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Fibonacci is All Around
I. UNIT OVERVIEW & PURPOSE:
The overall purpose of this activity is to explore the many wonders of the Fibonacci Sequence and see how the sequence is related to the Golden Ratio in our own natural habitat. The main focus group is for Algebra 1 or Geometry students to build a better understanding of finding patterns and relationships between patterns and how they can be used with real-world application. II. UNIT AUTHOR: Lynn Miller-Jones, Staunton River Middle School, Bedford County Public Schools III. COURSE: Mathematical Modeling: Capstone Course IV. CONTENT STRAND: Algebra, Geometry V. OBJECTIVES: Students will explore and investigate how to generate the Fibonacci sequence and discover how its unique attributes produce the Golden Ratio. Students will then use the Golden ratio created from the Fibonacci Sequence to identify how it appears in nature. Finally students will explore the use of a Fibonacci Gauge to help create “golden” materials. VI. MATHEMATICS PERFORMANCE EXPECTATION(s): MPE. 1 The student will solve practical problems involving rational numbers (including numbers in scientific notation), percent, ratios, and proportions.
MPE. 3. The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation.
MPE. 7. The student will use similar geometric objects in two- or three-dimensions to solve real-world problems about similar geometric objects.
MPE. 10. The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve real-world problems, including writing the first n terms, finding the n th term, and evaluating summation formulas. VII. CONTENT: Lesson 1 will involve having the students explore the Fibonacci Sequence and then using an excel file to generate the Golden Ratio. Lesson 2 will involve the students gathering information regarding how the Golden Ratio appears in nature. Lesson 3 will involve students creating a Fibonacci Gauge and using it to identify items within the room that meet the Golden Ration and then using the gauge to create a drawing.
VIII. REFERENCE/RESOURCE MATERIALS:
Students will need to have access to a computer for internet research and excel computations; rulers and grid paper; and various nature objects such as pine cones and flowers. IX. PRIMARY ASSESSMENT STRATEGIES: Students will be assessed based on research data, computation of individual works, observation of work habits and overall project completion. X. EVALUATION CRITERIA: Grading rubric is included with each lesson. XI. INSTRUCTIONAL TIME: Lesson 1: One 90 minute class period Lesson 2: One 90 minute class period Lesson 3: One 9 0 minute class period
Materials/Resources (per group) Domino style sets of ten tiles for each pair of students Grid paper Access to an excel program Exploring Fibonacci worksheet
Assumption of Prior Knowledge Student must have an understanding of how to create an array for multiplication.
Students must be able to analyze a pattern and produce a rule.
Students need to have an understanding of how to create formulas in an Excel
program.
Introduction: Setting Up the Mathematical Task
This activity is designed for students to explore the Fibonacci Sequence and make a conjecture about what ratio the sequence produces.
Duration: This project will take approximately one 90 minute class.
Student Exploration 1:
Introduction: (10 minutes) To introduce the activity, have students explore the beginning of the sequence for the existence of a pattern: 1, 1, 2, 3, 5, 8, 13 … and then extend the pattern to the next 5 numbers in the sequence. Discuss the findings of the students and have them explain how they got the remaining numbers in the sequence.
Small Group Work (30 minutes)
- Place students into pairs and provide a set of ten domino style rectangles. Distribute the “Exploring Fibonacci” worksheet
- Explain to the students they will be exploring how many possibilities there are to arrange a rectangle that measures 2 x 10. Make sure to inform students that the positioning of the bricks (tiles) makes a difference.
- The students will begin by working with a 2 x 1 rectangle and work their way up to a 2 x 10. Have the students explore the existence of a pattern in the chart.
- Challenge the students to find Fibonacci sequence in the following examples: a. Pascal’s Triangle b. One octave level in a set of piano keys. c. Set of branches on a tree
Whole Class Sharing/Discussion
Discuss findings of students. Possibly have students display their grid arrangements under a document camera. Have students explain where they see the sequences in each of the problems above.
