Title
Breeding Rabbits (Eric Knuth)
Goals
(1) Students will determine terms of a sequence defined recursively.
(2) Students will use recurrence relations to solve the problem.
Abstract
Students investigate a famous problem invented by the Italian mathematician Fibonacci almost 800 years ago. The problem gives rise to a sequence called the Fibonacci sequence, a sequence useful in describing phenomena as varied as patterns of sunflower petals, pine cones, and spiral galaxies.
Problem Statement
Suppose that a certain breed of rabbit is infertile during its first month of life, but after two months and after every month thereafter each male-female pair of rabbits produces one additional male-female pair. Starting with one newly-born male-female pair and assuming that no rabbit dies:
(1) How many rabbit pairs will there be at the beginning of the 7th month?
(2) Describe, in your own words, how this population of rabbits is growing?
(3) How many rabbit pairs will there be at the beginning of the nth month?
(4) Write a recurrence relation for the number of rabbit pairs at the beginning of the nth month.
(5) Can you find an explicit formula (closed form) for the number of rabbit pairs at the beginning of the nth month?
Instructor Suggestions
(1) Make sure students understand what is happening in the problem.
(2) It may be useful to suggest ideas for data organization.
(3) Separate students into small groups to work on the problem.
(4) Each group should present its results to the class.
(5) Discuss students' solutions in terms of their data organization and their solution methods.
Materials
Breeding Rabbits worksheet describing problem situation
Time
Problem introduction (5 minutes), group work (20 minutes), small group presentation (15 minutes), class discussion (15 minutes)
Mathematics Concepts
Discrete Mathematics Concepts:
recursion
Related Mathematics Concepts:
data representation, sequences
NCTM Standards Addressed
Problem Solving, Communication, Reasoning, Connections (within mathematics and across disciplines), Algebra, Discrete Mathematics
Colorado Model Content Standards Addressed
Number Sense (1), Algebraic Techniques (2), Data Collection and Analysis (3), Problem Solving Techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This activity could be integrated into (1) a Precalculus class as students begin to investigate sequences; (2) a Prealgebra class in which students investigate number patterns.
Further Investigation
Derive the explicit formula for the sequence.
Variations/Comments
References/Resources
Eves, H. (1990). An introduction to the history of mathematics. Orlando, FL: Saunders College Publishing.
UCSMP. (1992). Precalculus and discrete mathematics. Glenview, IL: Scott, Foresman.