Begin class with the Fibonacci Rabbits Activity Sheet.
The first page shows the problem posed by Fibonacci 800 years
ago. The second page shows one strategy for solving the problem. You
may wish to distribute the first page only.
A pair of rabbits cannot bear young until they are two
months old. But once a pair reaches maturity, they will produce one new
pair of rabbits each month.
If you start with one pair of new‑born rabbits, how many pairs of
rabbits will you have at the beginning of each month thereafter?
Use the second page of the activity sheet with rabbit pictures,
photocopied as a handout for each student to track their work. In the
first month, you have 1 pair. In the second month, there is still only
1 pair since they aren't old enough to reproduce. In the third month,
the first pair reproduces, so there will be 2 pairs. In the fourth
month, only the first pair is old enough to reproduce, so there will be
3 pairs. By the fifth month, the original and the next pair are able to
reproduce, so there will now be 5 pairs. By the sixth month, one more
additional pair is old enough to reproduce, so we will be adding 3 more
Students should take about 5-10 minutes to individually think
about ways they could solve this problem and record those strategies on
the first page of the activity sheet. Then, they can meet with a
partner to discuss these strategies.
This leads to the Fibonacci Sequence of 1, 1, 2, 3, 5, 8, 13,
21, 34, 55.... The students should keep the chart up to date with these
solutions. Use the second page of the activity sheet.
A physical model of the pairs will also work, keeping track of which pairs are old enough to reproduce.
Have students verbalize the pattern (add previous two numbers to
create the next number). Once students see that the pattern is adding
the previous number to each number—for example, the number after 5
is 8, because 3 + 5 = 8—have them predict the next several numbers in
Next, students should open a new spreadsheet file, using Microsoft Excel® or another program.
Label the first column Item Number; then, in A2, enter "1."
In A3, enter "=A2+1." Select the cell that contains the formula, and
fill down (using the edit pull‑down menu). Title the next column
Fibonacci Number. In B1, enter "1." In B2, write "1." In B3, enter the
formula "=B1+B2." The computer will add the numbers from B1 and B2.
Again, select the cell with the formula, and fill down (using the edit
pull‑down menu). Students will enjoy seeing how many Fibonacci numbers
can be generated just by entering just one formula! (Note: The width of the column will determine how many digits will be displayed before showing scientific notation.)
Have students make a scatterplot of the data with the
spreadsheet. Then, have them describe the shape. [At first it increases
very slowly; then, it increases very quickly.] Have students determine
if this is linear or non-linear. [Non-linear. There is no constant rate
of change. The rate of change, or differences between terms, is in the
Fibonacci numbers can be found in many places. Use the Fibonacci in Nature Overhead.
Fibonacci in Nature Overhead
In nature, many plants are in the Fibonacci sequence. The
Colorado state flower, the Columbine, has 5 petals. The black‑eyed
Susan has 13. As a demonstration, you can also cut open an apple and
count the number of seeds, or you may count the sections in a sliced
Measure the distance from the floor to one's waist (navel),
and then measure from the navel to the top of the head. What is the
ratio of these measurements? Similarly, find the ratio of the distance
from the neck to the top of the head to the distance from the neck to
the navel; then, find the ratio of the distance from the knee to the
floor to the distance from the navel to the knee. How are these ratios
related? In an adult, these ratios are approximately equal to the
Golden Ratio (as discussed below). For students in varying stages of
growth, they may not hold true. Students can work in pairs to help
measure each other and record the measurements on a piece of paper.
Next, have students return to the spreadsheet, and find the
ratio of one Fibonacci number to the previous Fibonacci number. That
is, have students divide one Fibonacci number by the Fibonacci number
in the cell below. In C2, have students enter the formula "=B2/B1."
Once again, have students fill down, and notice that all values
converge to 1.618! This is called the Golden Ratio. Why is it special?
Next, have the students, in D2, enter the formula "=B1/B2," and fill
down, to find the ratio of a Fibonacci number divided by the next
Fibonacci number. This will give the reciprocal of the previous ratio.
Yet, it converges on 0.618! The Golden Ratio is the only number that
differs from its reciprocal by 1.
Rectangles built in the Golden Ratio are said to be pleasing
to the eye. Have students measure the length of a switch plate cover
and divide it by the width. Often, the ratio is 1.618 : 1. Do the same
with a credit card (or student ID card). The ratio will likely
be 1.618 : 1 each time. In ancient times, many famous buildings—such as
the Parthenon—were built to these dimensions. As time permits, or as an
extension, students can research other objects which appear in the
Using Geometer's Sketchpad, or a similar geometry tool, have
students draw rectangles that are in the Golden Ratio, or draw
rectangles on graph paper with sides of sequential Fibonacci numbers.
What is the ratio of the sides? Examine all of the rectangles. How are
they all related? [They are all similar, meaning that the sides are in
proportion; they are enlargements of one other. Notice that the ratio
of the side lengths of the two rectangles below are 34/21 ≈ 1.619048,
and 55/34 ≈ 1.617647. As the Fibonacci numbers get larger, the ratio
gets closer and closer to the Golden Ratio.]
Have students explore where the golden rectangle is used. Artists
often divide their canvas into a rectangle and a square. This is called
the Golden Rectangle. Find objects and paintings that contain the
Golden Rectangle. Compare them with objects that are in other ratios.
How are they different? Which is more pleasing to the eye? [If a canvas
is divided into the Golden Rectangle, the eye is drawn to the line at
the right, and it often is the focal point of a painting.]
Naylor, Rhonda. "NTTI Lesson: I Am Golden." http://www.thirteen.org/edonline/nttidb/lessons/dn/golddn.html (accessed May 8, 2006).
In the Classroom