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Maths History

Fibonacci rabbits and number patterns

Turn the classic rabbit problem into a pattern hunt for older primary learners.

Quick answer

The Fibonacci sequence is a special number pattern where each number is the sum of the two numbers before it. It starts with 0 and 1, and goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. It was discovered by Leonardo Fibonacci, a 12th-century Italian mathematician, when he was exploring how rabbits might breed.

Learning outcome

By the end of this page, you will understand how the Fibonacci sequence works and see how it appears in nature, art, and everyday life. Success signs:
  • You can explain what the Fibonacci sequence is
  • You can continue a Fibonacci sequence from any starting point
  • You can find examples of the Fibonacci sequence in nature or art
  • You can use the Fibonacci sequence to solve a simple problem

Why this matters

The Fibonacci sequence is more than just a fun pattern. It shows up in nature, from the number of petals on flowers to the spiral of a shell. Understanding it helps you see the maths around you, and it's a great way to practice addition and number patterns. This knowledge can also help with more advanced maths later, like ratios, geometry, and even computer programming.

How it works

The Fibonacci sequence starts with 0 and 1. Then, to get the next number, you add the two numbers before it:
  • First number: 0
  • Second number: 1
  • Third number: 0 + 1 = 1
  • Fourth number: 1 + 1 = 2
  • Fifth number: 1 + 2 = 3
  • Sixth number: 2 + 3 = 5
  • Seventh number: 3 + 5 = 8
  • Eighth number: 5 + 8 = 13
This pattern continues forever. You can keep adding the last two numbers to find the next one.

Examples you can see

Let's look at some real-life examples of the Fibonacci sequence. Flowers and plants: Many flowers have a Fibonacci number of petals:
  • Lilies have 3 petals
  • Buttercups have 5 petals
  • Wild roses have 8 petals
  • Delphiniums have 13 petals
Shells: The nautilus shell grows in a spiral that follows the Fibonacci pattern. Each new section is about 1.6 times bigger than the last one, which is close to the golden ratio, a number that appears often in nature and art. Tree branches: If you look closely at how branches grow on a tree, you'll often see that the number of branches follows the Fibonacci sequence. Building a Fibonacci garden: Imagine you are planting sunflowers in a garden. You plant one sunflower in the first row, and then you plant one more in the second row. In the third row, you plant the total of the first two rows. In the fourth row, you plant the total of the second and third rows. If you keep doing this, you'll be creating a Fibonacci pattern in your garden.

Try it as a mini adventure

Let's go on a Fibonacci hunt! Here's your mission: Adventure: Fibonacci Flower Hunt You are a nature detective. Your job is to find flowers with Fibonacci numbers of petals. 1st time: Look for a flower with 3 petals. Write it down. 2nd time: Look for a flower with 5 petals. Write it down. 3rd time: Look for a flower with 8 petals. Write it down. 4th time: Look for a flower with 13 petals. Write it down. 5th time: Look for a flower with 21 petals. Write it down. 6th time: Look for a flower with 34 petals. Write it down. If you can't find 34 petals, don't worry. You're still a great detective! Challenge: Draw your own Fibonacci spiral. Start with a square of 1 cm. Then draw a square of 1 cm next to it. Then draw a square of 2 cm next to that. Then draw a square of 3 cm. Then draw a square of 5 cm. Then draw a square of 8 cm. Connect the corners with a smooth curve to make a spiral.

Common mistakes

  • Mistake: Thinking that the Fibonacci sequence starts with 1 and 1, instead of 0 and 1.
    Why it happens: Sometimes people remember the sequence as 1, 1, 2, 3, 5, 8, 13, and so on, and forget that 0 is also part of the pattern.
    Fix: Remember that the pattern starts with 0 and 1. Then you add them to get the next number: 0 + 1 = 1.
  • Mistake: Adding the wrong numbers together.
    Why it happens: When you get to larger numbers, it's easy to make a mistake.
    Fix: Always double-check that you're adding the two previous numbers. For example, to get 13, add 5 and 8, not 5 and 7.
  • Mistake: Forgetting that the pattern can go on forever.
    Why it happens: Children often think there's a last number.
    Fix: Explain that the pattern never ends. You can always add the last two numbers to get a new one.

Parent and carer guide

As a parent or carer, you can help your child understand the Fibonacci sequence by:
  • Playing Fibonacci games together. For example, you can count how many petals are on flowers or leaves on a plant.
  • Looking for Fibonacci patterns in everyday items like pinecones, pineapples, or even your own fingers.
  • Creating a Fibonacci spiral with a ruler and compass or drawing it on paper.
  • Using the sequence to practice addition. Ask your child to add the last two numbers to find the next one.
  • Encouraging your child to look for Fibonacci patterns in nature, art, or even in music.
If your child gets confused, remember that it's okay to start small. Begin with just the first few numbers and gradually build up. You can also use physical objects like blocks or coins to help them visualize the pattern.

Practice questions

  1. What are the first five numbers in the Fibonacci sequence?
    Answer: 0, 1, 1, 2, 3
  2. What is the next number after 5 and 8 in the Fibonacci sequence?
    Answer: 13
  3. What is the next number after 13 and 21 in the Fibonacci sequence?
    Answer: 34
  4. What is the next number after 21 and 34 in the Fibonacci sequence?
    Answer: 55
  5. What is the next number after 34 and 55 in the Fibonacci sequence?
    Answer: 89

FAQ

What is the Fibonacci sequence?

The Fibonacci sequence is a series of numbers where each number is the sum of the two numbers before it. It starts with 0 and 1, and goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on.

Who discovered the Fibonacci sequence?

The Fibonacci sequence was discovered by Leonardo Fibonacci, an Italian mathematician who lived in the 12th century. He introduced it in his book "Liber Abaci" in 1202.

Why is the Fibonacci sequence important?

The Fibonacci sequence appears in nature, art, and architecture. It helps us understand patterns in growth, design, and beauty. It's also a great way to practice addition and number patterns.

Where can I find the Fibonacci sequence in nature?

You can find the Fibonacci sequence in the number of petals on flowers, the arrangement of leaves on a stem, the spiral of a shell, and the branching of trees.

How can I teach my child the Fibonacci sequence?

You can teach your child by using physical objects like blocks or coins to show the pattern, looking for Fibonacci numbers in flowers or leaves, or creating a Fibonacci spiral with paper and a compass.