Learning Objectives 

Students will: 

• Identify elements of the Fibonacci sequence in nature and arts. 
• Recognize sequences as mathematical functions. 
• Photograph examples of the Fibonacci sequence.
• Plan and design a gallery experience.

Standards Alignment

National Core Arts StandardsNational Core Arts Standards

Common Core State StandardsCommon Core State Standards

Recommended Student Materials

Editable Documents: Before sharing these resources with students, you must first save them to your Google account by opening them, and selecting “Make a copy” from the File menu. Check out Sharing Tips or Instructional Benefits when implementing Google Docs and Google Slides with students.

Websites

• The Fibonacci Sequence: Nature's Code
• 14 Interesting Examples of the Golden Ratio in Nature
• The Work of M.C. Escher
• The Wall Mosaic Alhambra
• Works of Symmetry
• The Hidden Beauty of Mathematics
• Fibonacci Sequence and Golden Ratio
• Formal Visual Analysis: The Elements & Principles of Composition

Videos

• The Fibonacci Sequence in Nature
• Fibonacci Spiral Animation

Additional Materials

• Camera or smart device with a camera
• Foam board, cardboard, or sturdy paper

Teacher Background

Teachers should familiarize themselves with the definition and examples of the Fibonacci sequence as it applies to art and the golden ratio. Previewing the TED Talk, The Magic of Fibonacci Numbers with Arthur Benjamin, to learn how mathematics can inspire the arts. Reference the lesson, Photography: From Theory to Practice for photography tips.  

Student Prerequisites 

Students should be able to interpret functions and use of function notation.

Accessibility Notes

Assistive tools such as stabilizers, mounts, voice control (“Hey Siri, take a photo), and accessible cameras (ex. Vtech Kidizoom cameras) are effective resources for camera adaptations. Provide preferential seating for visual presentations and allow extra time for task completion.

Engage

1. Write the following sequence of numbers on the board: 0,1,1,2,3,5,8,13,21,34,55,89,144… Ask students: What pattern do you notice? What is the next number in the sequence? 

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. For example, the next number in the sequence after 144 would be 233 because it is the sum of89 and 144. You could also explain that in mathematical terms, in order to determine a given number F of the sequence, you could use the recurrence relation (F = F -1 + F - 2) when F = 0 and F = 1.

3. Share the history of the Fibonacci sequence with students. It was named after a Medieval Italian monk and mathematician, Leonardo de Pisa, who became 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. Show students The Fibonacci Sequence: Nature's Code to explain how the Fibonacci sequence appears in nature and, as a result, has highly influenced visual arts. To reinforce the Fibonacci sequence in nature, share the following resources with students: The Fibonacci Sequence in Nature video.

Build

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. Share the 14 Interesting Examples of the Golden Ratio in Nature article with students to see examples.

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 relationship between the Fibonacci sequence and golden ratio. A similar concept is the golden spiral, a logarithmic spiral whose growth factor is related to the golden ratio. Model for students how to construct a Fibonacci spiral or show them the Fibonacci Spiral Animation. 

4. 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. The Wall Mosaic Alhambra designs and Works of Symmetry are based on the golden spiral.

5. Have students explore patterns in other cultures or from their own culture. Students can examine blankets, paintings, fabrics, drawings, sculptures, structures, jewelry, etc. Encourage students to bring artifacts from home. Share The Hidden Beauty of Mathematics with students so they can explore patterns across the world. 

Apply

1. Have students brainstorm different representations of the Fibonacci sequence and golden ratio they’ve seen in art and nature. Have students research and identify pieces of art that use these themes or formulas. You may want to assign students specific pieces from the Fibonacci Sequence and Golden Ratio.

2. Tell students they are going to be photographers capturing the Fibonacci sequence at school, in their community, or at home. Share the resource, Formal Visual Analysis: The Elements & Principles of Composition, so students can review techniques prior to taking photographs.

3. Take photographs of the Fibonacci sequence. Have students bring in artifacts from their home or ones that represent their culture (blankets, home decor, objects from their yard or garden, etc.). Then take students on a field trip through the school, on-campus grounds, and if possible, around the community. Allow time for students to take photographs of the examples and provide evidence that the Fibonacci sequence was used to compose the example. 

4. Mount the photograph(s) on foam board, cardboard, or sturdy paper. Students should provide short descriptions of the photograph, including cultural facts, evidence of the Fibonacci sequence, as well as their name.

Reflect 

1. Tell students they will plan, design, and create a “Fibonacci Gallery” experience. Students will need to collaborate as a class to determine the themes found in their photographs. Ask students: What common themes are found in our photos? How can we group or arrange them to make visitors feel like they are having a real museum experience? 

2. Invite other classes or families to view the gallery. Students should be prepared to elaborate on their photos and explain the mathematical sequence and functions related to their photographs.