Our thoughts:
At the beginning of this process, we had no idea that the Golden Ratio was such a huge topic to explore. We found out during our long hours of researching that the Golden Ratio is so much more than a simple equation that one can memorize. We have learned that it is found everywhere in the world around us - in architecture, art, nature, music, and so many others!! It is connected to the golden rectangle, the fibonacci sequence, the golden triangle and the list continues. In addition, we came to realize after that yes, the Golden Ratio would be difficult to teach to children straight from a textbook with the defintion, numbers, equation and all. But, perhaps it could be taught through a music or science lesson for example in such a way that children could understand it.
Throughout this process we answered some of our questions but we also came up with more: How is the Golden Ratio directly related to the Fibonacci Sequence? Why would our students enjoy learning about this topic? How could we teach this topic to primary and elementary students in such a way that they will not become frustrated with it but understand it and enjoy it as well? Can it wait to be taught in higher grade levels? What would be the benefit or disadvantage of doing that? Would introducing it or teaching it at the lower grade level do injustice to the topic itself? How can this topic involve more problem solving for students?
Our learning experience about the Golden Ratio isn't over yet, we intend to learn more about this topic and many others as we continue our journey to become the best teachers we can be! We are also excited to start learning new things each day in our classroom right along with our future students as we discover the magic of mathematics and other subjects together!
Note: We could only be priviledged enough to scratch the surface of this exciting but extremely broad topic. Therefore, we strongly encourage you to get involved with the Golden Ratio and research it some more! Hope you enjoy as much as we did!! :)
Thursday, April 5, 2007
Tuesday, March 27, 2007
What's Golden?
What is this thing they call the Golden Ratio?
According to Wikipedia, the Golden Ratio expresses the relationship that the sum of two quantities is to the larger quantity as the larger is to the smaller. It is usually denoted as the Greek letter 'phi' (φ). However, it is more accurately represented by (√5+1)/2. It is also known as, or closely related to, the golden section, the golden number, and divine proportion.
If line AB is longer than the segment AC, the segment AC is longer than CB. If the ratio of the length of AC to that of CB is the same as the ratio of AB to AC, then the line has been cut in extreme and mean ratio, or in a Golden Ratio.
The Golden Ratio cannot be expressed as a fraction; in other words, the ratio of the two lengths AC and CB cannot be expressed as a fraction. For example, we cannot find some common measure that is contained, for instance, 31 times in AC and 19 times in CB. Two such lengths that have no common measure are called ‘incommensurable’.
Here is a link to show the calculation of the golden ratio:
http://en.wikipedia.org/wiki/Golden_ratio#Calculation
According to Wikipedia, the Golden Ratio expresses the relationship that the sum of two quantities is to the larger quantity as the larger is to the smaller. It is usually denoted as the Greek letter 'phi' (φ). However, it is more accurately represented by (√5+1)/2. It is also known as, or closely related to, the golden section, the golden number, and divine proportion.
The Golden Ratio is approximated as 1.6180339887... Therefore, it is considered an irrational number. Phi has the value √5 + 1 /2 and phi is √5 – 1 /2.
It is the only number, which, when diminished by unity, becomes its own reciprocal:
Φ – 1/Φ = 1 i.e., φ² - φ – 1 = 0
The golden ratio of a straight line can be viewed at the following site:
http://plus.maths.org/issue22/features/golden/
Here is an explanation of a line cut in Golden Ratio (as seen in the above link):
It is the only number, which, when diminished by unity, becomes its own reciprocal:
Φ – 1/Φ = 1 i.e., φ² - φ – 1 = 0
The golden ratio of a straight line can be viewed at the following site:
http://plus.maths.org/issue22/features/golden/
Here is an explanation of a line cut in Golden Ratio (as seen in the above link):
If line AB is longer than the segment AC, the segment AC is longer than CB. If the ratio of the length of AC to that of CB is the same as the ratio of AB to AC, then the line has been cut in extreme and mean ratio, or in a Golden Ratio.
The Golden Ratio cannot be expressed as a fraction; in other words, the ratio of the two lengths AC and CB cannot be expressed as a fraction. For example, we cannot find some common measure that is contained, for instance, 31 times in AC and 19 times in CB. Two such lengths that have no common measure are called ‘incommensurable’.
