MAM2000 (Essays/Dimension)

# Lengths and Perimeters

The fundamental geometry problem for one-dimensional phenomena is the determination of distance along a path. Key examples include calculation or comparison of perimeters of curves and polygons. There is one geometric number — p — that all students should learn to understand.

Despite its universal significance, most people do not know how to answer when you ask what p is. Most lay persons respond with a numerical estimate, 3.1416 or 22/7, without knowing in either case whether this approximation is too large or too small. Mathematicians will give a definition in terms of a geometric property, usually something like "the ratio of the circumference of a circle to its diameter" or "the ratio of the area of a disc to the area of a square with side equal to the radius." The fact that these two ratios are the same is, of course, a major theorem of mathematics. One can get a tremendous amount of mileage out of a continuing discussion of the estimation of p, from the first time a kindergarten student realizes that the belt around a can reaches a little more than three times across the top, to second-semester calculus where one studies integrals for arc length.

Finding the circumference of a circle is a one-dimensional problem, so its answer should have a representative on the number line. But where is it? How can we determine whether or not a given number is less than this length or greater? Comparisons with the circumference of circumscribed and inscribed polygons is an effective strategy for dealing with these questions. Although such comparisons cannot determine p exactly, they can convincingly show whether 22/7 is slightly above or slightly below p.

Certain counting games are especially important for developing in children facility in the arithmetic of algebraic quantities. Students can choose instruction cards saying "move forward two spaces 11" or "back three" (`T2` or `B3`), and they can follow the instructions with counters. Then they can be asked to trade two cards for a single card that accomplishes the same effect. By considering double or triple jumps, they gain experience with the idea of multiplying a signed number by a positive integer. The variations in the game are manifold. The operation of taking up three `B4` cards from one's hand is the same as taking up one `B12`; putting down three `B4`cards is the same as taking up one `F12`. One might introduce a symbolism: `P3B4` = "Put down three `B4` cards," which is the same as `T3F4` = "Take up three `F4` cards." Similarly, `PB5` = `TF5` and `PF2` = `TB2`, yielding a complete algebra of transactions.

The pedagogical trouble with signed numbers is that we use them both for locations and for operations. The rule that "the product of two negative numbers is positive" is one of the earliest stumbling blocks that convince many students that mathematics means memorizing, not reasoning. Appropriate experience with counting games can restore intuition to the rules of negative numbers. Board games help students appreciate the value of scoring, first with simple addition (especially where movement depends on the throw of a pair of dice) and later in more complicated games where the score can be positive or negative. Scoring experiences are generally one-dimensional. [an error occurred while processing this directive]