5/7, 17/12 and the Mangoes That Were Already Sold
Class 5 in a mandal primary school has 38 children and a 40-minute mathematics period. The teacher has just marked the unit test; the school's Class 5 examination is two weeks away and covers fractions and the unitary method. Three answers stand out. Ravi writes 3/4 + 2/3 = 5/7 and explains, 'I added the top numbers and the top numbers below.' Meena writes 3/4 + 2/3 = 17/12 and then, separately, 17/12 = 1 5/12, so her answer is correct. Bhavani answers the word problem 'A farmer has 240 mangoes. He sells 3/8 of them on Monday and 1/3 of the remaining mangoes on Tuesday. How many are left?' with the working 240 × 3/8 = 90; 240 × 1/3 = 80; 240 − 90 − 80 = 70. Twenty-two of the 38 children made the same error as Bhavani. The teacher has five remedial periods before the test and cannot reteach the whole unit to everyone.
1. Explain precisely what Ravi understands and what he does not, and write the single question you would ask him (not the rule you would state) to make the gap visible to him.
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What he understands: fraction notation as two counts in two columns, and addition as an operation applied to each column in parallel. That is a coherent generalisation, not random error - it is what the notation looks like if you have not yet built the idea of a fraction as one quantity with a size.
What he lacks: the magnitude sense that a fraction is a number that can be located on a line, which is why nothing in his working feels wrong. He also has no estimation check: 3/4 is nearly 1 and 2/3 is more than half, so the total must be more than 1 - and 5/7 is less than 1.
The single question: 'Is 5/7 bigger than 3/4? Show me on this paper strip.' Do not mention denominators. Folding or marking both fractions on the same strip forces a comparison that does not go through the addition algorithm, and the contradiction becomes his own observation rather than the teacher's correction. Once he says 'but 5/7 is smaller', the strip is available for building the common-denominator idea as a way of making pieces the same size.
2. Solve the mango problem correctly, explain why Bhavani's answer is arithmetically self-consistent but wrong, and name the concept the 22 children are missing.
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Correct solution: Monday: 3/8 × 240 = 90 sold, so 150 remain. Tuesday: 1/3 × 150 = 50 sold. Mangoes left = 100.
Why Bhavani's work looks right: every multiplication in it is correct. She computes 3/8 of 240 and 1/3 of 240 accurately. The error is not arithmetic but referential - she takes 1/3 of the original 240 instead of 1/3 of what remained after Monday. Her 70 is the correct answer to a different problem.
The missing concept: the fraction's referent - the 'whole' that the fraction is taken of, which changes after the first sale. 1/3 of what? is the question the children are not asking.
Diagnostic worth running: have them draw a single bar for 240, cut 3/8, and then mark where the second cut falls. Children who draw the second cut on the original full bar have the referent problem; children who cut inside the remaining part do not. The 22 will mostly draw on the full bar, which tells the teacher to teach 'of the remainder' language and bar modelling, not multiplication.
3. Allocate the five remedial periods across the three children's needs and the 22 others. Justify your allocation against the constraint that the test rewards procedural speed.
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Allocation of five 40-minute periods:
- Period 1 - whole class, 12 minutes only: one bar model for the mango problem, because 22 of 38 share the error. The remaining 28 minutes are mixed practice so the rest of the class is not held back. Whole-class reteaching of fraction addition is not justified: Meena's correct 17/12 shows the algorithm is not the class-wide gap.
- Periods 2 and 3 - Ravi and the small 5/7 group: strip and number-line work, comparison before addition, then unlike denominators, ending each period with an estimation exit ticket ('is the answer more than 1?').
- Period 4 - Bhavani and the 22: varied 'fraction of the remainder' problems, each with the prompt 'of what?' written in the margin, then fading the prompt.
- Period 5 - whole class: a timed paper in the test's exact format, then review.
Justification: the last period must look like the test, because speed is being assessed and unfamiliarity costs marks. But speed built on a wrong referent merely produces wrong answers faster, so the timed practice comes last, after the concept work. Grouping by error pattern rather than by 'weak/strong' avoids labelling the 22 as slow and lets Ravi's group be small enough to question individually.