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GRADE 9 ACADEMIC MATH RESEARCH STUDY

GRADE 9 ACADEMIC MATH RESEARCH STUDY. September 2000 - January 2001. GRADE 9 SECTION 01. THERE WERE THREE DESTINATIONS ON OUR INTEGRATED JOURNEY DURING THE SCHOOL SEMESTER: GOLDEN AGE OF ANCIENT GREECE ca. 500-200 B.C. HIGH RENAISSANCE IN ITALY & ENVIRONS ca. 1500-1527 A.D.

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GRADE 9 ACADEMIC MATH RESEARCH STUDY

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  1. GRADE 9 ACADEMIC MATHRESEARCH STUDY September 2000 - January 2001

  2. GRADE 9 SECTION 01

  3. THERE WERE THREE DESTINATIONS ON OUR INTEGRATED JOURNEY DURING THE SCHOOL SEMESTER: • GOLDEN AGE OF ANCIENT GREECE • ca. 500-200 B.C. • HIGH RENAISSANCE IN ITALY & ENVIRONS • ca. 1500-1527 A.D. • ESCHER’S EUROPE ca. 1898-1972

  4. GRADE 9 SECTION 02

  5. ASSIGNMENT # 1 RATIO, PROPORTION & THE GOLDEN SECTION

  6. HISTORICAL ERA: ANCIENT GREECE CHARACTER: PYTHAGORAS MATH/ART FOCUS: RATIO, PROPORTION & GOLDEN SECTION MATH STRAND: NUMBER SENSE & ALGEBRA

  7. A C B

  8. TEACHING HOW TO FIND THE GOLDEN SECTION [I] ALGEBRAICALLY x 1 = 1.61803 (Phi) NOW, BEGINNING WITH ANY GIVEN LENGTH (L): NEXT LARGEST SECTION L(1.61803) NEXT SMALLEST SECTION L(0.61803)

  9. TEACHING HOW TO FIND THE GOLDEN SECTION • [II] GEOMETRICALLY • BEGIN WITH A SQUARE; EXTEND ONE SIDE • FROM MIDPOINT, CUT AN ARC TO EXTENDED LINE • COMPLETE RECTANGLE & INTERNAL SQUARES

  10. A GOLDEN “SMARTWORK” (SECONDARY MATH & ART WORK) Based on the new learning from this assignment and using the handouts given, construct a SMARTWORK that features the Golden Ratio in some form. This project allows for great freedom of expression. Your piece may be 2- or 3- dimensional. It may be narrative (tells a story), literary (quotes or an illustrated poem), historical (perhaps a tie to math history?), abstract (no recognizable objects), a collage, a sculpture – your imagination is the limit! You may choose to simply draw a Golden Rectangle, or you may choose to incorporate more complex ideas from any of the Golden Triangle, the Golden Spiral, the Pentagram, the Lute of Pythagoras, the Fibonacci Number Sequence, etc. Once again, the final SMARTWORK must include the Golden Ratio somewhere in the composition and will also be assessed for creativity & complexity.

  11. SW 14

  12. SW 14 • A LOOK BACK AT THE LAST CENTURY • “I decided to link my golden rectangle with the last century because I think time is a very valuable power in math. So, starting with the early 1900s in the center square, this project represents my ‘golden century.’ Each square represents a character from the time being mentioned. • Starting in the center square is the Titanic, which sank in 1912 • 2nd is a male fashion from 1914 • 3rd is the development of new vehicles • 4th is a diskette, which represents the wide use of computers and technology • 5th is a blood-pumping machine, representing medical breakthroughs • 6th is the ever-changing fashion of the present day • 7th might be some of the new technology for the future The small numbers on each line represent the Fibonacci numbers in relationship with the Golden Rectangle. As each number is added with the one before it, it equals the two sides of different rectangles added together.”

  13. SW 15

  14. SW 16

  15. SW 16 “I decided to make a Golden Spiral for my SMARTWORK because when I was eating supper one night, I was eating spaghetti as I was thinking about the SMARTWORK. So I put the Golden Spiral and spaghetti together for my SMARTWORK. The Golden Sections can be found by the black dots, inside of the meatballs, which form the corners of the golden rectangles. In my SMARTWORK, there is a black line which makes up the Golden Spiral, and the spaghetti follows it to the end. Because my SMARTWORK was going to be fairly big, I decided that I would use pencil crayons, not markers, to colour it in.”

  16. SW 17 “I did my project on the Golden Rectangle. I put the names of snowboards in the squares because I like to snowboard and wish I could have at least one of each board. The background is done like that because that’s the picture that I would put on a skateboard.”

  17. SW 18

  18. SW 19

  19. SW 20

  20. SW 21

  21. SW 22

  22. SW 23 “My SMARTWORK is a mobile. The base is a Golden Rectangle in the Golden Ratio (1.61803). In the middle is the pentagram. I chose this because its framework clearly demonstrates the Golden Ratio. Also hanging, are pictures of the Golden Cut, which divides a line segment into two line segments. Also hanging are images of the Golden Section, the Golden Rectangle, and the Golden Spiral. The Golden Section is located many times on my mobile, in the base and in the pieces.”

  23. SW 24

  24. SW 24 “For my SMARTWORK I started out with a piece of paper cut into a golden rectangle. Then I got an atlas and found the circumferences of all the planets. I scaled down the planets by using the formula: Circumference / pi /4000. I used the answer from that formula as a radius to draw circles in the squares. I used a separate piece of paper that was the exact same to find the middle of all the squares by drawing two diagonal lines corner to corner. I put the paper on top of the good paper and poked holes with my compass through both paper where the diagonals intersected in each square. The planets are all scaled 1.2756 x 109 to 1. The golden cuts are shown by the straight black lines. The planets are in order from largest to smallest. The reason that I chose the planets of the solar system is that it represents ‘math is everything.’ When you think about it, everything that happens can be described with a mathematical equation. For example, me writing this sentence could involve math from the angle of the pencil to the muscle cells in my hand interacting with each other.”