Here is a link to show the calculation of the golden ratio:
http://en.wikipedia.org/wiki/Golden_ratio#Calculation
Where it all began...
The Golden Ratio is a proportion that has intrigued numerous intellectuals of diverse interests since antiquity. Although the first clear definition was derived by Euclid around 300 B.C., it was already used by the Egyptians in constructing the Pyramids at Giza.
Note: The Greeks usually attributed the discovery of the ratio to Pythagoras.
The common symbol for the Golden Ratio, in the professional mathematical literature, is the Greek letter 'tau' meaning "the cut" or "the section". However, the American Mathematician Mark Barr gave the ratio the name 'phi' at the beginning of the twentieth century to honor the great Greek sculptor, Phidias.
To View the timeline of the Golden Ratio, have a look at this link:
http://en.wikipedia.org/wiki/Golden_ratio#Timeline
The Golden Ratio frequently appears in geometry especially regular pentagrams and pentagons. It is this that fascinated ancient Greek mathematicians to first study the Golden Ratio. It is found in art and architecture, especially in ancient Greek temples. During the Renaissance in Italy, many scientists and artists such as Leonardo Da Vinci were drawn to mathematics and its relationship to the spiritual and physical order of the universe. This is why they called the Golden Ratio the "divine proportion".
Note: The Greeks usually attributed the discovery of the ratio to Pythagoras.
The common symbol for the Golden Ratio, in the professional mathematical literature, is the Greek letter 'tau' meaning "the cut" or "the section". However, the American Mathematician Mark Barr gave the ratio the name 'phi' at the beginning of the twentieth century to honor the great Greek sculptor, Phidias.
To View the timeline of the Golden Ratio, have a look at this link:
http://en.wikipedia.org/wiki/Golden_ratio#Timeline
The Golden Ratio frequently appears in geometry especially regular pentagrams and pentagons. It is this that fascinated ancient Greek mathematicians to first study the Golden Ratio. It is found in art and architecture, especially in ancient Greek temples. During the Renaissance in Italy, many scientists and artists such as Leonardo Da Vinci were drawn to mathematics and its relationship to the spiritual and physical order of the universe. This is why they called the Golden Ratio the "divine proportion".
What is all this fuss about?
“The Golden Ratio’s attractiveness stems first and foremost from the fact that it has an almost uncanny way of popping up where it is least expected.” (Livio 7) It may seem complicated to the average person, but it naturally arises in the most simplistic situations ranging from the position of rose petals to the compositions of priceless paintings.
Architecture:
The Golden Ratio is found in many architectural structures. Classical buildings or their elements are proportioned according to the Golden Ratio. It is not known whether they directly designed them in this manner or if they used their own good sense of proportions.
Examples:
- The Parthenon
- The Great Mosque of Kairouan
- Book Covers (such as Le Modulor by Le Corbusier)
Art:
The Golden Ratio is seen in many famous paintings by a variety of prominent artists. Leonardo Da Vinci’s illustrations in De Divina Proportione and The Mona Lisa suggest that he integrated the Golden Ratio into his works of art. This is highly debatable because the secretive Da Vinci rarely disclosed the bases of his art and therefore the proportions can never be conclusive.
Salvador Dali also suggests the Golden Ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the painting (105 ½” x 65 ¾”) are in Golden Ratio to each other which creates a Golden Rectangle.
In addition to paintings, the Golden Ratio is also used by the Australian sculptor, Andrew Rogers. His creation is made of 50-ton stone and gold and entitled Golden Ratio which is located outdoors in Jerusalem. The height of each stack of stones starting from either end and moving toward the middle is the beginning of the Fibonacci sequence: 1, 1, 2, 3, 5, 8.
Examples:
De Divina Proportione:
The Sacrament of the Last Supper:
The Golden Ratio:
Nature: (A few examples)
“Nature's surface beauty conveys no more than a hint of the loveliness hidden within” (Huntley 151). Adolf Zeising, found the golden ratio expressed in the arrangement of branches along the stems of plants, and of veins in leaves. He later extended his research to include the skeletons of animals and the branchings of their veins and nerves, chemical compounds, and the geometry of crystals.
Nature also offers the Golden Ratio through an ordinary apple. When the girth is cut, the seeds are arranged in a five-pointed star pattern or pentagram. This forms a ratio in which the length of the longer side to the shorter side is equal to 1.618, in other words, the Golden Ratio.
