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 pairs.
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 stratgies.
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 the pattern.
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 sequence!]
Fibonacci numbers can be found in many places. Use the overhead, Fibonacci in Nature.
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 lemon.
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 golden ratio.
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.]
