Download Lesson Procedures

Date: 22 December, 2014
Teacher: Athena KOLUKISA
Number of Students: 15
Grade Level: 11DPSL
Time Frame: 40 minutes
Fibonacci Sequence
1. Goal(s)
 In this lesson students will learn about the Fibonacci sequence and the golden ratio. They will see the appearance of these numbers in art, architecture, and nature..
2. Specific Objectives (measurable)
 Students will able to describe the Fibonacci number pattern sequence.
 Students will be able to write Fibonacci sequence’s formula recursively.
 Students will be able to derive the Fibonacci sequence from the “rabbit” problem.
 Students will be able to explain some ways that it appears in nature.
 Students will be able to determine how close a ratio is to phi and determine that phi is an irrational number.
 Students will be able to explain some connections between golden ratio and nature, architecture, human body etc.
3. Rationale
 The topic of Fibonacci sequences is part of algebra. Students will learn more about Fibonacci number sequence.
 Fibonacci sequence is seen in nature, architecture also human body.
 Fibonacci sequence is a good example for recursive sequences. Students will understand recursive sequences with Fibonacci sequence.
 Also the sequence uses for calculating golden ratio (Phi). So students will calculate golden ratio with the numbers.
4. Materials
 Board markers (at least two colors)
 Fibonacci Rabbits Worksheets( 15 copies for students and they are copied before the lesson)
 Computer and projector
5. Resources
 Hease and Harris Mathematics for the International Students SL
 http://www.discoveryeducation.com
 http://illuminations.nctm.org/
 http://www.youtube.com/watch?v=X1L8XMTi_Vw
6. Getting Ready for the Lesson (Preparation Information)
 Make sure worksheets are distributed in exploring part.
 Make sure projector and computer are ready for the video.
 Make sure students bring and prepare their TI’s.
7. Prior Background Knowledge (Prerequisite Skills)

Students will be expected to know that a sequence can be finite or infinite and can grow towards both infinities.
 They know the formula for arithmetic progression and the role of the common difference.
Lesson Procedures
Transition (5 minutes): Good morning! Today we work on enjoyable topic. Today’s topic is Fibonacci number sequence or Fibonacci numbers and Golden
Ratio.
Teacher ask to students:
 Do you know Fibonnaci?
 Have you ever heart Fibonacci numbers?
Now I want to tell Fibonacci life.
8A. Engage (5 minutes)
Leonardo Bonacci (1170–1250) known as Fibonacci and also Leonardo of Pisa, Leonardo Pisano, Leonardo Pisano Bigollo, Leonardo Fibonacci was an
Italian mathematician, considered as "the most talented Western mathematician of the Middle Ages."
Fibonacci introduced to Europe the Hindu–Arabic numeral system primarily through his composition in 1202 of Liber Abaci. He also introduced to Europe the
sequence of Fibonacci numbers (discovered earlier in India but not previously known in Europe), which he used as an example in Liber Abaci.
Liber Abaci posed, and solved, a problem involving the growth of a population of rabbits based on idealized assumptions. The solution, generation by
generation, was a sequence of numbers later known as Fibonacci numbers.
Transition: Now, we will work on Fibonacci problem about growth of a population of rabbits. I will distribute worksheets and I will give you 5 minute for
working. And I want you to study on the worksheets individually. Please prepare your pencils and work on it.
8B. Explore (10 minutes)
 Teacher distributes the exploration worksheet and gives students 5 minutes to work individually.
 If students do not understand the problem, teacher explains the problem.
After 5 minutes students share their ideas with student who sits next him.
ASK students to the questions:
a) Do you find a rule for pair numbers of the rabbits? (Analysis)
b) How number of rabbits increase? (Analysis)
Transition: Now we watch a video about the problem.
C. Explain (10 minutes). Show the video.( http://www.youtube.com/watch?v=X1L8XMTi_Vw)
First month we have one youth pair. Second month we have one mature pair. And third month we have one mature and one youth pair. So we have two pairs.
The forth month we have two mature pairs and one youth pair. So totally we have 3 pairs. And the population continues with the growth.
If we show the solution with table, the table is
Month
1
2
3
4
5
Pairs of rabbits
1
1
2
3
5
Ask students:
a) Do you notice any rule about the numbers? (Analyze)
b) What is the rule for the numbers? (Analyze)
Teacher writes equalities on the board:
First number is 1, second number is 1.
Third number is 1+1=2
Forth number is 1+2=3
Fifth number is 2+3=5
And teacher ask students
 So what is sixth number?
 So what is seventh number?
 So what is eighth number?
And teacher writes formula on the board:
If an is a nth term and an-1 is a (n-1)th term in Fibonacci sequence, (n+1)th term is an+an-1 .
an+1=an-1+an
D. Extend (10 minutes)
Transition : Now we will continue with a proportion with the Fibonacci numbers. I will distribute another worksheet about the activity. Please work on it
individually and prepare your pencils. I will give 3 minutes for the worksheet, now you can start!
Teacher explains the number is called Phi and the number is 1.618….Other names for it is the golden ratio.
And teacher asks to students:
 Is phi a rational number?
 How do you understand that phi is rational or irrational?
Teacher explains phi is irrational because the decimal never ends and it never repeats.
E. Evaluate (3 minutes)
 Students are evaluate with their abilities on worksheet 1 with criteria A,C and D.(Fibonacci Rabbits)
 Students are evaluate with their abilities on worksheet 2 with criteria A,C and D.(Finding Golden Ratio)
9. Closure & Relevance for Future Learning (2 minutes)
Remind students that they have learned about Fibonacci sequences today:
ASK, what specifically was learned today? Take notes on the board as a list of points.
10. Specific Key Questions:
1.
2.
3.
4.
5.
6.
Do you notice any rule about the numbers? (Analysis)
What is the rule for the numbers? (Analysis)
Do you find a rule for pair numbers of the rabbits? (Analysis)
How number of rabbits increase? (Analysis)
Is phi a rational number? (Knowledge)
How do you understand that phi is rational or irrational? (Knowledge)
Worksheet 1: Fibonacci Rabbits
Worksheet 2: Finding Golden Ratio
If we look at the ratios made by dividing each Fibonacci number by the one before it, we get
1/1,2/1,3/2,5/3,8/5…
1. Find the next five fractions in the new sequence.
2. Using a calculator, find the exact value of each fraction( including the five you found in the previous question)
Fraction
1/1
Decimal
2/1
3/2
5/3
8/5
3. If you continue in this way, you will see that the decimals are getting closer and closer to a certain number. What will be the number?