While returning homework at the start of the class I discovered that I mislaid one student’s paper. I apologized, said I’d look for it, and suggested that it might be a good idea (given my occasional carelessness) for students to keep a copy of their homework. A student asked jokingly if we should all trek to the public copy machine so they could make copies of the one due today. I said no – that I’d be careful. But that led me to wonder out loud how much photocopying there was on campus – a Fermi problem I said we could figure it out. The class was skeptical. I noted my quandary (and defined the word for those who needed it) – should I proceed with the class I’d prepared, or answer the how-much-photocopying question? I decided to do both, starting with unit calculations (prepared). Told the class to interrupt me when we were 20 minutes from the end.
I didn’t follow Chapter 4 of the text closely – they can read that. I began with currency conversion – first changing a particular number of euros to dollars, then thinking about the conversion rate, in euros/dollar. (Can’t write the fractions here with a horizontal bar, but told the students to whenever they could.) Then I talked about rates in other examples – miles/hour, hours/mile (better converted to minutes/mile), miles/gallon and the reciprocal gallons/mile (better converted to gallons/(100 miles). (That anticipates a homework problem assigned for next week.) With those examples, I returned to dollars/euro.
I had hoped to do fuel cost, converting $/gallon to euros/liter, thus introducing the metric system, but moved instead to the Fermi problem. A quick poll showed that roughly half the class thought the answer was on the order of 10oo reams/week, half on the order of 10,000 reams/week. When a student asked if we were thinking about the whole campus, or just the students, I answered that it was up to us to decide which question we were going to try to answer. We opted for just students. I was then surprised (and disappointed) when we redid the poll and the estimates didn’t decrease.
To solve the problem someone suggested that we start from the fact that there are about 13,000 students. Someone else then chipped in the information that each student got 2oo free pages of copying per semester. I hadn’t known that, so the method I’d had in mind was much more complicated than what we came up with – essentially
13000 students * 200 (pages/student)/semester
= 26(5 zeroes) pages/semester
= 26(5 zeroes) pages/semester * (1 ream)/(500 pages)
= (26/5)(3 zeroes) reams/semester
= 5,000 reams/semester
I made two observations along the way.
First, formally multiplying by the number
1 = (1 ream)/(500 pages)
is better than “divide by 500″ since it does not require figuring out whether to divide or multiply.
Second, in the last calculation I suggested 26/5 = 5. The extra little bit is meaningless in this kind of estimation, so round dramatically to keep the arithmetic really easy.
Then we needed to convert 5000 reams/semester to reams per week. That’s just multiplying by 1 = (1 semester)/(14 weeks):
5000 reams/semester * (1 semester)/(14 weeks).
I tried to say that 5/14 was about 1/3, so the answer was about 1/3 of 1000 reams/semester, or 300 reams/semester. The class wasn’t comfortable with quite that much gymnastics, and actually worked out 5000/14. The answer was the same, of course (that is, the same order of magnitude). We all agreed that the correct answer might be 150 reams/week, or 600 reams/week, but probably not many thousands of reams per week – we have the right number of zeroes. (I think our answer is probably high, since when polled most of the students said they didn’t use their 200 free copies. Very few said they paid for extra. And reams/week is probably higher at the end of the semester when people are printing term papers. I offered to try to find out the answer.)
Just as the class was ending another student suggested we might have done the problem by estimating the number of public copy machines, how fast they worked and how busy they were. I agreed, but we had no time to follow up.
I think this was a good exercise to do. We started with a question that seemed impossible – “no where to start” – and found an estimate that everyone agreed was reasonable. The fact that we could do that was eye-opening. I hope the belief in the process lasts.
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