Class 5 fractions: why 1/3 beats 1/2 and 2/3 + 1/4 makes 3/7
Class 5 mathematics, 40 children. After a fractions unit the teacher marks 40 scripts. Common answers: 2/3 + 1/4 = 3/7 from 21 children; 3/4 of 8 answered correctly by 28 children but as '6/32' by 9 others; comparing 1/2 and 1/3, 19 children say 1/3 is larger because 3 is greater than 2. Yet when shown one paper strip folded into halves and another into thirds, most children correctly say the half-strip piece is bigger. The textbook now moves to adding unlike fractions with denominators up to 12, and the annual examination is in six weeks; the school expects the syllabus to be finished. The teacher has paper, scissors, a blackboard, and a number line printed on the classroom wall.
1. Diagnose the pattern. What do these children hold, what do they lack, and is the error procedural or conceptual?
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Three distinct patterns sit in the numbers.
- 2/3 + 1/4 = 3/7 (21 children) is a whole-number procedure transferred to fractions: add the tops, add the bottoms. It is a procedural error produced by the absence of a magnitude concept - the child has no way to see that 3/7 is smaller than 2/3, so nothing checks the answer.
- 1/3 > 1/2 because 3 > 2 (19 children) is the same whole-number bias applied to comparison.
- '6/32' is a rule applied to 'of' with no meaning attached.
The paper-strip result is the key evidence: most children can see that the half piece is bigger than the third piece. So the concept is available when the material is concrete; the failure is representational. They can compare with folded paper and they can compare with symbols, but the two have never been connected in their heads. That is a teaching-sequence gap, not a developmental limit, and moving straight to unlike denominators will multiply all three errors rather than cure them.
2. Design a three-lesson sequence that addresses the 1/3 > 1/2 error and say what evidence you would look for at each stage.
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Lesson 1 - concrete comparison. Pairs fold two equal strips, one into halves and one into thirds, shade one part of each, then lay the thirds strip over the halves strip. Record '1/2 > 1/3' on a class chart. Repeat with 1/4 vs 1/3 and 3/4 vs 2/3. Evidence: children say which is bigger and point to the overlap without being told.
Lesson 2 - the number line. On the wall line from 0 to 1, children place their folded strips at the matching points; then mark halves, thirds and quarters together and see that 1/3 sits to the left of 1/2. Ask: which is bigger, 2/3 or 3/4? Require a strip or a point on the line as evidence. Evidence: children place an unseen fraction correctly without folding.
Lesson 3 - symbols, then the addition. Only now: 2/3 + 1/4 with the strips, replacing each with twelfths, folding and counting 8/12 + 3/12 = 11/12. Check on the line that 11/12 is close to 1, so 3/7 cannot be right. Evidence: a child who rejects 3/7 unprompted and reaches for the line or a strip to show why.
3. The head teacher wants to skip the sequence and push on to unlike denominators. Mount the argument for the sequence, and offer a workable compromise.
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Argue in the currency the head teacher cares about: examination performance, not pedagogical purity.
Say plainly: 21 of 40 children currently add across; if we move on to unlike denominators now, those children will produce fifteen wrong answers per paper in the annual examination, and the class average will fall. Three lessons on magnitude pay for themselves within six weeks. Also note that the paper-strip evidence shows the children are close - this is a short repair, not a restart.
Compromise on offer: run the sequence in week one, and in parallel keep a short daily drill in the previous unit's exam format so the syllabus record shows movement. Agree a checkpoint at the end of week two - a five-item quiz; if 30 of 40 children can correctly compare 2/3 with 3/4, the sequence has done its job and the remaining content can be compressed into the time saved. Record the decision and the quiz result. That is defensible in a staff meeting and honest with the children.