[Introduction] [Task] [Process] [Resources] [Evaluation]
[Teacher Information] [Back to the class Web Sites]
Introduction
"[The Universe] cannot be
read until we have learnt the language and become familiar with the characters
in which it is written. It is written in the language of mathematics."
- Galileo Galilei
"The pleasure we obtain from
music comes from counting, but counting unconsciously. Music is nothing but
unconscious arithmetic."
– Gottfried Wilhelm von Leibniz
"The mathematician’s best
work is art, a high perfect art, as daring as the most secret dreams of
imagination clear and limpid. Mathematical genius and artistic genius touch one
another."
– Gosta Mittag-Leffler
What are these people talking about? You mean there is more to math than arithmetic and algebra? Is Galileo saying that I can understand myself, and the world around me better by learning math? And what about Leibniz and Mittag-Leffler? What does art and music have to do with it?
Task
It is time to uncover the real reason as to why you are learning math. In the following activities, you will be learning how a good understanding of mathematics can help you make sense of the world around you. You will be exploring the relationships between mathematics in art, architecture, biology, and music. Your investigation will focus on the mystical golden ratio.
Working with at least three other students, you will first discover the golden ratio in biology. Then, you will be using several Internet resources to investigate the history of the golden ratio, the mathematics of the golden ratio, and the applications of the golden ratio. In the final project, you will be given the opportunity to illustrate how mathematics is present in your life by creating a demonstration for the class.
Process
Activity #1 - Discover the Golden Ratio
(90 minutes)
Go to "Discover the Golden Ratio" and complete this activity with your group.
Proceed to Activity #2 only if you are convinced that you discovered the golden ratio. If not, do not pass go. Proceed directly to Mr. Opfer for some extra help, so that you can keep up with the rest of your group.
Activity #2 – Using the Internet
(90 minutes)
First, you must decide on the roles each of your team members will take. You may either be a historian, mathematician, or observer. Next, you are to answer the following questions in your own words using the resources provided below. Don't just copy and paste, because you will have to explain the concepts to the rest of your team. Provide detailed answers. As you complete this activity, remember that you will be presenting your answers to your team in Activity 3, and will be evaluating your contribution to the team in Activity 4.
Historian
- Trace the development of the golden ratio by answering the following
questions in your own words. Remember you will have to present your
findings to the rest of your group.
- What is the golden ratio? The golden mean? Phi?
- How did the Pythagoreans use the golden ratio?
- How did the Ancient Egyptians use the golden ratio?
- How did the Greeks use the golden ratio?
- How did the artists during the Renaissance use the golden ratio?
Mathematician
- Explore the mathematics behind the golden ratio by answering the following
questions in your own words. Remember you will have to present your
findings to the rest of your group.
- Give at least two numerical expressions that represent phi (the golden ratio).
- Write a quadratic equation that when solved produces phi. Solve the equation using any process.
- What is the Fibanocci sequence? Where is the Fibanocci sequence present in nature? What does it have to do with the golden ratio?
- How is the golden mean present in three-dimensional shapes?
- Using paper and pencil construct at least two golden shapes (rectangle, spiral, triangle, star, line, graveyard cross, pentagon, pentagram, Penrose Tilings, etc.) and include an explanation as to how the golden ratio is present.
Observer
- Find out all the applications of the golden ratio by answering the
following questions in your own words. Remember you will have to present
your findings to the rest of your group.
- Where is the golden ratio used in architecture? (Give at least three examples.)
- Where is the golden ratio used in art? (Give at least three examples).
- Why have artists and architects used the golden ratio in their work? (Give at least two reasons.)
- In Activity #1 we saw how the golden ratio is present in a developed human body. How is the golden ratio present in human embryos?
- Where is the golden mean evident in animals, plants, or flowers? (Give at least three examples)
- Why would a dentist use the golden ratio?
- Where else might you find the golden ratio?
Activity #3 – Teach your Team
(90 minutes)
It is now time to organize your thoughts, and present your findings to the rest of your team. Your goal is to have your team members understand the concepts you discovered. To achieve this objective, you may use multimedia software to help organize your mini-presentation. Give your group members ample time to ask questions. The objective of this activity is to share and discuss your findings with the rest of your group. Make sure all of your team members understand the answers to each section (historian, mathematician, observer). Each team member is to turn in answers to all three sections written in your own words. In other words, copying and pasting your group members answers into your own document is unacceptable.
Activity #4 – Complete the Group Participation Rubric
(10 minutes)
Analyze the performance of each member of your group, including yourself for activities 1, 2, and 3. Use the following scale as a means of measuring the performance.
 | 4= Excellent |
| 3= Above Average |
| 2=Average |
| 1=Below Average |
| 0=No Meaningful Participation |
|
Group Members |
Effort |
Leadership |
Quality Of Work |
Cooperation |
Total |
|
|
1. |
||||||
|
2. |
||||||
|
3. |
Please answer the following in the space provided:
1. Did you think the group succeeded? Why? Why not?
2. If you were to do this over again, what would you do differently? Why?
Activity #5 – Donald in Mathmagic Land
(30 minutes)
Let’s take a break, and go on a visit with Donald Duck. Remember, you are never too old for Donald Duck. This award-winning film will no doubt inspire you to complete a creative, exemplary final project. What final project you ask? See Activity #6.
Activity #6 – Demonstrate the Golden Ratio
(120 minutes)
The instructions here are simpleÖBy yourself, creatively demonstrate your knowledge of the Golden Ratio. Some ideas may include: a painting, a musical composition, a multimedia presentation, a demonstration of the construction of a golden shape using geometer's sketchpad, a model of an architectural structure...
