FunctionsIs Fibonacci Repeated or Recursive?

Created by: Rachel Oakley

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Understand the concept of a function and use function notation.

Interpreting Functions

FUNCTIONS

http://www.fairfaxhs.org/ourpages/auto/2011/10/6/47466711/Mathematics%20-%20Functions.pdf

Sequences

Things to know about: Recursion“A very important way of generating sequences of numbers xn is by RECURSION. Basically recursion is a

well-defined procedure or formula of going from a given number in the sequence to the next number. One usually needs an initial set of numbers to get things moving. In view of modern

computing capabilities, recursion is a very powerful tool.”

Note: Did you know you have been looking at and collecting sets of numbers like this since you were in 2nd grade?

We are now just learning a more formal way for YOU to apply what YOU know!!! HOW AWESOME!

As an example suppose we start with the number x1 = 1 and the recursion formula for values f n starting

with n = 2. This generates the sequence of numbers.

Fibonacci

“Fibonacci rediscovered or discovered manyresults, proofs and examples WITH or WITHOUT adequate explanation. His contributionsclearly show that he had great talent, imaginative power and the unique mathematical ability in finding solutions of diverse problems of mathematics.”

Debnath, L. (2011). A short history of the Fibonacci and golden numbers with their applications. International Journal Of Mathematical Education In Science & Technology, 42(3), 337-367. doi:10.1080/0020739X.2010.543160

Today you are going to think like Fibonacci!

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...Fibonacci Numbers:

Task 1 Directions: Place Fibonacci Numbers in the map you were given. The first term is placed for you. Continue the sequence until you get to the top. See if you can figure out whether this has a recursive or repeated pattern. Color code the map to show how the numbers relate (loop). Answer the questions after all numbers are placed in the map, and it has been color coded.

What type of patterns does it create?

How would you describe the relationship between these numbers?

Team Member 1

Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8

How would you describe the relationship between these numbers? What type of patterns does it create?

If students are struggling with concept for more than a class period, they can be given this map to help guide their thinking. They have had sufficient enough time to reason with the problem. The goal is for the activity to be challenging, not frustrational.

AnswerRecursive & Repeated0+1= 1 0+1= 11+1=2 1+1=22+3=5 1+1+3=55+3=8 1+1+3+3=8

1

Fibonnaci Sequence: 0, 1, 1, 2, 3, 5, 8

23

5

8

I can decompose the data and to look at data vertically and horizontally.

Objective: Students should decompose the data which will helps them in the future be able to look at data vertically and horizontally.

The Fibonacci Sequence has been filled in for you. Work with your partner to answer the following questions.

Task 2

When you are finished you need to create your own Fibonacci sequence starting with a new number other than 0. The sequence must keep the same proportional changes.

Describe in additive terms, how the terms within the square are repeating. Label your graph, and write a brief description of what you know.

Describe in multiplicative terms, how the terms within the square are repeating.

Describe a pattern of proportion. (ratio) Draw a picture to show the different proportional values that match the pattern. (Assign a color to each different value of the sequence)

Singles Repeated Groups Repeated

Square 1- 1+0 +1

Square 1- 1+0 +1

Square 2- 1+1+1+1=4 2+2

Square 3- 1+1+1+1+1+1+1+1+1=9 3+3+3

Square 5 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1=25

5+5+5+5+5

Square 8 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+=64

8+8+8+8+8+8+8+8

Simplest Repeated

More Complex

Square 1- 1x1 1ⁱ

Square 1- 1x1 1ⁱ

Square 2- 2x2 2²

Square 3- 3x3 3³

Square 5 5x5 5⁵

Square 8 8x8 8⁸

Task 2Directions: Look at the picture that has been given to you along with this page. Work with a partner to fill in the charts and answer the following questions.

11 12 21 1 3 3 35 5 5 5 51 1 3 3 3 3 3 3

Answer0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...

3

1

8

00

3

1

2

5

1

Recursive Repeated0+1= 1 0+1= 11+1=2 1+1=22+3=5 1+1+3=55+3=8 1+1+3+3=8

Task 3: Music & FibonacciYou will be given your papers from Task 1 & Task 2, a sheet that has piano keys labeled and an ipad.

You will need to go to http://www.virtualpiano.net/

Based on how the keys are labeled, play each additive pattern of the Fibonacci sequence. Based on what you know so far, sketch a graph of what you think each would look like. BE SURE TO LABEL!For example: For the square 2, you would hit the 1 key 4 times.

Next, play the multiplicative patterns that you created with the Fibonacci sequence. Sketch a graph of what you think this would look like.

How are the graphs you sketched comparable? How are they different?

Now, rather than using numbers, let the following letters be used:

C=1 D=2 E=3 F=4 G=5 A=6 B=7 C=8Now, instead of playing the traditional

Fibonacci sequence, play the one you created.

http://www.youtube.com/

watch?v=sM1VLYfi

Blw

www.youtube.com/watch?v=pOwMDO0-zBw

www.youtube.com/watch?

v=ZUWbeep9sC4http://

www.youtube.com/

watch?v=SUxcFA0

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Fibonacci Music

Task 4: Continued Music & Fibonacci

Play the original Fibonacci Sequence. Now, instead of playing the traditional Fibonacci sequence, play the one you created.

Did they sound different? Why do you think this is based on what you know?

Extension Idea: Students could go through sequence with the different pedals. They could create a short piece of music using the different pedals applying the Fibonacci sequence.

Task 5: Finding ProofMathematicians work to figure out why something happens and develop proofs/and or theorems.

In order to do this, take the original Fibonacci Sequence, and the one you developed and find the function that explains the relationship between the two. Find the intervals where there is fluctuation in the two sets of numbers and explain the increasing and decreasing pattern between the two sets of numbers by finding the function between. Is it a positive relationship or negative? If it is positive, could it ever be negative, or vice versa? If so, can you explain? What is the maximum difference between the two sets of data? What is the minimum? How are you going to represent this?

Develop your own proof statement that explains develops from what your very first experience was to your most recent experience. Go strictly with the development of the information.

Task 6: Proof Statement

Develop your own proof statement that states there will always be the same proportion between the sequenced numbers. Develop your response based on your understandings from the very first experience to your most recent experience. Go strictly with the development of the information.

Resource for Studentshttp://faculty.up.edu/wootton/Calc1/Section1.1.pdf

Resource for Parentshttp://pta.org/files/HS%20Math%20June20.pdf

http://www.curriculumsupport.education.nsw.gov.au/primary/mathematics/assets/pdf/helpchnwith/graphs/enggr.pdf

http://www.curriculumsupport.education.nsw.gov.au/primary/mathematics/assets/pdf/helpchnwith/shapes/eng_shape.pdf

Interpret Functions

Representing Functions- http://faculty.up.edu/wootton/Calc1/Section1.1.pdf

Domain & Range of Functions- http://www.ck12.org/algebra/Domain-and-Range-of-a-Function/