Overview

Key Staff Primary instructor. If primary instructor is not a visual art teacher, it would be a good idea to enlist the assistance of a visual art expert.

Key Skills
Developing Arts Literacies:
Analyzing and Evaluating - Critique
Making Art:
Producing, Executing and Performing

Summary
The Fibonacci Sequence manifests in nature and visual arts. Through videos, lectures and images, students will learn what the Fibonacci Sequence is, where and how it appears in nature, and its role in the visual arts in various cultures. Students will also learn about the Golden Mean/Ratio and Golden Spiral, an important concept in art history. Students will identify works of art in which the Golden Spiral or Golden Ratio appear.

Learning Objectives
Students will:

Learn about the origin of the Fibonacci Sequence
Learn about the Golden Mean/Ratio and Golden Triangle, important concepts in art history
Be able to identify the Fibonacci Sequence in nature and visual arts

Teaching Approach
Thematic
Project-Based Learning
Arts Integration
Teaching Methods
Discovery Learning
Discussion
Experiential Learning
Reflection
Assessment Type
Observation

Preparation

What You'll Need
Materials
Resources
Required Technology
Television
DVD Player
Projector
Lesson Setup
Teacher Background

Teachers should familiarize themselves with the definition and examples of the Fibonacci Sequence as it applies to art and the Golden Ratio .

Prior Student Knowledge

Students should have some background in geometry and visual arts.

Physical Space
Classroom

Staging
Since images to be shown to students are on the computer, make sure your computer is connected to an LCD projector and that there is a screen in the room.

Accessibility Notes
Students who are deaf or hard of hearing will need handouts/reading materials to explain the Fibonacci Sequence and Golden Ratio.

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Instruction

Engage
1. Write the following sequence of numbers on the board and ask students if they know what it is. Ask them if they can identify the pattern and ask what the next number in the sequence would be.

0,1,1,2,3,5,8,13,21,34,55,89,144,….

2. Explain that this sequence is called the Fibonacci Sequence . The first two numbers of the sequence are always 0 and 1. Each following number is the sum of the previous two (so, the next number in the sequence after 144 would be 233 because it is the sum of 89 and 144).

You could also explain that in mathematical terms, in order to determine a given number F_{n} of the sequence, you could use the recurrence relation (F_{n} = F_{n} -1 + F_{n} -2) when F_{0} = 0 and F_{1} = 1.

3. Describe the history of the Fibonacci sequence. It was named after a Medieval Italian monk (and mathematician) named Leonardo de Pisa (known as Fibonacci --- a combination of filius Bonaccio, which means “son of Bonaccio” in Italian). The Fibonacci sequence was introduced to Western Europe in his 1202 book, Liber Abaci. Though the Fibonacci sequence is attributed to Leonardo de Pisa, it also appeared in other cultures in India and Northern Africa before the publication of his book.

4. Explain that the Fibonacci sequence appears in nature and, as a result, has highly influenced visual arts. Using your computer and an LCD projector, show students this video .

5. To reinforce the notion of the Fibonacci sequence in nature, you may want to pull some examples from this site to share with students.

Build Knowledge
1. Explain that artists have always been influenced by what they see in nature. The Fibonacci Sequence is related to the Golden Ratio, a concept that appears in both nature and visual arts.

2. Have students calculate the Golden Ratio with calculators: 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, 89/55 ...They should get 1, 2, 1.5, 1.6666..., 1.6, 1.625, 1.615384615..., 1.619047619..., 1.617647059..., 1.618181818...

Explain that these ratios will eventually reach the Golden Ratio, (√5 + 1)/2, a number approximately equal to 1.6180339887498948482. The Greek letter Phi is used to refer to this ratio. In the Fibonacci Sequence (0, 1, 1, 2, 3, 5, 8, 13 ...), each term is the sum of the two previous terms (for instance, 2+3=5, 3+5=8 ...). As you go farther and farther to the right in this sequence, the ratio of a term to the one before it will get closer and closer to the Golden Ratio.

3. Show students the graph showing the relationship between the Fibonacci sequence and golden ratio .

4. A similar concept is the Golden Spiral, a logarithmic spiral whose growth factor is related to the golden ratio .

5. Show students this video of how to construct a Fibonacci Spiral: http://library.thinkquest.org/27890/theSeries6a.html

6. Explain that the Golden Ratio and Golden Spiral have informed the work of artists throughout history. Talk about the ways in which the Fibonacci Sequence informed the work of M.C. Escher. M. C. Escher went to Alhambra, Spain and was inspired by the tile patterns he saw there to develop his signature style of images repeating and creating themselves (Islamic design). These tile patterns are also examples of geometric repetition and the Fibonacci Sequence in visual art. Tiles at Alhambra . These designs are based on the Golden Spiral.

For repetition in Escher’s work: http://www.mcescher.com/ (go to “picture gallery” and then “symmetry”)

Recommended Resources

Apply
1. Ask students to think about the different representations of the Fibonacci Sequence and Golden Ratio they’ve seen. Ask them to identify pieces of art that use these themes or formulas. Assign them to find a piece online that corresponds to these ideas.

You may want to assign students specific pieces. Examples of pieces of art that use the Fibonacci Sequence and Golden Ratio can be found at:

http://www.geom.uiuc.edu/~demo5337/s97b/art.htm or http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibInArt.html#daVinci

2. Ask students to prepare a short presentation on the piece they have been asked to research. The presentation should include the history of the piece, how the Fibonacci/Golden Ratio is used, and information about the artist.

Recommended Resources

Reflect
1. Have students present their findings about the Fibonacci sequence in an existing work of art. They should have an image of their piece to show the class as well as the information listed above (history of the piece, relation to Fibonacci Sequence/Golden Ratio, information about the artist).

2. End the lesson with a discussion about now the Fibonacci sequence has influences art across cultures and time periods (use student projects to exemplify this).

Assessment
Assess the students' final projects. Do they reflect an understanding of the Fibonacci Sequence, Golden Mean and Golden Spiral?

Key Vocabulary
Fibonacci Sequence
Nature
Golden Mean/Ratio
Golden Spiral
Visual Art
Pattern
Extending the Learning

Arrange a field trip to a local art museum. Assign students to research different pieces found in the museum and identify those that use the Fibonacci Sequence or Golden Mean/Ratio. Have students serve as “guides” and talk about these pieces while you are at the museum.

Engage students in a lesson about engineering and architecture. Identify the presence of the Fibonacci Sequence or Golden Mean/Ratio in processes related to these disciplines.

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Standards

Throughout the nation, standards of learning are being revised, published and adopted. During this time of transition, ARTSEDGE will continually add connections to the Common Core, Next Generation Science standards and other standards to our existing lessons, in addition to the previous versions of the National Standards across the subject areas.

The Arts Standards used in ARTSEDGE Lessons are the 1994 voluntary national arts standards. The Arts learning standards were revised in 2014; please visit the National Core Arts Standards (http://nationalartsstandards.org ) for more. The Kennedy Center is working on developing new lessons to connect to these standards, while maintaining the existing lesson library aligned to the Common Core, other state standards, and the 1994 National Standards for Arts Education.

A

rts E

dge Lessons connect to the National Standards for Arts Education, the Common Core Standards, and a range of other subject area standards.

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