DONALD IN MATHMAGIC LAND IES SIERRA DE STA. BÁRBARA Donald in Mathmagic Land MATHEMATICS 1º ESO IES SIERRA DE SANTA BÁRBARA The Pythagoreans Pythagoreans were students of the mathematical, philosophical, and religious school started by Pythagoras (c. 580 B.C. – c. 500 B.C.). Some historians think that Pythagorean students were expected to listen but not contribute during their first five years at school, and that they were to credit any mathematical discovery to the school or to Pythagoras. The Golden Section The lines of a pentagram can be divided into four different lengths. Lines #1 and #2 exactly equal line #3. This partition is a Golden Section. And lines #2 and #3 exactly equal line #4. This partition is also a Golden Section. The ratio of the lengths of the two Golden Sections is (square root of 5 + 1)/2, approximately 1.618. When this ratio is used to create the length and width of a rectangle, the result is called a Golden Rectangle. The Golden rectangle A Golden Rectangle is a rectangle whose ratio of length to width is approximately 8 to 5, or 1.618. These proportions were greatly admired by the Greeks, and Golden Rectangles are found in classical architecture and art. The Magic Spiral A magic spiral isn’t magic at all. It is a spiral that repeats the proportions of the Golden Sections of a Golden Rectangle into infinity. Magic spirals can be seen in many of nature’s spirals, such as the shell of the sea snail. BACKGROUND INFORMATION
Transcript
DONALD IN MATHMAGIC LAND IES SIERRA DE STA. BÁRBARA
Donald in Mathmagic
Land
2012-13
MATHEMATICS 1º ESO IES SIERRA DE SANTA BÁRBARA
The Pythagoreans Pythagoreans were students of the mathematical, philosophical, and religious school started by Pythagoras (c. 580 B.C. – c. 500 B.C.). Some historians think that Pythagorean students were expected to listen but not contribute during their first five years at school, and that they were to credit any mathematical discovery to the school or to Pythagoras.
The Golden Section The lines of a pentagram can be divided into four different lengths. Lines #1 and #2
exactly equal line #3. This partition is a Golden Section. And lines #2 and #3 exactly equal line #4. This partition is also a Golden Section. The ratio of the lengths of the two Golden Sections is (square root of 5 + 1)/2, approximately 1.618. When this ratio is used to create the length and width of a rectangle, the result is called a Golden Rectangle.
The Golden rectangle A Golden Rectangle is a rectangle whose ratio of length to width is approximately 8 to 5, or 1.618. These proportions were greatly admired by the Greeks, and Golden
Rectangles are found in classical architecture and art.
The Magic Spiral A magic spiral isn’t magic at all. It is a spiral that repeats the proportions of the Golden Sections of a Golden Rectangle into infinity. Magic spirals can be seen in many of nature’s spirals, such as the shell of the sea snail.
BACKGROUND INFORMATION
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Porch of the Caryatids The Caryatids are six female statues supporting the south porch roof of the Erechtheum temple. The temple, located on the Acropolis in Athens, Greece, was built between 420 B.C. to 406 B.C. The Caryatids that adorn the temple today are copies, but four of the six originals are housed in the Acropolis Museum.
Venus de Milo This famous ancient Greek sculpture depicts the goddess Venus. The sculpture was named for the Greek island of Melos where it was discovered in 1820 as it was about to be crushed into mortar. The sculpture was restored, but its broken arms were lost and never replaced. The sculpture is now housed in the Louvre Museum in Paris, France.
Cathedral of Notre Dame of Paris The Cathedral of Notre Dame in Paris, France, is regarded as the greatest masterpiece of Gothic architecture. It was constructed between 1163 and 1250 and was dedicated to Mary, the Mother of Jesus (“Notre Dame” means “Our Lady” in French). It was restored after the French Revolution ended in 1799.
Mona Lisa This portrait of a Florentine woman was painted between the years 1503 and 1506 by Leonardo da Vinci (1452-1519). The painting was stolen from France’s Louvre Museum in 1911, but was found in a Florence hotel room two years later and returned to the Louvre.
United Nations Secretariat
Building This 39-story building is one of several United Nations (U.N.) buildings located on the 18-acre U.N. complex in New York City. John D. Rockefeller Jr. donated the land and design began in 1947. The Secretariat building was completed in 1950. The U.N. Secretary General’s offices are located on the 38th floor.
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1. Pythagoras was a Greek mathematician. What are some of the mathematical contributions he made?
a. Parthenon.
b. Billiards.
c. Music.
2. What is weird about the trees Donald sees in Mathmagic Land?
a. The trees are black.
b. Their roots are square.
c. Their fruits are numbers.
3. Where is Donald going?
a. Europe.
b. Egypt.
c. Greece.
4. Where is Maths found in nature?. Give three examples.
a.
b.
c.
5. What does Donald mean when he says: “There’s a lot more to mathematics than two times two”?
6. We have seen in the video that there are many games that are developed in geometric spacesIn what games you can see maths? Give three examples.
a.
b.
c.
POSTVIEWING QUESTIONS
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7. Answer the following questions:
a. What is the only game in which there are circles?
i. Baseball
ii. Basketball
iii. Billiards
b. How many squares form the chess board?
i. 100
ii. 48
iii. 64
8. In the game of billiards mathematical calculations are essential. Say if the following statements are true or false.
a. To calculate the position where I shoot you only need a sum.
i. True
ii. False
b. The position of the diamond is marked with integers.
i. True
ii. False
9. List 10 geometry terms and shapes that you have seen in the film.
10. Write down three places where you can find the pentagon in nature.
a.
b.
