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Mathematics

Sample Extended Task: Golden Ratio

This task was administered in the 1999-2000 school year as part of the Rhode Island Skills Commission's Certificate of Initial Mastery pilot.


The Task


It has been proposed that many 18th and 19th century painters used the "golden ratio" in their paintings due to its "pleasing to the eye proportions." Could that same theory be used today in print advertising to catch the eye of consumers?

  • A. Investigate the golden ratio / golden rectangle. How are they defined? Why is it more pleasing to the eye?
  • B. Using newspapers or magazines, analyze at least 25 advertisements and photographs in terms of the golden rectangle model. Discuss any interesting patterns/relationships that you discover. What conclusions can be drawn from your investigation?
  • C. Using the golden ratio/golden rectangle model, prepare one or more sample pages showing the photographs, advertisements and text that would appear on full-size newspaper pages.

  • You will have two weeks to investigate this and to present your findings with examples drawn from actual newspapers and magazines.

     

    Sample Performance: Achieved the Standard (8 out of 10 points)

    A. Investigate the golden ratio / golden rectangle. How are they defined? Why is it more pleasing to the eye? 

    The Golden Ratio is the ratio, or proportion of the sides of a visually-appealing rectangle, called the Golden Rectangle. This proportion approximately equals 1.6. the actual formula of the golden ratio is (1 + (square root of 5)) / 2. The golden rectangle is the shape in which its sides proportionally equal approximately 1.6. 1.6 is not the true value of the golden ratio. Rather, it is an approximation or a rounded number of the original irrational number. (An irrational number is a number which cannot be expressed in its own fraction.)

    Interestingly, the golden ratio is related to the Fibonacci sequence of 1,1,2,3,5,8,13,21, etc. The ratio of the consecutive (two) terms equals the golden ratio. For example: 8/5 = 1.6.

    One main reason the golden ratio became so important was because ancient Greeks have found the golden rectangle to be pleasing to the eye. It seemed to be the model for the perfect rectangle. Since then, artists, architects, photographers, and many others have used the golden ratio/rectangle to enhance their projects.

    The golden rectangle may be pleasing to the eye for a couple of reasons. First, since it was composed of an interesting proportion, there are rectangles inside of it which are also golden rectangles. Within those golden rectangles are more golden rectangles until infiniti. This piece of information may have shocked the ancient Greeks. Everything about the rectangle is proportional. To build sculptures or buildings, the ancient Greeks needed their architecture to remain proportional throughout. This would prevent buildings from collapsing. Second, by choosing a unique rectangle with a ratio of its sides to be approximately 1.6, it was easy to create a model for the perfect rectangle. This one unique rectangle would serve as the basis for other rectangles. The Greeks may have forced themselves to like this rectangle so that they could have a model for everything else. Other rectangles were unattractive because they were not as proportional throughout like the golden rectangle. Therefore, if an unproportional rectangle was used, then a building may collapse or artworks would not look right. Finally, ancient Greeks were known for their artistic ability. They have eyes for beauty. When they spotted this rectangle, they saw it as beautiful and made it golden rectangle.
     

    B. Using newspapers or magazines, analyze at least 25 advertisements and photographs in terms of the golden rectangle model. 

    Discuss any interesting patterns/relationships that you discover. 

    What conclusions can be drawn from your investigation? 

    I have noticed a few things about the pictures and advertisements I have cut out. Many of them had similar ratios. The lowest ratio was 1.05 and the highest was 1.79. The few pictures and ads that were the golden rectangle were visually appealing. They fit nicely onto the page and made it seem like it always belonged there.

    I also noticed a few things about the rectangles that did not have the golden ratio of 1.6. They looked as if they appeared to be golden rectangles, but they really were not. Also, it seemed as if these rectangles were shrunk or blown up to fill empty space on the page.

    Basically, if someone wants to create something professional and clean, golden rectangles are the ways to follow.
     

    C. Using the golden ratio/golden rectangle model, prepare one or more sample pages showing the photographs, advertisements and text that would appear on full-size newspaper pages.
     
     
     

    Sample #1 Sample #2

     

    Sample Performance: Achieved the Standard with Honors (9 out of 10 points)

    A. Investigate the golden ratio / golden rectangle. How are they defined? Why is it more pleasing to the eye? 

    The golden ratio can also be known as the golden section. It is the division of a line segment in such a way that the ratio of the whole segment to the larger part is equal to the ratio of the larger part to the smaller part (it's a little tongue twisting). The ratio is approximately 1.618033989 to 1. A rectangle whose length and width that agrees with this ratio is called a golden rectangle. A golden rectangle has the interesting property that, if you create a new rectangle by swinging the long side around one of its ends to create a new long side, the new rectangle is also golden. Rectangles that look like golden rectangles are more "pleasing to the eye" than other rectangles, though no one knows why. However people in my opinion use the golden ratio/rectangle because they want to attract attention to their ad, building, or text. Many golden rectangles and golden ratios appear in famous paintings, sculptures, and architecture. The Parthenon for example, was constructed in Athens in the 400's BC, and buildings designed in the 1900's by the French architect Le Corbusier mostly incorporate the golden ratio and the golden rectangles.
     
