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Fibonacci Everywhere?

Handsome Fibonacci?
If you were even taught about the Fibonacci number sequence in school, you probably don’t remember much about it. As with other higher levels of math, many aren’t sure how Fibonacci could possibly be relevant to their real lives so why should they even attempt to remember him or his sequence?  In reality, Fibonacci numbers are something you come across practically every day.  Even so, let’s go back and start at the beginning.


The Fibonacci number sequence is named after Leonardo of Pisa (1175-1240), who was known as Fibonacci. (I love to say that name because it sounds like I know a foreign language.)  In mathematics, Fibonacci numbers are this sequence of numbers:

As you can see, it is a pattern, (all math is based on patterns).  Can you figure out the number that follows 89?  Okay, let's pretend I waited for at least 60 seconds before giving you the answer….144.  By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.  For those who are still having difficulty (like my daughter who is sitting here), it is like this.
  

The next number is found by adding up the two numbers that precede it.
F The 8 is found by adding the two numbers before it (3 + 5)
F Similarly, the 13 is found by adding the two numbers before it (5 + 8),
F And the 21 is (8 + 13), and so on!
It is that simple!   For those who just love patterns, here is a longer list:

 
Can you figure out the next few numbers?

The Fibonacci sequence can be written as a "Rule “which is:   xn = xn-1 + xn-2   The terms are numbered from 0 forwards as seen in the chart below.  xn is the term number n.   xn-1  is the previous term (n-1) and xn-2   is the term before that (n-2)


Sometimes scientists and mathematicians enjoy studying patterns and relationships because they are interesting, but frequently it's because they help to solve practical problems. Number patterns are regularly studied in connection to the world we live in so we can better understand it. As mathematical connections are uncovered, math ideas are developed to help us be aware of the relationship between math and the natural world.

As stated previously, we come across Fibonacci numbers almost every day in real life.  For instance, many numbers in the Fibonacci sequence can be linked to ordinary things we see around us such as the branching in trees, the arrangement of leaves on a stem, the flowering of an artichoke, or the frutlets of a pineapple. In addition, numerous claims of Fibonacci numbers are found in common sources such as the spirals of shells or the curve of waves. 


Fibonacci numbers can also be seen in the arrangement of seeds on sunflower heads, the state flower of Kansas (where I live). If you look at the seed arrangement in the center, you will observe what looks like spiral patterns curving left and right or clockwise and counter clockwise. Incredibly, if you count these spirals, the total will be a Fibonacci number. Divide the sunflower spirals into those pointed left and right, and you'll get two consecutive Fibonacci numbers. ­

Many other plants in nature also illustrate this sequence.  For instance, buttercups have 5 petals; lilies have 3 petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21 while daisies can be found with 34 or 55 or even 89 petals.

Pine cones clearly show the Fibonacci Spirals. On the right is a picture of an ordinary pine cone seen from its base where the stalk attaches to the tree. Can you see the two sets of spirals going left and right? How many are in each set?

 
Here are two questions to think about:  

1)      How might knowing this number pattern be useful?
2)      What kinds of things can the numbers in the Fibonacci sequence represent?

I want to close this discussion with a cartoon.  It is written by Bill Amend for his cartoon strip Fox Trot which appeared in the newspaper on February 8, 2009.  Just think!  Now that you know something about Fibonacci numbers, you can understand the humor in the cartoon.




Dots Lots of Fun

I am always looking for ordinary items that can be used in the classroom as manipulatives. I'm a firm believer in the Conceptual Development Model which advocates teaching the concrete (using manipulatives) prior to moving to the pictorial before even thinking about the abstract.  When I was at the Dollar Store (a great, inexpensive place to purchase school stuff) I saw sets of dominoes for $1.00 each.  Since they were inexpensive and readily available, I decided to create several math activities and games to introduce, reinforce, or reteach math concepts.

The Number 52
Think about it; if you lay a domino horizontally, you have a two digit number.  Put two dominoes side-by-side, and a four digit number is created.  Now you can work with place value, estimation, or rounding.  How about lining up dominoes in a column, and working on addition (with or without regrouping) or subtraction (with or without renaming)? 

Another perfect domino activity is practicing addition or multiplication facts.  How about adding the two sides of the domino or multiplying the two sides together?

The Fraction 1/4
If a domino is placed vertically, you immediately have a fraction.  Placed one way it is a proper fraction, but rotated around, it is an improper fraction which can then be reduced.  A fraction can also be changed into a division problem, a ratio, a decimal, or a percent.

So think outside that box of dominoes and use them as an inexpensive math manipulative because Dots Lots of Fun!

Check out all my Domino Resources available on Teachers Pay Teachers.
  1. Dots Fun for Everyone - FREE  Three math activities and one game for the intermediate grades.
  2. Dots Fun - FREE  Three math activities and one game for the primary grades.
  3. Dots Fun   A 24 page resource for grades 1-3 that includes 13 math activities and four games.
  4. Dots Fun for Everyone  A 29 page resource that features 15 math activities and three games for grades 3-6.
  5. Dots Lots of Fun  Seven math games that use dominoes for grades 2-5.

In the Hood

Occasionally I submit an article to the blog entitled: Critters in the Classroom.  Since I teach remedial math on the college level, the only critters I have in my classroom are a few stray insects, nothing exciting to write about.  On the other hand, my husband teaches science; so, many critters pass through his room either alive or dead!

Each year, my husband's science students read My Side of the MountainIn the book, Sam makes jesses (leg straps), leashes and a hood out of deer skin for Frightful his peregrine falcon. If Sam could construct a falcon hood, then maybe my husband's students could as well.

Head on over to Critters in the Classroom to read the full article and find out what his students created. I think you will be surprised by their creativity and ingenuity.






Here are two supplementary resources for My Side of the Mountain.
  1. Two Word Searches - The first puzzle is more challenging.  It lists clues instead of the hidden words so the student must determine what the word is before finding it on the grid.  Also, in this puzzle, is a hidden message.  When the 14 words are found, the hidden message appears from left to right.  The second puzzle is a standard word search where a list of 17 words must be located in the grid.  After the words are uncovered, a hidden message can be read from left to right. 
  2. A Crossword Puzzle - This puzzle highlights 17 different birds which appear in the book My Side of the Mountain.  The 17 clues are based on the bird’s unique characteristics, color, and song.  Page numbers where the answers may be found in the book are added to most of the clues so that the book can be used as a reference.