Title
Multiplying Everlasting Rabbits
Problem Statement
A newly born rabbit is
capable of reproducing at one month old (when it matures). Suppose the
rabbit never dies, and it continues reproducing one new rabbit every month.
So, when the rabbit is born, it has one member in its own family. After a
month, it matures, and by the second month it adds a new born member to its
family. In the third month, the rabbit produces another offspring; its
first child also matures and will be ready to have an offspring in the next
month.
The sequence named by Fibonacci (1,1,2,3,5,8,13,21,...)
can describe the number of members in the rabbit's family at each month.
Explain how.
Problem setup
After one month the rabbit
matures. It begins to give birth to one new rabbit per month. Once the rabbit is born it never
dies. This pattern continues as each rabbit is born. Determine how many rabbits (new and
mature) exist at any given month.
Plans to
Solve/Investigate the Problem
Begin the set of the problem
with one new rabbit. Create a
family rabbit tree to expand down for each month. Look for a pattern.
Investigation/Exploration
of the Problem
Begin at 0 month with one ‘new
rabbit’. At 1 month show
that rabbit has ‘matured’.
At month 2 rabbit gives birth, therefore, there are now a ‘mature
rabbit’ and a ‘new rabbit’. At month 3 ‘mature rabbit’
gives birth again, and new rabbit ‘matures’. Therefore, there are now 3 rabbits. The pattern continues. Each mature rabbit gives birth to a
new rabbit. The pattern for
number of rabbits is 1, 2, 3, 5, 8, 13, 21, 34, 55, …At each month to
determine the number of rabbits add the previous to amounts of rabbits (1 +
2 =3), (2 + 3 = 5),…Or you can create a formula to determine the
total number of rabbits for the next month. [(# of mature rabbits x 2) + # of
new rabbits]
Next input information into
Excel. Column A will be number
of months. Column B will be the
pattern for number of rabbits (1, 2, 3, 5, 8,…). Formula to input in Column B (=B1 +
B2), copy down. Column C will
be ratio (=B2/B3).
Several mathematical concepts
will be involved in this problem:
exponents, ratios, patterns, formulas, scientific notation, Pi,
sequence.
Extensions of the Problem
Compare the ratio of
consecutive numbers in this sequence. For example 1/ 1, 1/ 2, 2/ 3, 3/ 5,
5/ 8, .... What do you notice after a while?
Author & Contact
llaster@rockdale.k12.ga.us
Link(s) to resources, references, lesson plans, and/or other
materials
Link 1
Link 2
