TheGolde

nRatioBy Rachel

Lewis

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

Goal

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.

Objectives

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

Materials Golden Ratio

worksheet Ruler Any rectangular

index card Calculator Pencil

ProcedureFrom 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 TableMeasurements

Item Length Width Ratio

Favorite Rectangle

2.8  1.2   

Index Card  20.2 12.8   

Worksheet 27.9   21.5  

Finger  5.6  3.2  

 Text Book  23.9 16.6  

 Pencil Case  18.9 11   

 Notebook  29.2 25.1   

 Computer Screen  38.1  24.1  

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

Sample Ratios Index Card

=

Worksheet =

*These values should be the same for each student

Complete Sample TableMeasurements

Item Length Width Ratio

Favorite Rectangle

2.8  1.2  2.333333 

Index Card  20.2 12.8  1.578125 

Worksheet 27.9   21.5 1.297674 

Finger  5.6  3.2  1.75

 Text Book  23.9 16.6  1.439759

 Pencil Case  18.9 11   1.718181

 Notebook  29.2 25.1   1.163347

 Computer Screen  38.1  24.1  1.580913

Questions1. What do you notice about your ratios?2. Take the average of the 8 ratios you found.

Record this number on the chart on the board.

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

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

5. Find the average of the values on the board.

The Golden RatioThe 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 3/2 = 1.5 5/3 = 1.678/5 = 1.6 13/8 = 1.625 21/13 = 1.61534/21 = 1.619 55/34 = 1.618 89/55 = 1.618

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…)