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.

GOLDEN

Fibonacci Hunt (Review)

Fibonacci numbers show up often in nature. The Fibonacci sequence is named for Leonardo Fibonacci, an Italian mathematician who lived from 1170-1250. The sequence begins 1, 1, 2, 3, 5, 8, 13, 21… with each term being the sum of the previous two terms (8 = 3+5, 13 = 5+8, 21 = 8+13, etc.).

Give students the first eight terms of the sequence and challenge them to extend the pattern. Ask them where these numbers show up often in ( plants, pinecones, sunflowers, artichokes, celery, lilies and daisies).

Bring in a pineapple and demonstrate the Fibonacci numbers. Looking at the bottom of the fruit, you can easily see spirals in the patterns. A pineapple makes for an excellent demonstration – the pattern of polygons on the husk forms several spirals that can be followed in different directions. Count the number of spirals in different directions (masking tape helps), and you'll find they are Fibonacci numbers. You'll probably find 13 spirals in one direction, eight in another and five in a third direction.

Golden Ratio Sequence

Demonstrate how students can find the ratios between consecutive Fibonacci numbers:
1/1 = 12/1 = 23/2 = 1.55/3 = 1.666...8/5 = 1.613/8 = 1.615384...... Etc.

Point out that the ratios are going up and down but getting closer and closer to some value. That value is about 1.61803…an irrational number that is called "the golden ratio." The number has fascinated mathematicians and artists for millennia.

Now have students create their own "Fibonacci-like" pattern by choosing two starting numbers and applying the same rule to build a sequence. For example, if one chooses 3 and 7 as the starting numbers, the sequence is 3, 7, 10, 17, 27, 44, 71, 115, etc.

Ask your students to find the ratios between the terms of their own sequence. They will be surprised to find that their sequence, and everyone else's in the room, also converges to the golden ratio! Wow!

Golden Rectangle

There are many ways to construct golden rectangles; here's a fun way that uses Fibonacci numbers. Supply students with graph paper. Have them outline a 1 x 1 square near the center of the paper. Add a 1 x 1 square adjacent to the first, then a 2 x 2 square so that it touches both of the 1 x 1 squares. Continue to add squares, circling around the outside of the figure. In each square, students write in the side length – the lengths form the Fibonacci sequence.

Golden Spiral

When the page is filled, students can draw a quarter circles in each square in order to form an approximate "golden spiral" as shown above. This spiral is related to the shape of ram's horns, sea- shells and other spirals found in nature.

This is just a small taste of the wonders of mathematics in nature.

Art Project and Demonstration:

Introduce a book about Patterns and Relationships:

Mathematics is the science of patterns and relationships. Part of the sense of beauty that many people have perceived in mathematics lies not in finding the greatest elaborateness or complexity but in finding the greatest simplicity of representation and proof.

Activity: Homemade spirals.

The photo above shows washers laid out in a string, starting at a center. Each washer touches the previous one, and each wrap around the center just touches the previous wrap. No pattern is obvious at first, but after a number of wraps, a pattern of additional spirals emerges.

The sunflower seed head is an example of botanist William Hofmeister's 1868 observation that primordia form preferentially where the most space is available for them. They also must form where they attach efficiently to the rest of the plant, and this is a geometric consideration. The pattern can also be modified by moisture and nutrient conditions that affect the size of forming seeds.

You don't need biology to produce spirals similar to those found in sunflower seeds. Here hardware store washers have been laid out. After about six or seven wraps additional spiral patterns develop, just as in the sunflower, spiraling steeply outward, curving away from the center. The reason is simple. The growth pattern of the seed head (and our constructed spiral) is such that it is biased to povide reasonably close packing of the seeds (or washers) consistent with the growth processes.

Mathematical thinking often begins with the process of abstraction—that is, noticing a similarity between two or more objects or events.