Fabulous Fibonacci (2024)


Fibonacci numbers are an interesting mathematical idea. Although not normally taught in the school curriculum, particularly in lower grades, the prevalence of their appearance in nature and the ease of understanding them makes them an excellent principle for elementary-age children to study.

Learning Objectives
After completing the lessons in this unit, students will be able to:

  • Explain Fibonacci numbers and their origin.
  • Identify Fibonacci numbers in nature and art.
  • Generate the next numbers in the Fibonacci sequence.
  • Create an original number sequence.
  • Create a Fibonacci rectangle and spiral.
  • Write an acrostic Fibonacci poem.


  • Read through the lessons carefully.
  • Gather materials.
  • Print lessons with a color printer.

Lesson 1: Introduction to Fibonacci numbers

Have you ever pulled the petals off of a daisy? If you look closely at the center of a daisy, you will find that the yellow center is not solid. It is made up of sets of spirals that go out from the center. It’s not just daisies! Nature is all about math.

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Look at the pictures of a pinecone. It has those same kinds of spirals. They don’t go around and around in a circle — they go out like fireworks. Look at the pictures below to see what that looks like. How many spirals go in the clockwise direction (green lines)? How many spirals go in a counter-clockwise direction (yellow lines)? Isn’t that strange? Wouldn’t you expect that they would be the same?

To understand the spirals in pinecones, pineapples, daisies and lots of other things in nature, we have to meet a mathematician named Leonardo de Pisa. Most people call him Fibonacci (pronounced fib-o-nawch-ee). About 800 years ago, he wrote a book in which he included a math problem that went like this:

“A certain man put a pair of rabbits in a place surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair from which the second month on becomes productive?”
(Liber abbaci, pp. 283-284)

(Isn’t it weird that they had word problems 800 years ago?) Fibonacci’s work on this problem led him to this sequence of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 …

Can you figure out what the next number in the sequence will be?

We call this the Fibonacci sequence, and the numbers are called Fibonacci numbers. To get the next number in the sequence, you add the previous two numbers together. Now go back and look at those pinecone spirals. What do you notice about the number of spirals in each direction, now that you know about Fibonacci numbers?

Lesson 2: Finding Fibonacci

Now that you know what Fibonacci numbers are, you’re ready to go on a Fibonacci hunt.

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Look at this flower (it’s a Blizzard Bay daylily, image courtesy of Barossa Daylilies). How many petals does it have? Do you see six? Are you thinking that six is not a Fibonacci number? Look more closely. Do you see that there are really two sets of three petals? The outside petals with the straight edges are not really petals. They’re called sepals. So how many actual petals are there?

Not all flowers have three petals. Some have lots more. Some are hard to count because there are so many petals or they are in rows. And there are a few flowers that don’t have Fibonacci numbers for petals. See if you can find a drawing of a daisy. Are the petals a Fibonacci number?

Seed heads
The seed heads of flowers are in Fibonacci sequence spirals like you saw in the pinecones. Look at the seed head of this sunflower. Do you see how packed in the seeds are? But none of them gets smashed. The spiral pattern lets the flower fit the most seed heads in the least space.

How do you think you could use this information in your life? Do you think it would work to try to pack a suitcase like this? Why or why not?

Look at this picture of a purple cauliflower. If you look carefully, you can see the center of it where the little flowers (florets) begin.

Can you see the spiraling?

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Nature is full of Fibonacci numbers and the Fibonacci sequence. Look at this Forestry Service poster of wildflowers. How many examples of Fibonacci numbers can you find in the poster? (Don’t worry that the words are fuzzy; just look at the pictures of the flowers.)

Lesson 3: Working with Fibonacci


  • Large piece of paper
  • Graph paper (If you need graph paper, print some at enchantedlearning.com/math/graphs/graphpaper)
  • Ruler
  • Compass
  • Colored pencils

Fibonacci Rectangles
Let’s draw some rectangles using Fibonacci numbers. This will take us to an amazing place. Complete the Fibonacci sequence below (try to do it without help!):

0, 1, 1, 2, 3, 5, __ , __ , __ , __ ,

Great! You’re ready to start. First, we’re going to do an easy one using graph paper.

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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. This ratio gives us rectangles we call the “Golden Rectangle” because they are said to be the most beautiful rectangles to look it. The ratio is called the Golden Ratio. You can find it by dividing the long side by the short side. So if you have a rectangle that is 3 × 5, you would divide 5 by 3. This will give us a number right around 1.61. (Learn more about the Golden Ratio.)

Can you find some Golden Rectangles around you? How about this piece of paper?

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!

  1. First, draw two squares side by side. Use your ruler to make them 0.5 cm square.
  2. Now make a 2 × 2 square on top of the first square. So if the first square was 0.5 cm, the 2 × 2 square would be 1 cm square, right?
  3. Continue this pattern, making each square the next size in the Fibonacci sequence. So after the 2 × 2 square, you would make a 3 × 3 square (1.5 cm × 1.5 cm), then a 5 × 5 (2.5 cm × 2.5 cm), and so on. This is just like what you did with the graph paper, only using a ruler. Keep going until you have made a square that is 21 × 21 (10.5 cm × 10.5 cm).
  4. Each square will have an edge that is the sum of the two squares before it, just like in the Fibonacci sequence.
  5. 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?

