Fibonacci Series

The Fibonacci series dates back to the beginning of the thirteenth century, when Leonardo of Pisa (aka Fibonacci) studied the population growth of rabbits in a controlled (and hypothetical) environment. He noted that in every new generation, the number of rabbit pairs would grow by 1, 1, 2, 3, 5, 8, ... That is, the N^th population was the sum of the (N-1) and (N-2) generations: F(n) = F(n-1) + F(n-2). In the ensuing centuries, many examples of the series have been discovered in nature. See the thinkquest article for more information.

As a follow-on activity, see how the Fibonacci series converges to the golden ratio. Actually, the series does not converge, but rather the ratio of two consecutive pairs of Fibonacci numbers. For example, for ..., 5, 8, 13, ..., the two pairs are 8:5 and 13:8; 8 / 5 = 1.6, 13 / 8 = 1.625. Further out, we have 377, 610, 987 producing 610 / 377 = 1.61803713528 and 987 / 610 = 1.61803278689. The ratios are converging to the golden ratio, which we find in Greek architecture and Renaissance art (interesting article). As the challenge, use Scratch to compute the ratio out to six places; that is, stop when two consecutive ratios are within 0.000001 of each other. Display the ratio, the three Fibonacci numbers used to compute the ratio, and their positions in the series.

Besides practice with the Fibonacci Series, the students have been introduced to a major component in computer programming: asynchronous / synchronous communications. In the Fibonacci case, synchronous; that is, each sprite waits to execute until it is told by another sprite that it is its turn to execute.

Scratch incudes a "broadcast" command. A sprite broadcasts a message defined by the programmer to any sprite that is listening with the "When I receive" command. The sprite computing the first Fibonnaci number broadcasts "one is done," which is understood only by the "number two" sprite, which is listening for the "one is done" broadcast. When it receives the message, it computes the second Fibonnaci number. When two is done, it boradcasts "two is done," which is understood only by the third sprite. And so on. One second grader was introduced to the broadcast / "when I receive" commands, shown one example, and then took it from there.

The program will take a minute to load. Make sure you turn up your volume so you can hear the students call out their numbers. Use the space bar to start the program.