Fraction Comparison: The Model Works, the Reasoning Doesn't Transfer
Ms. Reyes gives her fourth graders a quick task: circle the greater fraction, 2/3 or 3/5. Of 26 students, 18 circle 3/5. In interviews with six of them, four say 'three is more than two,' and one says 'fifths are bigger than thirds.' Two days later she gives the same class fraction strips and asks them to build and compare 2/3 and 3/5; 22 of 26 now choose 2/3 correctly. But when she takes the strips away and asks students to explain, 9 of the 22 revert to '3/5, because 3 is bigger.' The pacing guide says the class begins adding fractions with unlike denominators next week.
1. What does the gap between the strip task and the explanation task tell you students do and do not know?
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What they can do: compare fractions when the comparison is carried by the materials. With strips in hand, 22 of 26 get the right answer - they can match the shaded lengths and see which is longer.
What they cannot do: reason about why. The strips supply the answer, so the answer is read off the model rather than generated from an idea about unit size. When the model is removed, 9 students fall back on whole-number logic ('3 > 2'), and one holds an inverted unit-size idea ('fifths are bigger than thirds').
Those two errors are different and both are informative:
- Whole-number thinking: treating numerator and denominator as two separate whole numbers to be compared.
- Reversed unit reasoning: knowing that a bigger denominator means something, but flipping the direction - not yet grasping that fifths are smaller than thirds because the same whole is cut into more pieces.
Neither is fixed by more strip practice, since strip practice already produces correct answers. The missing piece is the idea that the denominator names the size of the part and the numerator counts how many of those parts - and that comparing requires the parts to be the same size.
2. Make the case for or against following the pacing guide next week, and describe a two-day bridge.
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Recommendation: delay unlike-denominator addition by two days and teach the unit-size idea explicitly. Adding with unlike denominators demands exactly the understanding that just failed - that 2/3 and 3/5 must be expressed in a common unit before they can be counted or compared. Moving on now guarantees that students will learn a procedure (find a common denominator, convert, add) that they cannot connect to magnitude or check for reasonableness.
Day 1 - unit fractions as the building block. Students build 2/3 as two 1/3 pieces and 3/5 as three 1/5 pieces from the same whole and say the sentence aloud: 'I have two pieces, but my pieces are bigger.' Sort a set of fractions against 1/2 as a benchmark with justification, no strips on the second half.
Day 2 - common units before common denominators. Compare 2/3 and 3/5 by subdividing both strips into fifteenths; students see 10/15 versus 9/15 and state the quantity did not change, only the number and size of the pieces. Then a set of comparison problems where students must justify in writing without materials - this is the transfer check the strips task never demanded.
Assessment of readiness: students can compare 3/4 and 5/7 by reasoning (both are 1/4 and 2/7 away from 1, or by benchmark against 1/2) and can explain in writing why 5/8 is less than 3/4. Until at least 20 of 26 can do that without materials, adding with unlike denominators will be procedural only.
3. What evidence would convince you the class is ready to add fractions with unlike denominators, and what common error would tell you they are not?
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Ready when students can:
- Decompose a fraction into unit fractions - write 3/4 as 3 × 1/4 and explain what the 3 and the 4 each tell you.
- Rename without changing quantity - produce an equivalent fraction and defend it with a model or a multiplication argument ('multiplying numerator and denominator by the same number cuts each piece into 2, so I have twice as many pieces but each is half as big').
- Estimate before computing - say 2/3 + 3/5 is a bit more than 1 before calculating, which is the habit that catches wrong answers later.
Warning sign: if students add numerators and denominators across the board (2/3 + 3/5 = 5/8) and cannot see why that is wrong by estimating, the unit-size idea has not landed - 5/8 is less than 2/3, so the estimate check alone exposes it. If they catch the error only because a teacher says so, readiness is not there.
Second warning sign: students who can only justify with manipulatives in hand. The strip task in this class showed exactly that profile - correct answers, no transfer - so readiness must be judged from explanations produced without materials.