9-12 > Mathematics
 Grade level: 9-12 Subject: Mathematics Duration: Two class periods
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Students will
 1 understand what the Fibonacci sequence is; and 2 understand how the Fibonacci sequence is expressed in nature.

The class will need the following:
 • Computers with Internet access (optional but very helpful) • Pencils and paper • Ruler • Compass • Copies of Classroom Activity Sheet: Finding Fibonacci Numbers in Nature • Copies of Take-Home Activity Sheet: Creating the Fibonacci Spiral • Answer Sheet for the Take-Home Activity Sheet (for teacher only)

 Work on the Classroom Activity Sheet together in class. Help students find the Fibonacci numbers in each illustration. Students may also enjoy working on the Take-Home Activity Sheet as a whole-class project.

 1 Imagine that scientists in the rain forest have discovered a new species of plant life. Where might they look for the Fibonacci sequence? 2 Suppose that you’re shooting baskets with a friend. After a few practice shots, you decide that you want to keep score. The first basket either of you makes is worth one point. Just to make things interesting, you suggest that every time either of you makes another basket, you add your previous two scores to get a new total. To make the game even more appealing, you offer to start from zero, while your friend can start from one. What sequence of numbers would emerge after shooting eight baskets? What is the difference in points between you and your friend? What pattern has emerged from the point difference? 3 Explain that numbers missing from the Fibonacci sequence can be obtained by combining numbers in the sequence, assuming that you’re allowed to use each number more than once. For example, how could the number 4 be obtained from the sequence? How about 11? 56? Think of a number not in the sequence and try to figure out what numbers to combine to get it. 4 At first glance, the natural world may appear to be a random mixture of shapes and numbers. On closer inspection, however, we can spot repeating patterns like the Fibonacci numbers. Are humans more apt to perceive some patterns than others? What makes certain patterns more appealing than others? 5 Try to solve this problem: Female honeybees have two parents, a male and a female, but male honeybees have just one parent, a female. Can you draw a family tree for a male and a female honeybee? What pattern emerges? Are they Fibonacci numbers?(The male bee has 1 parent, and the female bee has 2 parents. The male bee has 2 grandparents, and the female bee has 3 grandparents. The male bee has 3 great-grandparents, and the female bee has 5 great-grandparents. The male bee has 5 great-great-grandparents, and the female bee has 8 great-great-grandparents. The male bee has 8 great-great-great-grandparents, and the female bee has 13 great-great-great-grandparents.)

 You can evaluate students using the following three-point rubric: Three points:active participation in classroom discussions; ability to work cooperatively to complete the Classroom Activity Sheet; ability to solve all the problems on the sheet Two points:some degree of participation in classroom discussions; ability to work somewhat cooperatively to complete the Classroom Activity Sheet; ability to solve three out of five problems on the sheet One point:small amount of participation in classroom discussions; attempt to work cooperatively to complete the Classroom Activity Sheet; ability to solve one problem on the sheet

 Finding Ratios Suggest that students measure the length and width of the following rectangles: a 3” × 5” index card an 8.5” × 11” piece of paper a 2” × 3” school photo a familiar rectangle of their choice Have students find the ratio of length to width for each of the rectangles. Then have them take the average of all the ratios. What number do they get?(1.61803). Tell students that this ratio is called thegolden ratioand that it occurs in many pleasing shapes, such as pentagons, crosses, and isosceles triangles, and is often used in art and architecture. An Algebraic Rule Encourage students to try to develop an algebraic formula that expresses the Fibonacci sequence. The formula is described below.Represent the first and second terms in the sequence withxandy. Then the first few terms would be expressed as follows: First term =x Second term =y Third term = (x+y)Fourth term = (x+y)+y= 1x+ 2y Fifth term = (x+2y) + (x+y) = 2x+ 3y Sixth term = (2x+3y) + (x+2y) = 3x+ 5y Seventh term = 3x+5y+ 2x+3y= 5x+ 8y Ask the students whether they notice anything familiar about the coefficients.(They’re numbers in the Fibonacci sequence.)

 Life By the Numbers Keith Devlin. John Wiley & Sons, 1998.Written as a companion volume to the PBS series of the same name, this book focuses on the role mathematics plays in everyday life. Each chapter examines a different aspect of the world we live in and how mathematics is involved: patterns appearing in nature, the curve of a baseball, the chance of winning in Las Vegas, the technology of the future. Lots of pictures round out this clear and exciting presentation. Designing Tessellations: The Secrets of Interlocking Patterns Jinny Beyer. Contemporary Books, 1999.For generations, people have created designs using repeating, interlocking patterns—tessellations. In this slightly oversized, beautifully illustrated book, the author shows how the combination of pattern and symmetry can result in stunning geometric designs. While this unique book uses quilt making as the focus of the design process, it could easily be applied to other arts as well.

 algorithm Definition:A step-by-step procedure for solving a problem. Context:Thealgorithmfor obtaining the numbers in the Fibonacci sequence is to add the previous two terms together to get the next term in the sequence. logarithmic spiral Definition:A shape that winds around a center and recedes from the center point with exponential growth. Context:The nautilus shell is an example of alogarithmicspiral. sequence Definition:A set of elements ordered in a certain way. Context:The terms of the Fibonaccisequencebecome progressively larger. term Definition:An element in a series or sequence. Context:The mathematician Jacques Binet discovered that he could obtain each of thetermsin the Fibonacci sequence by inserting consecutive numbers into a formula.

 This lesson plan may be used to address the academic standards listed below. These standards are drawn from Content Knowledge: A Compendium of Standards and Benchmarks for K-12 Education: 2nd Edition and have been provided courtesy of theMid-continent Research for Education and Learningin Aurora, Colorado.  Grade level:9-12 Subject area:Mathematics Standard: Understands and applies basic and advanced properties of the concepts of numbers. Benchmarks: Uses discrete structures (e.g., finite graphs, matrices, or sequences) to represent and to solve problems. Grade level:9-12 Subject area:Mathematics Standard: Uses basic and advanced procedures while performing the processes of computation. Benchmarks: Uses recurrence relations (i.e., formulas that express each term as a function of one or more of the previous terms, such as the Fibonacci sequence and the compound interest equation) to model and to solve real-world problems (e.g., home mortgages or annuities). Grade level:9-12 Subject area:Mathematics Standard: Uses basic and advanced procedures while performing the processes of computation. Benchmarks: Uses a variety of operations (e.g., finding a reciprocal, raising to a power, taking a root, and taking a logarithm) on expressions containing real numbers.

 Chuck Crabtree, freelance curriculum writer.