Reasonable guesses

Originally published 12 June 1989

Prob­lem: A per­son wish­es to build a square house with an area of 500 square feet. What should be the length of the side of the house?

That’s easy. Get out the cal­cu­la­tor. Punch in 500. Push the square-root but­ton. The answer: 22.36067977 feet.

The Cal­cu­la­tor Mon­ster has struck again!

Every teacher will rec­og­nize the mon­ster’s track. Answers that run to 10 dig­its, most of them meaningless.

OK kids, put away the cal­cu­la­tor. Now fig­ure out the answer? 20 times 20 is 400. Too small. 25 times 25 is 625. Too big. Split the dif­fer­ence. Try 22.5 feet for the side of the house. That’s 506 square feet. Close enough. The per­son who lives in the house will nev­er know the difference.

Now there’s noth­ing wrong with cal­cu­la­tors. The cal­cu­la­tor is prob­a­bly the great­est boon to quan­ti­ta­tive think­ing since the inven­tion of Ara­bic numer­als. Every kid should own one and know how to use it.

When I went off to col­lege to study engi­neer­ing my dad gave me his well-worn K+E slide rule. A thing of beau­ty. A pre­ci­sion instru­ment. “Wear it with pride,” he might have said. And off we went to class, engi­neer­ing nerds, slip-sticks dan­gling at our sides.

If some­one had told us then that kids today would car­ry in the palm of their hand a device cost­ing less than my slide rule that could do arith­metic and a host of high­er func­tions instant­ly and accu­rate­ly (to ten sig­nif­i­cant fig­ures!) we would have said, “impos­si­ble.” In those days we were just begin­ning to use gigan­tic com­put­ers that were bare­ly as pow­er­ful as today’s hand-held calculators.

Judicious approximation

But slide rules had one advan­tage over cal­cu­la­tors: They round­ed off by neces­si­ty. They lent them­selves to back-of-the-enve­lope cal­cu­la­tions. They encour­aged judi­cious approximation.

Yes, every kid should own a cal­cu­la­tor and know how to use it. But kids should also be taught the art of round­ing off. And of mak­ing rea­son­able guess­es. Too much pre­ci­sion can some­times obscure under­stand­ing. A lot of good sci­ence is done with a vocab­u­lary of “let’s assume,” “more or less,” and “to a first approximation.”

In one episode of Philip Mor­rison’s tele­vi­sion series The Ring of Truth, Mor­ri­son makes a clever order-of-mag­ni­tude mea­sure­ment of the size of a mol­e­cule by watch­ing oil spread out on a pool of water. One tea­spoon of oil poured on water will spread to cov­er half-an-acre. Assume that the oil slick at max­i­mum extent is one mol­e­cule thick; how big is a molecule?

Every high-school stu­dent should be able to solve this prob­lem, but, alas, few will have the con­fi­dence to try. It helps in solv­ing prob­lems like this to think met­ri­cal­ly, and of course the Unit­ed States is the only coun­try in the world (more or less) that con­tin­ues to hob­ble the brain with inch­es, ounces, tea­spoons, and acres.

The first thing an enter­pris­ing, met­ri­cal­ly-inclined stu­dent will want to know is how many cubic cen­time­ters there are in a tea­spoon and how many square meters in an acre. So, make rea­son­able guess­es. The side of a sug­ar cube is about a cen­time­ter; how many crum­bled sug­ar cubes would fill a tea­spoon? How big is the lot your house sits on, in acres? An adult’s pace spans about a meter; how many paces to walk the length and breadth of your lot?

What else do we need to know? The for­mu­la for the vol­ume of a sphere. Some junior-high geom­e­try. A lit­tle quick pen­cil work on the back of an enve­lope and—voila!—we have the size of an oil mol­e­cule. And not a bad esti­mate at that.

Now, kids, for a fol­low up ques­tion. Assume that the 240,000 bar­rels of oil spilled from the Exxon Valdez spreads out into a slick of max­i­mum extent. What is the area of the slick? How many bar­rels of water are there in the world’s oceans to dilute the oil? Assum­ing com­plete mix­ing, how many Exxon Valdezes would have to emp­ty their con­tents into the oceans to con­t­a­m­i­nate the water to one part in a bil­lion with crude oil?

Now for the quiz

There’s no end to the instruc­tive ques­tions stu­dents can answer with noth­ing but rea­son­able guess­es and rough-and-ready cal­cu­la­tions. How many trees must be cut down to sup­ply the junk mail that enters Amer­i­can house­holds each year? How large a land­fill would be required to receive the junk mail that enters your town in ten years? How long would it take for all the cars and trucks on Earth to cycle the entire atmos­phere through their engines? If a warm­ing cli­mate melts the mile-thick ice caps on Green­land and Antarc­ti­ca, by how much will the lev­el of the the oceans rise?

I would be pleased to hear from high school class­es who can pro­vide back-of-the enve­lope answers to these ques­tions. Don’t look up any­thing you can rea­son­ably guess, and watch out for those insignif­i­cant fig­ures. Cal­cu­la­tors (or slide rules) are per­mit­ted, but only as a tool. Microchips are no sub­sti­tute for brains.