Title
The Golden Ratio (Dan Snook)
Goals
1. Students will find a specific value for the Golden Ratio.
2. Students will develop a recursive and a closed form for the Fibonacci sequence.
3. Students will compare the Golden Ratio to the ratio between Fibonacci terms.
Abstract
This activity allows students to investigate patterns found in the natural world, specifically the Golden Ratio and the Fibonacci sequence, and the relationship between the two.
Problem Statement
Students will diagram the situation presented by the Golden Ratio, set up a proportion, and use the quadratic formula to specifically find the Golden Ratio. Students will then investigate the Fibonacci sequence, finding a recursive form, attempting to find a closed form, and finding where the ratios of successive terms converges. Students will compare this limit with the Golden ratio.
Instructor Suggestions
1. Review where examples of the Golden Ratio occur. (i.e. in architecture, sculpture, etc.)
2. Find the Golden Ratio together (Part A of the worksheet) as a class.
3. Separate students into small groups to work on Part B of the worksheet.
4. Each group should present its results to the class.
5. Discuss the relationship between these ratios.
Materials
"The Golden Ratio" worksheets, dry erase board
Time
Discussion of Golden Ratio (5 minutes), Finding the Golden Ratio Part A (as a class) (15 minutes), Small Group Work Part B (25 minutes), Discussion of relationship between Fibonacci and Golden Ratio (10 minutes)
Mathematics Concepts
Discrete Mathematics Concepts
Closed Form Solutions, Recurrence Relations, Finite Differences
Related Mathematics Concepts
Fibonacci Sequence, Golden Ratio, Quadratic Equations, Exponential Growth
NCTM Standards Addressed
Problem Solving, Reasoning, Communication, Connections, Algebra, Discrete Math
Colorado Model Content Standards Addressed
Number Sense (1), Algebraic Techniques (2), Data Collection and Analysis (3), Problem Solving Techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This activity would fit well into an Algebra I or Algebra II class, after work with the quadratic formula. I would use this activity after showing the film "Donald in Mathmagicland". It is a good film to show on a day that you will have a sub! It also shows many examples of how the Golden Ratio occurs in nature and in architecture.
Further Investigation
Have students find real examples containing the Golden Ratio, or a Fibonacci sequence and specifically explain the relationship. For example, take a picture of the Parthenon and make specific measurements. Do the same with a nautilus shell or a sunflower.
Variations/Comments
It may be beyond the scope of most students to find the closed form for the Fibonacci sequence. This, however, is not necessary. Focus instead on the ratio between successive terms.
References/Resources
Colorado Model Mathematics Standards Task Force. (1995) Colorado model content standards for mathematics.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author: