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Fibonacci Numbers (Grades 5-8)

Master Teacher

Sarah Lee

Overview

In this lesson, students will learn about the mathematician Fibonacci and his sequence of numbers. This special sequence of numbers will be found in vegetables, fruits, plants, pine cones, pine needles, pineapples, Pascal's tri-angle, and much more. Many secrets about math in nature will be revealed by these Fibonacci numbers.

SC Math/Science Standards Met

Math Achievement Standards

Patterns, Relationships, and Functions/Middle Grades

Science Achievement Standards
Area I—Living Things

Strand 1: B

Learning Objectives

The students will be able to:

calculate the Fibonacci sequence of numbers
identify Fibonacci numbers in flowers, leaves, fruits, vegetables, pine cones, and other forms of nature
construct the Fibonacci sequence from Pascal's Triangle

Materials

For stations:

leaves
flower(s) (daisy, lily, iris, buttercup, aster, black-eyed susan)
vegetable(s)/fruit(s) (cucumber, tomato, pear, apple, lemon)
pine cones and pineapples
OPTIONAL: pussy willow branch, pine needles, artichokes

For each student:

Pascal's Triangle

ITV Series

Math Vantage: Patterns, Lesson 3, "Sequences and Ratios"

Previewing Activities

1. Introduce the lesson with the story of Fibonacci (Leonardo of Pisa), the mathematician.

Leonardo was born in Pisa, Italy in 1175. This is the same town and the same year when they began to build the Leaning Tower of Pisa. During this time, people did not have first and last names like we do now, so they identified themselves with their hometown. This is why Leonardo is sometimes referred to as Leonardo of Pisa.

Leonardo loved mathematics and decided to write books about math and teach other people about numbers. He decided to give himself a new name to put on his books. Since he was the son of Bonacci, he combined part of filius, the Latin word for "son," with "Bonacci." Hence the name, Fibonacci.

Fibonacci discovered a sequence of numbers to explain math in nature. These numbers became known as Fibonacci numbers, or Fascinating Fibonaccis.

2. Provide sequences for students to complete and find the pattern. (For example: 1, 4, 9, 16, ___, ___, ___) Reiterate that Fibonacci discovered a special sequence of numbers named after him which appear many places in nature.

Focus for Viewing

To give students a specific responsibility while viewing, ask them to find which numbers make up the Fibonacci sequence and how they are derived. Students will also identify the appearance of Fibonacci numbers in nature and in the golden rectangle.

Viewing Activities

Ask students to answer the following questions in their journals as they watch the video:

What is the Fibonacci sequence?
What pattern is formed by these numbers?
How do Fibonacci numbers appear in rabbits?

Start the video at its beginning and pause after the girl states the Fibonacci sequence can be found all over—even in flowers. The music plays and she places a flower in her mouth.

Have students share what they wrote for the questions above.

Instruct students to make a list of examples where Fibonacci numbers occur in nature in the video. Begin the video and pause it after she shows the artichoke and pine cone and she states the Fibonacci sequence is not only seen but heard in music. Make a class list of their examples. Ask students to raise their hands whenever someone gives an example from their list.

For the last portion of the video, ask students to raise their hands whenever an example is given of the golden ratio or golden rectangle. Play the rest of the video and then discuss what the golden ratio and golden rectangle are.

Post-Viewing Activities

1. Set up the following Fibonacci exploration stations for students to explore the appear- ance of Fibonacci numbers in nature. At each station, have students write about what they discovered. Encourage students to include sketches of the objects and a written explanation of how the Fibonacci number appeared.

Flower/Leaf Station: Provide a variety of flowers and leaves for students to count their petals and leaf sections. Instruct students to find which flowers and leaves revealed a Fibonacci number.
Vegetable/Fruit Station: Provide sections of cucumbers, tomatoes, pears,lemons, apples, etc. Instruct students to count the sections in the seed cavity. Which revealed a Fibonacci number?Sketch the section and give the Fibonacci number.
Pine Cone/Pineapple Station: Provide a variety of pine cones and a pineapple. Instruct students to find the rows of bracts (pine cone leaves) on the pine cones and the rows of hexagonal scales on the pineapple. Count the gradual rows and steep rows of spirals. You can make the count easier by marking the rows with markers. Write two numbers together as a ratio. The numbers will reveal the golden ratio.
Plant/Branch Station: Provide plants with several branches spiraling from them. Count the branches from one branch position to the next branch in the same position. Also, you can count the buds on a pussy willow branch. Identify the Fibonacci numbers that were found.

2. Provide each student with Pascal's Triangle (See Activity 1 & Teacher Supplement) or have them construct their own. Ask them to draw in diagonal lines from left to right. Have students add up the numbers in each diagonal and summarize what they dis-covered.

Action Plan

1. Students may explore other occurrences of Fibonacci numbers in nature. They may also research the topic on the Internet and investigate its connection with the golden ratio.

Possible Internet sites:

Extensions

1. Science: Students may investigate more species of flowers and plants to see how frequently the Fibonacci numbers appear. Students may also investigate the rabbit problem introduced at the beginning of the video.

2. Math: Students may research the golden ratio and golden rectangle.

3. Art and Language Arts: Students may design a journal that contains sketches of objects which reveal Fibonacci numbers. Ask them to explain how the Fibonacci number appears in each object.

4. Technology: Students may generate the Fibonacci sequence on the computer and look for other patterns in this sequence.

Activity Sheets (PDF)

Activity Sheet 1