11. Indicate whether the following statements are true or false:
a. The voice that speaks with Donald is "the spirit of mathematics"
b. Donald does not like mathematics.
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12. Fill the gaps:
Pythagoras is the father of and . He invented the musical scale
using a . He tightened the rope and it in two equal parts
13. Look at the pictures and fill in the gaps:
a. The objects represented in these images are obtained from the section of a
b. In this case the objects are obtained from a
c. Finally, these are obtained from a
14. Try to imagine your world without numbers. How would you telephone your friends? How would you change your TV channel? How would you know what size you are?
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15. What instruments appear in the video?
a. Drums
b. Recorder
c. Piano
16. How do you get the pythagorean star from a pentagon?
a. By joining consecutive vertices.
b. By joining every two vertices.
c. The pythagorean star and a pentagon are not related.
17. Where can we find the Golden rectangle?
a. The Parthenon (Athens)
b. The sculptures.
c. The cathedral of Burgos.
d. Some skyscrapers.
e. Donald duck.
18. The strange creature which recites the digits of
the PI number is made of:
a. A circle.
b. A triangle.
c. A square.
d. A rectangle.
19. In which scenes of the film do numbers appear?
a. Animals
b. Footprints
c. Trees
d. The river
20. What do you get when you spin a circle?
21. What do you get when you spin a triangle?
22. What examples does the spirit give to explain that the circle has been the basis of many human inventions?
23. What’s the meaning of the closed doors of the film?
a. The past.
b. The present.
c. The future.
24. What is the key?
a. Imagination.
b. Luck.
c. Mathematics.
25. Who said “Mathematics is the alphabet with
which God has written the universe”
a. Pythagoras.
b. Galileo Galilee.
c. Donald.
26. In the video it is said that Pithagoras claimed: “ Everything is ruled by numbers and mathematical shapes”. Do you agree with Pithagoras? Why?
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THE GOLDEN RATIO IN ART AND ARCHITECTURE
The appearance of this ratio in music, in patterns of human behaviour, even in the proportion of the human body, points to its universality as a
principle of good structure and design.
Parthenon (Athens)
The Last Supper (Leonardo da Vinci)
APPENDIX A: THE GOLDEN RATIO
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Mona Lisa (Leonardo da Vinci)
CN Tower (Toronto)
Taj Mahal (Agra)
Notre Dame (Paris)
Status of Athena
Eiffel Tower (Paris)
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VISUAL POINTS OF INTEREST INSIDE A GOLDEN RECTANGLE
Any square or rectangle (but especially those based on the golden ratio) contains areas inside it that appeal to us visually as well. Here’s how you find those points:
1. Draw a straight line from each bottom corner to its opposite top corner on either side. They will cross in the exact center of the format.
2. From the center to each corner, locate the midway point to each opposing corner.
These points—represented by the green dots in the diagram above—are called the “eyes of the rectangle.”
One strategy often used by artists is to locate focal points or areas of emphasis around and within these eyes, creating a strong visual path in their compositions.
Let’s see some examples:
Edward Hopper’s composition, below, sets the sailboat right on the lower right eye (with the tip of the sails extending nearly to the upper right eye).
DONALD IN MATHMAGIC LAND IES SIERRA DE STA. BÁRBARA
APPENDIX B: CONSTRUCT A GOLDEN RECTANGLE
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PI (π) Pi represents the ratio of the
circumference of a circle to its diameter (approximately 3.14).
PENTAGRAM. A pentagram is a five-
pointed star. Formed by five straight lines, a pentagram connects the vertices of a pentagon and encloses another pentagon in the complete figure.
INFINITY. Infinity is an immeasurably
large amount that increases indefinitely and has no limits.
PENTAGON. A pentagon is a polygon
with five sides.
CONE. A cone is a pyramid-like object
with a circle-shaped base.
ANGLE. An angle consists of two rays
with a common end point.
SPHERE. A sphere is the set of all
points in space at a given distance from a given point called the center.
GOLDEN SECTION. The Golden
Section refers to a ratio between two dimensions of a plane figure, observed especially in art.
RATIO. A ratio is the comparison
between two numbers.
APPENDIX C: GLOSSARY TERM AND DEFINITION
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“Phi and the Golden Section in Architecture” [online]. Phi 1.618 The Golden Number. May 11, 2012. http://www.goldennumber.net/architecture
“LAB: The golden rectangle” [online]. Math Bits. http://mathbits.com/MathBits/MathMovies/LABGolden%20Rectangle.pdf
MIZE, Dianne. “A guide to the Golden Ratio for Artists” [online]. http://emptyeasel.com/2009/01/20/a-guide-to-the-golden-ratio-aka-golden-section-or-golden-mean-for-artists/
FREITAG, Mark. "Phi: That Golden Number." 15 June, 2005. [online] http://jwilson.coe.uga.edu/emt669/Student.Folders/Frietag.Mark/Homepage/Goldenratio/goldenratio.html
ELLIOTT, Ruth. “The golden rectangle (and the golden spiral)”, 2008. [online] http://www.edudesigns.org
STEWART, Ian. The magical Maze: Seeing the World Through Mathematical Eyes. Wiley & Sons, Inc., 1997
PERDIGUERO, Eva María. “Donald en el país de las matemágicas” [online]. http://edu.jccm.es/proyectos/cuadernia-descartes/eXe/donald-mates/index.html
