The Golden Ratio

TheGoldenRatioBy Rachel Lewisadapted from http://www.geom.uiuc.edu/~demo5337/s97b/discover.htm

GoalGiven a ruler and various rectangles found in the classroom, students will measure the length and width of each rectangleUsing 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.ObjectivesStudents will calculate the Golden Ratio and discover where it exists in the world around them.MaterialsGolden Ratio worksheetRulerAny rectangular index cardCalculatorPencil

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 cardlength and width of the worksheetdistance 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 TableMeasurementsItemLengthWidthRatioFavorite Rectangle2.81.2Index Card20.212.8Worksheet27.921.5Finger5.63.2Text Book23.916.6Pencil Case18.911Notebook29.225.1Computer Screen38.124.1Find the RatioSample RatiosComplete Sample TableMeasurementsItemLengthWidthRatioFavorite Rectangle2.81.22.333333Index Card20.212.81.578125Worksheet27.921.51.297674Finger5.63.21.75Text Book23.916.61.439759Pencil Case18.9111.718181Notebook29.225.11.163347Computer Screen38.124.11.580913QuestionsWhat 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 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 DaVincis famous drawing to the left shows a man drawn within a pentagon, suggests that the Golden Ratio exists in the human form.

Some Other ThoughtsHave 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)