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FASCINATING FIBONACCI!
MASTER TEACHER Terri Salas
GRADES 7-9
MATH/SCIENCE
OVERVIEW
This math/science lesson is an interesting follow up after a
math unit on arithmetic and geometric patterns and sequences in math. Other math
concepts reinforced by the Fibonacci numbers are similar figures and
proportions. The Fibonacci sequence stimulates exploration of patterns and
sequences found in the real world. Fibonacci is the nickname for the Middle Ages
(c.1200) mathematician Leonardo of Pisa who contributed much to algebra,
arithmetic, and geometry. The sequence highlighted in the video is named after
him and scientists have discovered that it occurs naturally in countless natural
phenomenon. For example in this video, a math problem dealing with the
reproduction cycles of rabbits, illustrates that patterns, such as the Fibonacci
sequence, exist in nature. In turn, scientists may use these patterns to make
predictions. The ratios made by comparing Fibonacci numbers are found in our
body, harmony with music, architecture and engineering such as the design of a
bicycle. The hands on activities allow the students an opportunity to practice
and experience patterns, problem solving, and calculator skills as they find the
missing numbers in the sequence. They may also apply the Fibonacci numbers
to their bodies and environment via the golden triangle ratio. This lesson
may be extended into other disciplines such as art or history and the hands on
activities can easily become student projects.
ITV SERIES
Math Vantage: #3 Sequence and Ratio
MATERIALS
5 or 6 fresh pineapples
blank overhead transparency
overhead transparency pens (for teacher use)
(per student groups of 2 to 4)
4 sets of Multi-Links - two colors per group (100 pieces
per color); any brand tiles or cubes such as unifix cubes may be used in place
of the Multi Links
(per student)
meter sticks, centimeter rulers and/ or
measuring tapes
worksheets "Busy Bees" and "The Rabbit
Problem"
calculator (TI-Explorer)
pencil
2 sheets of notebook paper
VOCABULARY
Fibonacci
pattern
sequence
ratio
similar figures
genealogy
drone
spiral
LEARNING OBJECTIVES
The students will be able to:
*identify the Fibonacci sequence as being arithmetic, geometric
or other
*write a short paragraph support their answer using appropriate
math terms
PREVIEWING ACTIVITIES
If the cost is not prohibitive, have several pineapples
available to pass around among the groups of students for examination. Ask the
students to look at the pineapple from all sides and comment on any math ideas
or patterns they may find in this fruit. (The students may notice that the eyes
are six sided shapes (hexagons) or they may count the eyes themselves. They may
also mention the spherical shape of the pineapple. Accept any answers and
write them on the overhead transparency.) The students record the brainstorming
answers on their own sheet of paper. Tell the students that after watching the
video the class will look at the brainstorming comments again, and compare their
initial responses to the knowledge acquired in the video.
Optional: Another connection may be made to nicknames.
Present several famous persons' names known to the students by first names or
nicknames. For example, list personalities in current events like Madonna,
Batman, Selena, Fabio, Pocahontas, etc. Again ask the students to state what
these words have in common. (They may answer that the persons are all famous or
that they appear in movies or TV.)
Tell them that all these persons are known by one name only
or a nickname and that the mathematician to be highlighted on the video is also
known today by a nickname, even though, he lived about eight hundred years
ago. Tell the students that his name is connected with some math pattern that is
found in many places in the natural and manufactured world. Challenge them
to find math connections with this mathematician and the pineapple as they view
the video.
FOCUS FOR VIEWING
To give the students a specific responsibility for viewing, tell
the students to use their listening and note taking skills to identify the
famous mathematician and the sequence or pattern found in nature. Remind the
students that they may add notes to their brainstorming list as they watch the
video.
VIEWING ACTIVITIES
Start the video after the credits with the scene of the
rock group. Let the video run through the rabbit scene and the explanation of
Fibonacci's name. The word "Fibonacci" will appear on the screen. Pause
right after the word "Fibonacci" is taken off the video screen. Ask
the students to tell in their own words what they learned about Fibonnaci.
