The Golden Ratio

James Sauls, ggleason, Oct 26, 2018

During our Geogebra search for Golden Ratio worksheets, James and I found random interactive worksheets that were interesting yet provided no direction or method to easily incorporate into a teacher's class curriculum. In addition, given how many ways the Golden Ratio exsits in math and nature, we wanted to make a Geogebra book that would feel complete and whole unto itself. Lastly, we also wanted to study the ethnomathematic question of whether humans are cultured to find the Golden Ratio aesthetically appealing. Thus, in order to create curriculum that is "classroom ready", feel like a completed lesson plan, and analyze this ethnomathematic question, we decided to structure our chapters by incorporating an 5 minute engaging video (Donald Duck's Mathemagic Land) and arranging our worksheets according to the movie script. This allowed our lesson plan to feel complete as it covered all the topics in this 5 minute portion, while also giving us time to give students a historical lesson about the aesthetic allure of Golden Ratio. James created the National Geographic Logo worksheet, and I created the Pentagon/Golden Triangle worksheet.

Table of Contents

  1. Teacher Lesson Plan
    1. Final Teacher Lesson Plan
  2. National Geographic Logo
    1. National Geographic Logo
  3. Golden Ratio
    1. Finding the Golden Ratio
  4. Golden Triangles in Pentagon
    1. Golden Triangles in Regular Pentagon
  5. Fibonacci /Golden Rectangle
    1. Fibonacci Numbers and the Golden Spiral - demonstration
  6. Golden Ratio in 3 Dimensions
    1. Golden ratio in regular icosahedron

1.

Teacher Lesson Plan

In order to quickly help teachers decide if this resource is for them, James and I chose to create a teacher "lesson plan" that provides overviews of all of our chapters. This lesson plan also provides links for the video resources we use in our chapters, including: 1. Donald Duck's Mathematic Land: https://www.youtube.com/watch?v=AJgkaU08VvY 2. The (Ab)surd Golden Ratio Ted Talk by Rob Enzmann https://www.youtube.com/watch?v=0vVxL60YFJU&t=307s 3. An Ethnomathematics Study of the Golden Ratio by Gustav Fechner http://goldenratiomyth.weebly.com/phi-in-psychology.html

  • 1. Final Teacher Lesson Plan

1.1.

Final Teacher Lesson Plan

Final Teacher Lesson Plan
– ggleason

2.

National Geographic Logo

In this chapter we let students examine and adjust the National Geographic logo until they create a rectangle that they find appealing. This serves as a survey to see if students have been culturally adapted to recognize logos that follow the Golden Ratio as more aesthetically appealing. This question was influenced by a study done by Gustav Fechner in the 1800's, which showed that out of a random assortment of 10 rectangles, "35.0% of subjects expressed a preference for the golden rectangle, and... none of the subjects selected the golden rectangle as their least-preferred choice. Fechner took these results to mean that the golden rectangle is most preferred" (Green 943) Historical Resource: http://goldenratiomyth.weebly.com/phi-in-psychology.html

  • 1. National Geographic Logo

2.1.

National Geographic Logo

Instructions
Imagine that you work for the marketing team of National Geographic and are on the team designing the corporate logo. The font has already been selected and there’s definitely going to be a bright yellow rectangle next to the words… But what should the dimensions of the rectangle be?![br][list=1][*]Use the sliders labeled [b]r[/b] and [b]s[/b] to change the rectangle until you find it most appealing.[br][/*][*]Check the box labeled “Show Corporate Logo.” This will bring up the logo used by the Nat Geo currently. (It may be hidden behind the rectangle you created.)[br][/*][*]Use the slider labeled [b]t[/b] to change the transparency of your rectangle and compare it to the corporate logo. Making [b]t[/b]=0 will make your rectangle entirely invisible while [b]t[/b]=1 will put your rectangle fully “in front of” the corporate logo. Don’t be discouraged if your rectangle doesn’t look anything like the one in use today, everybody has different taste![br][/*][*]Checking the boxes on the right will give the ratio of the side lengths of the rectangles and highlight these side lengths in the applet. The ratio is always given as the longer side to the shorter side. How do the ratios of the rectangles that you selected compare with those of the corporate?[br][/*][/list]
– James Sauls

3.

Golden Ratio

This Chapter serves as an introduction to the definition of a Golden Ratio. Students will begin class by watching a portion of Donald Duck's Mathemagic land (starting at 7:00 --12:40). URL: https://www.youtube.com/watch?v=AJgkaU08VvY

  • 1. Finding the Golden Ratio

3.1.

Finding the Golden Ratio

Instructions
Drag the point labeled "drag me" until the ratios [math]\frac{a+b}{a}[/math]and [math]\frac{a}{b}[/math] appear equal, then check the boxes to verify. How close can you get the two ratios to be to each other?
– ggleason, James Sauls

4.

Golden Triangles in Pentagon

Following the Donald Duck Mathemagic land video, students will examine the ratios of triangle side lengths in a pentagon with an inscribed pentagram.

  • 1. Golden Triangles in Regular Pentagon

4.1.

Golden Triangles in Regular Pentagon

Instructions
Before using the checkboxes, make conjectures about what the value of the ratios will be.[br][br]
– James Sauls

5.

Fibonacci /Golden Rectangle

From the Mathemagic video, students learned about a special property of a golden rectangle, where if you started with a golden rectangle and divided the rectangle into a square part and a rectangular part, the smaller rectangle would also be a golden rectangle. This process can continue indefinitely, creating a Golden spiral within our golden rectangle. For this next sections, students will conjecture if there is any easy way to construct a rectangle with such cool properties? How can we do it? One way to approximate a Golden Rectangle can be shown from Rob Enzmann's Ted Talk starting from 0:36 seconds until 1:22. URL: https://www.youtube.com/watch?v=0vVxL60YFJU&t=307s [b]Question: What differences exist between the Golden Rectangle created in Mathemagic Land, and in the rectangle animation from the worksheet? [/b] By the end of the lesson, students should know how to construct an approximate Golden Rectangle. Students should also recognize that the Fibonacci Spiral is an approximation of the Golden Spiral.

  • 1. Fibonacci Numbers and the Golden Spiral - demonstration

5.1.

Fibonacci Numbers and the Golden Spiral - demonstration

This applet demonstrates the Fibonacci Squares and the Fibonacci Spiral without going through all construction steps. It is a variation of the constructing applet [url]http://www.geogebratube.org/student/m161338[/url].[br][br]Click the Play button.
– Irina Boyadzhiev

6.

Golden Ratio in 3 Dimensions

We've learned about the special properties of Golden Rectangles and Golden Triangles. But what about 3 dimensions? This last "worksheet" provides an animated example of the Golden Ratio in 3-D space. The objective of this worksheet serves as a glimpse into how the Golden ratio has many facets and is not limited to 2-D examples.

  • 1. Golden ratio in regular icosahedron

6.1.

Golden ratio in regular icosahedron

Golden ratio in regular icosahedron (see colored rectangles)
– Mathieu Blossier

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