Transcript The Golden Ratio

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
Measurements
Item
Length
Width
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
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 27.9
=
𝑤𝑖𝑑𝑡ℎ
21.5
= 1.297674
*These values should be the same for each student
Complete Sample Table
Measurements
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
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…)