Comparing Fractions with Different-Size Parts
Ms. Ibarra's class is comparing 3/4 and 5/8. On a quick check she reads three responses. Jo wrote 5/8 > 3/4 and explained, '5 is bigger than 3 and 8 is bigger than 4.' Lila wrote 3/4 > 5/8 and explained, 'Fourths are bigger pieces than eighths.' Sam drew two number lines of the same length, partitioned one into fourths and the other into eighths, marked 3/4 at the point that matched 6/8, and wrote 3/4 > 5/8 with the explanation that 6 eighths is more than 5 eighths. The class has fraction strips and blank number-line strips. Ms. Ibarra has 25 minutes and wants students to reason about the size and number of parts rather than memorize a rule. The next unit introduces equivalent fractions on a number line.
1. Analyze Jo's and Lila's reasoning. Why is Lila's conclusion correct but her explanation incomplete?
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Jo treats numerators and denominators as independent whole numbers to compare, which is a whole-number strategy applied to a fraction. A fraction is one number built from two roles: the denominator fixes the size of the parts and the numerator counts them. Comparing 5 with 3 or 8 with 4 answers a different question. Lila's conclusion is correct, but 'bigger pieces' is only half the comparison: 3 pieces of size 1/4 must be compared with 5 pieces of size 1/8. Using a common partition, 3/4 = 6/8, so the real comparison is 6 eighths versus 5 eighths, and 6 > 5. Her unit-size reasoning is a useful start but does not by itself decide the comparison when the counts differ; she needs to express both quantities in same-size parts or place them on a common number line.
2. Design a 25-minute sequence using fraction strips and number lines that moves both Jo and Lila. Include the key question you would pose.
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First, 5 minutes: students fold strips to show 1/4 and 1/8 and label each part; ask, 'Which is larger, one fourth or one eighth, and why?' Establish that eighths are smaller because the same whole is cut into more equal parts. Second, 8 minutes: build 3/4 with fourths strips and 5/8 with eighths strips, then place both on parallel number lines of identical length. Third, 7 minutes: pose the key question, 'How many eighth pieces would cover the same length as 3/4?' Students lay eighth pieces over the fourths and find 6/8, and you record 3/4 = 6/8. Fourth, 5 minutes: compare 5/8 and 6/8 and write a sentence using the phrase 'same-size parts.' This sequence makes Jo's digit comparison visibly wrong and gives Lila the count she was missing.
3. Sam's work is correct. What follow-up would push Sam beyond this item, and how would you use Sam's number line with the class?
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Ask Sam to compare 2/3 and 3/5, where neither denominator divides the other, so the same-length number line must be partitioned into fifteenths or the fractions must be compared to a benchmark such as 1/2. That pushes Sam from matching partitions to a general strategy. With the class, display Sam's two number lines without the answer and ask students to locate 5/8 and 3/4 and explain which is farther from 0. Then ask Sam to describe the alignment of 6/8 and 3/4, which previews equivalent fractions. Evidence to look for: Sam explains that the number lines must be the same length and that equivalent fractions name the same point; if Sam says only that 6/8 equals 3/4 because that is how the drawing came out, ask for the strip overlay as proof.