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Now, with your compass, make an arc in the squares with a radius the size of the edge of the square. Don’t get nervous about these big words; they just mean that the arc will be one-quarter of a circle. 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?

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Art Connection
Now look at this painting by Mondrian:

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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?



Fabulous Fibonacci (8)

Fabulous Fibonacci (2024)


What is the Fibonacci sequence answer? ›

The Fibonacci sequence is a type series where each number is the sum of the two that precede it. It starts from 0 and 1 usually. The Fibonacci sequence is given by 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. The numbers in the Fibonacci sequence are also called Fibonacci numbers.

What is the easiest way to solve the Fibonacci sequence? ›

Remember, to find any given number in the Fibonacci sequence, you simply add the two previous numbers in the sequence. To create the sequence, you should think of 0 coming before 1 (the first term), so 1 + 0 = 1.

Whose real name is the first few Fibonacci numbers 0 1 2 3 5 8 these numbers are named after Fibonacci? ›

They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci.

What is a famous Fibonacci quote? ›

Fast Facts: Leonardo Pisano Fibonacci

Notable Quote: “If by chance I have omitted anything more or less proper or necessary, I beg forgiveness, since there is no one who is without fault and circ*mspect in all matters.”

How is Fibonacci used in real life? ›

The Fibonacci sequence, also known as the golden ratio, is utilized in architectural designs, creating aesthetically pleasing structures. In engineering and technology, Fibonacci numbers play a significant role, appearing in population growth models, software engineering, task management, and data structure analysis.

Why is 1.618 so important? ›

The golden ratio, also known as the golden number, golden proportion, or the divine proportion, is a ratio between two numbers that equals approximately 1.618. Usually written as the Greek letter phi, it is strongly associated with the Fibonacci sequence, a series of numbers wherein each number is added to the last.

What is the golden rule Fibonacci sequence? ›

The Golden Ratio is a relationship between two numbers that are next to each other in the Fibonacci sequence. When you divide the larger one by the smaller one, the answer is something close to Phi. The further you go along the Fibonacci Sequence, the closer the answers get to Phi.

Does the Fibonacci sequence ever end? ›

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, … This is the Fibonacci Sequence. It goes on infinitely and is made up of the series of numbers starting with 0, followed by 1, where each subsequent number is the sum of the previous two numbers.

How accurate is the Fibonacci sequence? ›

How Accurate Are Fibonacci Retracements? Some experts believe that Fibonacci retracements can forecast about 70% of market movements, especially when a specific price point is predicted.

Is Fibonacci a real person? ›

Little is known about the life of Leonardo of Pisa, also called Fibonacci. It is assumed that he was born around 1170 and died after 1240, probably in Pisa. There is an original document from 1241 in which the city of Pisa grants Fibonacci a pension. From this time onwards, little or nothing is known about his life.

Why did they call him Fibonacci? ›

The Italian mathematician, who was born around A.D. 1170, was initially known as Leonardo of Pisa. In the 19th century, historians came up with the nickname Fibonacci (roughly meaning "son of the Bonacci clan") to distinguish the mathematician from another famous Leonardo of Pisa.

What is the Rabbit problem Fibonacci? ›

Each pair is comprised of 1 male and 1 female and no rabbits die or leave the field. This is the classic rabbit problem Fibonacci used to generate the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… Ask students to work together in pairs and estimate the number of pairs they would have at the end of one year.

What is the strongest Fibonacci level? ›

The ratios form the support or resistance levels in Fibonacci Retracement analysis. The important levels are 61.8% (an-1 / an), 38.2% (an-2 / an), and 23.6% (an-3 / an). There are other important levels like 78.6% and 50%, which are not Fibonacci ratios but are nonetheless important.

What is the golden line in Fibonacci? ›

The golden ratio is derived by dividing each number of the Fibonacci series by its immediate predecessor. In mathematical terms, if F(n) describes the nth Fibonacci number, the quotient F(n)/ F(n-1) will approach the limit 1.618... for increasingly high values of n. This limit is better known as the golden ratio.

What is the golden pocket Fibonacci numbers? ›

The Golden Pocket is typically considered to be between the 0.618 (or -61.8%) and 0.65 (or -65%) levels.

What is the formula for the Fibonacci sequence? ›

What is the Formula for Generating the Fibonacci Sequence? The Fibonacci sequence formula deals with the Fibonacci sequence, finding its missing terms. The Fibonacci formula is given as, Fn = Fn-1 + Fn-2, where n > 1. It is used to generate a term of the sequence by adding its previous two terms.

What is the next term in the Fibonacci sequence 1 1 2 3 5 8? ›

The Fibonacci sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34... In this series, the next number is found by adding the two numbers before it. Hence, the next term in the series is 8 + 13 = 21.

When was Fibonacci born and died? ›

Little is known about the life of Leonardo of Pisa, also called Fibonacci. It is assumed that he was born around 1170 and died after 1240, probably in Pisa. There is an original document from 1241 in which the city of Pisa grants Fibonacci a pension.

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