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 Write-up

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