**The Golden Ratio and Pentagrams **

My first exposure to these topics was the “Donald in Mathmagicland” video by Disney. I viewed this video in my undergraduate math methods class, and have used it every year since with my students to introduce my geometry unit. In conducting this webquest, I learned more about these two concepts and some of the history behind them.

**Were there ideas or concepts you were not familiar with? What were they?**

*I was not familiar with the Golden Triangle — I had only heard of the Golden Rectangle and how it shows up in many applications like architecture, nature, and the human body.*

**What images did you find particularly striking?**

*I really like the image of the pentagram and how it is an infinite pattern within the shape. You can just keep making more and more pentagrams within the original. *

**Can you identify any manifestations of nonlinear patterns within your home or your workplace? What are they?**

Non-linear patterns are everywhere. Our daily schedules, while having a pattern to them, are not linear. The outcome of many events is non-linear (sporting event scores, number of channels a certain show is re-run on, number of times my husband will change TV channels in one evening). Manifestations of the golden ratio are all around our home — my son just did a statistics project last spring that investigated the number of household items that adhere to the Golden Ration, and the results were amazing! From windows to doors to mirrors to pictures hanging on the wall — it seems the Golden Ration is an aesthetically pleasing shape!!

**How can you adapt this webquest activity for your classroom?**

After showing the video of “Donald in Mathmagicland,” I could have my students go to the computer lab and investigate the Golden Ratio with the following websites. They would have to come up with a definition, explain the connection between the Golden Ratio and Pentagrams, and also find two interesting facts about the history of these topics. A brief biographical sketch of two “famous people” who employed these concepts would be extra credit.

Related websites I found interesting:

http://web.gnowledge.org/cw/index.php/Pentagrams_and_the_Golden_ratio

http://www.ldlewis.com/Teaching-Mathematics-with-Art/golden-ratio-instructions.html

http://www.contracosta.edu/legacycontent/math/pentagrm.htm

http://www.homeschoolmath.net/teaching/fibonacci_golden_section.php

http://vigilantcitizen.com/latestnews/donald-duck-in-mathmagic-land-video/

*(this last one contains the portion of the video “Donald in Mathmagicland” that pertains to Pythagoras and these two concepts of Golden Ratio and Pentagrams)*

**Fibonacci, Phyllotaxis, and Prime Numbers**

I also learned about these concepts as a student, in my History of Mathematics class while in pursuit of a Masters Degree in Mathematics and Computer Science Education. Much of the mathematics behind these concepts is above the middle school mind, but middle school students can certainly generate numbers in the Fibonacci sequence, and also can appreciate the beauty of mathematics as the sequence relates to the other topics.

The **FIBONACCI SEQUENCE** is the sequence that results in answer to this problemposed in Fibonacci’s great work *Liber abaci*:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begins a new pair which from the second month on becomes productive?

The number of pairs of rabbits in the *n*-th month begins 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …, where each term is the sum of the two terms preceding it. This sequence can be defined recursively as follows: *F*(1) = *F*(2) = 1, *F*(*n* + 1) = *F*(*n*) + *F*(*n* – 1) for *n* > 2, where F(*n*) is the *n*-th Fibonacci number. Johannes Kepler was the first to point out that the growth rate of the Fibonacci numbers, that is, *F*(*n* + 1) / F(*n*), converges to the golden ratio *φ*, which has the value √5 + 1)/2

**Were there ideas or concepts you were not familiar with? What were they?**

*I was not familiar with the term phyllotaxis, but after reading some of the websites, I was familiar with the concept — the arrangement of leaves, stems, etc in nature that correspond to the numbers in the Fibonacci sequence. I was also not aware that these concepts had a direct link to the golden ratio, which was the first webquest I conducted!* *Further, I did not know there were connections to prime numbers associated with the Fibonacci sequence. All very fascinating!*

**What images did you find particularly striking?**

I am always amazed by the natural spiral patterns in so many living things.

**Can you identify any manifestations of nonlinear patterns within your home or your workplace? What are they?**

*The concept of Phyllotaxis is a perfect example of math around us in the real world. Students could examine plants and other living things that follow this pattern.*

**How can you adapt this webquest activity for your classroom?**

As I stated earlier, the mathematics if much of this is over a middle school level, but introducing some basic concepts, allowing the students to read about the connections between the topics, and also allowing them to create the sequence (and learn about INFINITE sequences) are all valuable building blocks to set the stage for more sophisticated mathematics in the future. Some websites I found helpful would include:

http://maxwelldemon.com/2012/03/18/prime-phyllotaxis-spirals/

http://www.daviddarling.info/encyclopedia/F/Fibonacci_sequence.html

http://illuminations.nctm.org/LessonDetail.aspx?ID=L658

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibInArt.html