Fibonacci Sequence.

The Fibonacci sequence in terms of rabbits

The Fibonacci sequence in terms of rabbits (Photo credit: Wikipedia)

The Fibonacci sequence is a series of numbers in which each number in the series equals the sum of the two numbers before it.

It was described by Leonardo Fibonacci, an Italian mathematician, in his 1202 Book of Calculations.

He also introduced and popularized Hindu-Arabic numerals, which he learned when he traveled with his merchant father to North Africa. They eventually replaced Roman numerals for calculations and that’s what we use today, the numbers 1, 2, 3, instead of I, II and III.

The problem Fibonacci described, and why the number series was later named the Fibonacci sequence after him, was to suppose you have a male and female pair of rabbits.

They mate the first month and a month later have a pair of male and female rabbits.

How many rabbits will you have after three months? After six months?

Fibonacci explained that in the beginning, you have only one pair (1).

After the first month, you still only have one pair of rabbits because rabbits can’t get pregnant until they’re a month old and then it takes a month for the babies to be born (1).

But, then, they can keep having a pair of rabbits every month.

After the second month, you have the original pair and a new pair (1+1=2).

After the third month, you have the two pairs you had last month, plus one new pair because 2 babies are born to the rabbits that met two months ago (1+2=3).

After the fourth month, though, you have five, because you have new pairs from each of the two that met two months ago, plus another one from the pair that met three months ago (2+3=5).

By this method, you can figure out how many rabbits you are going to have next month by adding how many you had each of the last two months.

And, from now on, you can always predict how many rabbits you will have next month by adding the numbers from each of the last two months.

Let’s see if we can show the Fibonacci sequence of numbers in something we can count more easily than rabbits.

Materials:

  • Varied fruits, cut in bite-sized pieces, such as grapes, apple chunks, banana slices, kiwi or pineapple chunks
  • Skewers, one for each grandchild
  • Optional: two kinds of cereal in countable bites, like Cheerios or Rice Krispies
  • Optional: colored candies, like M&Ms or jelly beans, or colored beads

Instructions:

  • Have your grandchildren pick pieces of fruit to thread on the skewers.
  • Start with one chunk of one fruit, then one chunk of another, then two of the next kind, three of the next, and five of the next.
  • For example, one banana slice, followed by one apple slice, followed by two pineapple chunks, three strawberries and five blueberries.
  • Ask your grandchildren to add the number of pieces of the last two types of fruit to predict the next number they would need
  • The sequence 1, 1, 2, 3, 5 is the beginning of the Fibonacci sequence. Some mathematicians start with 0, giving 0, 1, 1, 2, 3, 5…. This helps in the beginning when you add 0 and 1 to get 1, the next number in the sequence.
  • When you tell your grandchildren that each number in the sequence (except the first two 1s) is the sum of the previous two numbers (2 = 1+1, 3 = 2+1, 5=2+3), then ask them if they can tell you the next number in the series.
  • The first few numbers in the Fibonacci sequence are:

1, 1, 2, 3, 5, 8, 13, 21…

  • Optional: Alternate pieces of cereal in a row, following the Fibonacci sequence. For example: one piece of Cheerios, followed by one piece of Rice Krispies, followed by two pieces of Cheerios, followed by three pieces of Rice Krispies. See how long your grandchild can make the row by adding the previous two numbers.
  • Optional: Use colored pieces of candy or beads to lay out the Fibonacci sequence in a row. For example: put down one red jelly bean, followed by one blue jelly bean, followed by two green jelly beans, followed by three red jelly beans, followed by five blue jelly beans. See how long your grandchild can make the row.

What Should Happen?

This is a Fibonacci sequence.

Each number in the series is the sum of the two numbers before it.

That is, starting with 1 and 1, add them together and you get 2, the next number in the sequence.

Add 1 and 2 and you get 3, the next number in the sequence.

Now, what are the next three numbers, after 5, 8, 13 and 21?

  • 13 + 21 = 34
  • 21 + 34 = 55
  • 34 + 55 = 89

Why Is This Important?

Fibonacci numbers, any number in the sequence, have been found in nature in a number of places, such as the number of petals on a flower.

But, it can also be used in practical situations. A computer scientist used it to develop a more efficient search technique than the then-current binary search.

A binary search splits an ordered list in half, checks to see if the entry you are looking for is in the upper or lower half and keeps searching until it finds the entry it wants.

For instance, if you know a book title you want to check out starts with the letter M, but, for some reason you can’t go directly to the M’s, you can decide if M is between A and L or M and Z and only look in the M-Z half of your list.

A Fibonacci search technique is used when the entries are not evenly divided within storage, as when magnetic tape was used and an entry might be near the beginning or the end of the tape.

Using Fibonacci numbers to aid in the search instead of binary numbers reduces the time it takes to find the entry.

Fibonacci cubes have been used to connect parallel and distributed computer systems and to graph chemical phenomenon.

Fibonacci coding can be used to compress data to take up less space in computer storage.

Thanks to annmccallumbooks.com for the Fibonacci snack stick idea. You can order a copy of her book from amazon by clicking on the title, Eat Your Math Homework.

This post first appeared on grandmotherdiaries.com

Carol Covin, Granny-Guru

Author, “Who Gets to Name Grandma? The Wisdom of Mothers and Grandmothers”

http://newgrandmas.com

 

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