The key here is to be creative and to demonstrate to me (and the rest of the class) that you learned something. This reminds me of yet another quote brought to you by one of the greatest scholars of all-time, Albert Einstein, "Imagination is more important than knowledge." Or if that didn’t make sense to you, try George Cantor, "The essence of mathematics is in its freedom." Anyway, you get the picture. Be creative and be thorough! Check out the individual final project rubric before you begin.
Resources
Thinkquest on the Golden Ratio
http://library.thinkquest.org/C005449/home.html
This is probably the most comprehensive resource on this list.
Complete coverage of the golden ratio is given here. The site explains what the
golden ratio is, and then, goes into various mathematical subjects associated
with the golden ratio. The geometrical aspect of the golden ratio is covered
including constructions of a star, rectangle, line segment, and spiral. You can
then go onto the biology, aesthetics (art and architecture), and history of the
golden ratio.
The Golden Section Ratio: Phi
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html
A very complete, text-based mathematical background of the golden
ratio.
The Golden Proportion through a Dentist’s Eyes
http://www.goldenmeangauge.co.uk/index.html
Believe it or not - a dentist provides the most engaging view of the
golden proportion. Through pictures he shows how the golden proportion is
everywhere (teeth, heartbeat, animals, fashion, flowers, birds, Picasso's
artwork, handwriting, etc.).
Phi’s Fascinating Figures
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi3DGeom.html
On this Site you will see some more marvelous math about the number Phi
itself, its multiples and powers. Explore how the golden ratio extends to more
than two dimensions.
Pentagonal Geometry and the Golden Ratio
http://www1.ics.uci.edu/~eppstein/junkyard/pent.html
This page includes links to many different types of geometric problems
involving regular pentagons and pentagonal angles. But, more importantly
for this project, this site contains several links to geometric problems based on the golden ratio (the ratio of diagonal
to side length in a regular pentagon).
Golden Ratio in the Arts
http://www.mikkeli.fi/opetus/myk/pv/comenius/kultainen.htm
How is the golden ratio used in geometry, architecture, and
artwork? This site can leads the way.
Scott's Phi Page
http://www.ga.k12.pa.us/academics/us/Math/Geometry/stwk98/SCOTTRK/Index.htm
A well book-marked site that has general information, applications of
phi, Fibonacci sequence, history of phi, golden figures, activities, and links
to other sites.
The Golden Number
http://goldennumber.net/
Dedicated to providing you with the 'phinest'™
information on the golden section, ratio, or mean, the divine proportion, the
Fibonacci sequence, or phi. The most complete resource.
Evaluation
Checklist for The Golden Webquest
Activity 1 - Discover the Golden Ratio
_____ Data Table (5 points)
_____ Line of Best Fit (10 points)
_____ Q & A (10 points)
Activity 2 - Using the Internet
_____ Complete your section using your own
words (20 points)
Activity 3 - Teach Your Team Members
_____ Present your research to your other
group members (25 points)
_____ Complete other two sections using
your own words (25 points)
Activity 4 - Complete the Group Participation Rubric
_____ Student Evaluation (20 points)
_____ Teacher Evaluation (20 points)
Activity 5 - Donald in Mathmagic Land
_____ Watch movie and answer questions (5
points)
Activity 6 - Demonstrate the Golden Ratio
_____ Depth of Knowledge (15 points)
_____ Creativity (15 points)
_____ Craftsmanship (15 points)
_____ Critical Thinking (15 points)
Total 200 points
A 180-200
B 160-180
C 140-160
D 120-140
F Below 120
Individual Final Project Rubric
|
Category
& Points |
Amazing! |
That’s Good
|
You can’t be serious
|
Depth of Knowledge
|
*
ALL required elements are
present and covered in detail *
Answers/explains ALL
questions/information in detail *
Uses a wide variety of source types *
Sources are cited in a standard format |
*
No More than 1 required element is missing * Elements have
acceptable level of detail *
Answers/explains most
questions/info with some detail *
Uses limited variety of source types *Sources
are listed |
*
More than 1 required element is missing * Elements lack
detail *
Answers few questions/info with
little or no detail *
Uses few sources, little variety of type *
Sources not listed |
Creativity
|
*
Original, interesting, unique, or insightful approach *
Insightful use of word play *
Display and presentation demonstrate unusual and/or humorous elements |
*
Some originality or insight apparent *
Display and presentation may or may not use humorous elements or other
techniques |
*
Little evidence of creative or original thought in topic/ approach *
Display and presentation choices demonstrate little creative/original
thought |
Craftsmanship
|
*
Products are eye-catching, appealing, and present a polished,
well-finished appearance *
Choices (supporting examples, visual/other elements, etc) are consistently
insightful or original *
Visual and creative elements enhance the factual presentation |
*
Products are attractive, neat, and organized *
Choices (supporting examples, visual/other elements, etc) demonstrate some
insight and/or original thought *
Visual and creative elements support the factual presentation |
*
Products are sloppy, unfinished, lack visual appeal *
Choices (supporting examples, visual/other elements, etc) demonstrate
little creative/original thought *
Visual and creative elements interfere with the factual presentation |
|
Critical
Thinking |
*
Significant/numerous
connections established between and among topic areas * Information is presented in a manner that demonstrates
ALL the higher level thinking skills (application, analysis, synthesis and
evaluation) |
*
Some
connections established between and among topic areas * Information
is presented in a manner that demonstrates at least 2 higher level
thinking skills(application, analysis, synthesis and evaluation) |
*
Few connections established between/among the topic areas * Information
is presented in a manner that demonstrates no more than one higher level
thinking skill(application, analysis, synthesis and evaluation) |