Salvatore Mosaico · 2026
Contents
- 1. A frequent answer: take the average
- 2. Let AI solve it
- 3. A simpler intermediate problem
- 4. Back to the original problem
- 5. Mathematical correctness and teaching choices
- 6. Why does the average fail?
- 7. Check the correct answer too
- 8. Another test
- 9. Automatic calculation without the least common multiple
- 10. The result is not our only goal
A teaching simulation inspired by “Calcolare tempo insieme”. The dialogue is adapted, not transcribed verbatim. AI correctly solved the original question using fractions; the incorrect average represents an answer the teacher often receives from students.
1. A frequent answer: take the average
Teacher: Aldo paints a room in 3 hours. Beniamino paints an identical room in 4 hours. How long do they take to paint one room together?
Assume they start together, maintain constant rates, can divide the work and do not obstruct each other. We consider the same complete job and ignore fixed preparation and drying times.
Anna: I would take the average: (3 + 4) ÷ 2 = 3.5 hours.
Teacher: I often get that answer. Before calculating further, describe what it means. Aldo takes three hours alone. He asks Beniamino to help, and now takes three and a half hours.
Luca: So getting help makes him finish later?
Teacher: Is that consistent with our assumptions?
Marta: No. If Beniamino contributes without getting in the way, they must finish in less than three hours.
Anna: Then the average cannot be the answer.
We can recognise that an answer is wrong before we know the correct one.
2. Let AI solve it
Teacher: Now let us ask AI.
AI: In one hour Aldo paints 1/3 of a room and Beniamino 1/4. Together they paint:
1/3 + 1/4 = 4/12 + 3/12 = 7/12
AI: One room takes 1 ÷ (7/12) = 12/7 hours, about 1 hour, 42 minutes and 51 seconds.
Davide: Less than three hours. It passes our first check.
Anna: Why add fractions instead of times?
Teacher: During the same time interval, their completed work adds up, not the times each would take alone.
Marta: The result is right, but I would like to understand without starting with fractions.
Teacher: Now our question is not only “what is the answer?”, but “which route makes the answer clear?”.
3. A simpler intermediate problem
Teacher: Present the data differently: Aldo paints 4 rooms in 12 hours; Beniamino paints 3 rooms in 12 hours. How many rooms do they complete altogether during those same twelve hours?
Luca: Seven: 4 + 3 = 7 rooms.
Teacher: How long do they take together for one room?
Marta: With a constant combined rate, divide time by rooms: 12 ÷ 7 hours per room.
Anna: The same answer without adding fractions.
Teacher: We have not eliminated the final division. We have avoided introducing fractions of a room and reciprocals immediately.
4. Back to the original problem
Teacher: We originally had three and four hours per room. Where do the four and three rooms come from?
Davide: Consider twelve hours, because twelve is divisible by both three and four.
| Painter | Hours per room | Rooms in 12 hours |
|---|---|---|
| Aldo | 3 | 12 ÷ 3 = 4 |
| Beniamino | 4 | 12 ÷ 4 = 3 |
| Together | To calculate | 4 + 3 = 7 |
Marta: We have reached exactly the simpler problem.
Teacher: Seven rooms in twelve hours: one room in 12 ÷ 7 hours.
Luca: Must they actually work for twelve hours?
Teacher: No. It is a convenient reference for comparing their working capacities. The requested room will be finished sooner, and we do not need seven real rooms. Our assumption that they can cooperate without obstruction lets us add their contributions to the same job.
5. Mathematical correctness and teaching choices
Teacher: In the original discussion, another AI was given the intermediate problem: four rooms and three rooms, both in twelve hours. It solved it correctly, but first calculated 12 ÷ 4 = 3 hours per room and 12 ÷ 3 = 4 hours per room. Then it returned to adding 1/3 and 1/4.
Luca: It turned the simpler problem back into the original one!
Teacher: Its method is valid, but does not immediately exploit the new presentation: both workers already have the same time interval.
Marta: We only needed to add four and three.
