Equivalent Fractions After the Exit Ticket
Mr. Okafor teaches fifth grade in a K–5 building with 26 students, including eight multilingual learners. The district pacing guide places equivalent fractions this week, and a 12-item quiz is scheduled for Friday. Tuesday's exit ticket asked students to compare 2/3 and 3/5 and explain using a representation. Fourteen of 26 students chose 2/3, but their explanations split into two patterns: seven added numerators and denominators and concluded 5/8; five argued 3/5 is larger because 3 and 5 are bigger numbers than 2 and 3. Only four students used a number line or benchmark. Mr. Okafor has 40 minutes before the quiz, fraction circles, blank number lines, and a colleague who recommends reteaching 'cross-multiply' as a quick fix. He also has two students who solved correctly but cannot explain why.
1. What conceptual understandings are missing, and what representation sequence would you use in the 40 minutes? Justify each step in terms of content, not just engagement.
Read the worked answer
The two wrong patterns are not the same. Seven students are using additive reasoning: they treat fractions as two independent whole numbers and add across, which ignores that the denominator names the size of the parts. Five are using whole-number magnitude: they compare 3 and 5 to 2 and 3 without attending to the relation between numerator and denominator. Both need experiences that keep the whole fixed and make part-size visible. I would sequence: (1) partition the same whole into thirds and fifths with fraction circles, naming each piece and checking that 2/3 and 3/5 are each made of equal parts of one whole; (2) place both on a number line marked 0, 1/2, 1, because a common error is thinking 3/5 is larger than 2/3; students see 2/3 is farther from 1/2 than 3/5; (3) partition both wholes into fifteenths so the comparison 10/15 versus 9/15 is generated from the model, not announced; (4) connect that partitioning to the common-denominator procedure. I would not begin with cross-multiplying. It can produce correct answers while leaving the additive and whole-number ideas intact; it becomes useful after students can explain why a common denominator makes the numerators comparable. The two students who solved but cannot explain would be asked to justify with the number line, not serve as answer keys.
2. How would you respond to the colleague's cross-multiply suggestion, and what evidence would you collect before Friday to decide whether to proceed or reteach?
Read the worked answer
I would tell the colleague that cross-multiplying is a valid shortcut, not a false method: 2/3 and 3/5 can be compared as 10/15 and 9/15, and the cross-products 2×5 and 3×3 are the numerators once both fractions are expressed in fifteenths. My concern is timing. If students have additive and whole-number misconceptions, a mnemonic can increase correct answers on Friday while leaving the misconceptions intact for later work with ratios and decimals. I would use it as a compression after the number-line and partitioning work, and I would ask students to explain the connection. For evidence, I would collect three things: a revised exit ticket with a number-line drawing, a 'compare and justify' item using unlike denominators, and a short interview with four target students asking 'How do you know which is larger?' I would look for language about equal parts and the same whole, not just the correct symbol. If most students still justify by number size or addition, I would adjust the quiz to include representation items and spend the next day on benchmark comparisons rather than proceed as planned.