by Jan von Stein

Introduction | The Task | Process and Resources | Evaluation | Conclusion

Geometry offers a means of describing, analyzing, and understanding the world and seeing beauty in its structures. Geometric ideas are useful both in other areas of mathematics and in applied settings. For example, symmetry can be useful in looking at functions; it also figures heavily in the arts, in design, and in the sciences. The Geometry Standard from the NCTM states that students should use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.

In the words of Leonardo da Vinci, "...no human inquiry can be called science unless it pursues its path through mathematical exposition and demonstration." It is with this thought in mind, combined with the statement by Johannes Kepler,

"Geometry has two great treasures: one is the Theorem of Pythagoras, the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold, the second we may name a precious jewel",

that the objective of this unit is to study the nature of the Golden Mean, Golden Rectangle and Golden Spiral from its historical development through its use in architecture and fine art.

The Task

Part I of this project involves cooperative work by students, using the internet as a research tool.
Your first task is to research the historical background of the golden ratio as well as investigate examples of the golden spiral in nature.

In mathematical analysis, you will calculate the golden ratio and then generate the Fibonacci Sequence and find the relationship between Fibonacci numbers and the golden ratio.

As a hands-on activity, you are to create a golden rectangle or golden spiral using any medium of your choice.

As a culminating project for the cooperative work, you are to prepare a class presentation, using power point, which will include the above work making connections between the golden ratio, golden rectangle and the golden spiral.

Part II of this project will allow each individual student to select one are of continued investigation from either:
1) Architecture
2) Abstract Art
3) Further Mathematical Investigation

Process and Resources

Part I - See Teacher Notes

A. Research - Golden Ratio (Week 1)
      See Fibonacci & The Golden Ratio
       1. Historical background
      2. Other Names for the Ratio

B. Mathematical Analysis
        1. Calculate the Golden Ratio
       2. Explain how to generate the Fibonacci Sequence and list the first 20 terms of the sequence.
       3. Find the relationship between Fibonacci Numbers and the Golden Ratio.

C. Construction (Week 2)
        1. You and your partner will construct either a golden rectangle or a golden spiral. Use any
              medium of your choice.
              These constructions will be used to decorate our classroom so please keep scale in mind.

D. Power Point Presentation (Week 4)
        1. Your power point presentation should be 8-10 slides maximum, including your title slide.
        2. As part of your presentation please include the following:
               a. Make a connection between the golden ratio, golden rectangle, and golden spiral.
               b. Give examples of the golden spiral in nature with explanations.

Part II - (Week 5-6) This part of the project will be done on an individual basis.
Select ONE of the following areas:

A. Architecture
1. Read the accompanying literature (excerpts from A Beginner's Guide to Constructing the Universe)
2. Design a facade to scale, of a building, using golden rectangles (Medium of your choice)

B. Abstract Art
       1. Study Composition with Red, Blue and Yellow by Piet Mondrian
       2. Prepare a one page report on Piet Mondrian
       3. Give examples of his work
       4. Discuss whether the golden rectangle may have influenced the piece studied in "1".

C. Calculus - Calculus by Howard Anton, John Wiley & Sons (ch. 2.4) An Intuitive Approach
          1. Read the accompanying literature
        2. Define the limit of a function
        3. Use Excel spreadsheet to graph ratios of the first 20 consecutive Fibonacci Numbers
        4. What does your graph suggest about the limit of these ratio?

Evaluation - There are several components to this project. Please review the rubric carefully for
how your project will be evaluated.

Rubric for Assessment

Part I

Research

Acceptable 1-2	Admirable 3-4	Awesome 5-6
Historical Background and/or other names for Golden Ratio are not included.	Historical Background and/or other names is minimal	Thorough historical background and other names for Golden Ratio are provided.
One source was used to provide information	Two sources used to provide information	A number of sources are utilized to obtain information
Information is not orderly	Information is somewhat orderly	Information is orderly and attractively presented
Sources are not documented	Not all sources are documented	All sources are documented

Mathematical Analysis

Calculation is Incorrect	Calculation is correct (1 error)	Calculation is correct
Explanation of Fibonacci Sequence is not included	Explanation of Fibonacci Sequence is unclear or absent	Explanation of Fibonacci Sequence is clear
First 20 Fibonacci terms not included	First 20 Fibonacci terms are included (1 error)	First 20 Fibonacci terms included

Power Point Presentation

Slides are disorganized
Slides have errors in grammar
Slides are not easy to read and/or do not flow logically
Information presented is incomplete or irrelevant
Slides do not appear attractive to the reader
No connection between the golden ration, golden rectangle and golden spiral is made
Examples and explanations of golden spiral in nature are not given or explanations are not accompanying example

Slides are essentially organized
Slides have minimal grammatical errors
Slides are essentially easy to read and flow logically
Information is relevant but could be more thorough or concise
Slides have some creative interest
Connection between G.R., golden rectangle and golden spiral is given
Some example of golden spiral in nature are made with minimal explanation

Slides are organized
Slides have no grammatical errors
Slides are interesting to read
Slides flow logically
Slides present a complete but concise history
Connection between G.R., golden rectangle and golden spiral is clearly made
Many examples of golden spiral in nature are provided with explanation.

Construction

Construction was not done or is incorrect or of low quality/effort

Construction is complete and correct Reasonably good effort and quality

Construction is complete and correct and demonstrate a high level of effort in development

Collaboration

Work was done by only 1 partner

Work was not evenly distributed

Work was equitably distributed

Part II
(Only 1 of the Following Categories Are To Be Selected)

	Architecture
Facade is not completed and/or is not to scale Workmanship is of poor quality	Facade is complete and done to scale Work is of moderate effort and quality is reasonably good	Facade is complete and done to scale Work is of superior quality and show high degree of effort in execution
	Abstract Art
Report is not completed Report does not include other examples Discussion of the G.R. influencing Composition is absent	Report is complete Report includes some other works by Mondrian Discussion of G.R. and Composition is minimal or without supporting explanation	Report is complete Report includes several other works by Mondrian Discussion of G.R. and Composition is thorough and scholarly.
	Calculus
Definition of limit of a function is not given Spreadsheet is not used to graph ratios Graphical interpretation is not made	Definition of the limit of a function is given but somewhat short sighted Spreadsheet is used to graph ratios Graphical interpretation is given	Definition of the limit of a function is clearly and thoroughly stated Spreadsheet is used to graph ratios Graphical interpretation is given with thorough understanding

Conclusion

What have you learned after completing this project? Was it easier or more difficult than you expected? Did completing the project give you a better understanding of the material? What would you like to explore further? Write a reflection paper on the above topics.

Created by Jan von Stein
The Masters School, Dobbs Ferry, NY
Grade: 9-12 Subject: Mathematics
Standards Addressed: Statistics & Probability
Date: September 26, 2000
Email :vonsteinj@worldnet.att.net