Teacher's Curriculum Guide

The Golden Ratio in Nature  next page>

             By Lettie McGuire

Educational Goal Educational Goal: 
The Goal of this Lesson is to introduce the mathematical concept of the Golden Ratio to middle and high school age students. In this beginning lesson, the Golden Ratio will be introduced by showing it’s occurence throughout nature.

Students will be able to connect the Golden Ratio to direct natural objects they might see in their everyday life.  This book can compliment the student's math and science curriculum.

Perceptual Options: 
Students are offered a variety of visual representations using computer images, photos, and correlating animations. All these visual aids help provide clear and diverse multiple representations of the Golden Ratio.

linguistic options


Linguistic Options:
Recorded audio, audio scaffolding (Pedro, Hali and Monty), and computer text-to-speech options provide a variety of linguistic options from which the student can choose from.

cognitive learning Cognitive Learning Methods:
The golden ratio concept is matched to student's prior knowledge of nature.  This type of matching
boosts each student’s working memory. Questions asked throughout the book demonstrate inquiry based instruction, using ‘self generated’ learning methods to increase students interest and attention. 

The repetition and repeating visuals of the golden ratio help store the lesson in the student’s long term memory and working memory. Identifying similarities and differences in of the golden ratio throughout nature allows students to consider and remember what was learned.  Clarifying animations assist in keeping students’ sustained engagement.

Reflection and review
at the conclusion allow the student to retain what they have learned.  Students are encouraged to continue research about the golden ratio. 

Applications to Neuroscience Research:
The concept of the Golden Ratio is also supported by 
developmental cognitive researchers and neuroscientists like Kristy VanMarle, PhD.  She currently researches whether humans perceive geometric figures that bear this ratio as more aesthetically pleasing than figures which bear different ratios.

In one of her current studies she is “trying to learn if babies (who cannot have had much experience with the Golden Ratio) prefer it over other "non-golden" ratios. In our task, babies are shown two rectangles, side-by-side, on computer monitors and we measure how long they look at each rectangle. For each pair of rectangles, one has the dimensions of a Golden Rectangle and one does not. If babies prefer the Golden Ratio, they are expected to look longer at Golden Rectangles than at non-golden rectangles. Evidence that babies prefer Golden Rectangles would suggest not only that this preference appears in humans, but also that it has an innate basis.”

Source of Cognitive Research Information: http://bengal.missouri.edu/~vanmarlek/

Musings of the author: http://mcguiret560.edublogs.org

Teacher's Curriculum Guide:
Introductory Information about the Golden Ratio

About the Golden Ratio: The Golden Ratio can be illustrated within special dimensions of Sprials, Triangles and Rectangles  where the ratio of the length of the short side to the long side is .618, was noted by ancient Greek architects as the most visually pleasing rectangle and its dimensions were used to construct buildings such as the Parthenon.

The Golden Ratio has also been used extensively in classical paintings where it was believed to produce the most visually pleasing figures. The ratio also appears all over nature, such as the number of petals on some flowers, biological forms like the nautilus shell, mollusks, animal antlers, leaves, human proportions, galaxy spirals, and the relations between harmonious tones in music.

The Golden Ratio
The Golden Ratio in a Spiral and Triangle

Page 1:  Welcome to the The Golden Ratio

The Golden Ratio begins with a simple ratio between a and b
     a        b

The Golden Ratio

begins with
a simple ratio

between a and b




Page 2: The Golden Ratio shown using squares and rectangles.
The formation of the Golden Ratio spiral pattern can be shown using squares and rectangles.

Page 3: The Golden Ratio shown in the snail shell

snail shell
Snail Shell

The Golden Ratio
is shown here

in the repeating

spiral shape

of the

snail shell




Page 4: The Golden Ratio is found in Sea Shells
seashell seashell

Next time you go to the Ocean,
pick up a

Can you find the repeating spiral pattern
of the
Golden Ratio?




Page 5: The Golden Ratio in the skin of the pineapple
pineapple pineapple

Look at the natural pattern in
the outer skin of a pinneapple.
Can you find the repeating spiral

pattern of the Golden Ratio




Page 6: The Golden Ratio found in the animal kingdom
horn horn

The Spiral Pattern of the
Golden Ratio
Can also be seen
in animal horns and tusks





Page 7:
In addition to natural patterns
found here on earth,
spiral pattern
of the golden ratio

can also be seen
in the pattern of
our galaxy

Page 8: The Golden Ratio is found in many places throughout nature.

See if you can find more examples of the golden ratio
in nature and design. The golden ratio can be identified
in other natural forms as well as the shape of the spiral.


  1. Does a pine cone show the pattern of the golden ratio?
  2. Go outside, look around at nature and discover more examples.
  3. Search for books about the golden ratio at your library.
  4. Ask your teacher for more examples of the golden ratio
    in nature and your environment. Enjoy, explore and have fun!

 The End.