Viatouch
Teacher Articles
Teacher Created Materials
The Fibonacci Sequence
by Jodie Wells-Slowgrove
| Subject: |
Math |
| Grade Level: |
5-7 |
| Objective: |
Students will participate in a number of practical activities designed to give a better understanding of the Fibonacci numbers and how they occur in the natural world. |
| Time Needed: |
3 - 45 minute lessons |
| Materials: |
Blank paper and pencils
Internet access
Calculators
Rulers
Compasses |
Instructions:
Background Information:
The discovery of the Fibonacci sequence is attributed to Leonardo of Pisa (also known as Fibonacci) as it was first introduced to Europe through his book Liber Abaci, published in 1202. Leonardo was born in Italy during the time of Roman numerals. He learned about decimal notation and the Arabic number system while studying in North Africa and brought this knowledge back to Europe. Today the Fibonacci numbers are commonly known due to the part they played in Dan Brown's novel, The Da Vinci Code.
Lesson 1
| |
1. |
Have students discover the first 7 numbers in the Fibonacci sequence by using the rabbit problem below |
In February you are given a pair of newborn rabbits. One month later the rabbits reach maturity. In another month they have a pair of babies. If every pair of rabbits has a pair of babies at 2 months of age and another pair every month after that, how many pairs will there be in each month from January to July?
Below is one example of how this could be illustrated.
 |
— Newborn pair |
 |
— Mature pair |
Each pair of rabbits has its own color arrows. A new color indicates a new pair of rabbits.
| Month |
|
|
|
|
|
|
|
|
Rabbit
Pairs |
| Jan |
|
|
|
X
|
|
|
|
|
|
| |
|
|
|
|
|
|
|
|
|
| Feb. |
|
|
|
|
|
|
|
|
1
|
| |
|
|
|
|
|
|
|
|
|
| Mar. |
|
|
|
|
|
|
|
|
1
|
| |
|
|
|
|
 |
|
|
|
|
| April |
|
|
|
|
|
|
|
|
2
|
| |
|
|
|
|
|
|
|
|
|
| May |
|
|
|
|
|
|
|
|
3
|
| |
|
|
|
|
|
|
|
|
|
| June |
|
|
|
|
|
|
|
|
5
|
| |
|
|
|
|
|
|
|
|
|
| July |
|
|
|
|
|
|
|
|
8
|
The clipart used in this diagram was taken from http://islandgems.net/clipart/15.html
| |
2. |
Have students write the first seven numbers in the sequence (0 1 1 2 3 5 8) in a row. In small groups, using only these numbers, see if they can find the key to generating Fibonacci numbers. Adding the previous two numbers produces each new number.
|
| |
4. |
Fibonacci numbers are also found in the honeybee family tree. Male bees are born from the queen bee's unfertilised eggs and have no father. Female bees are produced when the queen bee mates with a male and so they have a father and a mother. Show students the illustration of a bee family tree. This example uses a male bee, but can also illustrate a female family tree. Simply cover the bottom line. Have students continue the illustration to see if previous generations of bees also represent Fibonacci numbers. |
| |
Male—
|
|
|
Female—
|
|
|
|
|
| |
|
|
|
|
|
|
|
|
| |
|
|
|
|
|
|
|
|
| |
|
|
|
|
|
|
|
|
| |
|
|
|
|
|
|
|
|
| Great Great Grandparents |
|
______
|
|
|
|
______
|
|
5 |
|
|
|
|
|
|
|
|
|
| Great Grandparents |
|
|
|
|
_______
|
|
|
3 |
|
|
|
|
|
|
|
|
|
| Grandparents |
|
|
___________
|
|
|
|
2 |
|
|
|
|
|
|
|
|
| Parent |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| Child |
|
|
|
|
|
|
|
The clipart used in the above illustration was taken from http://www.free-clipart-pictures.net/bee_clipart.html
Lesson 2
| |
1. |
There are many other places in nature where Fibonacci numbers occur. Have the students construct some perfect rectangles using Fibonacci squares. Then using a compass, construct a Fibonacci spiral within these rectangles. Compare the Fibonacci spiral to the image of the Nautilus shell beside it. Where else can this shape found? Could it be found in a backyard or playground? In outer space? Have a discussion on where this shape occurs. |
|
|
|
|
|
Fibonacci Rectangles
|
Fibonacci Spiral
|
Nautilus
|
| |
2. |
The Fibonacci spirals also occur in the world of plants. The pictures below are of a Romanesco broccoli. Use the pictures to show the students how to trace the Fibonacci spirals. Have them count the spirals in each direction. What do they notice? See if they can trace and count the spirals in a pinecone, using either a real one or the picture provided. Is there a Fibonacci number of spirals? |
Lesson 3
| |
1. |
The Fibonacci numbers are also found in the petals of flowers. Show the students the examples below. Then have them search the playground or their backyard for flowers. Encourage the students to create a graph showing numbers of petals in the flowers found. How many are Fibonacci numbers? |
|
|
|
|
|
Calla Lily (1)
|
Day Lily (2 x 3)
|
Buttercup (5)
|
|
|
|
|
|
McKana (2 x 5)
|
Ragwort (13)
|
Black-Eyed Susan (13)
|
| |
2. |
Use the picture below to show the Fibonacci leaf arrangement. A leaf was chosen at the top (x). The other leaves were counted as they spiralled down the plant. The leaves directly under the original leaf are Fibonacci numbers. The number of turns it takes to reach these leaves is also a Fibonacci number. Have students count the spiralling leaves of a plant. Is there a Fibonacci leaf arrangement? |
| |
3. |
Play the music below. Tell the students it was inspired by the Fibonacci number sequence.
* Bartók's Dance Suite
* Track 1.618 from This Binary Universe by Brian Transeau
* Per Nørgård's 'Canon'
* Fibonacci's Random Walk from Playing the Market by Emerald Suspension
|
| |
|
|
| |
|
For more information on the relationship between Fibonacci numbers and music, visit http://goldennumber.net/music.htm. |
Acknowledgments
The pictures used in the above lessons can be seen in their original context on the following websites:
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html
http://www.union.edu/PUBLIC/PHYDEPT/jonesc/scientific_photos.htm
http://scienceblogs.com/chaoticutopia/2006/11/friday_fractal_xxv.php
http://belaray.com/blog/index.php/category/dermatology-news/
http://www.gracefulgardens.com/ deerresist.htm
http://www.rivernen.ca/plant_7.htm
http://milan.milanovic.org/math/english/fibon/pages/13.White%20calla%20lily%20with%201%20petail.html
http://healing.about.com/od/floweressences/ig/Flower-Essence-Gallery/Buttercup.htm
http://www.dpi.vic.gov.au/dpi/vro/vrosite.nsf/pages/weeds_herbs_perennial_ragwort
http://www.montgomeryschoolsmd.org
http://www.haughtyculture.com.au/reflections.htm
Jodie Wells-Slowgrove is a freelance writer and teacher-librarian. She lives in a pretty house by a creek on the outskirts of Sydney with her husband, two children and numerous pets.
Top
of Page
