Learning Activity 4-C-1: Non-Linear Pattern Web Quest

For this Learning Activity, I conducted a web-quest on “Fractals and Nature.”  I chose this particular web-quest because, although I’ve heard of fractals before, all I really knew about them was that they were patterns.  I didn’t know the formal definition of the word, nor did I know how the patterns were created.  Since I started this process with a clean slate, everything I learned was entirely new to me.  For instance, one of the most fascinating things I learned was that fractals can be computer-generated with the appropriate software and mathematical formula.  I also found it interesting that some artists make a living from drawing fractals, whereas others are paid to create computer-generated fractals and set them to music.  Initially, many of the websites I visited provided very detailed and mathematically confusing definitions of fractals.  However, I found the links below to be very helpful and simplistic.

http://player.discoveryeducation.com/index.cfm?guidAssetId=710F2EBB-27F0-439A-AB92-ECFD1F6879D1&blnFromSearch=1&productcode=US

http://tiger.towson.edu/~gstiff1/fractalpage.htm

Here are a few fractal pictures I found particularly striking.  From left to right, the first illustration is a mountain range found in nature, the second is a fractal that was created on someone’s back as a result of being struck by lightning, and the last fractal was found by slicing a head of cabbage.

At my school we have a large student population of Caribbeans, Caribbean Americans, and African-Americans.  Thus, I think it would be nice to have students identify fractal patterns within the various braided hair styles many of them wear to school.  Student’s could also find braided fractal patterns on-line, or in print.

In my home, I have non-linear patterns on my curtains, counter-top, and within a crocheted bed spread I am in the process of making.  In my classroom, there are non-linear patterns on my bulletin board, based on the way the student’s work is staggered and displayed, and there are graphing posters that are displayed throughout the room that have non-liner patterns, as well.

In addition to conducting a web-quest on “Fractals and Nature,” I also conducted a web-quest on “Fibonacci and Prime Numbers.”  Although I was quite familiar with prime numbers, I was not so familiar with Fibonacci’s Sequence, although I’ve heard about it before.  Initially, I didn’t know how to solve the sequence, until I analyzed it thoroughly.  However, I did eventually notice that two consecutive numbers could be added together to  obtain the third number.  Another concept I learned was that within any group of three numbers, the first and the third number could be multiplied, and the middle number could be squared; the difference between the two numbers is always one.  With specific regard to prime numbers, I learned that every number in the sequence is either prime, or is the product of the prime factorization of the number.  For example, although the number eight is not prime, it is the product of four and two, and two is a prime factor of eight.  The websites that assisted me with understanding this new insight are listed below.

http://player.discoveryeducation.com/index.cfm?guidAssetId=80B45D3D-F904-4560-BE00-BA479DF7511E&blnFromSearch=1&productcode=US

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

Within the school setting, students could identify Fibonacci’s Sequence on the piano keys in the music department, or they could use manipulatives to re-create the production of the rabbit pairs.  Within my home, I could probably find Fibonacci’s sequence applied in some of the knitting & crochet books I have; especially those patterns that have petals and spirals.  The photos below are striking photos of Fibonacci’s Sequence found in architecture, nature, needle work and the human body.

 

Learning Activity 4-A-2: Working with the Definition of Linear Patterns

Non-Traditional Patterns (Formal Definition): Patterns that do not follow a repetitive format.

Linear Patterns (Kid-Friendly Version): Patterns that have constant rates of change.  The patterns may constantly increase, constantly decrease, or constantly stay the same.  For example,  the amount of money one makes for babysitting may constantly increase by $10 every hour; the temperature may constantly fall/decrease by -5 every hour; Kim may constantly receive an allowance of $20 each week, no matter how many chores she has or has not completed.

Linear Patterns (Formal Definition): Linear patterns are patterns that have the same distance between each term; the pattern repeats indefinitely along a line.

Definition Comparisons:

• My definition references constant rates of change, while the formal version references distances between terms.
• My definition mentions that the constant rate may be positive, negative, or the same; the formal definition has no mention.
• We both mention that linear patterns look like straight lines.
• The formal definition makes reference to the line being indefinite; mine does not.
• Both definitions make reference to the pattern being repetitive or constant.
• I provided examples of real-world constant rates of change, even although it wasn’t a requirement.

I think students can learn the formal definition of linear patterns by providing them with visual representations first.  For example, if students were provided visuals of linear patterns that demonstrated positive rates of change, negative rates of change, and a zero rate of change for X and Y, I am confident they could work backwards to  come up with their own definition of a linear pattern.  Once they analyze the data and see the patterns for their self, the formal definition would make more sense, and it wouldn’t have to be memorized.  Rather, the abstract definition would be  more easily understood as a result of the prior visual analysis and informal interpretation.