Brianna Kallman
3690 Reading Strategy Lesson Plan
21 May 2012
Lesson Overview
This lesson will give students the opportunity to learn about the Fibonacci sequence of numbers
and about how the Fibonacci sequence is displayed in nature. After experiencing this lesson,
students will be able to apply the concept of the Fibonacci sequence to various situations and
identify the sequence in nature.
Sources
Crabtree, C. (2012). Numbers in Nature [Lesson Plan]. Retrieved May 20, 2012, from Discovery
Education website: http://www.discoveryeducation.com/teachers/free-lessonplans/numbers-in-nature.cfm#cre
Images:
http://www.forbes.com/sites/ericsavitz/2011/05/13/apple-analyst-says-no-lte-in-iphone-5-to-addsprint-t-mobile/
http://amazingdata.com/fibonacci-sequence-illustrated-by-nature-pics/
http://www.world-mysteries.com/sci_17.htm
Materials
Calculators (useful for classroom discussion)
Pencils and paper
Graph paper (for homework)
Ruler (for homework)
Compass (for homework)
Classroom Activity Sheet: Finding Fibonacci Numbers in Nature
Take-Home Activity Sheet: Creating the Fibonacci Spiral
Standards Addressed
These standards were taken from Content Knowledge: A Compendium of Standards and
Benchmarks for K-12 Education: 2nd Edition provided by Mid-continent Research for Education
and Learning in Aurora, CO.
Understands and applies basic and advanced properties of the concepts of numbers.
Uses discrete structures (e.g., finite graphs, matrices, or sequences) to represent and to solve
problems.
Uses basic advanced procedures while performing the processes of computation.
Uses recurrence relations (i.e., formulas that express each term as a function of one or more of the
previous terms, such as the Fibonacci sequence and the compound interest equation) to model and
to solve real-world problems (e.g., home mortgages or annuities).
Uses a variety of operations (e.g., finding a reciprocal, raising to a power, taking a root, and
taking a logarithm) on expressions containing real numbers.
Instructional Objectives
After discussing the history and origin of the Fibonacci sequence and identifying it in various
aspects of nature (condition), the student will be able to describe the Fibonacci sequence, apply it
to specific real-world examples, and graph a representation of the Fibonacci sequence (action)
with specific Fibonacci numbers and by using a ruler, graph paper, and compass (criterion).
(Bloom’s Cognitive Domain – Knowledge and Application)
Rationale Statements I believe this lesson is important because the Fibonacci sequence (and the
related Golden Ratio) occur everywhere. From the design of a sea shell, to the reproduction
patterns of animals, to the number of seeds produced by various types of plants, the Fibonacci
numbers can be observed in nature. The students will gain an understanding of the Fibonacci
sequence and be able to identify the sequence in nature. They will ultimately be able to create a
graph representing the Fibonacci sequence. Direct instruction will be used to introduce the
students to the Fibonacci sequence, provide vocabulary terms and definitions, and provide an
example of rabbit reproduction patterns. I am choosing direct instruction for the beginning of the
lesson because it allows for teacher-directed instruction and practice, and the careful supervision
of student progress as the students discover the Fibonacci sequence itself. Indirect instruction will
be used as students use the pictures of various elements of nature to discover the presence of the
Fibonacci sequence in plants. I am choosing indirect instruction for the second part of the lesson
because it allows for student discovery; it is student-centered and student-directed. Students have
the opportunity to examine the pictures of nature and investigate whether plant design has
anything to do with the Fibonacci number sequence.
“Have you ever noticed patterns in nature, like the design of leaves on the trees or the
shapes of seashells? Have you ever wondered what math has to do with the real world?
Today I am going to show you how to find mathematical patterns in flowers, fruit plants,
and other parts of nature and we will discover what math has to do with the real world.”
Procedures
Anticipatory Set
Find the first ten terms of the following sequences:
1, 4, 7, …
2, 4, 8, 16, …
Draw and/or describe two different patterns you have observed in nature, art, or architecture.
Write down any ideas you have on how they were created.
Objective and Purpose
After discussing the history and origin of the Fibonacci sequence and identifying it in various
aspects of nature (condition), the student will be able to describe the Fibonacci sequence, apply it
to specific real-world examples, and graph a representation of the Fibonacci sequence (action)
with specific Fibonacci numbers and by using a ruler, graph paper, and compass (criterion).
(Bloom’s Cognitive Domain – Knowledge and Application)
The purpose of this lesson is to teach students about the Fibonacci sequence and its origins.
Students will learn how to identify the Fibonacci sequence in various elements of nature and be
able to apply that knowledge to a future discussion about the Golden Ratio. Many relevant
examples will be provided on the Classroom Activity Sheet in class, and in classroom discussion.
