Golden Ratio Lesson

Overview

This lesson will look at a text exploring the Golden Ratio, phi, or ϕ, in different contexts. The text is from a website called Math is Fun, and goes through many different ways to think about phi, including:

• different ways to think about phi geometrically, including a piece of interactivity
• two simple algorithms which produce a value that approaches phi the more each is iterated
• two ways to calculate the formula for phi
• some talk about phi being thought beautiful and it’s use in art and architecture
• two different opportunities to use the Pythagorean theorem

Context for the Reading

I would use this lesson with 8th grade students to explore phi as an irrational number. Specifically, this lesson will address section 8.NS.1 of the CCSSM (Common Core State Standards for Mathematics) following a lesson talking about rational numbers. This standard requires that students know there are numbers that are not rational, and that they are called irrational, and to know how to approximate irrational numbers with rational numbers so as to be able to compare irrational number sizes. My thinking is that phi will be a great way to introduce these irrational numbers. And besides just approximating phi as an irrational number, this lesson will actually present a way to formulate phi to provide an exact representation.

Students will already have a strong association with algebra, which will help greatly with the section on calculating phi with the infinite fraction. Even though they will struggle to find the trick to the expression’s calculation, the struggle will in the end benefit their learning. They will have already had a lesson dealing with the Pythagorean theorem and that part of the lesson should come easily. Also, the fact that this is a ratio brings forth previous knowledge they have gained about ratios, proportions and unit rates in the 6^th and 7^th grades.

There are also other interesting discussions to be had, like talking about why phi is the “most irrational” irrational number. Also, there is a deep connection between phi and art and architecture, and even religion (indeed, many religions use this proportion in their art and architecture).

Text Complexity/Accessibility

Flesch Reading Ease:  81.9/100

Flesch-Kincaid Grade Level:  Grade 6.0

Math is fun provides very readable and interactive explanations of anything mathematical. The structure of this site is wonderful. Everything is broken down and portioned into short sentences with pictures and it uses concrete examples, avoids irrelevant details, and provides many different ways to think about and produce this ratio. The necessity of a student needing prior knowledge is low, as the page starts with an introduction before moving to more.  

Other notes

I will use three other texts with this lesson:

The first text will be a presentation of a problem that has the students calculate the formula for phi by examining an infinite fraction. I will present this first as an opening exercise, even though it would be more challenging than typical opening exercises. Considering this, I will give them hints to think about through their struggle. This will formally be my guiding question that will lead nicely into exploring everything included with the Math is Fun investigation.

The second text will be a Wolfram Demonstration of iterating the infinite fraction while building a Golden rectangle. 

The third text will come after the Math is Fun investigation. I will have the students look at the proportions of the human body using what we have explored about the golden ratio, specifically how to calculate the “golden length” in proportion with each total length they will look at.

The Lesson

This is now the meat of the gravy, the creme de la creme.

Golden Ratio Lesson Plan

Enjoy!

An Attempt to Analyze Texts

My blog was created to look at "Sacred Geometry." I first looked at the Platonic Solids and sought to spark interest in their symmetry and potential connection to the natural world. This was a loose connection to the natural through a look at a sacred tradition called Ayurveda.

Now, I would like to take a slightly different approach. I will examine the Golden Ratio and also look at its connection to the natural world, but we will see this is a little less "sacred" and more scientific, since we can see much evidence that the natural world "uses' this ratio to order itself (though I don't know about you, I certainly see the sacred in the natural). Hopefully you will see what I mean!

I will look at a collection of 6 texts and make an attempt to analyze their readability. This means I will analyze their complexity and accessibility by taking a basic look at certain quantitative and qualititative measures. My quantitative measures will be from the Flesch Reading Ease and the Flesch-Kincaid Grade Level measures, both which look at word a sentence length. My qualitative measures will include thoughts of text structure, writing style, language, and prior knowledge needed for understanding.

With the quantitative measures, I will take about a paragraph from each source and plug it into a website, StoryToolz, that measures readability and outputs a score according to various measures. With the Flesch Reading Ease measure, a lower number means a higher difficulty in readability. The Flesch-Kincaid Grade Level measure is more straight forward, giving an explicit grade level where this text would be comfortably readable.

Enjoy!

