Minggu, 04 Desember 2011

Golden Ratio Lessons for Middle School

The golden ratio is found throughout nature, including in the spiral shape of some shells.

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The golden ratio is a mathematical concept that, along with a string of numbers known as the Fibonacci sequence, is commonly found in nature. Middle school students can find the golden ratio interesting because of its appearance in everyday life, and teachers can use it as a way of introducing ratios and number sequences into their curriculum.

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The golden ratio is a ratio between two different lines or dimensions that is often considered to be aesthetically appealing to the eye. Technically, the golden ratio is the ratio between the mathematical constant Phi and 1. The value of Phi is approximately 1.62, so a golden ratio is any ratio in which the relationship between the two numbers is close to 1.62-to-1.

Introducing the Golden Ratio to Students

The golden ratio, or various approximations thereof, can be found in many different objects, especially rectangles. Rectangles that have a length of 1.62 times their width are called "golden rectangle." One example of a near-golden rectangle that middle school students may be familiar with is the 3-by-5-inch note card. Golden rectangles are also thought to be more aesthetically appealing to the human eye, so you can have students either draw a sample rectangle or pick their "favorite" rectangle from a collection of rectangles. Golden rectangles are also found in many works of art and architecture, including many pieces by Leonardo DaVinci.

Fibonacci and the Golden Ratio

You can use the golden ratio to introduce another mathematical concept: the Fibonacci sequence. With the Fibonacci sequence, the next number in the sequence is the sum of the previous two numbers; the beginning of the sequence is 1, 1, 2, 3, 5, 8, 13. As the Fibonacci sequence extends, the ratio between each number in the sequence and the number before it, such as 13 and 8, gets closer and closer to the golden ratio. This can serve as another interesting aspect of the golden ratio and can also introduce middle school students to the concept of sequences.

Applying the Golden Ratio

One way you can expand upon a golden ratio lesson plan is to have students try to find more examples of golden rectangles, either in common objects or in nature. You can also show students how the golden ratio appears in shell spirals, and how the Fibonacci sequence appears in flower petals. Once students understand the concept of the golden ratio you can also then use it as a launching pad for working with ratios. For example, you could give students a worksheet in which they are given the value of one dimension of a rectangle and are told to determine the other dimension in order to make it a golden rectangle.

ReferencesHomeschoolMath.net: Fibonacci Numbers and Golden SectionUniversity of Minnesota, The Geometry Center: The Golden RatioUniversity of Surrey: Fibonacci Rectangles and Shell SpiralsPhoto Credit Jupiterimages/Photos.com/Getty ImagesRead Next:

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