One of the easiest ways to think critically about a quantitative problem is something I have never seen stressed in any math textbook: examining the units of measurement. I’ve blogged about this before. But once again the calculus homework set in front of me provides a good case in point.
The particular exercise I’m looking at now describes a population of bacteria growing, with n=f(t) being the expression which gives the size of the population after t hours. No formula, graph, or table is given. Students are asked to explain what f'(5) would mean and state the units. If you teach calculus, you know what’s coming: A good many students will say that f'(5) tells you the number of bacteria present after 5 hours. And then they will say that the units are in bacteria per hour.
Think about that. If you were working in the lab, and I popped in and asked you how many bacteria you had right then in the petri dish, would you say “Oh, about a million per hour“? No — an answer given in units of a rate can’t possibly be the right answer to a question about size. If a cop pulls you over on the interstate and says, “Son, I clocked you going about 80 miles back there,” why would that sound like nonsense? Because the units are off; you don’t measure rates in units of size, and vice versa.
The units of measurement in a problem are an invitation to think critically about the solution. If the units don’t match the description, either the solution is wrong or the units are; and since units are pretty easy to get right, it’s a red flag to go and debug your solution.