1) The document is about Donald Duck visiting Mathmagic Land and learning about mathematical concepts like the Golden Ratio, Golden Rectangle, Fibonacci sequence and how they appear in nature, architecture and art. 2) In Mathmagic Land, Donald sees trees with square roots for branches and fruits that are numbers, and learns from the Spirit of Mathematics about how math is all around us. 3) Donald travels to Greece to see famous historical sites that incorporate mathematical patterns, like the Parthenon and Venus de Milo statue.

1. DONALD IN MATHMAGIC LAND IES SIERRA DE STA. BÁRBARA
Donald in
Mathmagic
Land
MATHEMATICS 1º ESO IES SIERRA DE SANTA BÁRBARA
2012-13
BACKGROUND INFORMATION
Rectangles are found in classical architecture
The Pythagoreans exactly equal line #3. This partition is a
and art.
Pythagoreans were students of the Golden Section. And lines #2 and #3 exactly
equal line #4. This partition is also a Golden
mathematical, philosophical, and religious
Section. The ratio of the lengths of the two The Magic Spiral
school started by Pythagoras (c. 580 B.C. – c.
Golden Sections is (square root of 5 + 1)/2, A magic spiral isn’t magic at all. It is a spiral
500 B.C.). Some historians think that
approximately 1.618. When this ratio is used that repeats the proportions of the Golden
Pythagorean students were expected to
to create the length and width of a rectangle, Sections of a Golden Rectangle into infinity.
listen but not contribute during their first five
the result is called a Golden Rectangle. Magic spirals can be seen in many of nature’s
years at school, and that they were to credit
spirals, such as the shell of the sea snail.
any mathematical discovery to the school or
to Pythagoras.
The Golden rectangle
A Golden Rectangle is a rectangle whose
The Golden Section ratio of length to width is approximately 8 to
The lines of a pentagram can be divided into 5, or 1.618. These proportions were greatly
four different lengths. Lines #1 and #2 admired by the Greeks, and Golden

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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 United Nations Secretariat
crushed into mortar. The sculpture was
Building
restored, but its broken arms were lost and
This 39-story building is one of several United
never replaced. The sculpture is now housed
Nations (U.N.) buildings located on the 18-
in the Louvre Museum in Paris, France.
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.
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.
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.

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POSTVIEWING QUESTIONS
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.

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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? 20. What do you get when you spin a circle?
a. Drums
b. Recorder
c. Piano 21. What do you get when you spin a triangle?
16. How do you get the pythagorean star from a
pentagon?
22. What examples does the spirit give to explain
a. By joining consecutive vertices. that the circle has been the basis of many
human inventions?
b. By joining every two vertices.
c. The pythagorean star and a pentagon are
not related.
17. Where can we find the Golden rectangle? 23. What’s the meaning of the closed doors of the
film?
a. The Parthenon (Athens)
a. The past.
b. The sculptures.
b. The present.
c. The cathedral of Burgos.
c. The future.
d. Some skyscrapers.
e. Donald duck.
24. What is the key?
18. The strange creature which recites the digits of a. Imagination.
the PI number is made of:
b. Luck.
a. A circle.
c. Mathematics.
b. A triangle.
c. A square. 25. Who said “Mathematics is the alphabet with
which God has written the universe”
d. A rectangle.
a. Pythagoras.
b. Galileo Galilee.
19. In which scenes of the film do numbers appear?
c. Donald.
a. Animals
26. In the video it is said that Pithagoras claimed: “
b. Footprints Everything is ruled by numbers and
c. Trees mathematical shapes”. Do you agree with
Pithagoras? Why?
d. The river

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APPENDIX A: THE GOLDEN RATIO
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)

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Taj Mahal (Agra)
Mona Lisa (Leonardo da Vinci)
CN Tower (Toronto)
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).

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 J.M.W. Turner uses the angle of his waves to create an arch that circles through the lower right and lower left eyes.
 In this painting, Carolyn Anderson places her subject’s hands around that spot too.

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APPENDIX B: CONSTRUCT A GOLDEN
RECTANGLE

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APPENDIX C: GLOSSARY TERM AND
DEFINITION
PENTAGRAM. A pentagram is a five-
PI (π) Pi represents the ratio of the pointed star. Formed by five straight lines,
circumference of a circle to its diameter a pentagram connects the vertices of a
(approximately 3.14). pentagon and encloses another pentagon
in the complete figure.
INFINITY. Infinity is an immeasurably
PENTAGON. A pentagon is a polygon
large amount that increases indefinitely
with five sides.
and has no limits.
CONE. A cone is a pyramid-like object
ANGLE. An angle consists of two rays
with a circle-shaped base.
with a common end point.
SPHERE. A sphere is the set of all
GOLDEN SECTION. The Golden
Section refers to a ratio between two
points in space at a given distance from a
dimensions of a plane figure, observed
given point called the center.
especially in art.
RATIO. A ratio is the comparison
between two numbers.

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BIBLIOGRAPHY
 “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

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Teresa Martín Gómez
J. César Bárcena Sánchez