Calculating Phi

Yorgey, Brent. “Challenge #1” The Math Less Traveled. wordpress.com, 08 March 2006. Web. 25 July 2014 <http://mathlesstraveled.com/2006/03/08/challenge-1/>

This is not a word heavy text. It presents a problem asking the reader to find the solution to what is called an infinite fraction.

The three dots at the end means that the sequence goes on forever.

Text complexity/accessibility:

Flesch Reading Ease:  68.7/100 (plain English)
Flesch-Kincaid Grade Level: Grade 7.1

This text has a nice structure. The formatting is clear and pleasing to the eye. It uses visual aids and complex relationships are explicitly stated. The writing style utilizes a familiar voice and creates a readable narrative, with a lighthearted and creative approach. The language is not too complex, though some of the sentences are long. There is needed a "creative leap" to solve the problem, but the author explains explicitly that no more than basic algebra is needed once that leap is made.

Why did I choose this text?:

I like this text because it is something challenging yet doable to get you thinking, then provides a nice answer for the equation of the Golden Ratio, which will come up in all the other texts I will analyze.

Think about this...:

I am linking the solution, but I implore you to try and figure it out before taking a peek. It may be tough, but you'll like yourself better for trying! Challenge #1 Solution

Geometric and Fraction Expansions of the Golden Ratio

Kelchner, Adam. “Geometric and Continued Fraction Expansion of the Golden Ratio” Wolfram Demonstrations Project. WolframMathematica. Web. 25 Jul 2014 <http://demonstrations.wolfram.com/GeometricAndContinuedFractionExpansionOfTheGoldenRatio/>

This site only includes 2 sentences, so to call it a "text" is not quite right. It is a tool that one can use to explore the iteration of the equation looked at with the previous text while providing a look at what is happening geometrically at the same time.

Geometric and Continued Fraction Expansion of the Golden Ratio from the Wolfram Demonstrations Project by Adam Kelchner

Text Complexity/Accessibility:

Flesch Reading Ease:  55.8/100
Flesch-Kincaid Grade Level: Grade 9.5

I love the interactivity of this program. One speed bump will be that the user needs to download a program to be able to interact with the demonstration, and this might limit some younger students from being able to use it. Though, once through that, the visual is appealing. It allows the user to take a step-by-step look at what is happening geometrically as the infinite fraction is iterated. I would include this after using the challenge from above, so the student will already be familiar with the fraction, and will gain some insight into its connection with the sequence of rectangles, which I feel many students will have seen. If not, then it will be a great introduction!

Why did I choose this text?:

I chose this text because it is interactive and allows the user to explore the fraction and geometric iterations in tandem, to help further understanding of this thing called the "Golden Ratio."

Think about this...:

Does it make sense why this fraction and this geometric representation are the same? Hint: think about what length the "golden" length is in relation to the length that is 1 unit.

The Golden Ratio

Pierce, Rod. "Golden Ratio" Math Is Fun. Ed. Rod Pierce. 8 Feb 2014. Web. 25 Jul 2014 <http://www.mathsisfun.com/numbers/golden-ratio.html>

This is an in-depth look at the Golden Ratio. It is a different look than that taken so far with the equation, and provides many other ways to think about this ratio. It also has a interactive part, but also has much more! I see this sight as the last in this series exploring just what is this "Golden Ratio."

Text Complexity/Accessibility:

Flesch Reading Ease:  81.9/100
Flesch-Kincaid Grade Level: Grade 6.0

Math is fun is a great website providing very readable and interactive explanations of anything mathematical. The structure of this site is wonderful. Everything is broken down and portioned into short sentences with pictures. It uses concrete examples and avoids irrelevant details, providing many different ways to think about and produce this ratio. The necessity of a student needing prior knowledge is low, as the page starts with an introduction before moving to more. 

Why did I choose this text?:

I debated if I should include this as an introduction before the challenge from above, as this goes through it all then ends up with the thought about the same equation. But, my thought is, the equation is doable with a simple understanding of algebra (and a creative leap!) so I liked putting that first, building up some mystery, then rounding it up in the end by seeing the same thing, now with all of this added knowledge. I chose this website precisely for how rounded and thorough it's look is at the Golden Ratio.

Think about this...:

What else can you think about that just looks beautiful but you don't know why? Can you maybe see some of the Golden Ratio in there?

