Even lotteries have winners

Lucky Larry hits a grand slam

It has been noted that state lotteries are basically a tax on innumeracy. You're better off dealing with the house percentage at a casino in Nevada. Nevertheless, even lotteries have winners. Just don't expect it to be you. Lightning is not going to strike you.

Of course, sometimes it strikes near by.

One of my students won the calculus lottery during the exam on integration, beating very long odds indeed. The result was the most bizarre “Lucky Larry” of my years as a math teacher. My colleagues were as flabbergasted as I was when I shared the student's “solutions” with them. Her work was nonsense, yet her answers were correct. Three times in a row. Of course, when that happens one suspects a hidden underlying pattern that produces valid results, contrary to all expectations. In this case, though—no. It was a giant fluke.

Or, rather, three flukes in a row. My flabber, she is as gasted as possible.

The problem on the integration exam was one of my “conceptual” exercises. One of my tasks as a calculus teacher is to clarify the meaning of the definite integral, ensuring that my students grasp its significance. Of course, one of the most common (and visual) interpretations of the definite integral is as the area under a curve. Surely any first-year calculus student must understand at least that much.

Accordingly, I presented my students with the graph of a simple function and asked them to evaluate three definite integrals of that function by inspection of the graph. I did not forbid them to use antidifferentiation and the fundamental theorem of calculus, but I emphasized that the simplest of calculations would suffice.

What, pray tell, is the value of the definite integral of f(x) from x = 1 to x = 2? A cursory examination of the trapezoidal region spanning the space between the x axis and the graph of the function reveals the area (and thus the definite integral) to equal 1.5. Easy! Not satisfied, however, with such a trivial computation, one of my students rolled out the big guns:

Damn! What a coincidence! The calculus is bogus, but the result is accidentally correct. No one expects lightning to strike twice, of course.

Brace yourself.

What if we ask for the definite integral from 1 to 3 instead? We get a little more area now. Take a look at the new graph, in which a second trapezoid now joins the first. We get an additional 2.5 square units which, added to the original 1.5, gives us 4. My student swung into action and unlimbered her surreal calculus calculation again:

I was now quite beside myself, shaking my head in astonishment as my red pen hovered over the page. Twice in a row! (What were the odds?)

Fortunately, I knew that I could count on part (c) to set the record straight and demonstrate to my student the error of her ways. It was, in fact, the simplest part of the problem. A kind of gift to the student possessing a clue. Can you find the area of a rectangle measuring 2 by 4? Of course! The answer must be 8.

My student presented her solution:

Nooooooooo!

Time to hit my head against the desk a few times.

In a million years, this will never happen again. (For one thing, this problem is going straight into the waste can, never to be recycled.)

I need to go lie down for a few minutes.