Copyright 1993, NCTM
Teachers of algebra have spent adequate amounts of time teaching two major representations of mathematics: verbal and algebraic, but have spent too little time on two other critical representations: graphical and numerical. Furthermore, the infusion of computer and calculator technology into algebra teaching and learning has not been realized. The study of the Golden Ratio allows teachers and learners to explore a well-known problem through four important representations as well as to incorporate technology in a manner that illuminates different aspects of the problem. The setting for the activities described below is any algebra class in which quadratic equations are studied. Interest in the Golden Ratio problem may be generated by consulting one of the many articles and books on the subject. (The Bibliography offers a few suggestions.) A computer algebra system provides the perfect technological environment for algebraic, graphical and numerical representations of the problem though electronic spreadsheets, and graphing utilities including graphics calculators may be used to reach the same end.
The verbal representation of the Golden Ratio allows one to appreciate mathematics as a means of communication. Consider the definition given by Eves: "A point is said to divide a line segment in ... golden section, when the longer of the two segments formed is the mean proportional between the shorter segment and the whole line. The ratio of the shorter segment to the longer segment is called the golden ratio" (Eves, 1983). A simpler statement can be found in the NCTM Student Notes: "[The Golden Ratio] is found by dividing a segment into two parts so that the length of the smaller part is to the length of the larger part as the length of the larger part is to the length of the entire segment" (Yunker, 1986). Regardless of which statement one examines, one conclusion is reached rapidly: the verbal representation of this problem is difficult to understand. Things improve if we translate the definitions into algebraic language: labeling the shorter part of the segment as x and the longer part as the unit length, 1, (see Figure 1) we conclude that the whole segment has length 1 + x and then the Golden Ratio may be defined in algebraic terms by means of the proportion: .
The simple elegance of the algebraic expression stands out in glaring contrast to the mind numbing English language expression of the same idea. Why do we study algebra? Because it provides us with an effective and efficient means of communicating certain ideas. Given the definition of the Golden Ratio in algebraic language, one can now investigate methods of finding the numbers satisfying the statement through other representations.
The algebraic analysis takes the form of solving the equation: . This can be done by multiplying the equation by 1 + x and solving the resulting quadratic equation using the quadratic formula. This type of analysis yielding two solutions: is familiar to algebra teachers.
The graphical analysis of the original problem can be accomplished by again manipulating the original equation into the form x^2 + x - 1= 0 and graphing the relation y = x^2 + x - 1. To solve the equation one can "zoom in" on the point where the curve crosses the x-axis (where the curve y = x^2 + x - 1 crosses the line y = 0). Figure 2 illustrates how the computer algebra system Deriveª (Rich and Stoutemyer, 1988) can be used to perform the algebraic manipulation and graphical analysis.
If the students are familiar with systems of equations and rational functions, the same analysis may be performed by graphing the system and approximating the points of intersection.
The numerical analysis of the Golden Ratio problem is performed by systematically examining the results of evaluating the expression x^2 + x -1 for different values of x. Figure 3 illustrates the result of the command TABLE -10 10 using the Mathematics Exploration Toolkitª (IBM, 1988). We wish to find when x^2 + x -1 equals 0; therefore we focus on the values of x that produce a sign change in the column labeled RANGE.
The strategy for finding the solution is based on the assumption that our expression is continuous and if there is a change from positive to negative (or vice versa), then at some intermediate point the expression must be zero. This strategy gives algebra students informal and intuitive experience with the Intermediate Value Theorem usually first encountered in a calculus course. In this example there is a sign change between -2 and 0 as well as between 0 and 2. To investigate further we enter the command TABLE 0 2 (see Figure 4) and find the new values of interest are between 0.6 and 0.8. The process can be repeated to yield more precise approximations of the Golden Ratio.
A final analysis can be performed using a calculator that possesses a reciprocal key:. The process is described in Figure 5.
Students may continue the process and note that by 18 iterations, a calculator with an eight digit display, will produce: 0.61803400 and its reciprocal 1.68103400 which differ by exactly 1. (The same analysis may be performed with more difficulty on calculators without reciprocal keys.)
To analyze what the calculator is disguising, write an algebraic expression for steps 2 and 3: . Since that result is being used in the next iteration, the next algebraic expression is: and the seventh iteration is .
Setting this final expression equal to x and solving (a symbol manipulator is of
great value here) produces the desired result. The simplification is illustrated
in Figure 6 with a screen output from muMATH-80ª, a first generation symbolic
mathematics system for Apple II computers, (Dickey and Rich, 1987) and in Figure
7 with output of the same problem solved by a second generation system, Derive.
Each of the representations of a mathematical problem may, for an individual student, provide a unique insight; each technique used within a representation becomes part of a student's repertoire of problem solving skills. Teachers can never be certain which representation is best suited for a particular student or a specific problem. Consequently, teachers should make available tools for exploring multiple representations of problems and, when possible, investigate problems with their students using different techniques. Besides its historical and aesthetic appeal, the Golden Ratio problem is one that can easily be investigated through various representations. Regardless of whether we investigate the Golden Ratio or any other problem, teachers of algebra should look for opportunities to examine beyond verbal and algebraic representations and to exploit the potential offered by computer and calculator technology in these explorations.