# Focus on Math

## Helping children become mathematicians!

### Fibonacci Numbers: A Fascinating Sequence January 10, 2012

I was recently given the gift of this delightful interactive book written by Emily Gravett. Although it appears to be a children’s book, The Rabbit Problem can be appreciated on the adult level as well. This tale of Lonely Rabbit and Chalk Rabbit is actually a retelling of a scenario that, according to Wikipedia, first appeared in the book Liber Abaci (1202) by Leonardo of Pisa, known as Fibonacci. Fibonacci considered the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?
•    At the end of the first month, they mate, but there is still only 1 pair.
•    At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
•    At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
•    At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.
The question, of course, is how do we know how many pairs of rabbits there will be at the end of any given month??***
The answer lies in the Fibonacci sequence of numbers, a fascinating set of numbers that keep popping up in nature in amazing ways. The sequence begins with 0 (or should I say “can begin”?), then add 1, and from there the next number in the sequence is always derived from adding the two previous numbers. So the third number is 0 + 1 or another 1, the fourth number is 1 + 1 or 2, then 1 + 2 or 3, then 2 + 3 or 5 and so forth, giving this sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 …
I recommend this short  video http://youtu.be/ahXIMUkSXX0 by Vi Hart to jumpstart your thinking about these numbers (it’s about 6 minutes long). She builds the topic of Fibonacci numbers off the topic of spirals, so be patient and the number part will come. As you watch, keep in mind the kinds of explorations that you and your class can pursue with pinecones, pineapples, artichokes, cactus fruit, flowers, and such. Often the number of seeds that show up in fruit and vegetables is a Fibonacci number. Try counting the seeds of the next apple or orange you eat!
This website http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html has a whole host of information about connections to the Fibonacci series. Note especially the section about plants. You might be able to do some of your explorations from the photos included here is real fruit, pine cones, etc are not readily available.
Exploring Fibonacci numbers can be a great “hook” to grab students’ interest about numbers and mathematics.
Mathematically yours,
Carollee
***By the way, the answer the the rabbit question is this: at the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs (which is the number of pairs in month n − 2) plus the number of pairs alive last month (n − 1). This is the nth Fibonacci number.