(Students may tell he lived 800 years ago or that he was the son of Bonnaci.
Another response may be that he acquired his nickname from his friends calling
him son of Bonnaci.) Rewind only a small segment if you want the students
to hear the segment again to self check. Resume the video telling the
students to listen for the problem Fibonacci created. The narrator states the
rabbit problem and the word "huh" appears on the screen. Pause
shortly after the word "huh" disappears from the screen and the
narrator states the rabbit problem. Ask the student to restate Fibonacci's
problem. (Fibonacci asked, "How many rabbits would you have at the
end of each month if each pair of rabbits gave birth to another pair of rabbits
and each pair of rabbits gave birth to another pair after they become two months
old?") Since the question is complex, be prepared to replay the segment
several times. Then take the opportunity to have students make predictions of
the amount and record them on their notes. Resume the video and watch the
rabbits multiply as the Fibonacci sequence is displayed on the screen. The
teacher in the video is throwing pair after pair of rabbits on the hay. Her
comments are "And on and on and on........". Pause as she says,
"There is a patterns here." Instruct the students to use the
calculators to compute the next unknown number in the sequence.
Encourage them to write the pattern numbers already displayed on the screen and
then problem solve for the next number. Facilitate the problem solving by
walking among the groups to check for understanding and answer questions. Ask
the students to rename the different types of patterns and their operations.
(Arithmetic patterns use addition and subtraction, geometric patterns use
multiplication and division while, "other" patterns use a
combination of operations.) Resume the tape so that the answer is
displayed and the students may self check their answer. (The answer is
"21".) Pause after the answer of "21" is displayed on
the screen. Give the students time to state how the pattern works. (The previous
two numbers are added to get the next number.) Allow time to compute the next
four numbers. Ask, "What type of pattern is this?" (This
pattern is an "other". The numbers being added keep changing.) Mention
again if anyone wants to make changes on the brainstorming notes. Resume
as the teacher continues to explain that the Fibonacci pattern is found
everywhere in the real world. Pause as she says, " The Fibonacci
sequence can be found all over, even, in flowers." At this point, instruct
the students to notate the examples from nature where the Fibonacci numbers are
found along with the Fibonacci numbers. A T chart may serve this
purpose well and it is also a problem solving tool for gathering and organizing
information. (A sample is provided.)
Resume the video for recording of data about the
Fibonacci examples. The students may record the numbers up to the segment with
the pineapple. The Fibonacci numbers are illustrated on the screen with purple
highlighting of the spiral in the pineapple. Pause when she says, "Fibonacci
numbers are found in artichokes and pine cones, too." The frame shows her
holding an artichoke and pine cone. The class may spend some time checking the
list of numbers and discussing the Fibonacci numbers occurrence in nature. The
students may go back to the original brainstorming guesses and make a statement
in their own words with their new knowledge of the Fibonacci numbers in the
pineapple. (The hexagons in the pineapple make spirals of hexagons around the
pineapple. When you count the spirals they make Fibonacci numbers. The numbers
are 8,13,and 21. A key with the examples is provided.) This is a good
place to stop for the first day if time warrants it and, assign the hands on
activity with the multi link blocks to illustrate the Fibonacci sequence.
Resume the video and let the students continue to
record the numbers on the piano and then continue with the ratios of the golden
triangle. The video will show similar rectangles with their ratios. Then she
says, " I change the ratio to a decimal number that means the same
thing." Pause when she says, "I divide the top number by
the bottom number to find the decimal." Give the students enough time to
use their calculators to divide the ratios and find the decimals. (The decimals
are 8/5=1.6; 13/8=1.625; 21/13=1.615.) Tell the class that these Fibonacci ratio
decimals have a special name known as the golden rectangles. Alert them to
listen for this information in the next segment. Ask them to record the real
world examples of the Fibonacci numbers now seen as "golden
triangles". Resume the video showing all the examples of golden
triangles. The students may continue to record the real world examples of
Fibonacci numbers know as golden triangles. (The examples are included in the
attached T chart.) Stop the video after the two basketball players
illustrate the body ratios. The teacher will say, " Try it with a
friend!" Allow time for the students to check answers on the T chart and
ask questions about the Fibonacci numbers.
