# Relationship between the golden ratio and fibonacci sequence for kids

### Fabulous Fibonacci - Mensa for Kids The spiral happens naturally because each new cell is formed after a turn. is a special relationship between the Golden Ratio and Fibonacci Numbers (0, 1, 1. This is a lesson plan about the Fibonacci sequence and the Golden ratio, Ask your students, how do they think this sequence of numbers continues: In the links below you can learn more and see many pictures about Fibonacci Should our children or students even learn about Fibonacci numbers or the golden ratio?. Although there is definitely a specific relation between Fibonacci and golden ratio that's not the end of the story. In fact the beauty is not exclusively in Fibonacci.

The interesting thing about making rectangles like this is that the ratio the number that shows how the sides relate to each other stays the same, no matter how big the rectangle gets.

The ratio is called the Golden Ratio. You can find it by dividing the long side by the short side. This will give us a number right around 1. Now you are going to create a Golden Rectangle on your own on your large piece of paper, not the graph paper and see something really amazing!

First, draw two squares side by side. Use your ruler to make them 0. So if the first square was 0. Continue this pattern, making each square the next size in the Fibonacci sequence. This is just like what you did with the graph paper, only using a ruler. Each square will have an edge that is the sum of the two squares before it, just like in the Fibonacci sequence. See the diagram below to find out how it should look. The last one is not done yet.

Can you see where it would go? Now, with your compass, make an arc in the squares with a radius the size of the edge of the square. The arcs in the first squares will be really, really tiny. But look how they grow! Look at this picture of a nautilus shell. What do you notice?

• Fibonacci numbers and the golden section
• Fibonacci Numbers and Nature
• Understanding the Fibonacci Sequence and Golden Ratio

Art Connection Now look at this painting by Mondrian: What connections do you find between what you drew and what Mondrian painted? Do all of his rectangles look like Fibonacci-based rectangles to you? Extension Printable poster of Fibonacci numbers in nature Assessment This series of lessons was designed to meet the needs of gifted children for extension beyond the standard curriculum with the greatest ease of use for the educator.

The lessons may be given to the students for individual self-guided work, or they may be taught in a classroom or a home-school setting.

Assessment strategies and rubrics are included at the end of each section. Check also this nice animated illustration of the golden rectangle and a spiral inside it.

Why study Fibonacci numbers? Let's consider something in the end. Should our children or students even learn about Fibonacci numbers or the golden ratio? It isn't any standard fare in math books. My opinion is yes, students should know about them.

I think it's important that our young people learn a few math topics that show how math appears in nature. It is about math appreciation. Children study "Art Appreciation" so they can appreciate human works of art. Oh, how much more should we appreciate the "artworks" in nature, such as flower petals, seedheads, and spirals in animal shells!

And, once you understand a little bit about the math behind them, you will appreciate them even more. Studying Fibonacci numbers and how they appear in nature could be done in middle school. The golden ratio is an irrational number so it fits better high school math.

## Fabulous Fibonacci

Studying about the Fibonacci sequence and the golden ratio makes an excellent project for high school to write a report on. Do you use a different name for the sister of your parent's? In law these two are sometimes distinguished because one is a blood relative of yours and the other is not, just a relative through marriage. Which do you think is the blood relative and which the relation because of marriage? How many parents does everyone have? So how many grand-parents will you have to make spaces for in your Family tree? Each of them also had two parents so how many great-grand-parents of yours will there be in your Tree? What is the pattern in this series of numbers? If you go back one generation to your parents, and two to your grand-parents, how many entries will there be 5 generations ago in your Tree?

The Family Tree of humans involves a different sequence to the Fibonacci Numbers. What is this sequence called? Looking at your answers to the previous question, your friend Dee Duckshun says to you: You have 2 parents. They each have two parents, so that's 4 grand-parents you've got. They also had two parents each making 8 great-grand-parents in total So the farther back you go in your Family Tree the more people there are.

It is the same for the Family Tree of everyone alive in the world today. It shows that the farther back in time we go, the more people there must have been. So it is a logical deduction that the population of the world must be getting smaller and smaller as time goes on! Is there an error in Dee's argument? If so, what is it? Ask your maths teacher or a parent if you are not sure of the answer! Fibonacci numbers and the Golden Ratio If we take the ratio of two successive numbers in Fibonacci's series, 1, 1, 2, 3, 5, 8, 13.

It is easier to see what is happening if we plot the ratios on a graph: The ratio seems to be settling down to a particular value, which we call the golden ratio or the golden number. What happens if we take the ratios the other way round i.

Use your calculator and perhaps plot a graph of these ratios and see if anything similar is happening compared with the graph above. You'll have spotted a fundamental property of this ratio when you find the limiting value of the new series! It is often represented by a Greek letter Phi. Fibonacci Rectangles and Shell Spirals We can make another picture showing the Fibonacci numbers 1,1,2,3,5,8,13,21. We can now draw a new square - touching both a unit square and the latest square of side 2 - so having sides 3 units long; and then another touching both the 2-square and the 3-square which has sides of 5 units. We can continue adding squares around the picture, each new square having a side which is as long as the sum of the latest two square's sides.

This set of rectangles whose sides are two successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, we will call the Fibonacci Rectangles. Here is a spiral drawn in the squares, a quarter of a circle in each square. The spiral is not a true mathematical spiral since it is made up of fragments which are parts of circles and does not go on getting smaller and smaller but it is a good approximation to a kind of spiral that does appear often in nature. Such spirals are seen in the shape of shells of snails and sea shells and, as we see later, in the arrangement of seeds on flowering plants too. The spiral-in-the-squares makes a line from the centre of the spiral increase by a factor of the golden number in each square. So points on the spiral are 1. In a whole turn the points on a radius out from the centre are 1. Cundy and Rollett Mathematical Models, second editionpage 70 say that this spiral occurs in snail-shells and flower-heads referring to D'Arcy Thompson's On Growth and Form probably meaning chapter 6 "The Equiangular Spiral".

Here Thompson is talking about a class of spiral with a constant expansion factor along a central line and not just shells with a Phi expansion factor. Below are images of cross-sections of a Nautilus sea shell. They show the spiral curve of the shell and the internal chambers that the animal using it adds on as it grows. The chambers provide buoyancy in the water.

### Understanding the Fibonacci Sequence and Golden Ratio

Click on the picture to enlarge it in a new window. Draw a line from the centre out in any direction and find two places where the shell crosses it so that the shell spiral has gone round just once between them.

The outer crossing point will be about 1.