieces than eighths.' Luis says 3/8 is less than 1/2 because 4/8 equals 1/2, and 3/5 is more than 1/2 because half of 5 is 2 1/2. The class has fraction strips, number lines marked in fifths and eighths, and circular fraction pieces. Ms. Okafor wants students to explain why a comparison rule works, not just state the correct answer. She also wants to anticipate the common error of applying whole-number reasoning to fractions. The lesson objective is to compare fractions with the same numerator by reasoning about the size of the parts.
1. Diagnose each of the three claims. Which are valid mathematical reasoning, which are incomplete, and which are misconceptions?
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Jamal's claim is a misconception based on whole-number thinking. He treats the denominator as a count of pieces and concludes that 8 pieces must be more than 5 pieces. But the whole is the same, so cutting it into 8 equal parts makes each part smaller than when it is cut into 5 equal parts. Eighths are smaller than fifths, so 3/8 is less than 3/5.
Priya's conclusion and reasoning are valid: for the same numerator, a larger denominator means smaller unit fractions, so 3/5 > 3/8. Luis's reasoning is also valid and more sophisticated: he uses the benchmark 1/2. Since 3/8 < 4/8 = 1/2 and 3/5 > 2 1/2/5 = 1/2, it follows that 3/5 > 3/8. His reasoning is not incomplete; it is a benchmark strategy that generalizes to many comparisons.
2. Design a number-line representation that would convince Jamal, and explain what he should attend to.
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Draw two number lines of exactly the same length from 0 to 1. Partition the first into 8 equal intervals and mark 3/8. Partition the second into 5 equal intervals and mark 3/5. Place the lines one above the other so the 0s and 1s align. The point for 3/5 lies to the right of the point for 3/8, so 3/5 > 3/8.
Jamal should attend to two features: the wholes are the same length, and the partitions are equal within each line. The denominator tells how many equal parts make one whole; it does not tell how many parts are shaded or how large the fraction is. The distance from 0 to 3/5 is three fifths of the whole, and each fifth is longer than each eighth, so the total distance is greater. Having Jamal physically fold or measure the strips reinforces that eighths are smaller pieces.
3. Write a general statement that captures Priya's reasoning and a counterexample task to test whether students overgeneralize it.
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General statement: For positive whole numbers a, b, and c, if b < c, then a/b > a/c. In words, when the numerators are the same, the fraction with the smaller denominator is larger because the unit fraction is larger.
The rule is limited to fractions with the same numerator. A counterexample task is to compare 2/3 and 3/5. A student who overgeneralizes 'bigger denominator means smaller fraction' may incorrectly focus only on the denominators and miss that the numerators differ. The correct comparison, 2/3 > 3/5, can be justified with a benchmark: 2/3 > 1/2 and 3/5 > 1/2, so a common denominator or cross-multiplication is needed. Asking students to state when the rule applies is the key assessment move.