Three Responses to 4 1/2 ÷ 3/4
Ms. Okonkwo's Grade 8 class is working on the problem 'How many 3/4-cup servings are in 4 1/2 cups of rice?' She posts three responses. A: sketches four and a half circles, tries to ring groups of three-quarters, loses track, and answers 4. B: writes 3/4 ÷ 4 1/2 = 1/6 and says that is the answer. C: writes 4 1/2 × 4/3 = 6 and says 'you flip the second one.' There are 27 students, nine of whom are still building English vocabulary for fractions, and the period ends in 25 minutes. A short quiz on fraction operations is scheduled for tomorrow; it was written by a colleague who expects the standard algorithm.
1. Identify the conception behind each of the three responses, then decide which response should anchor tomorrow's teaching given the quiz constraint. Justify your choice.
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- A is treating 3/4 as a countable object rather than a measure of a serving. Losing track signals weak unitizing: the student cannot yet coordinate 'how many groups' with 'how many quarters in each group.'
- B inverted the wrong quantity and produced a part-whole fraction (what fraction of the 4 1/2 cups one serving is). More telling than the inversion is that 1/6 was accepted as a serving count. There was no referent check: 4 1/2 cups must hold at least four 3/4-cup servings.
- C has the algorithm but no access to its meaning; remove the rule and the student is at A's level.
Anchor on B's missing referent check, paired with A's diagram. The quiz rewards exactly what C already has, so reteaching the algorithm would waste the lesson and would not help A or B. What A and B share is the meaning of the divisor as the size of the group. A quick estimate ('more than four servings, so 1/6 is impossible') fixes B's habit and gives A a target to check against. C's answer then becomes the object of study rather than the lesson's goal.
2. Sequence the three responses plus one additional representation into a 15-minute consolidation. Justify the order and say what you would deliberately leave out.
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Order: A, then a new representation, then B, then C.
- (4 min) Pairs repair A's sketch on a strip diagram. Concrete, language-light, and it makes 'groups of three-quarters' visible - the entry point for the nine students still building vocabulary.
- (4 min) Introduce the quarters count: 4 1/2 cups = 18 quarters; each serving is 3 quarters; 18 ÷ 3 = 6. This is new, not on the board, and it bridges picture and symbol with whole numbers the whole class owns.
- (3 min) Return to B: 'A serving count of 1/6 - what would that mean about our 4 1/2 cups?' Students reject it by estimation against the strip.
- (4 min) Write C's equation beside the quarters count and ask what the 4/3 is doing.
Leave out a general lesson on reciprocals and any drill on 'keep-change-flip.' The order moves from image to whole-number reasoning to symbol, and each step answers a question raised by the previous one rather than a rule the teacher supplied.
3. Student C asks why 'flipping' works. Write the explanation you would give at Grade 8, and name one thing you would not say.
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I would go back to the quarters count, because C already trusts it.
'We found 6 by turning both quantities into quarters: 4 1/2 = 18/4 and 3/4 stays as it is. Then 18/4 ÷ 3/4 asks how many groups of 3 quarters fit in 18 quarters - that is just 18 ÷ 3 = 6. The denominators cancelled because they were the same.'
Then the link to the symbol: 'Multiplying by 4/3 does the same job in one step, because 4/3 and 3/4 multiply to 1, and dividing by a number is the same as multiplying by its reciprocal.'
I would not say 'just flip the second one, that's the rule,' and I would not say 'flip the numbers' without naming that the second quantity is the divisor. C's original statement was correct by luck of position - B flipped the wrong one - and a rule stated without the divisor's role is what produced the confusion in the first place.