Three-Eighths and the Missing Model: Fraction Division in Class 8
In a Class 8 mathematics class of 45 in Kozhikode, the teacher tests fraction division with five items. Twenty-eight students correctly compute 3/4 ÷ 1/2 = 3/2 using 'multiply by the reciprocal'. Twelve write 3/8, multiplying numerators and denominators. Five write 3/2 but cannot explain why. On the word problem 'A 3/4-litre bottle is filled from a 1/2-litre cup; how many cups are needed?', only 9 students choose division, 19 choose multiplication and 17 leave it blank. The next chapter is rational numbers, and the teacher has two periods before the monthly test. The teacher must decide whether to reteach fraction division from Grade 5 or move on.
1. Diagnose the misconceptions. Why is 3/8 a reasonable error, and what does the word-problem pattern show?
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The 3/8 error is not random. The student applies the fraction-multiplication rule to a division sign: 3/4 × 1/2 = 3/8. The rule 'multiply numerators, multiply denominators' has been over-generalised because the surface form (two fractions and an operation) looks the same. The student also holds the whole-number intuition that 'division makes smaller' - and 3/8 is smaller than 3/4. Two faulty ideas reinforce each other.
The 3/2-but-cannot-explain group shows instrumental understanding: the procedure is memorised without a referent. They will pass a routine test and fail a novel one.
The word-problem pattern is the key diagnostic. Only 9 of 45 choose division. Nineteen choose multiplication because 'of' and 'fractions' cue multiplication, and because 'how many cups' has not been mapped to quotative division. Seventeen leave it blank, which indicates avoidance, not mere carelessness. The class has a procedure without a model.
Decision: do not move on to rational numbers yet. Two periods are enough to rebuild a meaning-based model for division, then connect it to the reciprocal rule. Moving on would carry the misconception into rational numbers, where it will be harder to dislodge.
2. Design a two-period reteach sequence using representations, including the quotative and partitive meanings of division.
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Period 1: Quotative division with a bar model (40 minutes)
- Warm-up: 'How many 1/2-litre cups in 1 litre?' Students shade paper strips and count. Answer 2, then 3 for 1½ litres.
- Core task: 3/4 ÷ 1/2. Each pair shades a 3/4 strip and marks half-litre jumps. Count the jumps: one full half, then half of a half, so 1½ or 3/2. Write the equation beside the diagram.
- Language: 'How many groups of 1/2 are in 3/4?' not 'divide 3/4 by 1/2'.
- Number line: the same jumps on a 0–1 line, showing the answer is greater than 3/4. This directly confronts 'division makes smaller'.
- Exit ticket: one bar-model item with a written sentence explaining the answer.
Period 2: From model to reciprocal (40 minutes)
- Revisit the exit tickets; display one 3/8 response and ask the class to find the error using the bar model.
- Partitive division: '3/4 litre shared equally into 3 bottles' - how much in each? Contrast the question with the quotative one.
- Pattern table: 3/4 ÷ 1/2, 3/4 ÷ 1/3, 3/4 ÷ 1/4; students predict and verify with strips, then notice the 'multiply by the reciprocal' pattern and explain why it works.
- Word-problem sort: eight cards; students classify as multiplication or division and justify with a diagram before computing.
Homework: two problems, each requiring a diagram and one sentence.
3. Write two assessment items that distinguish procedural from conceptual understanding, with expected responses.
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Item 1 - Conceptual justification (4 marks)
Without using the rule 'multiply by the reciprocal', explain why 3/4 ÷ 1/2 = 3/2. Use a diagram or a story.
Expected strong response: A bar showing 3/4 divided into halves; one full half and a further half of a half; answer 1½. Or a story: 'A 3/4-litre bottle filled with a 1/2-litre cup needs one full cup and half of another.'
Misconception signal: draws 3/4 × 1/2 or writes 'because we invert the second fraction' with no referent.
Item 2 - Is the answer reasonable? (3 marks)
Ravi says 3/4 ÷ 1/2 must be less than 3/4 'because division makes numbers smaller'. Is he right? Explain using a number line or an example.
Expected strong response: Ravi is wrong here because the divisor is less than 1; there are more than one half in 3/4, so the quotient 3/2 is larger than 3/4. Reference to the number of groups, not to the operation word.
Misconception signal: agrees with Ravi, or gives the rule without a counterexample.
Marking note: Item 1 rewards a representation; Item 2 rewards a comparison. A student who computes both correctly but explains neither should score at most half - that is precisely the distinction the teacher needs before the monthly test. Use one familiar and one unfamiliar context to check transfer.