NUMERICAL PATTERN

Objectives

Students will
1. understand what the Fibonacci sequence is; and
2. understand how the Fibonacci sequence is expressed in nature.

Materials
The class will need the following:
Paper
Pencils
Copies of Classroom Activity Sheet: Finding Fibonacci Numbers in Nature

Procedures

1. Begin the lesson by discussing the Fibonacci sequence, which was first observed by the Italian mathematician Leonardo Fibonacci in 1202.

2. Work with the class to see whether students can develop the sequence themselves.

3. Write the pattern that has emerged in step 2 on the board: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233. Discuss what “rule” is being followed to get from one number to the next. Help students understand that to get the next number in the sequence, you have to add the previous two numbers. Explain that this sequence is known as theFibonacci sequence. The term that mathematicians use for the type of rule followed to obtain the numbers in the sequence isalgorithm. As a class, continue the sequence for the next few numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987…

4. Tell students that the Fibonacci sequence has intrigued mathematicians for centuries. What’s more, mathematicians have noticed that these numbers appear in many different patterns in nature, often creating the beauty we admire. Tell students that they are going to look for Fibonacci numbers in pictures of objects from nature. Make sure that students understand that they are looking for specific numbers that appear in the sequence, not for the entire sequence.

5. Divide students into groups of three or four. Distribute the Classroom Activity Sheet: Finding Fibonacci Numbers in Nature. Tell students to work together to try to answer the questions on the sheet. Make sure that each student fills out his or her own sheet. For my information, the questions and explanations are listed below. We will work on the first example as a class so students understand what they are looking for.

Flower Petals:
Count the number of petals on each of the flowers. What numbers do you get? Are these Fibonacci numbers?(Lilies and irises have 3 petals, buttercups have 5 petals, and asters and black-eyed Susans have 21 petals; all are Fibonacci numbers.)

Seed Heads:
Each circle on the enlarged illustration represents a seed head. Look closely at the illustration. Do you see how the circles form spirals? Start from the center, which is marked in black. Find a spiral going toward the right. How many seed heads can you count in that spiral? Now find a spiral going toward the left. How many seed heads can you count there? Are they Fibonacci numbers?(The numbers of seed heads vary, but they are all Fibonacci numbers. For example, the spirals at the far edge of the picture going in both directions contain 34 seed heads.)
Cauliflower florets. Locate the center of the head of cauliflower. Count the number of florets that make up a spiral going toward the right. Then count the number of florets that make up a spiral going toward the left. Are the numbers of florets that make up each spiral Fibonacci numbers?(The numbers of florets will vary, but they should all be Fibonacci numbers.)

Pinecone:
Look carefully at the picture of a pinecone. Do you see how the seed cases make spiral shapes? Find as many spirals as possible going in each direction. How many seed cases make up each spiral? Are they all Fibonacci numbers?(The numbers will vary, but they should all be Fibonacci numbers.)

Apple:
How many points do you see on the “star”? Is this a Fibonacci number?(There are five points on the star.)

Ask students where else they see the spiral shape in nature. Would they guess that those spirals are also formed from Fibonacci numbers? Do they find this shape pleasing to the eye? To conclude, discuss other pleasing shapes and patterns in nature, such as those of waves, leaves, and tornadoes. Discuss whether these, too, may have a mathematical basis.

Activity:
Creating the Fibonacci Spiral. Discuss how rectangles with Fibonacci dimensions are used in art and architecture. Use the examples of painter Piet Mondrian, who used three- and five-unit squares in his art; the Egyptians, who used Fibonacci dimensions in the Gaza Great Pyramid; and the Greeks, who used these dimensions in the Parthenon.

Discussion Questions
At first glance, the natural world may appear to be a random mixture of shapes and numbers. On closer inspection, however, we can spot repeating patterns like the Fibonacci numbers. Are humans more apt to perceive some patterns than others? What makes certain patterns more appealing than others?

Evaluation
Evaluate students using the following three-point rubric:

Active participation in classroom discussions; ability to work cooperatively to complete the Classroom Activity Sheet; ability to solve all the problems on the sheet

Some degree of participation in classroom discussions; ability to work somewhat cooperatively to complete the Classroom Activity Sheet; ability to solve three out of five problems on the sheet

Small amount of participation in classroom discussions; attempt to work cooperatively to complete the Classroom Activity Sheet; ability to solve one problem on the sheet

Suggested Readings / Books To Use
Symmetry in Nature (Reading Essentials Discovering & Exploring Science) (Library Binding)by Allyson Valentine Schrier

Life By the Numbers Keith Devlin. John Wiley & Sons, 1998. Written as a companion volume to the PBS series of the same name, this book focuses on the role mathematics plays in everyday life. Each chapter examines a different aspect of the world we live in and how mathematics is involved: patterns appearing in nature, the curve of a baseball, the chance of winning in Las Vegas, the technology of the future. Lots of pictures round out this clear and exciting presentation.

Designing Tessellations: The Secrets of Interlocking Patterns Jinny Beyer. Contemporary Books, 1999. For generations, people have created designs using repeating, interlocking patterns—tessellations. In this slightly oversized, beautifully illustrated book, the author shows how the combination of pattern and symmetry can result in stunning geometric designs.

Vocabulary
algorithm: A step-by-step procedure for solving a problem.
bontanist: A Botanist is someone who studies plants.
primordia form:

Standards
This lesson plan may be used to address the academic standards listed below.

• These standards are drawn from Grade level:


• Subject area: MathematicsStandard


• Understands and applies basic and advanced properties of the concepts of numbers.


• Benchmarks: Uses discrete structures (e.g., finite graphs, matrices, or sequences) to represent and to solve problems.


• Uses basic and advanced procedures while performing the processes of computation.


• Uses recurrence relations (i.e., formulas that express each term as a function of one or more of the previous terms, such as the Fibonacci sequence and the compound interest equation).


• Uses basic and advanced procedures while performing the processes of computation.