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Home > Key Stage 3 (Ages 11 - 14) > Beautiful numbers: Recursive Fibonacci Series
Beautiful numbers: Recursive Fibonacci Series

The Fibonacci sequence is a series of numbers which appear throughout nature. This sequence can be used to calculate the golden mean which is represented by the Greek letter Phi. The Fibonacci sequence and Phi can be found visually in plants and seashells, and in the reproductive family trees of animals. Phi, also known as the golden mean, is the ratio between two sequential numbers in the Fibonacci sequence.

Can you see the pattern?

Screen_Shot_2012_10_05_at_15.42.16.png

Screen_Shot_2012_10_05_at_15.42.27.png

Screen_Shot_2012_10_05_at_15.42.37.png

Screen_Shot_2012_10_05_at_15.42.46.png

Screen_Shot_2012_10_05_at_15.42.56.png

Ok, so how does it work? 

The sequence starts out: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and continues ad infinitum.
Each new number in the sequence is created by adding the previous two numbers.
By definition the first two numbers are:

Fibonacci(0) = 0
Fibonacci(1) = 1
The next number is always the sum of the previous two. Fibonacci(n) = Fibonacci(n-1) + Fibonacci(n-2)
  • Fibonacci(2) = 0 (0) + 1 (1) = 1
  • Fibonacci(3) = 1 (1) + 1 (2) = 2
  • Fibonacci(4) = 1 (2) + 2 (3) = 3

How to model it?

To be able to start to model the Fibonacci series as an algorithm it is essential that they understand what a 'variable' is and how to use them. Also, students should know the standardised symbols for drawing flow charts.

more_efficient.gif

more_efficient_answer.gif

Modelling the Fibonacci series in Scratch with just variables and loops.

Basic algorithm in Scartch

Using selection with the Fibonacci series.

Screen_Shot_2012_10_05_at_15.43.22.png

Turning an algorithm into a program

`FibonacciSeries.jpg

Outcome from running the program:

 1.jpg
 2.jpg
 5.jpg
 8.jpg
 13.jpg

 

Using data inputs with the variables

Screen_Shot_2012_10_05_at_16.00.43.png

`FibonacciSeries_N.jpg
FibonacciSeries_N_Question.jpg
FibonacciSeries_N_21.jpg

 Using variable in a list

FibonacciSeries_N_List.jpg FibonacciSeries_N_List_Scratch.jpg

Teaching the basics of recursion

 

Screen_Shot_2012_10_05_at_16.09.57.png 

 
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