Twenty-Nine Right and Eight Right: Fractions in Class 5
Halfway through the year you take over Class 5 mathematics in a school in Bhagalpur: 36 students, six periods a week, one blackboard and one box of teaching aids. In the last unit test, 29 of 36 students correctly answered 3/4 + 1/4, but only 8 answered the word problem: 'Rehana ate 1/4 of a cake and her brother ate 2/4 of the same cake. How much did they eat together, and how much is left?' Seven students wrote 3/4 + 1/4 = 4/8. In the same test, 24 of 36 could not place 3/4 on a number line, and 19 wrote that 1/5 is bigger than 1/3 'because 5 is bigger than 3'. You have six periods before the term examination, and equivalent fractions must also be covered. A colleague says: 'Give them 50 sums a day; practice makes perfect.'
1. What do the four results, read together, say that students have and have not learned?
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Read them as one pattern: 29 of 36 can add same-denominator fractions; 7 write 4/8; 24 cannot place 3/4 on a number line; 19 order 1/5 above 1/3 by comparing denominators. Students have a symbol rule, not a sense of quantity. 4/8 is not carelessness - it is the only rule that makes sense if a fraction is just two numbers written above and below a line. The same reasoning produces '5 is bigger than 3, so 1/5 is bigger'.
Consequences: magnitude must come before procedure, and a fraction must be met as a length before it is met as a pair of numerals. Adding more sums makes the wrong rule faster and harder to unlearn. Also note which number is the real baseline - the 8 correct word-problem answers, not the 29 procedural ones.
2. Plan the six periods, using one box of teaching aids.
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- P1–P2, magnitude: fold strips of equal length into halves, thirds and quarters; build a fraction wall on the wall; place 1/2, 1/3 and 3/4 on a drawn number line. Exit: compare three pairs and justify each with the strip.
- P3–P4, equivalence (also the required topic): 'same length, different name' - 2/4 laid over 1/2 on the wall; then the multiplication rule, checked back against the strip.
- P5, addition: 3/4 + 1/4 with strips first, then symbols. Ask directly: 'Why does the answer have 4 in the denominator?' The seven students who wrote 4/8 must answer this aloud.
- P6, word problems: a fixed routine - draw, name the parts, write the number sentence.
Assess ten minutes each period with three items, marked the same day, so that P6 targets whatever is still wrong instead of what you assumed was wrong.
3. How do you answer the colleague who wants 50 sums a day?
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Agree with the principle, then be specific: practice fixes whatever is being practised, including 4/8. Fifty sums a day across 36 students is about 1,800 items - no teacher can read that, so nothing gets diagnosed and the same error is rehearsed at speed.
Eight items chosen to force a decision do more: compare 1/5 and 1/3; place 3/4 on a number line; draw 3/4 + 1/4; decide whether 2/4 and 1/2 are the same. The seven students who wrote 4/8 do not need repetition; they need a moment where their rule fails visibly - the paper strip that shows 3/4 + 1/4 is one whole strip, not four-eighths. Drill after that, on correct understanding, is exactly what you want.