Activity Sequence: The Fibonacci Sequence

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The Fibonacci Activity Sequence document can be downloaded here

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Learning snapshots, Pedagogical advice and teaching tips

General background

Leonardo of Pisa (also known as Leonardo Fibonacci), the most prominent mathematician of his century, in his book “Liber Abacci” which was published in the year 1202, posed the following problem: How many pairs of rabbits will be produced in a year, beginning with a single pair, if every month each pair bears a new pair that becomes productive from the second month on? The total number of pairs, month by month, forms the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on. Each new term is the sum of the previous two terms. This set of numbers is now called the Fibonacci sequence.

The Fibonacci sequence, just like Number Theory, has intrigued mathematicians for centuries and continues to provide a hot research area. This simple sequence has been  noticed to appear in many different situations in nature, often creating the beauty we admire, from the number of petals in various flowers to the number of scales along a spiral row in a pine cone,  the golden ratio, etc. They also arise in computer science, especially in sorting or organizing data.

Among the many interesting topics that Fibonacci sequence originates one  could also the mention the Golden Ration, the Golden Rectangle and the Golden Triangles. In architect ture, the golden rectangle seems to appear throughout in history. From the Parthenon, the United Nations Building and even the burial chamber of Ramses IV. In Art, Greek art including urns, vases, statues and building frequently reveals the golden ratio and rectangle. There are numerous examples of art in many different forms and cultures where the golden ratio or golden rectangle appears. In Music, the most obvious place to begin looking is at the keys of a piano. The black keys are grouped in arraignments of 2's and 3's. An octave consists of 5 black keys and 8 white keys, for a total of 13 - all Fibonacci numbers.

The Fibonacci sequence shows itself in innumerable situations, both natural as well as technological ones. Its occurrence is so frequent that it is difficult to know where to begin the descriptions. One of the most interesting things about the Fibonacci sequence is the interplay between the numbers themselves; how they interrelate and their "behaviours." Some of the interactions are simple fun tricks, while some others move forward and demand more complex observations. For instance, the sum of any ten consecutive Fibonacci numbers is always evenly divisible by 11, while no two consecutive Fibonacci numbers have any common factors. Moreover, amazingly, the ratios of successive terms of the Fibonacci sequence get closer and closer to a specific number, often called the golden ratio. It can be calculated as (1 + √5)/2, or 1.6180339887.... For instance, the ratio 55/34 is 1.617647..., and the next ratio, 89/55, is 1.6181818....

With the use of a computer programming environment, students will generate the Fibonacci sequence, discuss arithmetic and geometric sequences, use problem solving and critical thinking skills to look for patterns and analyze the Fibonacci number sequences.  

Knowledge Domain

The generation of Fibonacci number sequences using the computational environment of ToonTalk and the exploration of the many beautiful patterns in Fibonacci number sequences are among the core aims of this activity sequence. Starting with some simple and moving towards more complex sequences, students will explore the construction of number sequences, investigate and discuss sequences’ properties and share their ideas and models over the web. More specifically, students will work on and explore some of  properties  of the following sequences:

·        a_n=bn

The exploration of these sequences is very powerful in a visual programming environment, like ToonTalk. Students are able to manipulate directly and observe the process of generating the sequences, making conjectures and “seeing” if they are right or wrong and “why”. Additionally, the implementation of a problem solving strategy in the rich environment of ToonTalk, arises opportunities for students to make effective use of the technology, to create their own models and try them out. Students reflect on their solutions and provide evidence.

Among the core attitudes expected of students are the following (also connected with in elementary and secondary school mathematics standards) :

1. Mathematics is useful
2. Doing mathematics is more than following rules (They participate actively in generating and exploring the sequences)
3. Doing mathematics is communicating and discussing

Working on the Fibonacci sequence is connected to several Standards including:

               - Algebra Standard (understand patterns, relations and functions)

               - Connections Standard (recognize and apply mathematics in contexts outside of                            mathematics)

               - Representation Standard (use representations to model and interpret

                 physical, social, and mathematical phenomena)