Honors Algebra I Project: THE GOLDEN RATIO



Honors Algebra I Project: THE GOLDEN RATIO (Due: Thursday, 11/16/06 – No exceptions!)

Goal: To find the rectangles whose length-to-width ratios are equal to the golden ratio.

Exploring the Golden ratio:

In this project you will have to investigate the GOLDEN RATIO and apply it to GOLDEN RECTANGLES. The golden ratio, approximately 1.618034, also known as PHI (FEE), was used by the ancient Greeks in their construction and artwork. The golden ratio is believed to still be used in today’s architecture and art.

In this activity, your group will try to find examples of the golden ratio in rectangles. It is believed that Greeks felt a rectangle was very pleasing to the human eye if the ratio of the length to the width equaled the golden ratio. This kind of rectangle is called a golden rectangle.

Instructions:

1. Write an equation that relates the length and width of a golden rectangle to the golden ratio. Then solve the equation for the width. (For this project, use the longer side of a rectangle as the length)

2. Measure the lengths and widths of ten rectangles you find in the classroom or at home. Determine if the rectangle is a GOLDEN RECTANGLE according to their measurements (information should be presented and organized in a table).

3. If the rectangle is NOT a golden rectangle, use the equation from Step 1 to find the width that would make the rectangle a golden rectangle (i.e. use the rectangle’ length and the golden ratio to find the width that would make the rectangle a Golden Rectangle).

Analyzing the Results:

1. Discuss why you think the Greeks found a golden rectangle more pleasing to the eye than other rectangles?

2. Examples of the golden ration (also known as the Divine Proportion) can be found in nature. Find two more examples of the golden ratio and explain how the golden ratio is being used or can be seen. Include images.

Golden Ration Project Rubric:

|Equation |-Ratio of length and width for the golden ratio is correct and all proper variables are used |

| |-You solved for width in terms of length and the golden ratio. |

|5 points |-Everything is typed |

|10 Rectangles |- Must have the length and width of 10 different rectangle organized in a table (see example in attached sheet) |

| |- Name/description (i.e. where did you find the rectangle) of all 10 rectangles must be included in the chart |

| |- Table is neat, organized and visually appealing AND everything is typed |

| |- Lengths and widths must have proper units to them (cm, mm, in.) AND be rounded to the nearest tenth (Be sure BOTH lengths & widths are in the same units if they come|

| |from the same rectangle). |

| |- Table includes work & columns that answer whether the rectangle is a golden rectangle |

| |- Table is marked with proper headings |

|25 points | |

|Finding the Golden Ratio width |- Added to the table is the width of the given rectangle IF it were a Golden Rectangle |

| |- Must have its own column and marked with a proper heading & a column with its work |

|10 points |- All widths are correct for the given rectangle and its length. |

| |- All information is in proper units and in the table |

| |- All information is typed in |

|Discussion #1 |- A clear discussion, that is typed, well thought out and includes both your opinions |

| |- Complete sentences are used so I do not need to read the original question |

|5 points | |

|Examples of the Golden Ratio |- Two examples of the Golden Ratio are presented (NOT including the Parthenon OR the Pyramid of Giza (presented below)). |

|(Discussion #2) |- Examples include images that are clear |

| |- Examples are unique |

| |- Examples include a 3 - 6 sentence description on what they are and how the golden ratio is seen in them |

|10 points |- Examples and their description are typed and everything is NEATLY presented |

|Total Points: 55 |All projects that meet the above requirements to their highest potential will receive full credit |

EXAMPLE OF TABLE: Can vary but information presented should be the same!

