I remember very clearly the reaction that my classmates and I had one day when our teacher asked, "About how many ping-pong balls do you think would fit inside this room?" There was a general murmur of disbelief, but the repetition of the words "impossible" and "zillions" helped us penetrate the whispered conclusions that our teacher had "lost it for sure this time."
"More than a hundred?" he asked. Of course. "More than 10,000?" he went on. Certainly. "More than a million?" Well, probably - maybe a million, yeah. "Ten million? A hundred million? A billion?"Little debates began breaking out among various factions in the class concerning the upper limit of the pingpong-ball capacity of our room, which was precisely what our teacher had wanted. What we needed, he said, was some plan by which we could estimate this "unknowable" number with enough precision to put us at least in the right numerical ballpark. He finally educed from us the idea that if we could just estimate the number of pingpong balls that would fit along one wall, then the number along the adjoining wall, and the number from floor to ceiling, the volume of the room would be the product of our three estimates.
The ability to make reliable estimates is one that can benefit children in their school work and throughout their adult lives as well. Exercises in estimating the magnitude of enormous distances or weights or volumes may seem to be wholly unrelated to a child's understanding of the real world, but these exercises require children to bring different pieces of outside information and knowledge to bear upon each different problem. Our ping-pong ball estimate, for example, would have been impossible without the knowledge that volume
length x width x height. This not only reinforces the meaning behind such an equation, but it provides an immediate and useful application of the concept as well.
Would a stack of a million pennies reach to the top of the Empire State Building? Just look at the several pieces of information - the supposedly "common knowledge" - that children must put to use in making this estimate. A roll of pennies is about 3 inches long; there are 50 pennies in a roll; there are 12 inches in a foot; the Empire State Building is 1,250 feet tall. (A stack of a million pennies would reach four times the height of that building.)
Other fantastic problems (and the more fantastic the better, I think) can require children to call up, or look up, additional pieces of knowledge in order to make their estimates - the circumference of the earth, the number of feet in a mile, the distance to the moon, the length of a football field.
These exercises in estimating mammoth amounts are ideal for use in the home precisely because they have no "correct" answer. What parents should focus on is their children's plan for arriving at a reasonable estimate. "How should we go about figuring this out?" is much more important than "What is the answer?" You might have your children draw circles or boxes on a piece of paper to represent the information that they need, then map out what they would do with that information if they had it at hand. Last of all, where do they get that information to plug into their plan? What do your children think their dollar value would be if they were "worth their weight in gold"? More than a thousand dollars? More than a million? More than a billion?