Introduction

The Golden Ratio is the ratio of the length to the width of what is said to be one of the most aesthetically pleasing rectangular shapes. This rectangle, called the Golden Rectangle, appears in nature and is used by humans in both art and architecture. The Golden Ratio can be noticed in the way trees grow, in the proportions of both human and animal bodies, and in the frequency of rabbit births.

This WebQuest is designed to lead you to connections between the Golden Ratio and the Fibonacci sequence through the use of algebraic and geometric concepts. You will be absolutely amazed at the number patterns that exist in real-world situations! You will also be asked to integrate Art,Biology, or Music into your final project: creating your own lesson plan.

By the end of this WebQuest, you will know the answers to the following questions (Some of you will become experts on this topic!!):

• Who was Fibonacci?
• What is the numerical approximation of the golden ratio?
• How can you construct a golden rectangle (at least two different methods -one geometrically and one algebraically)? How can you prove (algebraically) that the methods works?
• What does the Fibonacci sequence have to do with the Golden Ratio?
• What other patterns can be derived from the Fibonacci Sequence?
• How can you show (graphically) the relationship between the Golden Ratio and the Fibonacci Sequence?
• What are some well-known examples of architecture, art, or nature that contain the Golden Rectangle?
• What is the Greek symbol used to represent the Golden Ratio?

In connection to answering the questions above, you will have an opportunity to use the Geometry Investigator of Logal Express and gain experience creating and formatting spreadsheets. Get set to exercise you brain and learn a lot of cool and interesting stuff!!

To help you get started, I have provided several links to websites dedicated to the Golden Rectangle and the Fibonacci Sequence. Of course, this is not a complete list, but it should be enough to get you on your way. As always, you may use a search engine such as Alta Vista to search on your own. You can use such key words as "golden", "rectangle", "ratio", Fibonacci, "spirals", etc.

The Fibonacci Numbers and the Golden Section is hosted at the Department of Computing, University of Surrey, UK where the author, Dr. Ron Knott, was a Lecturer in the Mathematics and Computing departments for many years. This site has a huge amount of information including the first 500 Fibonacci numbers, Fibonacci numbers and the Golden Section in nature, background information about Fibonacci himself, Fibonacci puzzles, and links and references to other sites. I highly recommend this site!

Fascinating Flat Facts about Phi gives some information on how to construct a golden rectangle. However, this is not the only method and may not be the easiest. You can determine this as you conduct your research. You can also learn how to make a paper knot to shoe the Golden Section in pentagons.

As you search to find information to connect the Fibonacci numbers and/or Golden Ratio to Biology (nature), Art, Architecture, or Music, try clicking on one of these links. You may also wish to try clicking here.

Want to know what rabbits have to do with the Fibonacci numbers? How about another way to construct a Fibonacci rectangle or a shell spiral? (keep scrolling down!!)

Your final project will be to create a lesson plan on the Golden Ratio and the Fibonacci Sequence. Your lesson should captivate your audience and be informative. Also, include something for your audience to DO! As you conduct this WebQuest, you will find tons of information for different levels of learners. Try to keep your lesson and explanations simple so that even the most "math phobic" person can learn something!! One group will be chosen at random to present your lesson to an actual class. I will not be able to help you present the lesson, so you must become well-versed on this topic and feel capable of answering any questions that are thrown at you -- just like real teachers do!! In addition to presenting your lesson, each group will turn in a portfolio containing the following information:

1. First you'll be assigned to a team of 4 students. After reading the rest of the steps, you need to decide on an equitable way to split up the work so that you are not duplicating efforts.
2. Gather the following background information: 1) Who is Fibonacci and what contributions did he make to mathematics? 2) Define the Golden Ratio and the Fibonacci Sequence, including an approximation for the golden ratio. 3) Find the Greek letter(symbol) that represents the Golden Ratio.
3. Find an example of the Golden Ratio or the Fibonacci numbers in nature, art, architecture, or music and prepare a visual presentation that shows the connection. Be able to explain the "whats" and "whys" of your example. Click here then click on "bees" to see an example. This is really cool!!
4. Choose a method of constructing the Golden Rectangle that your group feels is easy to explain and understand. You need to be convincing in your explanation. You will need to complete this homework assignment (print a copy of this assignment from the Internet). You should do your drawings in Logal. Make sure all your work (the algebra) is included.
5. Use spreadsheet software to present the following information.
6. ... and so on.

In the Process block, you might also provide some guidance on how to organize the information gathered. This advice could suggestions to use flowcharts, summary tables, concept maps, or other organizing structures. The advice could also take the form of a checklist of questions to analyze the information with, or things to notice or think about. If you have identified or prepared guide documents on the Web that cover specific skills needed for this lesson (e.g. how to brainstorm, how to prepare to interview an expert), link them to this section.

Describe to the learners how their performance will be evaluated. You can link to a separate rubric document from here, or you could briefly summarize your criteria on this page. Also specify whether there will be a common grade for group work vs. individual grades. Make sure the evaluation of your students evaluates the accomplishment of the objectives listed in the lesson.

Put a couple of sentences here that summarize what they will have accomplished or learned by completing this activity or lesson. You might also include some rhetorical questions or additional links to encourage them to extend their thinking into other content.