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Number Lines and Circles

Even at a very early age children can understand the significance of addresses. Anyone can appreciate the ordinary algorithm used for finding a specific location in terms of its street address: first go to the street, then find the number of some building. If it happens to be the one you are looking for, you are done. If not, go to a nearby building and check its number. If it is closer to the one you want, keep going in that direction. If it is farther away, go in the other direction. Stop when you get to the number you want.

Discussion of even this simple algorithm illustrates a number of important topics. We identify a location by a specific number, and we move along a one-dimensional path, in one direction or another, to get from one position to another. After the basic procedure is understood one can add refinements such as whether the address is on the even or odd side of the street. Estimating the distance one has to travel in order to get from one location to another is another refinement, leading to the geometric interpretation of subtraction as well as to the notion of absolute value.

Early exercises can take place on a number line with positive addresses or on real streets in a scavenger number hunt. Later the same notions can be used for scales with negative values, like temperature, where the vertical orientation of the thermometer emphasizes the directionality. "What happens when the temperature goes from 65 degrees to 40 degrees?" "It goes down by 25 degrees." Such observations can take place far in advance of introducing signed numbers.

Many cities use directional addresses in their street plan (e.g., in New York City there is both a West and an East 42nd Street). In this case the algorithm to find a building from knowledge of its address is slightly different but still easy enough to discuss at an elementary school level. The distance between two addresses on the same side is determined as usual, while the distance between two locations on different sides is the sum of their addresses. No memorization is required for such a statement! Signed numbers do not have to be mysterious.

A similar one-dimensional algorithm works in setting a clock, whether it is analogue or digital, depending on whether or not one can go backward as well as forward. Setting a watch is slightly different from a street address, even on a curving road that does not come back on itself. On a circular drive, however, the problem of locating a specific address is analogous to the problem of setting a watch: you can go in either direction and ultimately arrive at your destination. Of course one direction might be much easier than the other.

The problem of deciding on a strategy for locating an address on a circular drive is a good example of the kind of multistep problems that students should learn to attack. In this example, as in many others, there is no single answer — there are several strategies that will achieve the same result. The person facing the situation must decide first what the choices are and then what might be the advantages of each. The aim of minimizing effort is very easy to understand, easier than minimizing cost measured in money or some other quantity.

One-dimensional examples require just one number to locate any point. Directions for moving from one position to another are also one-dimensional: "Go three houses to the right" or "Go around counterclockwise five spaces" or "Go halfway around the circle to the opposite point." This last sort of instruction depends on the size of the circle and can form the beginning of an appreciation of angular measure.

Setting a clock, whether analogue or digital, provides an excellent example of "wrap-around". This phenomenon can also be viewed on a linear scale, for example, on the selector of many car radios. In many analogue devices the moving indicator stops at the extreme left or extreme right, while in the digital versions the indicator simply goes from the top value to the bottom. Finding a particular radio station then presents two different sorts of problems depending on the nature of the radio selector.

The dimensionality of gauges is an important concept that arises over and over again in mathematics as well as science. As students become more sophisticated in the kinds of numbers they use, they can introduce fractions or decimals into number lines and number circles. Locating a telephone pole along a road in a rural area requires a different kind of address, using fractions or real numbers representing actual distances. The numbers become more complicated, but the procedures remain the same.

Locating objects or addresses in a one-dimensional world can be accomplished efficiently by the bisection algorithm (or the variation of it that divides each interval decimally), a procedure with almost universal significance that is related, for example, to the informal technique used to find phone numbers. First you make a guess to divide your problem into two parts — by opening the phone book or by picking a number. Then you compare your guess with what you want and make a new guess that is in whichever part (above or below your first guess) that contains what it is that you are looking for.

A similar scheme can be used to find the "address" of the length of the diagonal of a square without requiring a calculator with a square root key. Finding the decimal equivalent of a fraction can be viewed as a more sophisticated version of the one-dimensional address problem. If we want to find 3/17, we can multiply different decimals by 17 to see if the product is bigger or smaller than 3. All decimals get put into one category or the other: it never comes out even. For 3/16 on the other hand one decimal does come out even, so there is a fixed location on the decimal line for the solution to this problem. [an error occurred while processing this directive]