Golden Ratio and Fibonacci Sequence

Monday, November 23 was, apparently, Fibonacci Day.  I Googled it, and it apparently is really an official "Day."  Who knew?  Someone on my Facebook feed acknowledged the special day by posting about Fibonacci numbers found in nature.

http://io9.com/5985588/15-uncanny-examples-of-the-golden-ratio-in-nature

Of course I saved it to show my little ones, who love this sequence.  What matter that it's bedtime when there is math to be explored?  The article mentioned the golden ratio too.  I haven't brought it up before, but since they now have the background in operations on fractions and decimals, they were ready. 

I think this is the first time we have brought out a calculator for math.  The one that comes on the latest version of windows even records your calculations as you go along.

We took consecutive numbers in the Fibonacci Sequence and divided the larger by the smaller:  2, 1.5, 1.66667, 1.6, and so on.  Every so often, we stopped to see what patterns the kids noticed. 

At first, they said things like, "They all have a six.  Well, most of them have a six."  While true, we discussed whether this was a mathematically useful observation in this situation.  Does that tell us anything about the whole number?

The answers all begin with 1.6.  Except for the first ones.  That's something, isn't it.

Then they noticed that the answer gets smaller, then bigger, and smaller, bigger, smaller, and so on.  That's interesting.  There's a pattern in the sequence of answers.  Anything else?

Spider Girl mused, "What's happening?"  A good question!  It's not just the resulting numbers, but there is a sequence, a chain of events. 

We decided to graph to get a better sense of "what's happening."  Off to Excel!

BatBoy pointed out that the graph has a couple of bumps at the beginning and then makes a flat line.  It looks like a flat line, yes.  But we know the numbers are not exactly the same.  What's happening?  They stared at the graph a moment as I tried to figure out how to expand it vertically.  (I never did.)  Oh!!  They're (the results are) getting closer to a number!

Does the pattern continue if we keep going?  SpiderGirl wanted to know.  I added a couple of columns and modeled how to use formulas on Excel.

The sequence really seems to tend towards a number.  We had to take more and more decimal places to see the differences between terms higher up in the sequence.  This number that we are getting closer and closer to is called the Golden Ratio.

BatBoy wanted to keep going to find the exact number.  No, I told him, the Golden Ratio doesn't repeat and it doesn't end, just like pi.  We aren't going to be able to find the exact value.  But we can approximate it.  Why is it interesting outside of the Fibonacci Sequence?

Because, shells and galaxies, hurricanes, the human face, human fingers and animal bodies, reproductive dynamics and health, animal flight patterns, even DNA molecules.   Wow.  Both SpiderGirl and BatBoy were floored.  It was worth staying up a few minutes late, I think.