EXPLORE: Fibonacci

Let's Explore!

This is a class activity for you to practice completing whole number patterns. It also provides an opportunity to explore a special pattern called the Fibonacci Sequence.

 

You will need the following materials:

• Compass
• Ruler
• Pencil
• Construction paper (a variety of colors)
• Scissors
• Glue

Analyze a pattern

Examine the following pattern of dots.

1st	2nd	3rd	4th	5th

Is this an increasing or decreasing pattern?

• Explain in words what the rule for the pattern is.
• Write the pattern in numbers and continue up until the 10th entry.

Compare with a friend.

• Did they explain the rule in a different way?
• Compare and contrast your answers.

 

The pattern you have just discovered is very similar to a famous pattern known as the Fibonacci Sequence.

The Fibonacci Sequence is named after the 13th Century mathematician Leonardo de Fibonacci of Pisa, Italy. It is a pattern that appears in nature - the branching of trees, the family tree of the honey-bee, the count of petals on flowers and many other instances.

This time, try to complete this pattern:

1, 1, ☐, ☐, ☐, ☐

Let's try for more!

 

Visual model of Fibonacci numbers

In this activity, you will create circles each with a diameter of the length of a number from the Fibonacci Sequence. The diameter is a line from one end of the circle to the other end of the circle and passes through the center of the circle. There will be one circle for each number in the Fibonacci Sequence up to the 6th term. The following are the steps that you should follow:

1. Use your ruler to set the compass so that it will create the desired radius. The radius is the distance from the center point to the edge of the circle. It is half of the diameter. For example, if I want to create a circle with a diameter of 1, I will set the distance between the compass legs, which indicate the radius, to be 0.5.

2. When your compass is set, use it to draw a circle on a piece of construction paper. Try to use a different colored piece of construction paper for each circle.

3. Cut out the circle you have just drawn with your scissors.

4. Repeat this until you have 6 circles; one for each of the first six terms in the Fibonacci Sequence. The radii of each of the circles will be: 0.5, 0.5, 1, 1.5, 2.5, and 4 centimeters. Or if you want slightly bigger circles, you could start with 1 centimeter, and your sequence of radii would be 1, 1, 2, 3, 5 and 8 centimeters. Glue the 6 circles you have cut out to a piece of paper in any design that you would like.

Reflect

Work on the following questions with a friend.

Question 1:

Compare and contrast the way you and your partner chose to arrange the circles on the piece of paper. Does it help you see the pattern in different ways?

Question 2:

Can you see the general pattern of the Fibonacci Sequence? Explain.

Question 3:

Use the internet to investigate where Fibonacci numbers can be found in nature.