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The Golden Ratio page is intended to arouse students' interest in Geometry by showing them some interesting applications and by uncovering the Geometry inherent in nature. This website is a basic introduction to the meaning of the Golden Ratio and is intended to help students discover how and where it appears in real life. In doing so, they will begin asking (and answering) the basic questions of who, what, when, where, and why, with regard to math concepts. Hopefully they will see that math is not something that is created by man but rather discovered to be as old as time.
Students need to understand how Geometry is used in the real world, but at the same time they must realize that math holds certain universal truths. In this manner, students must understand that the mathematics they are learning has not been "invented" but rather discovered. Also, understanding where the math we learn in the classroom comes from is important - students should learn who discovered the concepts and when, and also realize that certain universal truths have been discovered in different places and at different times. Finally, students need to understand that mathematics is connected to all subject areas and is present in the world surrounding us.
The Golden Ratio page was written with High School Geometry (9th to 10th Grade) students in mind, but the activities are simple enough to be applied to the Middle School (6th to 8th Grade) curriculum.
Students are expected to have some experience working with constructions, and to know the properties of basic shapes (rectangles, triangles, circles, etc.). They should understand the concepts of both linear and angle measure. They need experience in using the keyboard and mouse, and in navigating the Internet. Some experience with search engines would be useful, as well as the ability to download and save information and images from the Internet, but this is not absolutely necessary.
Students will begin by discovering how the Golden Ratio appears in everyday objects in the classroom or at home. They will move on to examining different works of architecture and finding the Golden Ratio within the structures. They will then use a search engine to find different Leonardo da Vinci masterpieces, download them, and dissect them to discover how he incorporated the Golden Ratio into his work. Students will also construct their own Golden Rectangles and Golden Spirals. They will then examine the Golden Ratio in nature. Finally, they will examine the faces of different celebrities to see if there is a connection between the Golden Ratio and human attraction. Those who are more artistically inclined may choose to sketch a portrait that exhibits Golden Ratio characteristics.
Many goals and objectives are incorporated into this website. Some of them are:
This site is meant to guide the student without the use of an instructor. The concepts and activities are simple enough so that students may do them on their own without the presence of a teacher. At the most, a teacher should be there to assist in web navigation and technical matters. However, the site could easily be adapted so that the activities could be done in the classroom. Rather than having the students read through the Introduction on the Home page, the instructor may choose to present that information in their own way. Some of the activities can be made simpler by printing out the tables or images ahead of time. This would prevent technical difficulties (printer problems) from getting in the way, save printer ink, and definitely save time. The strongest need for the teacher comes during Activity 3, where students are required to search the Internet to find images. The instructor should monitor the progress of the student in finding those images, and with search engine use where necessary. As always, teachers should monitor the students' navigation to make sure they are not making inappropriate use of the Internet
The material contained in this site cannot be completed in one class period. Here is a guide to how the curriculum web may be broken up, assuming 45 minutes to an hour per day:
Computers (obviously), rulers and pencils are needed. Some other measuring objects (tape measures, yardsticks) may be useful. A color printer would be useful if available.
A basic online quiz is provided. Students may fill out this quiz online, but a submit function is not yet available. The page is meant to be printed out and turned in to the teacher to be graded. If printing is not possible, then quizzes should be copied down and filled out on paper. If might be a good idea in this situation for the teacher to make copies of the quiz ahead of time and distribute them to the students. The questions are very basic and are meant only as a survey of general knowledge about the Golden Ratio. If further assessment is needed, it will have to be provided in the classroom. The solutions to the quiz may vary - some of the questions have multiple answers. If would be a good idea for the teacher to do the quiz ahead of time, determine the acceptable answers, and construct a grading rubric. To see the quiz, click here. The quiz will open in a new window; close that window to return to this site.
Resources
Please see the Links page.
Glossary
Golden Ratio - a special number that appears in many places in nature, art, architecture, and geometry. It is an irrational number that is approximately 0.618.
Golden Rectangle - a rectangle where the ratio of the length to the width is the Golden Ratio.
Golden Spiral - a special spiral drawn inside of a Golden Rectangle that appears in may places in nature.
Irrational Number - a number that cannot be written in the form a/b, where a and b are both integers.
Phi - a Greek symbol referring to the Golden Ratio
Sequence - a "list" of numbers where there is a special relationship between successive numbers in the list. For example: the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ...).
For examples of student work, please click here.
Last Updated December 1, 2002