GO IN DEPTH

# The Fibonacci Sequence

### Purpose

To appreciate and investigate a numerical pattern; to look for evidence of mathematical patterns in nature.

### Context

In this lesson, students will explore the Fibonacci sequence. They will identify the pattern among the Fibonacci numbers, look for applications of these numbers, and explore the ways that this pattern can be related to objects and shapes in both the natural and designed world.

Science and technology are rich and especially important contexts in which to learn the value of mathematics and to develop mathematical problem solving skills, but they are not the only ones. (Benchmarks for Science Literacy, p. 32.) This lesson uses examples from art and architecture, as well as nature, to reinforce the ideas in the central benchmarks. In grades 3-5, students should be encouraged to describe all sorts of things mathematically—in terms of numbers, shapes, and operations. In middle school, students should continue to have opportunities to reflect on the nature of patterns and relationships in a purely abstract way.

### Motivation

In a discussion, have students provide examples of patterns and some of the reasons why it might be helpful to study them. Tell students that they will investigate a numerical pattern and how it relates to the world around them.

Write the following number sequence on the blackboard: 0, 1, 1, 2, 3, 5, 8, _. Have students look at this series of numbers and allow them to guess the next number in the series. Ask students to explain the pattern or rule that they are following. You may wish to go on to the next activity before showing students the correct number.

### Development

Tell students that the Fibonacci sequence is a series of numbers developed by Leonardo Fibonacci as a means of solving a practical problem: Fibonacci wanted to determine the rate at which pairs of rabbits would reproduce.

Refer students to the Fibonacci Sequence student esheet, which will guide them through some online investigations of the Fibonacci sequence and it's appearance in nature. Some instructional suggestions regarding these resources are listed here:

• Fibonacci's Rabbits, found on the Fibonacci Numbers and the Golden Section in Nature website, provides text and an illustration related to the Fibonacci sequence. You may wish to print out the page and use the text and illustration to create a student worksheet depicting the rabbit problem. Distribute the worksheet to students and have them work with a partner to draw the next 1-2 lines of rabbits. Ask students to record and share their method for solving the problem.
• Honeybees and Family Trees is another example of the Fibonacci sequence.
• Petals on Flowers shows how on many plants the number of petals is a Fibonacci number.

Now have students explore Solve the Puzzle of the Seashell Spiral. After students have explored the puzzle, ask them to work with a partner to generate the next two (or more) numbers in the series, allowing students to use calculators if desired. Share results as a class. Create a classroom chart of the first ten numbers in the Fibonacci sequence for future use.

Ask students to answer these questions on the Fibonacci Sequence student sheet:

• How might knowing this number pattern be useful?
• What kinds of things can the numbers in the Fibonacci sequence represent?

Tell students: Sometimes patterns and relationships are studied simply because they are interesting, and sometimes because they help solve practical problems. Number patterns also can be studied in relation to the world in which we live, in order to help us better understand it. For instance, many of the numbers in the Fibonacci sequence can be related to the things that we see around us.

Refer again to Solve the Puzzle of the Seashell Spiral. Challenge students to draw a "perfect" spiral on a blank sheet of paper. Allow students to post their drawings and explain the strategy that they used to create the spiral.

One example of the golden spiral can be found on a seashell. Have students look for other natural examples of the golden spiral in a seashell, pinecone, pineapple, or cauliflower. For more information on designing this activity, go to Fibonacci Numbers and the Golden Section in Nature. Allow students to explore the outside world to look for examples of Fibonacci numbers in seed and leaf arrangements, flowers, and other natural objects.

Ask students to revisit their answers to the following questions, adding or refining their responses based on what they have learned about patterns in nature:

• How might knowing this number pattern be useful?
• What kinds of things can the numbers in the Fibonacci sequence represent?

### Assessment

The following activities can be used to assess student understanding:

• Ask students to record their answer to the following: Why study patterns? Give an example of how understanding a numerical pattern might be useful.
• Have students construct a golden spiral using the method of their choice. Then, have them write about the strategy that they used to construct the spiral and how this relates to Fibonacci numbers.
• Sometimes mathematicians study patterns in shapes and numbers because they explain how the world works or because they help to solve practical problems. Can you think of how we can use the Fibonacci sequence in this way?
• Collect a natural object that can be related to the Fibonacci sequence. Draw a sketch and write an explanation of how it relates to one or more of the Fibonacci numbers.

### Extensions

For an additional Nature of Mathematics lesson for grades 6-8, go to the Science NetLinks lesson entitled Finding Satisfactory Solutions.

In the Illuminations lesson Golden Rule, students explore the Fibonacci sequence, examine how the ratio of two consecutive Fibonacci numbers creates the Golden Ratio, and identify real-life examples of the Golden Ratio. In this lesson, students will use spreadsheet and geometry sketching programs to explore the numbers.

For further examples of how the Fibonacci numbers can be related to objects in the designed world, go to the Golden Section in Art, Architecture and Music

Go to Easier Fibonacci Puzzles for activities in which students manipulate objects such as bricks (substitute blocks and conduct as a hands-on activity), dominoes, and chairs in order to find numerical patterns and solve the puzzles. All of the puzzles have Fibonacci numbers as their answer.

Biographical information about Italian mathematician Leonardo Pisano, better known by his nickname, Fibonacci, can be found at the Leonardo Pisano Fibonacci page.

The Fibonacci Sequence for Visual Layout, on the Art Studio Chalkboard site, explains how the Fibonacci sequence is used in composition.