“Nature loves logarithmic spirals. From sunflowers, seashells, and whirlpools, to hurricanes and giant spiral galaxies, it seems that nature chose this marvelous shape as its favorite ‘ornament’” (Livio 117).
For example, as a mollusk’s spiral shell grows, the increments are proportionate to the Golden Ratio.
Picture of a mollusks spiral shell:
“Geometry has two great treasures: one is the theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we name a precious jewel” (Kepler 1571-1630)
Architecture:
The Golden Ratio is found in many architectural structures. Classical buildings or their elements are proportioned according to the Golden Ratio. It is not known whether they directly designed them in this manner or if they used their own good sense of proportions.
Examples:
- The Parthenon
- The Great Mosque of Kairouan
- Book Covers (such as Le Modulor by Le Corbusier)
Art:
The Golden Ratio is seen in many famous paintings by a variety of prominent artists. Leonardo Da Vinci’s illustrations in De Divina Proportione and The Mona Lisa suggest that he integrated the Golden Ratio into his works of art. This is highly debatable because the secretive Da Vinci rarely disclosed the bases of his art and therefore the proportions can never be conclusive.
Salvador Dali also suggests the Golden Ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the painting (105 ½” x 65 ¾”) are in Golden Ratio to each other which creates a Golden Rectangle.
In addition to paintings, the Golden Ratio is also used by the Australian sculptor, Andrew Rogers. His creation is made of 50-ton stone and gold and entitled Golden Ratio which is located outdoors in Jerusalem. The height of each stack of stones starting from either end and moving toward the middle is the beginning of the Fibonacci sequence: 1, 1, 2, 3, 5, 8.
Examples:
De Divina Proportione:
The Sacrament of the Last Supper:
The Golden Ratio:
Nature: (A few examples)
“Nature's surface beauty conveys no more than a hint of the loveliness hidden within” (Huntley 151). Adolf Zeising, found the golden ratio expressed in the arrangement of branches along the stems of plants, and of veins in leaves. He later extended his research to include the skeletons of animals and the branchings of their veins and nerves, chemical compounds, and the geometry of crystals.
Nature also offers the Golden Ratio through an ordinary apple. When the girth is cut, the seeds are arranged in a five-pointed star pattern or pentagram. This forms a ratio in which the length of the longer side to the shorter side is equal to 1.618, in other words, the Golden Ratio.
“Nature loves logarithmic spirals. From sunflowers, seashells, and whirlpools, to hurricanes and giant spiral galaxies, it seems that nature chose this marvelous shape as its favorite ‘ornament’” (Livio 117).
For example, as a mollusk’s spiral shell grows, the increments are proportionate to the Golden Ratio.
Picture of a mollusks spiral shell:
“Geometry has two great treasures: one is the theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we name a precious jewel” (Kepler 1571-1630)
In the Classroom...
Mathematcis is connected to all of the subject areas and is very present in the world around us. Hopefully our students will see that math is not something that is created or invented by people but rather discovered to be as old as time.
The Golden Ratio can be used to arouse students' interest in Geometry units. When introducing the Golden Ratio to students it may help to have a visual demonstration of it. Here is a link for students to further understand the golden rectangle:
http://nlvm.usu.edu/en/nav/frames_asid_133_g_3_t_3.html
Teachers may find these sites helpful when planning lessons on the Golden Ratio:
http://www.geom.uiuc.edu/~demo5337/s97b/worksheet.html
http://math.rice.edu/~lanius/Geom/golden.html
http://www.markwahl.com/golden-ratio.htm
http://cuip.uchicago.edu/~dlnarain/golden/activities.htm
http://www.geom.uiuc.edu/~demo5337/s97b/figures.htm
When researching this topic and creating this blog, we found out that it would be difficult to address the golden ratio in primary grades because it is such an advanced topic. They could be introduced to the Golden Ratio in the later elementary years of grades five or six when doing a geometry unit for motivation purposes for example. Most of the lesson plans we came across are designed for junior high and high school students. If a primary/elementary teacher decides to take this topic upon them to teach or introduce, he or she can possibly take a lesson plan designed for the higher grades and adjust the material to accomodate students at the lower grade level.
