1/2 + 1/3 = 2/5, and the rotis that say otherwise
Vinod's Class 5 has 36 children. In a fractions diagnostic he found that 19 wrote 1/2 + 1/3 = 2/5; 8 answered correctly; 6 wrote 3/5; 3 left it blank. Asked which is larger, 1/2 or 1/3, 22 chose 1/3, explaining that '3 is bigger than 2'. Yet when he gave the same children three rotis to share equally among four, 29 folded and tore them correctly and reported three-fourths. On a shading task, only 11 of 36 could shade two-thirds of a rectangle correctly when the rectangle was pre-divided into six equal parts. The textbook unit on fractions is scheduled for next month and the annual examination will test addition of unlike fractions. Vinod's colleague says the fix is ten drill sums a day on LCM.
1. Diagnose what the children understand and what they do not. Is this a careless error, a language problem, or a conceptual gap? Use the roti and shading evidence.
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Read the five pieces of evidence together.
1/2 + 1/3 = 2/5 in 19 of 36, and 3/5 in six more is a consistent rule: add the tops, add the bottoms (or add the tops, keep a bottom). That is the natural extension of whole-number addition - more of everything makes a bigger number - and it is not carelessness, because the wrong answers are produced confidently and repeatedly.
1/3 is larger than 1/2 'because 3 is bigger than 2' is the same whole-number schema applied to a notation the children read as two independent numbers.
The roti task shows the concept is not absent. 29 of 36 shared three rotis among four children correctly. They can partition and name the result when the quantity is visible and the goal is practical.
The shading task locates the break exactly: children cannot translate between symbol and region, and some may be counting parts of the six rather than relating two to the whole.
So: strong informal fraction knowledge, weak symbol–representation mapping, and an intact whole-number schema overriding both. Not a language problem. Not carelessness. A conceptual gap that is narrower than the test scores suggest, which is the reason to teach from the rotis inward rather than from the rule outward.
2. Design a two-period intervention and state what you would assess at the end. Explain why drilling LCM first would or would not work.
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Two periods, both starting from sharing.
Period 1 (40 min): pairs get paper strips; fold into halves, quarters and thirds and label them. Then 'which is more, 1/2 or 1/3?' - decide by laying one strip on the other, not by comparing 2 and 3. Record on a class chart. Close with three symbol–picture matching cards.
Period 2: the roti problem on paper - three rotis, four children - draw the cuts, write the share. Then 1/2 + 1/3 with strips: fold a strip into sixths, lay the halves and the thirds on it, and read 5/6 off the strip rather than being told it. Finally contrast 1/2 + 1/3 with 1/2 + 1/4 to show why 'add tops and bottoms' breaks.
Assessment (not the same sums): shade 3/4 on an undivided strip; place 1/2, 1/3 and 5/6 on a number line; explain in one written sentence why 2/5 cannot be the answer to 1/2 + 1/3; solve one novel sharing problem. Mark explanation and representation, not just answers.
Why not drill LCM first: the LCM procedure is a second rule laid over a wrong one. Children who already believe 1/2 + 1/3 = 2/5 will learn 'find the LCM, then add the tops' as a magic step and revert under pressure or when the numbers look unfamiliar. Drilling raises the score in the drilled format and leaves the misconception intact underneath.