POSTVIEWING ACTIVITIES
(OPTION 1 on day one )
Assign the students to groups of 2 to 4 (preferably 2). The
students are to model the Fibonacci sequence numbers as the amount of
blocks allow. The colors should alternate so that the distinct numbers may be
seen. This may be the model of the Fibonacci rabbit problem and the different
colors stand for the different generations.
For example if the colors are Yellow and White the blocks
should align like this:
W=White and Y= Yellow.
Y
Y
Y
W Y
W Y
Y W Y
W Y
W Y
W Y W Y W Y
1 ,1, 2, 3, 5,
8,...............
The student will be able to build the Fibonacci pattern
with the multi links and draw a visual model that illustrates what she or he
build with the multi Links.
The student will then write a short paragraph to explain
what type of pattern the Fibonacci numbers create and use math terms to explain
the computations involved in finding the next missing number.
Close the lesson with a class discussion on why this pretend
situation is not life like. (Other factors in the environment may bring
the numbers down such as food webs, natural deaths,, diseases, etc.) The
"Rabbit Problem" worksheet or "Busy Bee" may be assigned for
homework for added practice.
(OPTION 2 to reinforce the ratios of the golden
rectangles)
Have the students work in groups of two. They are to do what the
characters in the video did. Take the measurement from the waist to the floor
and compare to the measurement of the waist to the head. The
students help each other in measuring and calculating the ratio. The answers
should be close to the golden ratio but not exact since measurement is an
approximation and human beings come in all shapes and sizes. Assign the
pair to find other golden rectangles on their bodies and in the room such as the
door,the window, the rectangular chalkboard, a book, the calculator, etc.
Students may share their findings on a class chart. Have the students then
gather measurements of four items found in the home that illustrate the golden
rectangle (examples are a bicycle , door, window, a bed, a table, etc.)
The student must show the ratio, units used, and the division problem including
the rounding of the decimals.
ACTION PLAN
The students may invite a local agriculture extension agent to
bring samples of local vegetation and demonstrate the different growth patterns.
Then the students bring in a small branch from a local tree and demonstrate the
growth pattern (s) on that particular tree. The main branch and its subsequent
branches are labeled with the Fibonacci numbers. These models may fill a
Fibonacci bulletin board.
EXTENSIONS
a) Students dress up as Leonardo de Pisa and give biographical
information about the mathematician or teach a lesson on the Fibonacci pattern.
b) A fruit and vegetable tempera print is created after the
students examine different items for occurrence of the Fibonacci numbers.
Examples: the bell pepper cut crosswise reveals three chambers; an apple cut
horizontally has a five point star cross section; daisies have 13,21 or 34
petals and giant sunflowers have 89 to 144 spirals.
c) Explore sums of the Squares of the First n
Fibonacci numbers
n
1: 1^2=1
=1X1
2: 1^2 + 1^2=2
=1X2
3: 1^2 + 1^2 + 2^2= 6 =2X3
4: 1^2 + 2^2 + 3^2=15 =3X5
5: 1^2 + 1^2 +2^2 + 3^2 + 5^2 = 40 =5X8
d) Use a calculator to help write the ratios of the Fibonacci
numbers as decimals to the nearest thousandth. Find the real world
examples of these ratios. Extend to the examples of the golden ratio (1.6).
e) Study the work of Sonya Kovalevsky (1850-1891) who was
fascinated by infinite sequences.
f) A student project may be to teach another student how to
compute arithmetic or geometric patterns on a calculator.
g) A student project may be a slide presentation of Fibonacci
numbers found in nature.
h) Weather data may be recorded, organized and illustrated on
a mathematical model to demonstrate patterns and their use in making
predictions.
T CHART
items
numbers
Updated: April 01, 2008
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