Input
1. Visual Display: Display the Fibonacci Poster on the computer, SMART board, or
projector. Give the students about 1-2 minutes to discuss and brainstorm how the visual
display might relate to today’s lesson. Ask the students to answer the question on the
poster: What might those images have in common? Allow this to lead to a discussion of
the Fibonacci sequence. Note that the Fibonacci sequence was first noticed by Italian
mathematician Leonardo Fibonacci in 1202. He was studying the reproduction patterns of
rabbits under ideal circumstances.
2. Knowledge Rating Scale: Distribute Fibonacci Knowledge Rating Scales to each
student. Students will rate their prior knowledge of the terms algorithm, logarithmic
spiral, sequence, and term. Throughout the lesson, students will add the definitions of
these terms to their knowledge rating scale.
3. Give the students the assumptions of Fibonacci’s experiments.
 We start with one female and one male rabbit. Rabbits can mate when they reach
one month in age, so by the end of the second month, each female will produce
offspring.
4.
5.
6.
7.
 Our rabbits never die.
 The female produces a pair of rabbits—one male and one female—every month.
Work with students to see whether they can develop the Fibonacci sequence themselves.
Remind them that they are counting pairs of rabbits, not individuals. Walk students
through the first few months of the problem with illustrations on a marker board or
SMART board.
 We begin with one pair of rabbits (1).
 At the end of the first month, there is still one pair (1).
 At the end of the second month, the female produced a second pair (2).
 At the end of the third month, the original female has produced another pair (3).
 At the end of the fourth month, the original female has produced another pair and
the female born two months ago has produced her first pair (5).
Write the discovered pattern on the board: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
and so on. Discuss the “rule” that is being followed to find the next number. Help
students understand that to get each successive number in the sequence, you have to add
the previous two numbers. Explain that this is known as the Fibonacci sequence. At this
point, students will fill in their Knowledge Rating Scales with the definition of sequence
and term. A sequence is a set of elements ordered in a certain way. A term is an element
in a sequence or series.
The term used to describe the “rule” followed to obtain the numbers in the sequence is
algorithm, a step-by-step procedure for solving a problem. At this point, students will fill
in their Knowledge Rating Scales with this definition. As a class, students can use their
calculators and we will continue the sequence for the next few numbers: 1, 1, 2, 3, 5, 8,
13, 21, 34, 55, 89, 144, 233, 377, 610, 987, and so on.
Tell students that the Fibonacci sequence has intrigued mathematicians for centuries.
What's more, mathematicians have noticed that these numbers appear in many different
patterns in nature, often creating the beauty we admire. Tell students that they are going
to look for Fibonacci numbers in pictures of objects from nature. Make sure that students
understand that they are looking for specific numbers that appear in the sequence, not for
the entire sequence.
Divide students into groups of three or four. Distribute the Classroom Activity Sheet:
Finding Fibonacci Numbers in Nature. Tell students to work together to try to answer the
questions on the sheet. Make sure that each student fills out his or her own sheet. For
your information, the questions and explanations are listed below. It may be helpful to
work on the first example as a class so students understand what they are looking for.
Flower petals. Count the number of petals on each of the flowers. What numbers do you get? Are
these Fibonacci numbers? (Lilies and irises have 3 petals, buttercups have 5 petals and asters and
black-eyed Susans have 21 petals; all are Fibonacci numbers.)
Seed heads. Each circle on the enlarged illustration represents a seed head. Look closely at the
illustration. Do you see how the circles form spirals? Start from the center, which is marked in
black. Find a spiral going toward the right. How many seed heads can you count in that spiral?
Now find a spiral going toward the left. How many seed heads can you count there? Are they
Fibonacci numbers? (The numbers of seed heads vary, but they are all Fibonacci numbers. For
example, the spirals at the far edge of the picture going in both directions contain 34 seed heads.)
Cauliflower florets. Locate the center of the head of cauliflower. Count the number of florets that
make up a spiral going toward the right. Then count the number of florets that make up a spiral
going toward the left. Are the numbers of florets that make up each spiral Fibonacci numbers?
(The numbers of florets will vary, but they should all be Fibonacci numbers.)
Pinecone. Look carefully at the picture of a pinecone. Do you see how the seed cases make spiral
shapes? Find as many spirals as possible going in each direction. How many seed cases make up
each spiral? Are they all Fibonacci numbers? (The numbers will vary, but they should all be
Fibonacci numbers.)
Apple. How many points do you see on the "star"? Is this a Fibonacci number? (There are five
points on the star.)
Modeling
We will work through the first question on the Classroom Activity Sheet as a whole group. Also,
students will have the rabbit situation modeled for them on the board at the beginning of class.
Check for Understanding
Think-Pair-Share: Pose these questions to students. Have students think individually at first.
