Here’s a little idea I just thought of, that I’d like to try. I’m always thinking about ways to get students to ask “does this make sense?” “Is this answer reasonable?” Well, what about putting that into the statement of the question? So for example, take this typical calculus problem:
A plane flies horizontally over a radar station at an altitude of 1km at a speed of 500 km/hr. How fast is the distance to the radar station changing when the plane is 2km away from the radar station and heading away from it?
…and replace it with this one:
A plane flies horizontally over a radar station at an altitude of either 1.00m, 1.00km, or 1000km (whichever makes sense) at a speed of either 5.00 km/hr, 500 km/hr, or 50,000 km/hr (whichever makes sense). How fast is the distance to the radar station changing when the plane is either 2.00m, 2.00km, or 20,000km away from the radar station (whichever makes sense) and heading away from it? Before doing any calculation, (a) what units will the answer have, (b) will it be positive, negative, or zero, and (c) is it likely to be in the range [-1000,-500],[-500,0],[0,500],[500,1000], or none of the above? Justify your reasoning. Then analyze the problem carefully and give a numerical answer, to appropriate precision.
A more creative option than the “multiple-choice question” format above would be to have every student (or every group) make up their own problem, making sure that they can justify the reasonableness of the numbers they put in. But that might be unwieldy, with many different answers that would be hard to check and hard to have a whole-class discussion about.
We’ll see if I can find time to put this into practice.