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In the year 1202, using the pen name Fibonacci, Leonardo of Pisa wrote Liber Abaci, "Book of the Abacus." This book introduced the Western World to the Arabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). Fibonacci used a variety of examples to show these numbers as superior to the Roman numeration commonly used at the time. One of the problems (see figure 1) prompted Edouard Lucas, in the nineteenth century, to name the resulting sequence of numbers (1, 1, 2, 3, 5, 8, 13, 21,...) the Fibonacci Numbers.
Many examples of the Fibonacci numbers appear in art, music and nature. Information on the Fibonacci sequence and their occurrence in the natural world can be found in the book Fascinating Fibonaccis by Trudi Hammel Garland. One of the sections in this book refers to the fact that male bees develop from unfertilized eggs (they have no male parent) while female bees develop from fertilized eggs (they have both male and female parents). This peculiar trait leads to the generation of the Fibonacci sequence when tracing back in a male bee's family tree.
Figure 2 "A Goblin's Family Tree," was developed based on a problem from the calendar in the October 1994 Mathematics Teacher magazine which utilized this special attribute of bees. Remember that in the next generation up the family tree each male goblin branches to one female and each female branches into two, a male and female. The numbers in the right column of the chart in figure 1 generate the Fibonacci sequence.
This problem demonstrates an enjoyable way to introduce middle school students to Fibonacci numbers or to review and extend previous knowledge. Students become very excited if they are allowed to find the pattern for themselves. The difficulty comes in trying to restrain eager students who want to shout out the answers before others have the time needed to make their own discoveries. A few well chosen extension questions can help alleviate this problem.
A word of caution, depending on the age and sophistication of your students, prepare for the inevitable comment, "I didn't know that goblins were real." Yes, it happened to me!
Extension Questions:
1. Without extending the family tree, can you predict how many goblins will be in the tenth generation?
2. At what generation will the number of goblins be a square number?
3. Count the number of female goblins in the first 6 generations. Predict the number of female goblins in the tenth generation.
4. Count the number of male goblins in the first 6 generations. Predict the number of male goblins in the tenth generation.
5. Starting with the 3rd generation, what can be said about the number of male and the number of female goblins in each generation?
Garland, Trudi Hammel, Fascinating Fibonacci. Palo Alto, CA: Dale Seymour Publications, 1983.
Runion, Garth E. The Golden Section. Palo Alto, CA: Dale Seymour Publications, 1990. Tannenbaum, Peter and Robert Armold. Excursions in Modern Mathematics. Englewood Cliffs, NJ: Prentice-Hall, 1992.
Wahl, Mark, A Mathematical Mystery Tour. Tucson, Arizona, Zephyr Press Learning Materials.