GOLDEN

Fibonacci Hunt (Review)

Fibonacci numbers show up often in nature. The Fibonacci sequence is named for Leonardo Fibonacci, an Italian mathematician who lived from 1170-1250. The sequence begins 1, 1, 2, 3, 5, 8, 13, 21… with each term being the sum of the previous two terms (8 = 3+5, 13 = 5+8, 21 = 8+13, etc.).

Give students the first eight terms of the sequence and challenge them to extend the pattern. Ask them where these numbers show up often in ( plants, pinecones, sunflowers, artichokes, celery, lilies and daisies).

Bring in a pineapple and demonstrate the Fibonacci numbers. Looking at the bottom of the fruit, you can easily see spirals in the patterns. A pineapple makes for an excellent demonstration – the pattern of polygons on the husk forms several spirals that can be followed in different directions. Count the number of spirals in different directions (masking tape helps), and you'll find they are Fibonacci numbers. You'll probably find 13 spirals in one direction, eight in another and five in a third direction.

Golden Ratio Sequence

Demonstrate how students can find the ratios between consecutive Fibonacci numbers:
1/1 = 12/1 = 23/2 = 1.55/3 = 1.666...8/5 = 1.613/8 = 1.615384...... Etc.

Point out that the ratios are going up and down but getting closer and closer to some value. That value is about 1.61803…an irrational number that is called "the golden ratio." The number has fascinated mathematicians and artists for millennia.

Now have students create their own "Fibonacci-like" pattern by choosing two starting numbers and applying the same rule to build a sequence. For example, if one chooses 3 and 7 as the starting numbers, the sequence is 3, 7, 10, 17, 27, 44, 71, 115, etc.

Ask your students to find the ratios between the terms of their own sequence. They will be surprised to find that their sequence, and everyone else's in the room, also converges to the golden ratio! Wow!

Golden Rectangle

There are many ways to construct golden rectangles; here's a fun way that uses Fibonacci numbers. Supply students with graph paper. Have them outline a 1 x 1 square near the center of the paper. Add a 1 x 1 square adjacent to the first, then a 2 x 2 square so that it touches both of the 1 x 1 squares. Continue to add squares, circling around the outside of the figure. In each square, students write in the side length – the lengths form the Fibonacci sequence.

Golden Spiral

When the page is filled, students can draw a quarter circles in each square in order to form an approximate "golden spiral" as shown above. This spiral is related to the shape of ram's horns, sea- shells and other spirals found in nature.

This is just a small taste of the wonders of mathematics in nature.

Art Project and Demonstration:

Introduce a book about Patterns and Relationships:

Mathematics is the science of patterns and relationships. Part of the sense of beauty that many people have perceived in mathematics lies not in finding the greatest elaborateness or complexity but in finding the greatest simplicity of representation and proof.

Activity: Homemade spirals.

The photo above shows washers laid out in a string, starting at a center. Each washer touches the previous one, and each wrap around the center just touches the previous wrap. No pattern is obvious at first, but after a number of wraps, a pattern of additional spirals emerges.

The sunflower seed head is an example of botanist William Hofmeister's 1868 observation that primordia form preferentially where the most space is available for them. They also must form where they attach efficiently to the rest of the plant, and this is a geometric consideration. The pattern can also be modified by moisture and nutrient conditions that affect the size of forming seeds.

You don't need biology to produce spirals similar to those found in sunflower seeds. Here hardware store washers have been laid out. After about six or seven wraps additional spiral patterns develop, just as in the sunflower, spiraling steeply outward, curving away from the center. The reason is simple. The growth pattern of the seed head (and our constructed spiral) is such that it is biased to povide reasonably close packing of the seeds (or washers) consistent with the growth processes.

Mathematical thinking often begins with the process of abstraction—that is, noticing a similarity between two or more objects or events.