After they have had a couple of minutes to write down their own answers, have students pair up
and share their answers. Choose a few students to share their partner’s answers with the class.
1. Ask the class which shape emerges the most often from clusters of seeds (the spiral).
Discuss whether there are any advantages to this shape. Explain that seeds may form
spirals because this is an efficient way of packing the maximum number of seeds into a
small area.
2. Ask students where else they see the spiral shape in nature (nautilus shell). Explain that
this is may be described as a logarithmic spiral, a shape that winds around a center and
recedes from the center point with exponential growth. Would they guess that those
spirals are also formed from Fibonacci numbers? Do they find this shape pleasing to the
eye? To conclude, discuss other pleasing shapes and patterns in nature, such as those of
waves, leaves, and tornadoes. Discuss whether these, too, may have a mathematical basis.
Guided Practice
Discussion Questions as follow-up to the Classroom Activity Sheet
1. Imagine that scientists in the rain forest have discovered a new species of plant life.
Where might they look for the Fibonacci sequence?
2. Try to solve this problem: Female honeybees have two parents, a male and a female, but
male honeybees have just one parent, a female. Can you draw a family tree for a male
and a female honeybee? What pattern emerges? Are they Fibonacci numbers? (The male
bee has 1 parent, and the female bee has 2 parents. The male bee has 2 grandparents,
and the female bee has 3 grandparents. The male bee has 3 great-grandparents, and the
female bee has 5 great-grandparents. The male bee has 5 great-great-grandparents, and
the female bee has 8 great-great-grandparents. The male bee has 8 great-great-greatgrandparents, and the female bee has 13 great-great-great-grandparents.)
Assessment/Evaluation
Independent Practice
Take-Home Activity Sheet: Students will follow the directions on the worksheet. Using a ruler, a
compass, graph paper, and a pencil, students will create a graph of a spiral representing the
Fibonacci sequence.
Students will be evaluated using the following three-point rubric:

Three points: active participation in classroom discussions; ability to work
cooperatively to complete the Classroom Activity Sheet; ability to solve all the
problems on the sheet

Two points: some degree of participation in classroom discussions; ability to work
somewhat cooperatively to complete the Classroom Activity Sheet; ability to solve
three out of five problems on the sheet

One point: small amount of participation in classroom discussions; attempt to
work cooperatively to complete the Classroom Activity Sheet; ability to solve one
problem on the sheet
Closure
Exit Slip: Write down the first ten terms of the Fibonacci sequence. Students may use their
calculators for this. Also, provide two examples of something in nature that represents the
Fibonacci sequence.
Vocabulary
Algorithm
Definition: A step-by-step procedure for solving a problem.
Context: The algorithm for obtaining the numbers in the Fibonacci sequence is to add the
previous two terms together to get the next term in the sequence.
Logarithmic spiral
Definition: A shape that winds around a center and recedes from the center point with exponential
growth.
Context: The nautilus shell is an example of a logarithmic spiral.
Sequence
Definition: A set of elements ordered in a certain way.
Context: The terms of the Fibonacci sequence become progressively larger.
Term
Definition: An element in a series or sequence.
Context: The mathematician Jacques Binet discovered that he could obtain each of the terms in
the Fibonacci sequence by inserting consecutive numbers into a formula.
Differentiation
Content- Verbal and visual instruction
Visual display of the Fibonacci Poster
Verbal instruction with visual representations of the rabbit example, visual examples in
the Classroom Activity Sheet
Process-individuals, partners (neighbors) in Think-Pair-Share, small group work (groups of three
to four) for the Classroom Activity Sheet, whole group work on the rabbit example at the
beginning of class
Product-if there is enough time, students may prefer to work on the Take-Home Activity Sheet as
a group project
Extension Activities
Find the length and width of various rectangles (a 3x5 index card, an 8.5x11 piece of paper, a 2x3
school photo, and other familiar rectangles) and find the ratio of the length to the width. Have the
students find the average of all the ratios. This leads to the Golden Ratio, a concept related to the
Fibonacci number sequence. Discuss the occurrence of the Golden Ratio in pleasing shapes such
as pentagons, crosses, isosceles triangles, and other forms of art and architecture.
Develop an algebraic formula for the Fibonacci sequence, representing the first term as x and the
second term as y. Students will find that the coefficients for the algebraic formulas for each term
are Fibonacci numbers in order.
Knowledge Rating Scale
Vocabulary
Term
Know It
Well
Have seen
or heard it
No
clue
What It Means
Algorithm
Logarithmic
Spiral
Sequence
Term
Knowledge Rating Scale
Vocabulary
Term
Algorithm
Logarithmic
Spiral
Sequence
Term
Know It
Well
Have seen
or heard it
No
clue
What It Means