Teacher: For the learning route we want, yes. Fractions are not wrong: mathematical correctness and teaching choices are different aspects.
Anna: How could we improve the request to AI?
Build a lesson for students who are not yet comfortable adding fractions. Start with a common time interval, add the quantities of work, then return to the original problem. Do not begin with reciprocals.
Davide: We are asking not just for a result, but for a particular understanding.
6. Why does the average fail?
Anna: I calculated the average correctly. Why was it the wrong operation?
Teacher: It answers a different question: the average of individual times, not simultaneous working time. Imagine two painters who each take three hours. What does the average give?
Luca: (3 + 3) ÷ 2 = 3 hours.
Teacher: But under our assumptions, two equally fast painters finish in half the time.
Marta: An hour and a half. The average does not recognise the benefit of doubling the workforce.
Before choosing an operation, understand the relationship it must represent.
7. Check the correct answer too
Teacher: 12 ÷ 7 ≈ 1.7142857 hours. Beware: 1.714 hours does not mean 1 hour and 71 minutes. After the whole hour, 5/7 of an hour remains.
(5/7) × 60 = 300/7 ≈ 42.857 minutes
Teacher: About 1 hour and 43 minutes; more precisely, about 1 hour, 42 minutes and 51 seconds.
Davide: Can we check each worker’s contribution?
Aldo: (12/7) ÷ 3 = 4/7
Beniamino: (12/7) ÷ 4 = 3/7
4/7 + 3/7 = 1
Marta: Now the fractions describe something we already understand.
Teacher: We are not avoiding them forever: we are choosing when to introduce them.
8. Another test
Teacher: Aldo takes four hours for a job. Beniamino takes six. How long together? What do we know before calculating?
Anna: It must take less than four hours.
Luca: In twelve hours, Aldo completes three jobs and Beniamino two.
Marta: Five jobs in twelve hours together. One job takes 12 ÷ 5 = 2.4 hours.
Teacher: Multiply the decimal part by sixty: 0.4 × 60 = 24 minutes. So 2 hours and 24 minutes, not 2 hours and 40 minutes.
9. Automatic calculation without the least common multiple
Anna: Must I always find the least common multiple for an automatic calculation?
Teacher: No. Let a and b be the numerical values of the individual times in the same unit, here hours, with a > 0 and b > 0. Choose a × b hours as the reference interval; it need not be the smallest common interval.
Aldo: (a × b) ÷ a = b
Beniamino: (a × b) ÷ b = a
Together: b + a = a + b
Marta: During that interval, Aldo completes b jobs and Beniamino a jobs: a + b altogether.
Teacher: Divide the common time by total work:
t = (a × b) ÷ (a + b)
Teacher: This is the general formula for two workers under our original assumptions. It needs no least-common-multiple calculation and also works for positive non-integer times, provided work is divisible and rates are constant. Here a and b are numerical hour values: we are not treating hours squared as a time interval.
Davide: For four and six hours:
(4 × 6) ÷ (4 + 6) = 24/10 = 12/5 = 2.4 hours
0.4 × 60 = 24 minutes
Anna: We previously chose twelve hours; now twenty-four. The result is unchanged.
Teacher: Exactly. The comparison scale changes, not the completion time. For three and four hours, (3 × 4) ÷ (3 + 4) = 12/7 hours. If a = b, we get a/2, as expected. For positive a and b the result is always smaller than either individual time.
Marta: Can we automate it in a spreadsheet?
Teacher: If A1 and B1 contain positive times in hours, use:
=(A1*B1)/(A1+B1)Teacher: The result is still in decimal hours. The formula automates calculation, but does not check whether the assumptions fit the real job.
10. The result is not our only goal
AI supplied a correct solution. The teacher’s contribution was to challenge the average with a concrete objection, introduce an accessible intermediate problem, choose the order of explanations and finally derive a general formula suitable for automatic calculation.
AI can help us solve a problem. Educational collaboration also means choosing a route that allows students to understand it.