Most students are familiar with the “dirt equation“: D = R x T. But applying this simple equation to a nontraditional problem leaves most kids (and adults) wondering what happened.
Take the case of average speed – if you take a round trip going in one direction at 20 m.p.h. and returning at 30 m.p.h., the large majority of people will figure that the average speed is 25 m.p.h.
But this is incorrect – the actual average rate is 24 m.p.h. The calculation is quite simple, but it takes logical, step-by-step thinking; this is the specialty of mathematics. Let’s take a concrete case. Suppose there are two towns, A and B, 60 miles apart.
Then the trip from A to B at 20 m.p.h. takes 3 hours, but the return trip at 30 m.p.h. takes only 2 hours. So since our definition of average speed is Total Distance ÷ Total Time, we get 120 miles ÷ 5 hours, or 24 m.p.h.
This disturbs students the first time they see it. This is actually a good thing because you will then have their attention when explaining the seeming logical fallacy. In simple terms, what happens is that the traveler spends more time moving at the slower speed than at the faster speed, thus causing the average rate to be closer to the slower speed.
When explaining this concept in class I have had to go over this scenario several times with differing rates and distances until the kids get the idea. In each case, the average rate of speed is less than simply taking the average of the two rates.
This makes a wonderful project for class. I’ve included two for you to look at. At conferences and workshops teachers report that the student projects I share are one of the highlights – they like to see what students are capable of doing, and they like doing the projects back at their school with their own kids. I’ve trained my students with some simple concepts to do projects on the computer. After they see the basic ideas of formatting their work, their creative juices take over and the results are really quite nice. I hope you and your students enjoy them; if you have a projector in class you can display them for students to see and discuss as a source of curriculum.
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