Fibonacci Pattern (Investigation)

Objectives

• To practice completing whole number patterns.
• To explore different representations of patterns.

Materials

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

Pre Investigation Questions:

1. Examine the following pattern of dots and explain in words what the rule is for the pattern.
   

    

2. Express the pattern in numbers and continue it up until the 10th entry.
3. Is this an increasing pattern or a decreasing pattern?
4. Can you think of a way to represent the 100th entry in this pattern?
5. Compare with a friend! Did they represent the 100th entry in a different way? Compare and contrast your representations.

Investigate!

1. The pattern you have just discovered is very similar to a famous sequence known as the Fibonacci Sequence.
2. Look up the mathematical definition of a sequence. How is it similar to a number pattern?
3. Use this website to look up the number(s) that are missing from the pattern above to make it the Fibonacci Sequence.

Create a Visual Representation of the Fibonacci Sequence

In this investigation you will create circles each with a diameter of the Fibonacci Sequence. There will be one circle for each number in the Fibonacci Sequence up to the 6th term. 

1. Use your ruler to set the compass so that it will create the desired radius. 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.  Or if you want a slightly bigger one you could start with 1, and your sequence of radii would be 1,1,2,3,5,8.
5. Glue the 6 circles you have cut out to a piece of paper in any design that you would like.
   

    

Post Investigation Questions

1. Work on the following questions with a friend.
2. 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?
3. Can you see the general pattern of the Fibonacci Sequence? Explain.
4. Together, use the internet to investigate where Fibonacci Numbers can be found in nature.