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Plant Mathematics and Fibonacii

Oak Leaves
We continue to look at Fibonacci numbers and how nature continually exhibits this pattern. As stated in my last post, this number pattern can be linked to ordinary things we see every day such as the branching in trees, the arrangement of leaves on a stem, the flowering of an artichoke, or the fruitlets of a pineapple. BUT were you aware that scores of plants, including the elm or linden trees, grow their leaves, twigs and branches placed exactly half way (1/2) around the stem from each other?

Similarly, plants, like beech trees, have leaves located 1/3 of a revolution around the stem from the previous leaves. In the same way, plants like the oak tree have leaves positioned at 2/5 of a rotation. Plants like the holly continue this pattern at 3/8, while larches (conifers) are next at 5/13. The sequence extends on and on. Looking at these fractions side by side (see below) do you see a number pattern in the numerators?

Likewise pay attention to the precision of a similar pattern in the denominators. Interestingly, in the numerators and denominators, if you add the two sequential numbers together, you create a Fibonacci series where all numbers in the series are the sum of the two preceding numbers.

Mathematicians recognize this unique pattern as the Fibonacci sequence. Since patterns such as this one are commonplace in botany as well as other areas of science, they are regularly studied so we can better understand the relationship between mathematics and our world. In my opinion, such mathematical precision and accuracy can only be the product of an intelligent Designer. "Through Him all things were made; without Him nothing was made that has been made." (John 1:3 - NIV)


Fibonacci Numbers Are Everywhere?

Handsome Fibonacci?
Even if you were 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.
  • The 8 is found by adding the two numbers before it (3 + 5)
  • Similarly, 13 is found by adding the two numbers before it (5 + 8),
  • 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 fruitlets 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. If you look at the seed arrangement in the center, you'll observe what looks like spiral patterns curving left and right or clockwise and counter clockwise. Incredibly, if you count the 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.

A Go Figure Debut for a Book Nerd who is new!

Marypat's TPT Store
Mary Pat has been teaching (in one way or another) for all of her adult life. She has been a classroom teacher for 20+ years, teaching 4th grade through college (primarily writing and reading). However, the majority of her teaching experience is in middle school, and she says she loves those middle school kids! Is it a challenging age? Absolutely! But she declares that you won’t find more honest, lively and funny students anywhere else!

Additionally, she tutors and acts as a mentor teacher. She enjoys working with both teachers and students and thinks the best part about teaching is figuring out how to reach every student. While this is a challenge for any teacher, it does keep the job interesting, exciting and valuable. She finds inspiration in Emily Dickinson’s poem “If I can stop one heart from breaking.” She believes that by helping just one person on his or her journey through this world gives meaning to her own life. And what better way to do that than through teaching!

Marypat loves to read and write, so she fills her classroom with lots of books. She posts student work on the walls when she can (since that’s always more interesting than her attempts at cute bulletin boards!). She also likes to post a “graffiti” board where students can write titles of books they have enjoyed reading. This also makes a handy reference for those who are looking for something to read.

As you can tell by the title of this post, she is a total book nerd, so her idea of a perfect day is reading a current middle school book at the beach! She also likes to knit, walk her dog (a rescue lab who loves to eat soap!!), and play Settlers of Catan with her family.

Free at TPT
She would love for you to download one of the 11 freebies she has in her store called Just Add Students. Her favorite free item is the reading badge (see visual on your right) because she loves getting students motivated about reading, and reading badges really get them excited! (Plus they’re cute!)

Only $4.00
In addition, Marypat feels strongly that teachers need to teach students how to navigate the Internet! Out of 108 resources, she has several in her store that teach digital literacy.  As teachers, it is so important to help students to think critically about what they read and see on the Internet. Her resource “How to Identify Bias Online” will help any teacher work on sharpening digital literacy skills in the classroom…and beyond.

Marypat also blogs. In fact, she has two blogs, one by herself called Just Add Students (what else?) and another where a group of middle school teachers collaborate to offer teaching ideas. They call themselves the Middle School Mob. She invites you to pop in sometime and say hello.

Measuring Snow

I am not a very crafty person; so, I am always looking for items that are easy to make that I can give to my grandchildren. One year, I gave them a snowman making kit that included buttons, a carrot, six rocks and two sticks. This year, I am giving them a Snow Measuring Tool.  Not is it only fun to use, but it also helps them to practice using a ruler. Here is how you can make one!

Here is the list of supplies you will need:  

1) A paint stick - free at most paint stores
2) A permanent marking pen
3) Something to glue at the top of the stick (You can make it, or be like me and purchase one from a craft store.)

First, using a ruler, mark off every inch along the paint stick. I was able to make nine marks. (Notice I used the plain side of the paint stick and not the side with all of the advertising.) Now write the inches beside each corresponding mark.

When that is completed, glue the item you have chosen at the top of the stick.  I really wanted to use a snowflake, but my local craft store didn't have any; so, I settled on using one of Santa's reindeer.  Which one, I'm not sure since it didn't come with a name.(Hint: My husband used Gorilla Glue so the reindeer wouldn't fall off.)

When it snows, venture outside and stick the Snow Measuring Tool into the snow and read the number of inches that have fallen. If it isn't exactly on an inch mark, then have your child estimate using fractional parts.

While you are measuring the snow, think about this saying: "Ten inches of snow equals one inch of rain." I am sure you have heard that claim as it is a commonly shared belief that seems to be repeated every time it snows a few feet. But, is the saying true? The immediate answer is: Sometimes.

When the temperature is around 30 degrees, one inch of liquid precipitation (rain) would fall as 10 inches of snow, presuming the storm is all snow. But, the amount of moisture in each snowflake differs depending on the temperature which in turn changes the snow to rain ratio.

For example, if a big January snowstorm occurred with colder temperatures (such as 25 degrees), the snow ratio would be closer to 15 inches of snow to one inch of rain. In fact, weathermen take this into account when forecasting how much snow a location will receive. There have been storms with snow closer to 20 degrees, moving the snow ratio closer to 20 to one. And, when it's warmer, say 35-40 degrees, the ratio moves to 5" of snow to 1" of rain.

So, after your children measure the snow in your yard with their Snow Measuring Tool, try converting the inches of snow into inches of rain based on the 10":1" ratio. By doing so, you may become your neighborhood's weather forecaster or even better, a first rate mathematician!