Nature by Numbers

Vila, Cristobal. “Nature by Numbers Movie” etereastudios.com, 2010. Web. 25 Jul 2014 <http://www.etereaestudios.com/docs_html/nbyn_htm/nbyn_mov_youtube.htm>

This "text" is not a text, but a video. It is a piece of art that provides intrigue and wonder as it animates, very beautifully, where we can see the Golden Ratio, along with some other mathematics, in the spiral of a shell, the seeds of a sunflower, and the wings of a dragonfly.

Text Complexity/Accessibility:

The fact that this is a video doesn't allow me to apply an exact quantitative measure according to the measures I have been using. Though, I would say that this is slightly lower readability than the Math Is Cool website and slightly higher than the 2 sentences contained in the Wolfram Math website. Though, I can certainly speak to the video's structure. It is just a beautiful video. Even if a student comes away without a clear idea of what the Golden Ratio exactly means within the context presented, I feel that interest will be built nonetheless, and I at least hope that it will convince one that this proportion relates to the nature that is observed and animated. The language (mathematical language) is similar to that already presented, making it relevant to a student's prior knowledge.

Why did I choose this text?:

I have touched on this already a bit. My main goal is to build interest and (at least slightly) convince a student that this ratio is seen in the natural world. I want to create a sense of "aaahhhh" like a sigh, and a pleasant interest. It is a sort of scaffolding into the next text, which talks more in depth about the reality of this ratio in nature
.
Think about this...:

While watching this video, just let it sink in without hoping to understand everything. The Golden Ratio is about beauty in the end!

The Golden Proportion in Nature

O’Connor, Aidrian. “Phi / The Golden Proportion in Nature” Nature’s Word. 2010. Web. 25 Jul 2014
<http://www.natures-word.com/sacred-geometry/phi-the-golden-proportion/phi-the-golden-proportion-in-nature>

This text deals with much of what was seen in the video above. As we learned from higher up in this post, I like the mystery first then the closer look later. This text does this very much the same in relation to the video, looking at the angles presented and giving more words to the visuals seen in the video. It goes through many examples of plants and animals exhibiting the Golden Ratio, ending with a brief look at the human body.

Text Complexity/Accessibility:

Flesch Reading Ease:  61.7/100 (plain English)

Flesch-Kincaid Grade Level: Grade 12.7

This is the most complex of the texts I have chosen. Though, I hope that it is quite accessible because of the scaffolding I have provided with the previous texts. The grade level is high but the reading ease is "plain English." I think this means that it has long sentences but short words. Indeed, this is the case. Students shouldn't have much problem with the words in this text. If my job has been done, having built interest, I can only foresee a positive association with this information. Some of the math he talks about is confusing even for me, but it seems to make sense. Structurally, the paragraphs are short and the pictures are pleasant and helpful.

Why did I choose this text?:

I chose this text to build upon the knowledge and beauty shown in the video. The point of all of these texts is to talk about the Golden Ratio and to relate it to nature, which is another view of what is possibly "sacred." It talks about what is not talked about in the video, and provides more examples. It provides (I think) clear proofs for the claims that this nature "uses" this ratio and, further, that it in fact provides nature with the most efficient way to organize itself.

Think about this...:

I wonder, since we are a part of nature, would it follow that the things we create and find beautiful also conform to the structure provided by the Golden Ratio?

The Human Body and the Golden Ratio

Meisner, Gary. “The Human Body and the Golden Ratio” Phi 1.618 The Golden Number. PhiPoint Solutions, LLC. 31 May 2012. Web. 25 Jul 2014 <http://www.goldennumber.net/human-body/>

This text provides the finishing touch to this discussion of the Golden Ratio and nature, and what better way to connect to the information by relating it to something we all know so intimately well - our own bodies. The text doesn't calculate anything, but uses some knowledge, built already (which the reader can figure out I think), to create some lines that are then used to analyze the human body according to the Golden Ratio. At the end I really liked how it took a look at the number 5 and its prevalence throughout the human body, and even within the formula we looked at before! This also ties the whole thing in nicely with my infographic from post #3 and the discussion about there only being 5 Platonic Solids and (typically) 5 elements of the Universe. Really cool!

