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A Fibonacci Investigation
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A Fibonacci Investigation
Lesson Plan for the Rabbit Problem
Sep 1, 2009 © Harry P. Schlanger
The original rabbit problem was posed to Fibonacci in the 13th century and is presented here with its
solution. A lesson plan includes an investigation of the Fibonacci sequence.
The famous
Fibonacci sequence
is found to occur in nature in a variety of different contexts: art,
architecture,
plants, etc...,
even in financial markets.
This article reviews the original rabbit problem, its solution and
presents a methodology for students to study the Fibonacci sequence and report on findings.
Who was Fibonacci
Leonardo of Pisa (1170 - 1250) or Leonardo Bonacci was an Italian mathematician. His name is an adaption
of his father's name - Guglielmo Bonacci but he is now known simply as Fibonacci, where Fi Bonacci means "the
son of Bonacci".
This talented mathematician wrote many books and was responsible for introducing the decimal number
system in Europe. One of his books written in 1202, Liber Abaci, detailed the abacus and this
work persuaded many mathematicians to adopt the new decimal system. Fibonacci is also famous for a number
sequence named after him, the Fibonacci sequence, which is described below.
The Rabbit Problem
In the year 1202, Fibonacci was presented with a problem of how quickly the rabbit population will grow
in ideal conditions:
"A certain man put a pair of rabbits in a place surrounded by a wall. If the rabbits can breed during
January, how many pairs of rabbits can be produced in a year, if it is supposed that every month each pair
produces a new pair which can breed from the second month?"
The essence of this information can be broken down as follows:
- Rabbits take 1 month to grow up
- After they have matured, it takes a pair of rabbits 1 more month to produce another pair
of newly born rabbits
- Whenever a new pair of rabbits is produced, it is always a male and a female
- The problem begins with just 1 pair of newly born rabbits (1 male, 1 female)
The Fibonacci Number Sequence
The rabbit problem solution may be outlined as a Fibonacci number sequence as described in Figure 1 below
over the first five months of rabbit breeding.
Fig 1. Emergence of Rabbit Pairs form a Fibonacci Sequence
Description of number pattern generated:
- At the beginning of the experiment, there is only one pair of newborn rabbits. After one month, the
two rabbits have matured and mated but have not given birth. Therefore, there is still only one pair of
rabbits.
- After two months, the first pair of rabbits gives birth to another pair, making two pairs in all.
- After three months, the original pair gives birth again, and the second pair mate, but do not give
birth. This makes three pairs.
- After four months, the original pair gives birth, and the pair born in month #2 gives birth. The pair
born in month #3 mate, but do not give birth. This makes two new pairs, for a total of five pairs.
- After five months, there are eight rabbit pairs as shown
Lesson Plan for the Fibonacci Investigation
It is suggested that students begin by typing general information about Fibonacci and the rabbit problem
into a Word file. A nice heading and images of rabbits should be included. Next, students answer the
following Parts A & B questions. A table of rabbit pairs (Fibonacci numbers) may be done in Excel and
students may copy and paste the results from Excel to Word.
Fig 2. Table of Fibonacci Data to be calculated in Excel
Part A: Fibonacci sequence
- Extend the Fibonacci sequence to complete the table as shown in Figure 2, for the first year
- What are some assumptions that make the Fibonacci model for rabbit population growth a little
unrealistic?
- Describe the pattern of numbers and how they are generated
- Determine the number of pairs of rabbits there would be after two years
Part B: Rabbits were introduced in Australia and now populate most areas.
- Find out how many rabbits were released in the natural environment and the date they were released
- Research how many rabbits are estimated to live in Australia today
References:
- Leonardo of Pisa and the New Mathematics of the Middle Ages. Joseph and Frances Gies. New Classics
Library, GA. 1969.
- Essential Mathematics VELS Project. David Robertson, et al. Cambridge University Press, NY. 2006.
The reader may be interested in Fibonacci numbers that are found in a different context -
dynamic symmetry in financial markets.
The copyright of the article A Fibonacci Investigation: Lesson Plan for the Rabbit Problem is owned
by Harry P. Schlanger. Permission to republish in print or online must be granted by the author in writing.
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