     

    This picture shows a house that was built in France by Le Corbusier. The house was developed with the golden ratio. Therefore this strange and unusual "thing" has a ratio of 1.618033989... to 1. The Cathedral of Florence was an early achievement of Italian Renaissance architecture. Filippo Brunelleschi designed the cathedral. It said that all the rectangles in the structure were golden.

     
    Parthenon was constructed under the order of Pericles. He was the leader at the time it was constructed. This time was called the "Golden Age." The two architects that actually built it were Ictinus and Callecrates. Parthenon was built in honor of Athena. Athena was a goddess. It was built on the acropolis Athens. It took 9 years to build from 447-432 B.C. The temple was surrounded by 46 Doric columns. There is an outer wall and an inner chamber. After a while it was turned into a church. Then after that it was turned into a storing place for gunpowder. Around 1687 it was changed. The center was destroyed by a gunpowder explosion, and was then reconstructed. The picture above is a picture of Parthenon.

    Some historians state that the properties of the golden ratio aided the Pythagoreans in discovering irrational numbers, actually they're geometric equivalent to lines. It is certain, however, that since many philosophers, artists, and mathematicians have been intrigued by the golden ratio, which renaissance writers have called the divine proportion. It is widely accepted that a rectangle with sides in this ratio has a special beauty, which no one can describe.

    B. Using newspapers or magazines, analyze at least 25 advertisements and photographs in terms of the golden rectangle model. 

    Discuss any interesting patterns/relationships that you discover. 

    What conclusions can be drawn from your investigation? 

    The recommended number of newspapers or magazines that had to be analyze was only twenty-five, in my opinion that number would not give the assignment very good results/odds. To conduct a proper search of the golden ratio fifty-five newspaper-ads were used. The formula that was used was length plus width divided by length, then square root the result. Some patterns that I noticed while calculating the math part, was that it would not even come close to the ratio, but then after a few it would and some went over it as well. None of the results actually ended with the golden ratio as it was completed, however about thirty-one out of fifty-five came close or in the range of the golden ratio. An Internet site that I visited presented two ways of finding the ratio, one is mentioned above, and the second one divided the length into the width. There are five pages of calculations, four for the first method and four pages for the second. The web site also stated that there are two ways of presenting the golden ratio, one is 1.618033989 to 1, or 0.618033989 to 1, either way the golden ratio is achieved.

    In conclusion, not every rectangle in the world is golden. The assignment brought a new meaning of a rectangle to me, because I thought they were just silly little boxes on paper that no one put effort into, but some do in order to make the ratio balance. Also due to this assignment I have a better understanding of ratios and why ads use the golden ratio to attract attention to their ad. As Pythagoras said "The Golden Ratio manifests in the whole of creation. Take the ratio of the length of a man and the height of his navel. The ratio the sides of the Great Temple because the ratio of the whole to the Greater is the ratio of the Greater to the lesser." In a way this is true because the makers of these ads want to get the attention of potential buyers and in doing so they need an eye pleasing way of getting these people. This infamous eye-pleasing ratio has boggled the minds of many people since the time of the ancient world, but painters, sculptors, and others have used the golden ratio and it will always be used as long as people want to attract attention to their work.

    C. Using the golden ratio/golden rectangle model, prepare one or more sample pages showing the photographs, advertisements and text that would appear on full-size newspaper pages.

    Rubric

    Achieved the Standard with Honors
    Achieved the Standard
    Nearly Achieved the Standard
    Below the Standard
    Little Evidence of Achievement
  • Student may present a definition of golden ratio/rectangle which reflect an in-depth understanding by including applications and history.
  • Explanation of "pleasing to the eye" is adequate.
  • Student may analyze greater than 25 advertisements using the golden rectangle model and presents his/her findings in a clear but creative manner.
  • Student may identify 2 or more patterns and/or relationships in the data.
  • Conclusions are clear, logical, and valid.
  • Student may prepare multiple sample newspaper pages using concept of golden ratio / rectangle.
  • No math errors.
  • Student presents an adequate definition of golden ratio/rectangle.
  • Explanation of why it is "pleasing to the eye" is adequate.
  • Student analyzes 25 advertisements using the golden rectangle model and presents the findings in an organized manner.
  • Student identifies one pattern or relationship from the data.
  • Conclusions are valid, logical, and presented in a clear manner.
  • Student prepares one sample newspaper page incorporating the concept of golden ratio/rectangle.
  • No math errors.
  • Student presents an adequate definition of golden ratio/rectangle.
  • Explanation of why it is "pleasing to the eye" is adequate.
  • Student may have analyzed less than the required 25 advertisements. Findings may not be presented in an organized and clear manner.
  • Student may identify one pattern or relationship but conclusions are not valid or logical.
  • Student prepares one sample newspaper page incorporating the concept of golden ratio/rectangle.
  • Minor math errors may be present throughout project.
  • Student presents an incomplete definition of golden ratio/rectangle.
  • Explanation of why it is "pleasing to the eye" is not adequate.
  • Student analyzes less than 25 advertisements and findings are presented in a disorganized manner.
  • Student may fail to identify a pattern or relationship in the data.
  • Conclusions may be invalid or illogical.
  • Sample newspaper page does not incorporate concept of golden ratio/rectangle.
  • Major errors in math are evidence.
  • Major task elements are missing, incomplete, and/or incorrect. Major math errors evident throughout the task (including definitions, history, reasoning, and computation).