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!