  25. SW 25

  26. SW 26

  27. SW 26 “I chose to do my Golden Rectangle like this because I thought it would be different. I used the cut out magazine phrases and works because they are things I usually say, or think about. I used the fabric because it was a different approach to cover up the space. The black designs I saw in a book, and thought it looked cool. The ‘eye rectangle’ is my favourite part because it’s so unique. I got the idea from a lamp in my room that resembles that. The square with the bow represents a present. It’s like if you take off the bow there are more rectangles and squares that go on forever. To get the squares I multiplied the length by 0.61803.”

  28. SW 27

  29. SW 27 “The theme for my SMARTWORK is ‘The road to hell is paved with good intentions.’ I got the idea from a TV show. The golden section in my SMARTWORK is the road. It is the Golden Spiral. I got the Golden Rectangle by taking the length and dividing by the width until I got 1.61803. Then I just kept doing that until I couldn’t see it. I chose the colours for what the word represents (i.e. water - blue, fire - red, etc.).”

  30. SW 28

  31. SW 28 “I had a lot of fun completing this part of this Golden Section project. I tried to make my SMARTWORK as creative and neat as I could. I made a series of boxes out of bristol board. I measured each box accurately to be the Golden Ratio. The front, back and bottom of each box has the dimensions to be classified as a perfect Golden Rectangle. I was trying to prove that the Golden Ratio could be in a rectangle as big as the Earth and as small as a grain of sand.”

  32. SW 29 “I first cut a piece of bristol board into the Golden Rectangle, with length 56 cm and width 34.6 cm. After this, I decided to show a picture of a boy getting closer and closer until you can practically see inside his pupil. I chose to do this because I think it’s interesting how you can take a whole image and end up focusing on just one thing. I chose this colour of bristol board because I thought it might catch people’s eyes.”

  33. SW 30

  34. SW 30 “For my SMARTWORK I chose to do mine in the Golden Rectangle pattern. In the biggest square I wrote all the terms that I could think of that had to do with this project: Golden Section, Phi, pentagram, Golden Triangle, pentagon, Golden Rectangle, Golden Ratio, Fibonacci Sequence, Golden Section, Golden Cut, Phidias, and the Parthenon. Then I repeated these over and over. In the next square I drew four Golden Rectangles with the symbol for Phi in the center. One rectangle is red, one yellow, one green, and one blue. I chose these colours because they are bright. I put four because there are four sides on a rectangle. In the next square I drew three Golden Triangles, using the three primary colours. I put three because there are three sides on a triangle. In the next square I put the Fibonacci Sequence. I wrote it out until I ran out of room, up to 75 025. I wrote is because it’s in relationship to the Golden Rectangle. In the next square I put Phi,1.61803, which you get when you divide the length of a Golden Rectangle by its width. In the final square I drew two pentagons with pentagrams inside. All the things that I put in my SMARTWORK have to do with the Golden Ratio and the Golden Proportion.”

  35. SW 31 “I chose to represent my project this way because it gives the effect of how each square becomes smaller and smaller, to infinity. This style also gives a 3-D appearance. To figure out if the squares are golden you take the length of the largest square and divide by the length of the next smallest square.”

  36. SW 32 “I measured the length of the figure [on p. 241 of handout] and divided it by the length of my page to get the representative fraction for 1 cm. I did this so that my picture of the Lute would be more accurate. I wanted a challenge so I followed the Four-Colour problem, and made sure that no neighbouring colours were the same with the exception of the corners. The sheet of paper is also in the Golden Rectangle.”

  37. SW 33

  38. SW 34 “The SMARTWORK is in the shape of the Golden Rectangle. Around the pentagon there are little Golden Rectangles. Inside the pentagram there is the Golden Triangle and the Lute of Pythagoras. I used different colours to show different changes.”

  39. SW 35 “I cut the paper so that it was the golden proportion by measuring the length and multiplying it by Phi, then cutting off the extra. I then divided it into the Golden Section and coloured each square differently. In one square, for example, I used fingerpaint and put my feet and my little brother’s feet in different colours (I got that idea because my little brother walked across the page). I used simple designs for the others and in the biggest square I wrote math and art together.”

  40. SW 36

  41. SW 37

  42. SW 37 “For my golden SMARTWORK I chose a sculpture made out of Rice Krispie squares. I started with a Golden Rice Krispie Rectangle with the measurements of 25 cm by 15.45 cm. Then I decided to show that within this one large rectangle there were six other Golden Rectangles. If you measure each Golden Rectangle’s length and width you will see that the ratio of the length to the width is approximately the Golden Ratio (1.61803). Upon completing my structure of Golden Rectangles, I then decided to construct the spiral through the whole rectangle. I chose dark string for this. I chose to create this sculpture because I enjoy making things in the kitchen. I thought that using Rice Krispie squares would be an interesting way to show Golden Rectangles and would be fun to do. It was more challenging than I had anticipated because as I piled the rectangles and stuck them together, my measurements often changed, making it necessary to keep rechecking my work. The leftover pieces did make this project quite rewarding!”

  43. THERE IS GEOMETRY IN THE HUMMING OF THE STRINGS. THERE IS MUSIC IN THE SPACING OF THE SPHERES. PYTHAGORAS (C.A. 582-500 B.C.)

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