Twelve Workers, Twenty Days, and Three Answers
A Class 7 teacher poses: '12 workers build a wall in 20 days. How many days will 15 workers take, working at the same rate?' Of 36 students, 19 answer 16 days, 11 answer 17 days, and 6 answer 25 days. In discussion, the 17-day group explains: '15 workers are 3 more, so the wall will take 3 days fewer.' The 25-day group says: '15/12 × 20 = 25, because more workers means more days.' The teacher has one 40-minute period before the unit test. The textbook presents the unitary method: find the work done by 1 worker, then divide. A bright student asks why we cannot simply use the ratio 12:15 = 20:x. The teacher must decide which explanation to lead with, how to use the three answers, and what to put on the board so that the test does not reward memorised steps without understanding.
1. Diagnose the three answers. Which is correct, what reasoning error do the other two groups make, and why does the phrase 'same rate' matter?
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The correct answer is 16 days. The total work is 12 workers × 20 days = 240 worker-days. With 15 workers, 240 ÷ 15 = 16 days.
The 17-day group uses additive thinking: 15 is 3 more than 12, so they subtract 3 days. That would be valid only if one extra worker saved exactly one day regardless of the total, which is not how rates work. The 3-worker increase must be scaled against the original 12; the saving is not 3 days but 4 days.
The 25-day group uses direct proportion: 15/12 × 20. They correctly see a ratio but put the quantities in the wrong order. More workers should mean fewer days, so the ratio must be inverted.
'Same rate' matters because it fixes the product: each worker contributes the same amount of work per day, so workers × days is constant. Without that assumption, the problem has no unique answer. The teacher should make the constant product visible, not just the division step.
2. The bright student asks, 'Why can't we use 12:15 = 20:x?' Explain whether the equation is valid, how to correct it, and how to connect the ratio method to the unitary method.
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The equation 12:15 = 20:x is not valid for this inverse relationship. It pairs 12 workers with 20 days and 15 workers with x days, but it treats both quantities as increasing together. Since more workers shorten the time, the days must be placed in the opposite order. Correct forms are 12:15 = x:20, or 12/15 = x/20, which give x = 16; equivalently 15:12 = 20:x.
The unifying idea is the constant product: 12 × 20 = 240 worker-days, and 15 × x must also equal 240. The unitary method is one route: 240 ÷ 15 = 16. The ratio equation is another route to the same invariant. The teacher can write both on the board and ask the student to show where the 'inverse' appears in each. This validates the student's question while correcting the equation, and it prevents the test from rewarding a memorised 'cross-multiply' that works only for direct proportion.
3. Design a 40-minute board sequence and a 5-minute exit ticket that would reveal whether students are still using additive reasoning.
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Start with 5 minutes on a familiar contrast: 'If 2 workers take 10 days, what do 4 workers take?' Let students commit to 5 days or 8 days, then discuss. This surfaces the additive intuition before the formal method.
Spend 10 minutes building a table: workers 2, 4, 6, 12; days 60, 30, 20, 10; product 120 each time. Ask what stays constant. Spend 10 minutes on a rectangle diagram: area = total work, one side = workers, the other = days; doubling one side halves the other. Spend 10 minutes comparing the unitary method and the corrected ratio equation on the original problem. Reserve 5 minutes for the exit ticket.
Exit ticket: '8 workers take 15 days. How long will 12 workers take? Explain why the answer is not 11 days.' A student still using additive reasoning will write '4 more workers, so 4 fewer days'. A student with understanding will use 120 worker-days and get 10 days, and will explain that the saving must be scaled. The board should end with the invariant W = workers × days, not only the steps of the unitary method.