"Fibonacci" Sequence

Junior (Age 9 – 12)

Curriculum Goal

Junior: Patterning and Algebra

  • Identify and describe repeating, growing, and shrinking patterns, including patterns found in real-life contexts, and specify which growing patterns are linear.
  • Determine pattern rules and use them to extend patterns, make and justify predictions, and identify missing elements in repeating, growing, and shrinking patterns.

Junior: Data Management

  • Collect data, using appropriate sampling techniques as needed, to answer questions of interest about a population, and organize the data in relative-frequency tables.

Context

  • Individual or pair-based exploratory activity at desks followed by a whole-class debrief.

Materials

  • Number pattern cards (Appendix A)
  • Flower cue cards (Appendix B)
  • Pre-cut sections of banana, apple, and pineapple
  • Natural objects showing “Fibonacci” pattern (e.g., pinecones)
  • Recording t-chart (Appendix C)

Lesson

  • Hand out number pattern cards to students and have them discover the pattern of each sequence.
  • Give students a brief history on Fibonacci.
  • Have students explore the number of petals, sections, or spirals on the cue card flowers and the pre-cut sections of fruit; invite them to document their findings on a t-chart (Appendix C).
    • Go through the first exercises together to ensure all students know what a petal is and what is meant by ‘sections’ of a fruit.
    • Use tape to trace the spirals on pineapple. 
    • Have students draw the rows of spirals they see on the pinecones.
  • Debrief as a class. Put the documented numbers in order from smallest to largest.
  • Introduce the golden spiral and explain that it is drawn according to the numbers in the sequence: the number determines the length and width (not the number of square units).
    • Guiding them for just the first couple of squares, allow students to draw the spiral.

Look Fors

  • Can children identify the pattern rule in the Fibonacci sequence? Can they continue the pattern?
  • What information do children use to justify observation of the pattern rule?

Extension

  • The golden spiral is found in art and photography, where the focal point of the picture lies at the center of the spiral.
  • Introduce students to the relationship between the golden ratio grid and the golden spiral (shown to the right).
  • Have students take one photo where the focal point is at the center of the spiral and one without the use of the spiral.
    • The default camera apps on most Android phones have the golden ratio grid as a toggle in the settings. If not, the Open Camera app is a free alternative.
    • iOS users can use the free Golden Ratio – Camera app.
  • Compare photos and vote on which one is more aesthetically pleasing.

Note: The Robertson Program has chosen to place Fibonacci in quotes as per research by Rochelle Gutierrez: 

I place Fibonacci in quotes to highlight the presence of settler colonialism. That is, although the Italian Leonardo Pisano (Fibonacci) receives credit for the pattern, many cultures and persons throughout the world, including Pingala in 200BC in India, had already known/performed the same pattern many years earlier. In fact, if humans are no longer the center, we might credit Nautilus pompilius (Nautilus shell), Pinus coulteri (pinecone), or Helianthus annus (sunflower) with the “discovery” (p. 12, Footnote 6). 

Gutiérrez, R. (2017). Living mathematx: Towards a vision for the future. Philosophy of Mathematics 12 Education, 32(1). 

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