|Rectangle Description |Length |Width |Use the ratio you set up in question #1 & |GOLDEN RECTANGLE? |Use the equation you found in the second |Width need for GOLDEN |

| | | |see if it equals the GOLDEN RATION then | |part of question 1 to find the answer for|RECTANGLE |

| | | |answer the next question in the next | |the next column – show neat and clean | |

| | | |column – show neat and clean work HERE | |work HERE | |

|My little brother’s |[pic]inches |[pic] inches |SHOW WORK HERE TO EXPLAIN YOUR ANSWER FOR | |Show work in order to find the | |

|head | | |THE NEXT COLUMN |NO, this is not a golden |width needed to make this rectangle a |Width needed = |

| | | | |rectangle |golden rectangle | |

|My Large Screen TV |[pic]cm |[pic]cm | |No | |Width needed = |

EVERYTHING IS TYPED!!! - EVEN THE WORK YOU SHOW !! Ex. [pic]…. You can also write the fractions as improper fractions. However, avoid decimals unless you are using them for final answers….such as comparing numbers to PHI…1.618034.

Extra Credit: (6pts)

Everything above is to be included…..For extra credit, you and your partner are to…

- Create a Golden Rectangle and present it in the middle of a poster board

- A description of what a Golden Rectangle is and why it is special should be included below it or above it (You should make reference to the golden ration and why it is so important!!!)

- On the corners of the poster you should have 4 example of the golden ratio (you may include the two examples you found for the paper) with a description of what they are and how the golden ratio is seen in them

- Somewhere on your poster, you should have the symbol of PHI, the word PHI and the number PHI, 1.681….

- A clever title to the poster

- Name, date and period on the back of the poster

- You will be graded on the above requirements AS well as the effort and time your poster demonstrates you put in to it!!

Extra EXTRA Credit: (12 pts)

You are GREEK ARCHITECTS. You mission is rebuild some of our most famous artworks or architects (examples below).

You have to hand in the regular project as well as …..

- Recreate any of the building seen below using clay, toy block, straws, sugar cubes, popsicle sticks….

OR search for another example of ARCHITECT or ARTWORK where the golden ratio is seen and rebuild it in miniature size but your ……

RATIOS MUST STILL BE THE SAME TO THE GOLDEN RATIO!!!!!

- you must show how you stuck to the golden ratio, i.e. label your length, widths or spaces with their proper measurements AND then on a note card (or other piece of paper attached to your recreation) you show the calculations that show how the golden ratio is still present in your work!

Example 1: Ancient Egyptians, Mesopotamians and Greeks were aware of the beauty of the Golden Rectangle and used it to create many different buildings. One of the most famous and beautiful buildings, built in ancient Greece on the Acropolis, is called the Parthenon. It was created about 2500 years ago. The Parthenon - Ancient Greek Temple was built in 5th century B.C. by three architects: Iktinus, Callicrates and Phidias. (The symbol for golden ratio is the Greek letter [pic], named after the sculptor Phidias.)

Golden Ratio appears in several places in the design of the Parthenon. The dimensions of the front of the temple form a perfect golden rectangle, as shown on the diagram above. Also, the spaces between the columns are in proportion to the golden ratio. The golden rectangles can be seen in almost all exterior dimensions of this great masterpiece.

The Parthenon - Nashville, Tennessee

The Greek Parthenon is such an amazing building that it was rebuilt in Nashville, Tennessee. It is the world's only full-sized copy of the original Parthenon and is a museum, open to the general public.

Example 2: Another example of the golden ratio in architecture occurs in Egyptian pyramids. Ancient Egyptians used the Golden Ratio to build their pyramids about 2000 years before the Greeks built The Parthenon. In fact, the pyramids show one of the first examples of using the golden ratio in architecture.

Great Pyramids of Giza

The golden ratio was used to build these wonders of the world in about 2500 B.C. The pyramids have a square base, where the length of each side equals to about 230 meters. The height of the pyramids is about 146 meters. The ratio of these measurements gives us the golden ratio: [pic]Phi Pyramid

GOLDEN PARTNERS:

Liz Humphrey Thearse Symcz

Megan Koester Kendra Thiele

Michelle Michalski Portia Johnson

Yasmin Bennet Laura Jett

Brittany Phillips Arianna Artega

Ivette Manzo Antia Watkins

Ashley Carrion Ramsey Carey

Lauren Plicher Marie Huels

Kendra Hardy Dominque Brown

Candance Burke Lindsey Bollacker

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