For my webquest, I chose to explore the Fibonacci sequence and fractals. Both of these concepts provide great examples of observable mathematical patterns in nature. The following post will provide a brief description of both concepts, some interesting points and pictures from each, and ideas for how this webquest could be modified for middle school students.
Fibonacci Sequence
Fibonacci first described this now famous sequence of numbers during the early part of the 13th century. He posited this pattern by considering the following problem. Two rabbits are placed in a pen and made to reproduce. How many rabbits will there be after n months. Each consecutive number of rabbits in the sequence can be derived by finding the sum of the previous two entries. So the first two rabbits make a sum of two (1 + 1 = 2). They then reproduce and make three rabbits (1 + 2 = 3). These three rabbits can then make five if there are two pairs with one left over (2 + 3 = 5). The sequence reads 1, 1, 2, 3, 5, 8, 13, and so on.
When limited to promiscuous bunnies, this list of numbers is not very interesting. However, what is interesting are the items in nature that follow this pattern. Things like the number of ridges on a pine cone, the distance between leaves on plants, the centers of sunflower seeds, and the sides of pineapples all follow the Fibonacci sequence and something called a logarithmic spiral. The logarithmic spiral is a method for graphing points that represent the Fibonacci sequence on the Cartesian plane. When this is done, the shape followed by most plant life is followed. By orienting themselves this way, plants are able to receive the optimal amount of sunlight.
Fractals
Fractals are a relatively new concept in mathematics and science. A fractal is defined as anything that “self-symmetry.” In other words, if you zoom in on part of the object, you will find repetitions of the larger pattern. Prior to their discovery it was thought that all objects in nature have some amount of chaos. However, through close study it has been found that even the most chaotic objects seem to contain these fractals. One exercise that supports this theory is to take randomly generated numbers and plot them. After several repetitions of this activity the image of a fern “appears.” Fractals have also been observed in such “random” objects as a desert landscape, broccoli, and bacteria.
Reflective Questions
1. Were there ideas or concepts you were not familiar with? What were they?
While I was familiar with the concept of the Fibonacci sequence and its applications in nature, I had never taken the time to study fractals. I was only aware that there was such a thing, even though I wasn’t entirely sure of the definition. I was also unaware that Fibonacci may not have been the man’s actual name (it may have actually been Leonardo de Pisa), or of how he first came up with the idea of the Fibonacci sequence. I had never seen either graphing activity for the fractals (fern iterations) or the Fibonacci primes (graphing the primes derives the shape for optimal acquisition of lights by plants).
2. What images did you find particularly striking?
I found the graphing activities to be the most fascinating, because I could see the direct relationship between these mathematical patterns and the real world (kind of like John Nash in A Beautiful Mind). I also found the fractal pictures to be absolutely stunning. To see the patterns in something so small as bacteria and something so strange as a stalk of broccoli and to realize that there was math to be found there was amazing.
3. Can you identify any manifestations of nonlinear patterns within your home or your workplace? What are they?
As I look around I can see quite a few repeating patterns in my own home. For instance, the globes on my lights seem to follow a certain pattern. While I cannot specifically prove that the arrangement follows the Fibonacci sequence, it would make sense to use this pattern for such purposes. If light is most efficiently obtained in this arrangement, it would make sense that it would most efficiently be projected in this arrangement. It also seems that there may be some fractals in the patterns of my carpet and in the leaves that are quickly piling up in my yard outside (someday I may actually have time to rake them).
4. How can you adapt this webquest activity for your classroom?
I would most likely have my students search first for images of the Fibonacci sequence without looking at the definition. I would then have them trace the objects on a coordinate plane and see if they could find any patterns from the ordered pairs or the points’ distances from the axes. From this activity, students should be able to work together to derive the numbers in the sequence.
For fractals, I would have students download pictures of the Sepinski triangle (seen above) and make their own in a drawing program. I would then have them download pictures of snowflakes to see if they can find fractals within the object. I feel that both of these activities would provide students with hands on experience that would help them further understand these topics.
References
http://www.daviddarling.info/encyclopedia/F/Fibonacci_sequence.html
http://www.geom.uiuc.edu/~zietlow/defp1.html
http://www.lifeinitaly.com/heroes-villains/fibonacci.asp
http://www.miqel.com/fractals_math_patterns/visual-math-natural-fractals.html
