The Fibonacci Sequence in Gifted Classes
Math Extension Activities For Gifted and Talented Students
Aug 31, 2008
The Fibonacci Sequence is a series of numbers where each consecutive number is equal to the total of the previous two. It begins with the numbers 0, 1. When these numbers are added the answer is again 1, so the sequence becomes 0, 1, 1. This process is repeated and the sequence becomes 0, 1, 1, 2, 3, 5, 8, 13 and so on.
Connections have been found between the Fibonacci numbers and aspects of engineering, architecture, painting and music. They occur regularly within nature from the petals on a flower to the spirals of a pinecone. Trace back a honeybee’s family tree and each generation will represent a Fibonacci number. You can even see what is known as a Fibonacci spiral in the formation of galaxies.
The Discovery the Fibonacci Sequence
According to Robert B. Banks in his book Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics, published by Princeton University Press in 1999,the first written record of the Fibonacci Sequence was in the book Liber Abaci, written by Leonardo of Pisa and published in 1202. The Arabic number system was not commonly used in Europe at this time and it was noted by Banks that Leonardo may have acquired his knowledge while studying in North Africa.
Experiences and Outcomes
The Fibonacci Sequence allows gifted and talented students to experience the seemingly magical properties of numbers. By applying simple equations to this number sequence students will uncover unexpected connections. Opportunities are provided to generate and manipulate large numbers, to predict answers and to locate patterns within and between sequences of numbers.
Activities for Gifted and Talented Students
Many of the ideas below can be found in Tony Crilly's book 50 Mathematical Ideas You Really Need to know, published by Quercus Publishing in 2007. They are illustrated in worksheet form at the bottom of the page.
- Begin with 0, 1. Use the formula of adding the last two numbers in the sequence to generate the first 26 numbers in the Fibonacci Sequence.
- For each number in the Fibonacci Sequence, add all the previous Fibonacci numbers. Is there a pattern to the results?
- Add alternate Fibonacci numbers starting with 0, 1 and with 1, 2. Compare the results.
- Create a table to record the following results. Multiply each Fibonacci number with the next one in the sequence. Multiply each Fibonacci number with itself. Add the squared Fibonacci numbers. Compare the results.
- Create a table or list the first 30 numbers in the Fibonacci Sequence. Use different colours to shade the multiples of the Fibonacci numbers 2, 3, 5 and 8. Does a pattern emerge?
Ideas for Research Projects
Gifted and talented students are often fast finishers. A research project can keep them motivated and on task while they wait for others to catch up. Have students choose one of the research topics below and discover how it relates to the Fibonacci Sequence.
- The Golden Ratio
- The Lucas Sequence
- Perfect Rectangles
- Fibonacci Spirals
- The Mona Lisa
- Music - Bartok's Dance Suite
- Fibonacci's Rabbit Problem
Whole Class Activities
- Find flowers in the playground that have a Fibonacci number of petals.
- Count the number of spirals on a pinecone. Do they represent a Fibonacci Number?
- Create perfect rectangles using squares the length of Fibonacci numbers.
- Female honeybees have both a mother and a father. Male honeybees have only a mother. Create a honeybee family tree to show how the number of family members in each generation is always a Fibonacci number.
Study of the Fibonacci Sequence provides a unique opportunity for gifted and talented students to experience both the abstract and practical aspects of mathematics. It is an ideal topic for student-centred learning that would appeal to students with each of Gardner's Multiple Intelligences. e.g. a student with Visual/Spatial Intelligence may choose to study the Mona Lisa whereas a student with Musical/Rhythmic intelligence may prefer to study Bartok's Dance Suite.These varied applications of the Fibonacci Sequence make it an ideal platform from which to explore both pure mathematical concepts and the relevance of mathematics to everyday life.
Students who are gifted mathematicians may also enjoy learning about Perfect Numbers.
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