The Journal of Online Mathematics and Its Applications, Volume 8 (2008)

Modeling Spiral Growth in a GSP Environment, Dogan-Dunlap and Jordan

To carry out the suggested activities, each student needs a copy of the section How to Run GSP File along with the GSP file Spirals.gsp. After successfully entering the decimal and rational approximations of various irrational numbers (the value of `n`), students should be able to conclude that rational numbers such as 1/4 (see Figure 1) generate a predictable number of rays while some approximations of irrational numbers may not begin forming rays early in the process but form spirals. For instance, after forming a few spirals, the rational numbers 1/100 and 3/100 display 100 rays. Note that 100 is the denominator value for both rational numbers. Additionally, after entering rational numbers such as 2/50, the observation of the number of rays formed (25 rays for the case of 2/50) may lead to discussions on equivalent fractions.

For the case of irrational numbers, with a decimal approximation of the irrational number, for instance √2, students observe all spirals but no rays forming within the limits of the GSP window. Instructors may have students enter the decimal number 1.41, and observe spirals and eventually rays forming (100 rays!). Students can be further encouraged to compare and contrast the rays and spirals formed by the decimal numbers, 1.41421 (the decimal approximation that our version of GSP uses for √2), and 1.41 (as another decimal approximation for √2). This may lead to class discussions on exact value, approximate value, and accuracy issues with approximations of irrational numbers.

One may, in addition, have students study the behavior of the golden ratio (1 + √5) / 2, (see Figure 2 for the spiral formation). Instructors can have students observe that the decimal approximation (GSP uses) of golden ratio create spirals that mimic the spirals found on many plants such as sunflowers. They may also have students conduct a research on the characteristics of some of these plants displaying spiral formations. With the golden ratio one can furthermore observe that the generated points are well distributed and very little space is wasted. In addition, students may discover the Fibonacci sequence extending along the horizontal axis on the spiral formation of the golden ratio (See Naylor (2002) for a detail discussion). This may lead to activities on sequences, specifically on Fibonacci sequence.

See a sample worksheet that can be used in a mathematics classroom to have students investigate the characteristics of rational and irrational numbers.