# Fibonacci and Recursion

To introduce the fibonacci sequence using the recursive series to my Pre-Calc H class (although most of them know it), I came up with something on the fly that worked nicely. To set up the fibonacci sequence, I showed them a recursive program that calcuates the sequence in a very clear (but inefficient) way.

def fibonacci(n):
if n==1:
return 1
elif n==2:
return 1
else:
return fibonacci(n-1)+fibonacci(n-2)

for number in range(1,10):
print "fibonacci number " + str(number) + " is ",
print fibonacci(number)

This is probably the first time they’ve seen programming in person, so to make it real I gave a post-it to each student and then had them stand up in a line. I had the person at the start of the line write n=1 on their post-it, and then they continued to the end of the line, n=19. I gave the first two students in the line their fibonacci numbers $f_1$ and $f_2$ (1 and 1 respectively -haha).

Then I asked the person at the end of the line what his fibonacci number was ($f_{19}$). After a bit of questioning back and forth he agreed that he needed to know $f_{18}$ and $f_{17}$.

Well this continued all the way back to the start, and then after $f_{3}$ got her fibonacci number, the information went down the line and finally:

$f_{19}$ got 4,181, and I was desperately hoping that no-one made a mistake in their number, and went to check the result on wolfram alpha. Success! Before having the students sit down, have them calculate the ratio of the two fibonacci numbers that are under their number. So $f_{17}$ calculates the ratio: $f_{16}$ / $f_{15}$ and then have them write their decimal on the board in increasing order. Save this for later.

To show the class how inefficient the above algorithm is, I calculated $f_{19}$ and every time $f_{1}$ and $f_{2}$ were “asked” what their number was, it would print their names.
The Code:

def fibonacci(n):
if n==1:
print "Sage"
return 1
elif n==2:
print "Jessica"
return 1
else:
return fibonacci(n-1)+fibonacci(n-2)

print fibonacci(19)

The Output (Click to expand, but it’s very long. An interesting number of lines *hint*):

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So that algorithm is very inefficient for calculating fibonacci numbers because it has no memory of the previous fibonacci numbers. When $f_{19}$ asks $f_{17}$ for her number, she doesn’t remember it, and she needs to calculate it again when $f_{18}$ asks for it.

Finally to cap it off, talk about the ratio of the fibonacci numbers that you’ve written on the board. These numbers tend towards the golden ratio ($\phi$).
Awesome.
Continue deep into math teacher land talking about phi and fibonacci numbers. Possibly share an applet like the following which shows that phi is present in seed growth in plants like sunflowers.