# Fibonacci Sequence

This is a complete lesson plan for teaching students about the Fibonacci Sequence.  It has the history of Fibonacci himself, real-life examples of where to find the sequence, as well as downloadable worksheets for the students.  It includes discussion questions at the end for extentions of the lesson.

### Standards & Objectives

CLE 3102.1.6
Employ reading and writing to recognize the major themes of mathematical processes, the historical development of mathematics, and the connections between...
CLE 3102.3.1
Use algebraic thinking to analyze and generalize patterns.
GLE 0301.2.1
Continue to develop basic listening skills necessary for communication.

Alignment of this item to academic standards is based on recommendations from content creators, resource curators, and visitors to this website. It is the responsibility of each educator to verify that the materials are appropriate for your content area, aligned to current academic standards, and will be beneficial to your specific students.

Learning objectives:

Students will:

• understand what the Fibonacci sequence is; and
• understand how the Fibonacci sequence is expressed in nature
Essential and guiding questions:

Imagine that scientists in the rain forest have discovered a new species of plant life. Where might they look for the Fibonacci sequence?

Suppose that you're shooting baskets with a friend. After a few practice shots, you decide that you want to keep score. The first basket either of you makes is worth one point. Just to make things interesting, you suggest that every time either of you makes another basket, you add your previous two scores to get a new total. To make the game even more appealing, you offer to start from zero, while your friend can start from one. What sequence of numbers would emerge after shooting eight baskets? What is the difference in points between you and your friend? What pattern has emerged from the point difference?

Explain that numbers missing from the Fibonacci sequence can be obtained by combining numbers in the sequence, assuming that you're allowed to use each number more than once. For example, how could the number 4 be obtained from the sequence? How about 11? 56? Think of a number not in the sequence and try to figure out what numbers to combine to get it.

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?

Try to solve this problem: Female honeybees have two parents, a male and a female, but male honeybees have just one parent, a female. Can you draw a family tree for a male and a female honeybee? What pattern emerges? Are they Fibonacci numbers? (The male bee has 1 parent, and the female bee has 2 parents. The male bee has 2 grandparents, and the female bee has 3 grandparents. The male bee has 3 great-grandparents, and the female bee has 5 great-grandparents. The male bee has 5 great-great-grandparents, and the female bee has 8 great-great-grandparents. The male bee has 8 great-great-great-grandparents, and the female bee has 13 great-great-great-grandparents.)

### Lesson Variations

Blooms taxonomy level:
Understanding
Extension suggestions:

Finding Ratios

Suggest that students measure the length and width of the following rectangles:

a 3" ? 5" index card

an 8.5" ? 11" piece of paper

a 2" ? 3" school photo

a familiar rectangle of their choice

Have students find the ratio of length to width for each of the rectangles. Then have them take the average of all the ratios. What number do they get? (1.61803) . Tell students that this ratio is called the golden ratio and that it occurs in many pleasing shapes, such as pentagons, crosses, and isosceles triangles, and is often used in art and architecture.

An Algebraic Rule

Encourage students to try to develop an algebraic formula that expresses the Fibonacci sequence. The formula is described below.

Represent the first and second terms in the sequence with x and y . Then the first few terms would be expressed as follows:

First term = x

Second term = y

Third term = ( x + y )

Fourth term = ( x + y )+ y = 1 x + 2 y

Fifth term = ( x +2 y ) + ( x + y ) = 2 x + 3 y

Sixth term = (2 x +3 y ) + ( x +2 y ) = 3 x + 5 y

Seventh term = 3 x +5 y + 2 x +3 y = 5 x + 8 y

Ask the students whether they notice anything familiar about the coefficients. (They're numbers in the Fibonacci sequence.)

Materials needed:

The class will need the following:

• Computers with Internet access (optional but very helpful)
• Pencils and paper
• Ruler
• Copies of Classroom Activity Sheet: Finding Fibonacci Numbers in Nature
• Copies of Take-Home Activity Sheet: Creating the Fibonacci Spiral

Contributors: