I AM GOLDEN
Grades 6 - 8
This lesson will familiarize students with patterns, the Fibonacci
Sequence, and the Golden Ratio, and will show students the many places these
occur. Students will have practical applications for using the calculator
and making charts to extend patterns.
"Math Vantage 103: Patterns and Sequences"
Students will be able to:
- continue a pattern
- write the Fibonacci sequence
- list occurrences in nature of the Fibonacci sequence
- compute the Golden Ratio
- explore similar rectangles
Per group of 4:
- film for one Polaroid picture
- tape measures
- yard or meter sticks
- trundle wheels (if convenient)
- paper and pencil
- 3x5 index card
- 5x8 index card
- worksheet with similar rectangles
- worksheet on Fuller
- Foam board
- rabbit toothpicks
- Polaroid Camera
- Poster of Fibonacci
- model or real pineapple
- columbine (the Colorado state flower)
- transparency of the generations of bees,
- credit cards like Value card, frequent miles, hotel chains rather
than Visa or other charge cards
- Computer program "Geometer's Sketchpad" and spreadsheet
(may be shown with transparencies instead!)
- golden rectangle
Show a model of triangles made out of spheres: "1, 3, 6,
10, ... What comes next? " Help students to continue the pattern, and
write it on their papers. "These are called triangular numbers. Where
do we see these numbers used in games: 10 in bowling pins, 15 in the game
of pool or billiards. What happens if I put each triangle on top of the
previous one? These are called pyramidal numbers: 1,4,10, 20. What comes
next? " Again have students write this on their papers.
Write down the first few rows of Pascal's Triangle. Have the class look
for the pattern, and continue it for a few more rows. Show many of the patterns:
triangular numbers, pyramidal numbers, sum of the numbers in a row powers
of 2, reading numbers in a row powers of 11, coefficients of the products
of binomials, permutations and combinations, etc. "Today we are going
to study patterns. Write down everything you know about Fibonacci."
To give students a specific responsibility while viewing, ask
students to: "Listen for the problem Fibonacci, an Italian mathematician
who lived 800 years ago, posed about the rabbit population."
BEGIN tape PAUSE tape after Ellen Wicket poses
Fibonacci's problem but before she says, "I'll show you": Ask
the students if they can repeat the problem for the class. "How many
pairs of rabbits will there be at the beginning of each month if each month
each pair produces 1 new pair and begin to bear young 2 months after birth.
" Using a foam board, and toothpick rabbits, model the problem. Students
should record each month on their paper. Show 1 pair rabbits Jan 1, 2 pair
Feb. 1, as the pair reproduced, March 1 the original pair reproduced, but
the new pair wasn't old enough so there are 3 pair of rabbits; April 1 two
pair are reproducing, plus the one pair that isn't old enough, making 5
pair of rabbits, etc.
Focus for viewing
To give students a specific responsibility while viewing, ask students
to: "See how Ellen models this problem. Check our work to see if you
agree with her explanation."
RESUME tape. If the class had trouble with this, rewind to Ellen
posing the problem again, through her use of stuffed animals. Check for
understanding in the students' charts and drawings. Clear up misconceptions
before moving on.
PAUSE tape when Ellen says, "There's a pattern here. Do you
see it?" Ask your own students to explain the pattern if they see it,
and to continue the pattern. Refer to the poster of the Fibonacci sequence
in the classroom.
Ask students if they have ever noticed the Fibonacci sequence in music.
"How many notes is an octave?" 8, a Fibonacci number. Continue
to ask questions about the keyboard of a piano-- 8 white keys, 2 black and
3 black = 5, all Fibonacci numbers. When you play the first, third and fifth,
you hear harmony.
Focus for viewing:
To give students a specific responsibility while viewing, ask students
to: "Listen for more applications of the Fibonacci sequence in music."
Resume tape until you hear Ellen discuss Fibonacci in nature. Focus for
viewing: To give students a specific responsibility while viewing, ask students
to: "Remember the family tree of the bee family? Where else in nature
do we see the Fibonacci sequence? Make a list." (orchid, daisy, pineapple,
pine cones, artichoke).
RESUME tape. PAUSE after the music to measure some of
the rectangles in the room: switch plate, index cards, credit cards. If
"Geometer's Sketchpad" is handy in the classroom, try to construct
some golden rectangles. If not, visit the computer room later to do so.
RESUME tape until Ellen uses a calculator to divide the fractions.
PAUSE to try this yourself: If a computer with PC viewer is readily
available, program a spreadsheet to display the Fibonacci sequence. Then
divide to find the golden ratio.
Fibonacci number Ratio
Fibonacci number Ratio
RESUME tape until Ellen says, "the Golden Rectangle."
PAUSE tape. To give students a specific responsibility while viewing,
say: "Watch to list all the items in the Golden Ratio ." (football,
credit card, finger).
PLAY tape to end.
Use a transparency of the generations of bees: a male bee develops from
an unfertilized egg, but a female bee develops from a fertilized egg. Therefore,
a male bee has only a mother, whereas a female bee has both a mother and
a father. The first generation has Mr. and Mrs. Bee, 2 members. The generation
above has 3 parents, one for the male, and two for the female. The generation
above has 5 members, two for the mother of Mr. Bee, one for the father of
Mrs. Bee, and two for the mother of Mrs. Bee. There are 5 grandparents in
this generation. As we continue, there are 8 great grandparents, and 13
great-great grandparents. The Fibonacci sequence in science!
Each student should measure from their waist to the floor, and from their
waist to the top of their head. This should be in the golden ratio, just
as with Ellen and the basketball player. This is Le Corbusier's Study of
Human proportion: Measure from the neck to the top of the head, and the
neck to the waist. Also, from the knee to the floor, and from the waist
to the knee. This is called the Golden Proportion.
Discuss similar rectangles. Provide students with several drawings. Measure
each side. Which have the same proportions? An easy way to check is to draw
in the diagonals. If the rectangles are similar, the diagonals will match
List the first 20 Fibonacci numbers, and then find their prime
factorizations. No consecutive Fibonacci numbers have any common factors.
The 12th Fibonacci number is 144, twelve squared. A mathematician at the
University of London in 1963 proved that 144 is the only square number in
the entire infinite sequence except 1. The only cubic number is 8!
Add the sum of the first ten Fibonacci numbers. Add the sum of the second
ten numbers. The sum of any ten consecutive Fibonacci numbers is always
a multiple of 11. Teach how to add every other digit to see if divisible
by 11. Check with calculator.
Every 3rd Fibonacci number is divisible by 2, every 4th is divisible by
3, every 5th is divisible by 5, every 6th is divisible by 8, etc.
Bring in your favorite board game. Measure the size of the cards, and the
size of the rectangle spaces (like properties in Monopoly). Which games
use the golden ratio? Are they pleasing to the eye?
Accurately draw the hands of a clock at 10:10. With a ruler, draw a rectangle
around these hands. Measure the ratio of the sides.
Look again at Pascal's triangle, and try to see the Fibonacci sequence going
up diagonally as you add.
Visit the computer room to use the "Geometer's Sketchpad." Instruct
class to construct any rectangle. Highlight the large side and smaller side,
in that order, and then pull down the measure menu. Click on ratio. If the
ratio of the large side to the smaller is not exactly 1.618, click on one
side and move it until the ratio is 1.618. You have constructed a golden
rectangle! Now make another one, either around it, or inside it. Once again
find the ratio, until it is the golden rectangle. "Do you find this
shape rectangle pleasing to the eye like the ancient Greeks did?"
Invite an artist to come to class. Ask them to explain how they
use the golden mean and the golden spiral to draw the audiences eye to the
most important part of their work.
Contact the art teachers to find out their use of the Golden Ratio.
Take rulers, tape measures, trundle wheels outside of the classroom. Students
measure, measure and measure and make a list and sketch of any items they
find that are in the golden ratio.
The Polaroid camera is used by groups of 4 to photograph items found in
the golden ratio, and the pictures are immediately brought back to class.
Present these pictures to another class to explain the golden ratio.
Visit the Museum of Natural History, including the class on plant classification.
Examine flowers with Fibonacci numbers of petals as ways to classify the
Invite a science teacher to class to teach about plant classification by
number of petals (which are Fibonacci numbers?).
Read aloud The Math Curse, a book where the teacher is Mrs. Fibonacci.
Find out why artists use the golden ratio (visual harmony). Find examples,
such as Edgar Degas in his painting "Jockeys Before the Race"
use the golden ratio to draw your attention to the pole, which is the right
vertical line in the golden spiral.
A violin is constructed in the golden ratio Familiar songs have Fibonacci
number of notes: Mary had a little lamb, 5 notes; Row, Row, Row Your Boat,
8 notes; and most music today, 13 notes.
1600 BC, Rhind Papyrus refers to a "sacred ratio" used to build
the pyramids of Egypt 1000 years earlier! Golden ratio in ancient Greek
temples like the Parthenon, Grecian urns
Write a formula to generate the Fibonacci sequence on a spreadsheet. Write
a formula to find the ratio of one Fibonacci number divided by the previous.
Use Geometer's Sketchpad to explore golden rectangles.
The Fibonacci numbers are in Pascal's triangle, but are much easier
to spot in the Chinese triangle, where the ones are lined up down the left
side of the triangle.
Dichotomous Key Population Problems: Thomas Fuller was a mathematical
genius who was stolen from Africa at the age of 14, about the year 1732.
He calculated mathematics problems mentally. Fuller problems are attached,
with extensions to Fibonacci.
In physics, there are many figurate numbers, such as the sum of consecutive
odd numbers gives the distance traveled by a free falling body on a given
time. Da Vinci and Galileo graphed these number patterns.
Barnard, Jane. "Those Fascinating Fibonaccis!" NCTM Student Math
Notes. Reston, VA:NCTM, January 1996.
Benson, John et.al. Gateways to Algebra and Geometry. Evanston, Illinois:
McDougal, Littell, 1993.
Garland, Trudi Hammel. Fascinating Fibonaccis: Mystery and Magic in Numbers.
Palo Alto, California: Dale Seymour, 1987
Lumpkin, Beatrice and Dorothy Strong. Multicultural Science and Math Connections.
Portland, Maine: J. Weston Walch, 1995.
Math Vantage Project. Patterns Unit. Lincoln: Nebraska Mathematics and Science
Preble, Duane and Sarah. Art forms. New York: Harper Collins College, 1994.
Scieszka, Jon and Lane Smith. Math Curse. New York: Viking, 1995.
Master Teacher: Rhonda Wanger
Lesson Plan Database
Thirteen Ed Online