Further investigation into this topic can lead students into discovering and making connections through the relationship between the Golden Ratio and the Fibonacci Sequence. As well, as taking a closer look at the Golden Ratio through the golden rectangle, the golden triangle, the pentagram, art, nature, etc.
The Golden Ratio can be used to arouse students' interest in Geometry units. When introducing the Golden Ratio to students it may help to have a visual demonstration of it. Here is a link for students to further understand the golden rectangle:
http://nlvm.usu.edu/en/nav/frames_asid_133_g_3_t_3.html
Teachers may find these sites helpful when planning lessons on the Golden Ratio:
http://www.geom.uiuc.edu/~demo5337/s97b/worksheet.html
http://math.rice.edu/~lanius/Geom/golden.html
http://www.markwahl.com/golden-ratio.htm
http://cuip.uchicago.edu/~dlnarain/golden/activities.htm
http://www.geom.uiuc.edu/~demo5337/s97b/figures.htm
When researching this topic and creating this blog, we found out that it would be difficult to address the golden ratio in primary grades because it is such an advanced topic. They could be introduced to the Golden Ratio in the later elementary years of grades five or six when doing a geometry unit for motivation purposes for example. Most of the lesson plans we came across are designed for junior high and high school students. If a primary/elementary teacher decides to take this topic upon them to teach or introduce, he or she can possibly take a lesson plan designed for the higher grades and adjust the material to accomodate students at the lower grade level.
Further investigation into this topic can lead students into discovering and making connections through the relationship between the Golden Ratio and the Fibonacci Sequence. As well, as taking a closer look at the Golden Ratio through the golden rectangle, the golden triangle, the pentagram, art, nature, etc.
Works Cited
Websites:
http://cuip.uchicago.edu/~dlnarain/golden/teaching_guide.htm
http://en.wikipedia.org/wiki/Golden_ratio
http://mathworld.wolfram.com/GoldenRatio.html
http://mathforum.org/dr.math/faq/faq.golden.ratio.html
Books:
Livio, Mario. The Golden Ratio. New York: Broadway Books, 2002.
Huntley, H.E. The Divine Proportion: A Study in Mathematical Beauty. London: Dover Publications, 1970.
Walser, Hans. The Golden Section. Washington: The Mathematical Association of America, 2001.
Koshy, Thomas. Fibonacci and Lucas Numbers with Applications. New York: John Wiley & Sons, Inc, 2001
Dunlap, Richard, A. The Golden Ratio and Fibonacci Numbers. New Jersey: World Scientific Publishing Co. Pte. Ltd, 1997
Journals:
Breckenridge, Mary Beth. “Phi Rules with Pleasure.” Beacon Journal (2007).
Kimberling, Clark. “A Self-Generating Set and the Golden Mean.” Journal of Integer Sequences Vol.3 (2000)
Peterson, Ivars. “Sea Shell Spirals.” Science News Vol. 167, No. 14 (2005)
Williams, Kim. “Winter 2002 Special Issue: The Golden Section.” Nexus Network Journal Vol.4, No. 1 (2002)
http://cuip.uchicago.edu/~dlnarain/golden/teaching_guide.htm
http://en.wikipedia.org/wiki/Golden_ratio
http://mathworld.wolfram.com/GoldenRatio.html
http://mathforum.org/dr.math/faq/faq.golden.ratio.html
Books:
Livio, Mario. The Golden Ratio. New York: Broadway Books, 2002.
Huntley, H.E. The Divine Proportion: A Study in Mathematical Beauty. London: Dover Publications, 1970.
Walser, Hans. The Golden Section. Washington: The Mathematical Association of America, 2001.
Koshy, Thomas. Fibonacci and Lucas Numbers with Applications. New York: John Wiley & Sons, Inc, 2001
Dunlap, Richard, A. The Golden Ratio and Fibonacci Numbers. New Jersey: World Scientific Publishing Co. Pte. Ltd, 1997
Journals:
Breckenridge, Mary Beth. “Phi Rules with Pleasure.” Beacon Journal (2007).
Kimberling, Clark. “A Self-Generating Set and the Golden Mean.” Journal of Integer Sequences Vol.3 (2000)
Peterson, Ivars. “Sea Shell Spirals.” Science News Vol. 167, No. 14 (2005)
Williams, Kim. “Winter 2002 Special Issue: The Golden Section.” Nexus Network Journal Vol.4, No. 1 (2002)
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