Webquest

Webquest - The Golden Ratio

Introduction

Once upon a time, long before seashells, pine cones, Leonardo DaVinci, and the Greek Parthenon, there were two lonely number 1's. They were so bored that they decided to make a pattern using themselves as the first two members. So they added themselves together and made the number 2. So now the 1's had a friend. From then on, they walked the earth like this: 1, 1, 2. But then the 2 felt like a third wheel, so he added himself and the number 1 next to him and got the number 3. So now they traveled like this: 1, 1, 2, 3. Well, after a few days, 2 and 3 got into an arugument, so 3 decided to add himself to the 2 next to him and got the number 5. So now they traveled like this: 1, 1, 2, 3, 5. All was harmonious for some time until 5 realized he was too much of a social butterfly and needed more friends. So he added himself to the 3 next to him and got an 8. So now they traveled like this: 1, 1, 2, 3, 5, 8.
Well, the first number 1 said, "Hey, why don't we keep up this pattern and populate the whole world!" So they did. And here we are today with this huge lovely pattern which is now called the Fibonacci Sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89....

Before long, the Greeks found out that if you divide say the 3 by the 2, the 5 by the 3, the 8 by the 5, the 13 by the 8, etc, you get a number that comes closer and closer to the number 1.618... This number (or ratio) goes on forever non-repeating, which makes it an irrational number. It turns out that many things in nature, history, art, and science have this ratio built into them in some way. The most beautiful things in the world adhere to the principle of this ratio, hence it is called the Golden Ratio.

Task

Now, after all these numbers kept going on and on forever, you would think that our orginal numbers, 1 and 1, would not be bored anymore. But they have been depressed lately because there are so many numbers crowding their vision that they can't see the beauty that they've created. Your task is twofold:
To find an example of the golden ratio in art, science, or nature, and show our little buddies how the golden ratio exists in that example. Please be very specific and use mathematical equations to prove your findings.

Create your own Golden Ratio such as a golden rectangle or golden spiral, or pyramid, to show them how absolutely perfect they made the world. Search the internet to find out how to make a golden ratio figure.

Process

Groups 1 and 2, find a golden ratio in science and nature by clicking here. Golden Ratio in Science and Nature.

Groups 3 and 4, find a golden ratio in art and architecture by clicking here. Golden Ratio in Art and Architecture.

Each person in the group must write a letter to numbers 1 and 1 describing to them a golden ratio figure and why it is a golden ratio. There can be no repeats within a group. Help each other if someone gets stuck.

Each group must make 2 golden ratios to present to numbers 1 and 1. It can be written or molded or simulated on the computer. Use these links to help you. You Tube Video on Golden Spiral Google Images of Golden Ratios

Evaluation

Objectives Lacking: 1 point Poor: 2 points Developing: 3 points Accomplished or Exceeded: 4 points

Appropriateness of Item Chosen Not appropriate. Somewhat appropriate. Appropriate. Very Appropriate.

Written Report of Validity of Item Students did not follow instructions Students partially followed instructions. Students followed instructions most of the time. Students followed instructions exactly.

Appropriateness of Item Constructed Not appropriate. Somewhat appropriate. Appropriate. Very Appropriate.

Workmanship of Item Constructed Looked as though Very little time was spent on item. Item was constructed somewhat satisfactorily. Item was constructed satisfactorily. Item was constructed almost perfectly.

Objectives	Lacking: 1 point	Poor: 2 points	Developing: 3 points	Accomplished or Exceeded: 4 points
Appropriateness of Item Chosen	Not appropriate.	Somewhat appropriate.	Appropriate.	Very Appropriate.
Written Report of Validity of Item	Students did not follow instructions	Students partially followed instructions.	Students followed instructions most of the time.	Students followed instructions exactly.
Appropriateness of Item Constructed	Not appropriate.	Somewhat appropriate.	Appropriate.	Very Appropriate.
Workmanship of Item Constructed	Looked as though Very little time was spent on item.	Item was constructed somewhat satisfactorily.	Item was constructed satisfactorily.	Item was constructed almost perfectly.

Conclusion

Congratulations! You have convinced the numbers 1 and 1 that their efforts were not made in vain! Good job! If you would like to further read up on the famous golden ratio phenomenon, see the link below.

Pre-History of the Golden Ratio
Based on a template from WebQuest.org