Assessment
Students will be assessed through observation and peer cooperation, answers expressed on the Exploring Fibonacci worksheet, creation of excel program.
Grading Rubric
Participation and peer cooperation: 30 points Acceptable responses to worksheet: 30 points Creation of Excel File using rules: 20 points Fibonacci internet Exploration: 20 points
Strategies for Differentiation
The students could explore the arrays on grid paper. The excel file could be generated using calculators instead. Provide various pictures for students to explore and discuss the culture the picture may be from. Have the students explore the graphs of the ratios and then compare the sum of the squares of the ratios and discuss findings.
Exploring Fibonacci Worksheet
Student Exploration Part 1:
Introduction:
A list of numbers has been given. Find the pattern necessary to complete the remainder of the sequence.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, _____, _____, _____
Explain the pattern used to find the remaining numbers for the sequence.
- Suppose you are a craftsman and you are designing a 2 x 10 rectangle in honor or Mr. Fibonacci. You have been given ten 2 x 1 bricks to cover this rectangle. How many different ways can you lay the bricks. (You will be using domino tiles to represent the bricks.)
Solve the problem by arranging the bricks for a 2 x 1 rectangle first. How many possible ways can they be laid? Next, look at a 2 x 2 construction. How many arrangements are possible? Now try 2 x 3 arrangements. Continue to fill in the chart show to represent the possible ways for the bricks to be laid.
What pattern emerged within the chart?_______________________
Dimensions
Number of possible arrangements 2 x 1 2 x 2 2 x 3 2 x 4 (^) (answer is not 4) 2 x 5 2 x 6 2 x 7 2 x 8 2 x 9 2 x 10
Exploring Fibonacci Worksheet KEY
Student Exploration Part 1:
Introduction:
A list of numbers has been given. Find the pattern necessary to complete the remainder of the sequence.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610
Explain the pattern used to find the remaining numbers for the sequence. The final term in the sequence is added to the previous term to get the next term. 1+1 = 2; 2+1 = 3; 3+2 = 5; etc.
- Suppose you are a craftsman and you are designing a 2 x 10 rectangle in honor or Mr. Fibonacci. You have been given ten 2 x 1 bricks to cover this rectangle. How many different ways can you lay the bricks within this rectangle. (You will be using domino tiles to represent the bricks.)
Any particular pattern starting to emerge? Fibonacci Sequence
Dimensions
Number of possible arrangements 2 x 1 1 2 x 2 2 2 x 3 3 2 x 4 5 2 x 5 8 2 x 6 13 2 x 7 21 2 x 8 34 2 x 9 55 2 x 10 89
- Can you identify how the Fibonacci numbers are used in Pascal’s Triangle?
- Piano keys also take advantage of the famous sequence. Can you explain how? There are 13 notes in an octave span. From the scale of C to C there are 13 keys: 8 that are white, 5 black keys and they are split into groups of 3 and 2.
- Tree branching also makes use of the Fibonacci Sequence. Can you identify where? The number of branches in each section is creating the sequence.
The Golden Ratio in Nature
Strand Geometry
Mathematical Objective Identify how the Golden Ratio produced from the Fibonacci Sequence appears in nature.
Mathematics Performance Expectation(s)
MPE. 1 The student will solve practical problems involving rational numbers (including numbers in scientific notation), percent, ratios, and proportions.
MPE. 10. The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve real-world problems.
Related SOL
G.14 The student will use similar geometric objects in two- or three-dimensions to compare ratios between side lengths, perimeters, areas, and volumes and solve real- world problems about similar geometric objects.
A.7 The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including making connections between and among multiple representations of functions including concrete, verbal, numeric, graphic, and algebraic.
NCTM Standards
represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules; relate and compare different forms of representation for a relationship; solve problems involving scale factors, using ratio and proportion; use geometric models to represent and explain numerical and algebraic relationships; recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life; Recognize and apply mathematics in contexts outside of mathematics.
