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 Write-up

Title
Multiplying 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

How does the number of rabbits produced in a certain month relate to the number of rabbits produced in the months prior to that?

Plans to Solve/Investigate the Problem

Continue the diagram started in the problem to include the 4th, 5th and 6th month.  Look for a pattern.

Investigation/Exploration of the Problem

I set up the diagram that was included in the explanation of the problem.  I extended the pattern to include the number and types of rabbits found in months 4, 5 and 6.  In month 4 there were 3 mature rabbits and 2 newborn rabbits for a total of 5 rabbits. In month 5 there were 5 mature rabbits and 3 newborn rabbits for a total of 8 rabbits.  In month 6 there were 8 mature rabbits and 5 newborns for a total 13 rabbits.  Each month the total number of rabbits will be equal to the total number of rabbits in the month before in addition to the number of rabbits the month prior to that. This pattern is that of the Fibonacci Sequence.

Extensions of the Problem

In the Extension portion of this problem I compared the ratios of consecutive numbers in the sequence as suggested:  1/1, ½, 2/3, 3/4, etc…  After the 24th term number I noticed that the ratio became a continuous

0.618033989. This ratio is known as the Golden Ratio.

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Author & Contact
Insert name and contact information.
lparsons@rockdale.k12.ga.us

Link(s) to resources, references, lesson plans, and/or other materials

This page discusses real-world applications associated with the Fibonacci numbers and golden section.

http://www.ee.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

This worksheet from Mathematics Teaching in the Middle School leads students to discover the golden ratio by making various measurements and calculations.
http://www.nctm.org/mtms/1999/01/worksheets.htm

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