The Mysteries of Math: Fibonacci and Golden Ratio

I wanted to share out a really fun activity that you could make into as quick or as lengthy an activity or project as you would like. It incorporates two of what I like to call, the "mysteries of math", the Fibonacci Sequence and the Golden Ratio.

For those of you unaware, the Fibonacci sequence is generated by the rule of adding the two previous numbers in order to obtain the third and so on. The most famous version of this sequence is generated when you start with the numbers 1 and 1

The golden ratio (also known by the Greek letter phi)  is the irrational and transcendental number which is approximately 1.6180339887. One of the ways of obtaining the golden ratio is by dividing the larger number by the smaller number in the Fibonacci sequence (e.g. 5/3 = 1.66666, 13/8 = 1.625, 21/13 = 1.61538, etc). In calculus language, one would say that the limit as the number of terms in the Fibonacci sequence approaches infinity, the ratio approaches phi.

So before I talked with the students about the Fibonacci sequence or the Golden Ratio, we did a measurement activity where the students measure various parts of their bodies and calculate the ratio of the larger to the smaller. After completing this, they individually submitted their data to this Google Form (if you are not using this amazing and free technology in your science and math class, learn how at the Google Form Tutorial page)

I should mention that I was inspired to do this activity after reading this great post at the Republic of Math blog.

When we looked at the data we examined their average class data and individual data and realized how similar it was. We had a conversation about the Golden Ratio and I showed them this YouTube video describing where the Golden Ratio can be found in nature and in our world.

Just as I had hoped, one of the students asked, "Well why doesn't our data match up exactly with the Golden Ratio?"Some students predicted that it was because of measurement error. Another student had a great idea that perhaps since they are teenagers who are still growing, perhaps it throws off the ratio.

I love days like this, where I can have great conversations about the nature of science and math. Random topics pop up in these discussions like Pascal's Triangle and how many sides and/or vertices does a circle have.

I asked them if they could think of a python program that could come up with all of the Fibonacci numbers and Golden Ratio that they wanted. Since they had not written a program by themselves yet they were initially stumped, but next week I will show you the conversation that eventually led to their algorithm and program.

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