Materials/Resource Pictures or actual Flowers with various amounts of petals and various pine cones 3 petals: lily, iris 5 petals: buttercup, wild rose, larkspur, columbine (aquilegia) 8 petals: delphiniums 13 petals: ragwort, corn marigold, cineraria, 21 petals: aster, black-eyed susan, chicory Various Rectangular shapes or pictures of rectangles Ruler or tape measure Graph paper Exploring the Golden Ratio worksheet You Tube access
Assumption of Prior Knowledge Students explored the Fibonacci Sequence and discovered the Golden Ratio of 1. Students are capable of reading a ruler or tape measure accurately.
Introduction: Setting Up the Mathematical Task
This activity is designed for students to explore the Golden Ratio and Fibonacci Sequence and investigate how they apply to many things in the natural world. Each activity will be set up as a station where students investigate versions of the Golden Ratio or find the Fibonacci Sequence.
Duration: This project will take approximately one 90 minute class.
Small Group Work
- Separate the students into groups of 3 or 4 students.
- Station 1 Rectangles : Provide the students with various rectangular shapes and have them identify the ones they feel are pleasing to the eye. Once the rectangles have been chosen, have the students measure the horizontal length and vertical width of each rectangle and calculate the ratio of length to width. Students should then compare the ratios to the golden ratio and note the similarities. (Alternative for rectangles: have the students bring in rectangular shapes that they feel are eye appealing and then use those for comparison.)
- Station 2 Flowers and Pine Cones : Have various flower bundles and pine cones available for students to study. (Flowers and pine cones could be brought in by students.) Have them take note of the number of petals per flower and count the number of spirals going clockwise and counter-clockwise in a pine cone. Ask students to make a comparison about the petal numbers and spiral ratio of clockwise : counterclock-wise and make a conjecture.
- Station 3 Square Spirals : Using grid paper, have the students create the Fibonacci spiral by creating squares whose side measurements correspond to the Fibonacci numbers. First have the students draw a square that measures one square unit. Next have the students draw a second square of one square unit to the left of the square. Third, the students should draw a 2 x 2 square above the ones just drawn. Now, they need to draw a 3 x 3 to the right of the other three squares. The students should continue this pattern until they have filled up the graph paper with similar squares. To create the spiral, the students need to draw an arc starting on the inside of the initial square and have it pass from one corner to the next so that it is continuously passing each new square from corner to corner. The final result should look similar to the given picture.
Ask the students how the lengths of each square relate to the Fibonacci Sequence and where in nature they have seen the spiral that is produced.
Assessment
Students will be graded based on responses to observations at each station
Grading rubric
Participation at each station 10 points per station (40 points) Responses per station 10 points per station (40 points) Exit Ticket 20 points
Strategies for Differentiation
Have squares for the spiral square cut out to length and have the students reconstruct
the spiral.
Have the students research ways to produce the Golden Ratio and have them do a
presentation on their research.
Have students measure various rectangles found within the room.
Have students read the Elliott Wave Principle and explore how it relates today’s issues.
Exploring the Golden Ratio Worksheet
- Station 1: Rectangles Using the various rectangles provided, identify the ones that you feel are appealing to the eye. Measure the horizontal length and vertical width of the rectangles you chose. Calculate the ratio of the length to the width and share with your group members. Once all members have calculated the ratios, compare the results to the Golden Ratio.
- Station 2: Flowers and Pine Cones a) Using the bundle of flowers provided, create a chart that indicates the type of flower you observed and the number of petals on the flower.
b) Looking at a pine cone from the top view, count the number of spirals going clockwise and counter-clockwise. Find the ratio of clockwise spirals to counter-clockwise spirals for each pine cone. c) Note any observations you make regarding the flower petals and the ratios of the pine cones.