Text Complexity/Accessibility:

Flesch Reading Ease:  69.1/100 (plain English)
Flesch-Kincaid Grade Level: Grade 9.6

This text is not very complex. The only problem is that it is talking about human proportions, but we're all different right? I think this would create problems for some people, but my take on it is that it is just talking about a typically proportioned body, which I can accept. We can't talk about all possibilities all the time! The language is simple English and the grouped meanings are fairly short. It was nice that the author chose to only talk of 5 points of divine proportion, to keep with his idea of the prevalence of the number 5. 

Why did I choose this text?:

This text is all about the Golden Ratio and the number 5, which is why I love it. I have not talked a lot about the number 5 in this post, but in my previous post and infograph it was kind of all about the fact that there were only 5 Platonic Solids and 5 elements of the Universe. It is also very accessible by being about our own bodies, so we can experiment and see if it is all true. I really really loved the fact that the author was able to see the number 5 within the equation I started this post with, and to tie it all in further with the talk of the Platonic Solids, I mean, come on! What are the odds!

Think about this...:

While reading through this text, can you think of any other places where we can see these proportions and/or the number 5?

Metatron's Cube and the Platonic Solids

There are many nuances about Sacred Geometry that I could not get into with my project.  To keep it interesting and free from clutter, I had to make sure to limit what I wanted to say, though, I think this helped me to create something more meaningful.  Putting in too much would only have distracted from what should be a fun, informative infograph.

I knew I wanted to do an infograph, which is a sort of “I need to scratch this itch” kind of deal. I love to produce art when there is a purpose to and here was a great opportunity for me to do so. A simple infograph is what I ended up with, though there were certainly many challenges.

Oh Adobe Illustrator you dog!  I had planned on using Easel.ly, yet right when I was about to start, I decided to peruse the campus Mac I like to use and saw to my great surprise and delight that it has the program, Adobe Illustrator.  I had never used Illustrator, though I knew of its capabilities, and once the thought entered my mind I could not shake it.

Part of the problem with having such capability is that I now had the option to create every bit of the project. And, having never used the program before, much of my time involved figuring out just how to do what I wanted to do. I had to do it though once I set my mind on it. This added more time to the project, but it opened up the ability for me to do exactly what I wanted. The program is great! I suggest

using it.

So, I had to limit what I might have said with this infograph. There is just SO much to talk about with Sacred Geometry. All I knew was that I wanted to talk about the Platonic Solids, and originally was going to incorporate into it a talk of the Flower of Life. This is not exactly the conversation I ended up having.  I used something called Metatron's Cube, which is derived from part of the Flower of Life, called the Fruit of Life. The Platonic Solids can be seen within this mysterious cube. What I decided to focus on I think worked much better than a conversation about the Flower of Life, and I put together something I hope everyone can relate to and can find interest in.  You will notice part of the Flower of Life as the background, and the keen observer will notice that the dimensions of the infograph are quite close to the Golden Ratio.

Many religions talk about 5 elements. They are not all the same, though mostly they all include the 4 main ones: fire, earth, air, and water. I have seen the fifth being metal or wood, but what I chose to look at was from the Ayurvedic tradition, whose fifth element is the mysterious ether.

I could not fit a short description I had prepared about this connection to the Ayurvedic tradition in my infograph, so here I will. Ayurveda is a traditional Hindu medicine practice. In the Hindu religion, medicine and spirituality go hand in hand. Ayurvedic medicine, along with yogic practice, seeks to balance the body and mind. Within the Ayurvedic belief, these elements are considered fundamental because all that we can think of that exists in some way incorporates these five elements. Surely chemists scoff at this notion of “elements,” as was shown to me through my research, but the idea is not that they correlate with our chemical elements of our periodic table. The idea of these sacred elements is more of a general sense of relationship. Besides the characteristics I talked about, they represent the 4 states of matter – solid, liquid, gas, and plasma – and the “grid” upon which matter exists – the ether. "The body and mind."

Below are the websites I used to gather my thoughts and gain insight into how I wanted to create this project.

http://en.wikipedia.org/wiki/Platonic_solid

http://www.thelivingcentre.com/cms/earth/the-five-great-elements

http://www.pbs.org/wgbh/nova/blogs/physics/2011/12/beautiful-losers-platos-geometry-of-elements/

And this last link is a fun (at least for me) video, giving a look at Metatron's Cube:

Sacred Geometry 101: Metatron's Cube

"Sacred Geometry" - What Is It?

