By Rachel Lewis

The Golden Ratio

adapted from http://www.geom.uiuc.edu/~demo5337/s97b/discover.htm

Goal

Students will calculate the Golden Ratio and discover where it exists in the world around them.

Objectives

   Given a ruler and various rectangles found in the classroom, students will measure the length and width of each rectangle Using a calculator, students will be able to calculate the ratio of length to width for each rectangle.

Using a calculator, students will find the average of their results to estimate the Golden Ratio.

Materials

     Golden Ratio worksheet Ruler Any rectangular index card Calculator Pencil

Procedure

From this picture, students are asked to measure the length and width of their favorite rectangle in centimeters and to record these values in the table in their packets.

Then measure the:  length and width of the index card   length and width of the worksheet distance from the knuckle on the back of your hand to the next knuckle (use as length), and then from the second to knuckle to the next (use as width)

Students are then told to walk around the classroom and find various rectangles to measure. All data is to be recorded in the same table.

Sample Table

Item Favorite Rectangle Measurements

Length Width 2.8

1.2

Index Card Worksheet Finger Text Book Pencil Case Notebook Computer Screen

20.2

27.9

5.6

23.9

18.9

29.2

38.1

12.8

21.5

3.2

16.6

11 25.1

24.1

Ratio

Find the Ratio

Students are then asked to calculate the ratio of length to width for each rectangle the measure using the formula: 𝑙𝑒𝑛𝑔𝑡ℎ 𝑤𝑖𝑑𝑡ℎ

Sample Ratios

 Index Card 𝑙𝑒𝑛𝑔𝑡ℎ 𝑤𝑖𝑑𝑡ℎ = 20.2

12.8

= 1.578125

 Worksheet 𝑙𝑒𝑛𝑔𝑡ℎ 𝑤𝑖𝑑𝑡ℎ = 27.9

21.5

= 1.297674

*These values should be the same for each student

Complete Sample Table

Item Favorite Rectangle Measurements

Length Width 2.8

1.2

Ratio 2.333333

Index Card Worksheet Finger Text Book Pencil Case Notebook Computer Screen

20.2

27.9

5.6

23.9

18.9

29.2

38.1

12.8

21.5

3.2

16.6

11 25.1

24.1

1.578125

1.297674

1.75

1.439759

1.718181

1.163347

1.580913

Questions

1.

2.

3.

4.

5.

What do you notice about your ratios?

Take the average of the 8 ratios you found. Record this number on the chart on the board.

Do you think this number would change if you measured in inches instead of centimeters?

Measure this worksheet in inches and find the ratio. What do you notice?

Find the average of the values on the board.

The Golden Ratio

The number you have calculated should be close to 1.61803. This is called the Golden Ratio. Remember the Fibonnaci sequence we studied before? Well you will notice that if we find the ratio of consecutive numbers… 2/1 = 2.0

8/5 = 1.6

34/21 = 1.619

1.618

3/2 = 1.5

13/8 = 1.625

55/34 = 1.618

5/3 = 1.67

21/13 = 1.615

89/55 = the result gets closer and closer to the Golden Ratio! The number first got its fame in Ancient Greece when mathematicians noticed how frequently it appeared in geometry. This ratio is said to be used in architecture from the Parthenon in Greece to the Great Mosque of Kairoun in Tunisia. Leonardo DaVinci’s famous drawing to the left shows a man drawn within a pentagon, suggests that the Golden Ratio exists in the human form.

Some Other Thoughts…

 Have students measure the distance from their shoulder to elbow and elbow to wrist.

 Give them a picture of the Parthenon and see if they can find Gold Rectangles  Research other places where the Golden Ratio is apparent (art, architecture, etc…)