Estimated Time: Depending on students’ familiarity with Fibonacci Sequence, one to two 50-minute class periods will allow plenty of time to explore the main concept. In-depth exploration of vibrations per minute and other musical relationships will require additional time.
Objectives
- To demonstrate understanding of the relationships of the Fibonacci Sequence.
- To apply the Fibonacci Sequence and find its relationship to a piano keyboard.
- To explore various relationships between music and the Fibonacci Sequence.
Materials Needed
- Access to a piano keyboard or to a picture for reference
- One or more episodes of the PBS Ken Burns JAZZ documentary
- Access to the Internet
Standards
This lesson correlates to the following math standards established by the National Council of Teachers of Mathematics at http://www.nctm.org/standards/:
- Understand patterns, relations, and functions.
- Analyze and evaluate the mathematical thinking of others.
- Recognize and apply mathematics to contexts outside of mathematics.
Procedures and Activities
- Begin by asking students how many of them know how to play the piano. Challenge students to draw or describe an exact representation of a piano keyboard.
- Ask students to identify different types of music that prominently feature the piano. What types of music come to mind—rap? country? jazz? classical? etc.
- Explain to students that they’ll be learning how jazz musicans use the piano, and also about how the piano relates to a famous mathematical concept: the Fibonacci Sequence. View clips from various episodes of the PBS JAZZ documentary that emphasize the role of the piano in jazz and provide visual access to the piano keyboard.
Suggested clips:
- Episode Four, 49:12 – 50:02—Fats Waller
- Episode Four, 1:24:23 – 1:25:07 and/or 1:28:45 – 1:29:25—Art Tatum
- Episode Five, 1:00:35 – 1:01:21—Teddy Wilson
- Episode Ten, 25:13 – 25:45—Cecil Taylor
- Episode Ten, 47:22 – 48:20—Herbie Hancock
- Episode Ten, 1:36:00 – 1:36:47—Gonzalo Rubalcaba
- Remind students that the piano is technically a percussion instrument. What other instruments are grouped as percussion? What is the “responsibility” of the percussion section? Lead into a discussion of rhythm and tempo. How does a band establish rhythm and tempo? Who decides each? What is the difference between rhythm and tempo? Students might find useful information by accessing the PBS JAZZ Web site’s Jazz Lounge which features a “virtual piano” application, and the Music Theory section of the site which explains different musical terms like rhythm and tempo.
- Ask students how rhythm, tempo, and the piano keyboard connect to math. Brainstorm a list on the board. (Possible answers include: rhythm and tempo can connect to fractions and rates; the keyboard has a certain pattern of black keys and white keys at regular intervals, etc.) Tell students they’re going to learn about another relationship between music and math, and that it involves the famous Fibonacci Sequence.
- To learn more about the history of the Fibonacci Sequence and its different manifestations in the world around us, have students access the following web site (or print out prior to class if access to the Internet is limited): Isaac Newton Institute for the Mathematical Sciences
http://www.newton.cam.ac.uk/wmy2kposters/januaryThis Web site has a poster from the London Underground asking viewers to spot the pattern (Fibonacci Sequence). The Web site goes on to give the history behind the Fibonacci Sequence. For more advanced math classes, the site provides a more detailed analysis as well as additional links.
For more information concerning Fibonacci numbers you might check out the book Fascinating Fibonaccis: Mystery and Magic in Numbers by Trudi Hammel Garland (published by Dale Seymou Press) or the following Web site:
University of Minnesota: Technology in the Geometry Classroom
http://www.geom.umn.edu/~demo5337/s97b/fibonacci.htmlAsk students to share what they’ve learned about the Fibonacci Sequence. How did it get its name? What’s the pattern or rule behind it? What are some examples of the pattern?
- Given a visual of the piano keyboard, challenge students to find examples of the Fibonacci Sequence in the keyboard.
Students should easily see a repeating pattern with the keyboard: the pattern should begin with the white key immediately preceding the set of two black keys and continue to the white key immediately following the set of three black keys.
Hopefully students, perhaps with guidance, will find some or all of the following:
Starting at C and proceeding to the next C gives us an octave. In this octave you will find two black keys grouped together and then three black keys grouped together. Surrounding the two black keys are three white keys and surrounding the three black keys are five white keys. In the octave are eight white keys. Counting the white and black keys, you have 13 notes. These thirteen notes are known as a chromatic scale.
These numbers are the first numbers (after 1) in the Fibonacci Sequence.
- Students can go on to research (at the previously mentioned Web sites) the vibrations per second of the notes in the chromatic scale beginning with middle C on the keyboard. Middle C vibrates at 264 vibrations per second, while A (the sixth) vibrates at 440 vibrations per second. This ratio reduces to 3/5, a Fibonacci ratio.
- Students may also explore the Pentatonic, Diatonic and Chromatic scales. (Five, eight, and 13 notes respectively.) Additional research could include where these scales appear in the various styles of jazz—fusion, swing, blues, etc.
Assessment Suggestions
This activity can be used to assess the student’s ability to extend patterns as well as the student’s ability to see patterns in areas outside of mathematics.
The student’s contribution to class discussion, research into the history and manifestation of the Fibonacci Sequence, and ability to identify (verbally or in writing) other examples of the Fibonacci Sequence may be used to evaluate student understanding of the concepts presented.
Extensions / Adaptations
Students can explore other occurrences of the Fibonacci Sequence in nature, such as pine cones, flowers, etc.; in art; and architecture (such as the Golden Ratio). Additional exercises using the Fibonacci Sequence may be found at the following sites:
PBS Mathline: Challenges
http://www.pbs.org/teachersource/mathline/
challenges/jun00/puzzle1.shtm
PBS Mathline: Peddling Petals
http://www.pbs.org/teachersource/mathline/
lessonplans/esmp/petals/petals_extensions.shtm
PBS Mathline: Squares Within Squares
http://www.pbs.org/teachersource/mathline/
lessonplans/hsmp/squares/squares_extensions.shtm
The Golden Section in Art, Architecture and Music http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibInArt.html
About the Author
Carol Fisher is a veteran teacher within the Chicago Public School system. Winner of both the prestigious Golden Apple Award for Educators and the Illinois Presidential Awards for Excellence in Science and Math Teaching, she has been recognized as an innovative math educator. Her lifelong involvement in music, currently as principal bass clarinet with the internationally acclaimed Northshore Concert Band balances her right brain and left brain energies. Carol enjoys sharing ideas with other educators and welcomed the opportunity to intertwine two aspects of her life and share these activities with others.