"Sacred Geometry" - what is it?  That is a good question.  I know only a few things, but my knowledge does not include much in the way of understanding why in any way these mathematical relationships I would like to explore are in any way "sacred."  A quick wikipedia search reveals to me something in the way of an answer:

http://en.wikipedia.org/wiki/Sacred_geometry

This wiki page provides this sentence which is nice and concise: 

"The term Sacred Geometry has come to be known [as] an art form, generally digital geometric representations, closely aligned with mandala art."

So there's a start.  A mandala is a form of art associated with Buddhism and Hinduism and is believed to represent the universe, somehow or another.  There are geometric forms which religions have picked up on and seem to think are sacred, yet on the other hand, there is a line of thought which says that this geometry is representative of nature.  Indeed, is there religion in nature?  I think I should come back to this question of what is "sacred" later.

I know that within Sacred Geometry are some very interesting mathematical concepts. 
These are: 

the so called "Flower of Life":

http://www.crystalinks.com/floweroflife.html

the "Golden Ratio":

http://mathworld.wolfram.com/GoldenRatio.html

http://www.natures-word.com/sacred-geometry/phi-the-golden-proportion/phi-the-golden-proportion-in-nature

the Fibonacci sequence and the Fibonacci spiral (or Golden spiral):

http://www.mathsisfun.com/numbers/fibonacci-sequence.html

http://en.wikipedia.org/wiki/Golden_spiral

and the Platonic solids:

http://mathworld.wolfram.com/PlatonicSolid.html

http://www.mathsisfun.com/platonic_solids.html

The Platonic solids are a group of five 3-dimensional geometric shapes with specific properties, the Fibonacci sequence is created by adding the two preceding numbers to get the next and so on, the Flower of Life is created by overlapping congruent circles in a pattern, and the Golden Ratio can be found within all of these others.  There are claims of close connections among all of these things, that the Flower of Life "creates" all forms of nature, that the Platonic solids are found within the Flower of Life, that plants (and dare I say all that is natural) grow according to the Fibonacci sequence and spiral.  So much to explore!

With this blog, I seek to explore these connections and what they mean and hopefully come to some meaningful conclusions as to just why and how any of this math is in any way "sacred."

What is it All about

Hello all!  I am Aaron Beach and I study mathematics education.  I have not always studied math nor have I always wanted to teach.  I come from Lafayette, Indiana having come to Milwaukee to pursue my initial college career in the field of fine arts at the Milwaukee Institute of Art and Design, looking to focus on painting.  I like to produce art, yet it is not a passion of mine, and I decided it was not something I wished to pursue in college. But don't get me wrong, I love the process!  "Passion" is a hard word for me.  I often peruse thoughts of what my passion would be, and what I have found to be my passions I think would have to be: philosophy and communication.  I fall on these two because they are very broad, and my interests indeed are very broad.  But, they mostly fall in the categories of thinking and talking about what I have been thinking about.

Nonetheless, I do have many interests.  One of them is actually math, believe it or not. And another is spirituality.  I am very interested in the truth of things, and many seekers of truth shun spirituality, mostly I believe because of its link to religion. There is much to scorn for sure!  Though, the subject is fascinating to me.  I grew up going to church, yet when it came down to it I could not realistically affirm any belief in God.  But, I knew there was some truth there.  I have never wanted to shut out a thought just because my initial understanding said that I should, and especially with something that seems to be such a big part of many people who are dear to me.  It is a controversial topic and there is much that I do not agree with, but so many people that I respect affirm belief in God in one form or another, so I could not logically be so naive to think that all of these people are 100% wrong. I have investigated Buddhism and found with meditation much truth.  Having since left Christianity and returned to entertain any potential truth, I have found many connections between the two forms of thought.  I like to say I am spiritual, and all this means to me is that I seek to understand a deeper truth to my life and the world around me.  And to love people!

In any case, I want to bring up my connection to spirituality because of the topic I would like to pursue with this blog.  It is not something I know a lot about but through spiritual communities of which I am a part I have learned enough to spark my interest.  This topic is broadly: "Sacred Geometry".  More specifically, I would like to pursue the Golden Ratio, the Platonic Solids and the Flower of Life with other of its associated forms.  From my initial understanding, these are all intricately connected.  One side screams "spirituality!" while the other screams "mathematics!" I would like to explore these connections to see just what is going on.  I am very excited to share what I will find out!