Fractions That Work Only on Paper
A Class V mathematics teacher in a government primary school in Nellore has 35 students. In a diagnostic test, 28 students correctly computed 2/3 + 1/4 using the least common denominator, but only 9 could solve: 'Ravi ate 2/3 of a roti and Sita ate 1/4 of the same-size roti. How much did they eat together?' When the teacher asked the 28 to explain, most said, 'We did cross multiplication because it is addition.' The school has no fraction kit. There are six 40-minute mathematics periods before the unit test. The textbook introduces unlike denominators through pictures, but the teacher has been giving rules and practice sums. The head teacher wants at least 30 students to score above 60% in the unit test. The teacher must plan the next six periods so that the 28 do not simply repeat the algorithm and the 9 are not left behind.
1. What does the gap between 28 and 9 tell the teacher, and what two activities should she use in the next periods?
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The gap is procedural fluency without conceptual understanding. The 28 can execute a rule but do not recognise the operation in a situation; their explanation - 'cross multiplication because it is addition' - shows they have memorised a step, not the meaning of adding parts of a whole. The 9 may have fragile procedures but better context sense, or vice versa; the teacher should check both groups, not assume the 9 are 'weak'.
Activity 1 - paper fraction strips. Each pair folds two equal strips into thirds and quarters, shades 2/3 and 1/4, and places them end to end. They record what they see: the total is more than 3/4 but less than 1. This makes the denominator a unit size, not a number to cross-multiply.
Activity 2 - 'which operation?' sort. Give six short situations, three addition and three subtraction, with the same numbers but different actions (joining, comparing, removing). Students draw a bar model before computing. The teacher asks, 'What is the whole? What are the parts?'
Reasoning: the activities target the representation and the situation, not more practice of the same algorithm. The teacher should connect the strip to the symbolic method only after students can explain why the denominators are made the same.
2. How should the teacher plan the six periods and assess whether the intervention worked?
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Plan backwards from the unit test, but do not teach to it. Periods 1–2: concrete strips and bar models for unlike denominators, with oral explanation. Period 3: number-line placement and comparison. Periods 4–5: mixed word problems with drawing first, then computation; pair checking. Period 6: short review and a 4-item diagnostic, including one non-routine item.
Assessment: use three sources - a drawn model, an oral explanation, and a written word problem. The teacher should record who can (a) choose the operation, (b) represent it, and (c) compute. A student who computes correctly but cannot draw the situation is not yet secure. The unit test should include at least two context items so the 28 cannot pass by algorithm alone.
Watch for unintended effects: the 9 should not be given only easier sums; they need the same representation work with more scaffolding. The 28 should not be told they were wrong; their fluency is an asset to build on. Decision rule: if at least 25 students can draw and explain one unlike-denominator situation by Period 5, the teacher moves to mixed practice; if not, she spends Period 6 on one more concrete comparison and reduces the number of sums.