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Fibonacci numbers and golden section
Free lesson plan from HomeschoolMath.net

Let's consider one question before the lesson. Should your child or student even learn about Fibonacci numbers or golden ratio? Is this important to know? It isn't any standard fare in math books.

Fibonacci numbers appear in nature in many places. Golden ratio does too. My opinion is yes, he or she should know. But why? After all, that stuff is not needful in daily life.

I think it's important that youngsters learn a few math topics that show how math appears in nature. It is about math appreciation - (or better yet, appreciation of the Master Mathematician of the Universe...).

Kids learn Art Appreciation - so they can appreciate human works of art... Oh, how much better you can appreciate the "artworks" in nature such as flower petals or seedheads when you understand a little bit of the math behind them!

Studying Fibonacci numbers and how they appear in nature could be done in middle school. The golden ratio is an irrational number and requires solving a quadratic equation so fits better for high school.

However, studying about Fibonacci numbers and the golden ratio makes an excellent project for high school to write a report on. Besides algebra, it ties in with geometry, botany and art at least. Students do projects and reports in history and English and other school subjects - why not do one or two in math too?

To complement the lessons on these topics you might want to check out Spiral Science & Math Manipulative and their downloadable lesson plans. The manipulative consists of tiles with sides taken from the Fibonacci sequence, and it can be used to study Fibonacci numbers, the golden ratio, proportions, measuring, addition, and more.


Fibonacci numbers

Ask your students, how does this sequence of numbers continue:

tiles with sides taken from the Fibonacci sequence0, 1, 1, 2, 3, 5, 8, 13, 21...

The solution is here:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
-> Add two consecutive numbers from the sequence to get the next one following them.

This sequence is called Fibonacci numbers. And it isn't just any ole sequence of any ole numbers... it has some amazing properties, plus it's found in nature in many places.

For example, Fibonacci numbers are found in

Tree branches rotate around the trunk in a pattern based on Fibonacci numbers. A 13-year old student has built a "tree model" with solar panels and proven that it collects MORE sunlight than a flat array of solar panels!

The links go to a magnificent site about Fibonacci numbers with tons of information and pictures.


How to find Phi with Fibonacci numbers

Here's another amazing thing about these numbers:
Let's study the RATIOS when you take a Fibonacci number divided by the previous Fibonacci number, and make a list:

1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, 89/55, ...

So what's so great about that? you ask. (And your student might ask too when you do this with him.) It is not so visible when you see the ratios written as fractions, but let's take the decimal expansions of these (and you should have the student or students do this on their own):

1, 2, 1.5, 1.6666..., 1.6, 1.625, 1.615384615..., 1.619047619..., 1.617647059..., 1.618181818...

Do you notice something about that sequence?

It's something special. If you continue calculating the decimal expansions of the ratios, they will keep getting closer and closer to a certain number... they never reach it totally but they keep getting closer and closer and closer each time.

The ratios keep approaching the number (√5 + 1)/2 which is approximately 1.6180339887... if you write out some of its decimal expansion. I couldn't write out all of the decimal expansion because this number in itself is IRRATIONAL and it has the name Phi.

So what we did was: take the ratios of a Fibonacci number per the previous one, look at decimal expansions, and notice they keep getting close to something - and I told you (without proving it) that something is Phi, and it's an irrational number and it's exactly (√5 + 1)/2.

See also The Ratio of neighbouring Fibonacci Numbers tends to Phi - this page has a graph and also a proof for this fact.


The Golden Section

Phi is also called the golden section number. You might have heard about it. Even Euclid studied that in ancient times (he called it dividing the line in mean and extreme ratio).

This is how we get this golden section or golden cut:

line cut in golden section

Take a line and divide it into two parts, L (Long part) and S (short part). We want the ratio of short part to long part be the same as the ratio of long part to the whole line (W). In other words, as the short part is to the long part, so is the long part to the whole line.

S:L = L:W

From this can be solved that L = (√5 + 1)/2 × S or L ≈ 1.618 × S. This number (√5 + 1)/2 is Phi.
So if you divide the line so that longer part is Phi times (about 1.62) the shorter part, you've divided it in the golden section (or golden cut).

And the golden ratio is the ratio Phi:1.


Solving the equation - more details
Skip this box if you so wish.

Solving this simple-looking equation of golden cut requires using the formula for quadratic equations, so it is a nice exercise for high-schoolers.

S:L = L:W is usually written in the form of S/L = L/W

Since S+L = W, we can substitute that for W and get:

S/L = L/(S+L)

And another trick is, since this is just a general line, we can choose for the shorter part S to be 1. After that, the equation looks simple enough:

1/L = L/(1+L)

Solving that using the quadratic formula, and discarding the negative root, you get L = (√5 + 1)/2

Golden rectangle

Golden rectangle is one where the length and the width of the rectangle are in the golden ratio... the length is approximately 1.62 times the width.

here is one golden rectangle

Some people say this shape is an especially aesthetic rectangle, or that humans prefer golden rectangle over others; it hasn't been proven true so think what you like! I like that kind of rectangle okay. Next time try crop a photograph in that ratio and see what you think.

Check also this nice animated illustration of the golden rectangle and a spiral inside it.

And then you're ready to study where all golden section is found! The links below go to a fantastic website about Fibonacci numbers and golden ratio which is packed full of info - there is LOTS and LOTS more to study.

My list is just a suggestion of a few basic topics that could be included in a project in case you don't want to cover it all.


P.S. Some folks try to find golden ratio in everything in universe and make it some sort of mystical or sacred thing or "universal constant of design". It's true you can find it in nature in plant leaf arrangements and in seashells but not every statement you find on the internet about Phi or Fibonacci numbers has been confirmed scientifically. See for example this scientific study proving just the opposite: The Fibonacci Sequence: Relationship to the Human Hand.



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