




912 > Mathematics 










Grade level: 912 

Subject: Mathematics 

Duration: Two class periods 
















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 TakeHome Activity Sheet: Creating the Fibonacci Spiral 
• 
Answer Sheet for the TakeHome Activity Sheet (for teacher only) 










1.

Begin the lesson by discussing the Fibonacci sequence, which was first observed by the Italian mathematician Leonardo Fibonacci in 1202. He was investigating how fast rabbits could breed under ideal circumstances. In developing the problem, he made the following assumptions:
 Begin with one male and one female rabbit. Rabbits can mate at the age of one month, so by the end of the second month, each female can produce another pair of rabbits.
 The rabbits never die.
 The female produces one male and one female every month.
Fibonacci asked how manypairsof rabbits would be produced in one year. 
2.

Work with the class to see whether students can develop the sequence themselves. Remind them that they’re counting pairs of rabbits, not individual rabbits. You may want to walk them through the first few months of the problem:
 You begin with one pair of rabbits (1).
 At the end of the first month, there is still only one pair (1).
 At the end of the second month, the female has produced a second pair, so there are 2 pairs (2).
 At the end of the third month, the original female has produced another pair, so now there are 3 pairs (3).
 At the end of the fourth month, the original female has produced yet another pair, and the female born two months earlier has produced her first pair, making a total of 5 pairs (5).

3.

Write the pattern that has emerged in step 2 on the board: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233. Discuss what “rule” is being followed to get from one number to the next. Help students understand that to get the next number in the sequence, you have to add the previous two numbers. Explain that this sequence is known as theFibonacci sequence. The term that mathematicians use for the type of rule followed to obtain the numbers in the sequence isalgorithm.As a class, continue the sequence for the next few numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987… 
4.

Tell students that the Fibonacci sequence has intrigued mathematicians for centuries. What’s more, mathematicians have noticed that these numbers appear in many different patterns in nature, often creating the beauty we admire. Tell students that they are going to look for Fibonacci numbers in pictures of objects from nature. Make sure that students understand that they are looking for specific numbers that appear in the sequence, not for the entire sequence. 
5.

Divide students into groups of three or four. Distribute the Classroom Activity Sheet: Finding Fibonacci Numbers in Nature. Tell students to work together to try to answer the questions on the sheet. Make sure that each student fills out his or her own sheet. For your information, the questions and explanations are listed below. It may be helpful to work on the first example as a class so students understand what they are looking for.

Flower petals. Count the number of petals on each of the flowers. What numbers do you get? Are these Fibonacci numbers?(Lilies and irises have 3 petals, buttercups have 5 petals, and asters and blackeyed Susans have 21 petals; all are Fibonacci numbers.)

Seed heads. Each circle on the enlarged illustration represents a seed head. Look closely at the illustration. Do you see how the circles form spirals? Start from the center, which is marked in black. Find a spiral going toward the right. How many seed heads can you count in that spiral? Now find a spiral going toward the left. How many seed heads can you count there? Are they Fibonacci numbers?(The numbers of seed heads vary, but they are all Fibonacci numbers. For example, the spirals at the far edge of the picture going in both directions contain 34 seed heads.)

Cauliflower florets. Locate the center of the head of cauliflower. Count the number of florets that make up a spiral going toward the right. Then count the number of florets that make up a spiral going toward the left. Are the numbers of florets that make up each spiral Fibonacci numbers?(The numbers of florets will vary, but they
should all be Fibonacci numbers.)

Pinecone. Look carefully at the picture of a pinecone. Do you see how the seed cases make spiral shapes? Find as many spirals as possible going in each direction. How many seed cases make up each spiral? Are they all Fibonacci numbers?(The numbers will vary, but they
should all be Fibonacci numbers.)

Apple. How many points do you see on the “star”? Is this a Fibonacci number?(There are five points on the star.)

6.

Ask the class which shape emerges the most often from clusters of seeds(the spiral).Discuss whether there are any advantages to this shape. Explain that seeds may form spirals because this is an efficient way of packing the maximum number of seeds into a small area. 
7.

Ask students where else they see the spiral shape in nature(nautilus shell). Would they guess that those spirals are also formed from Fibonacci numbers? Do they find this shape pleasing to the eye? To conclude, discuss other pleasing shapes and patterns in nature, such as those of waves, leaves, and tornadoes. Discuss whether these, too, may have a mathematical basis. 
8.

Assign the TakeHome Activity Sheet: Creating the Fibonacci Spiral. Have students share their drawings in class. Discuss how rectangles with Fibonacci dimensions are used in art and architecture. You could use the examples of painter Piet Mondrian, who used three and fiveunit squares in his art; the Egyptians, who used Fibonacci dimensions in the Gaza Great Pyramid; and the Greeks, who used these dimensions in the Parthenon. Then brainstorm some animals with spiral features.(The shape is similar to the spiral of the snail, the nautilus, and other seashells).What function does the spiral serve?(Some scientists think it protects the animal inside the shell.)







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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 TakeHome Activity Sheet as a wholeclass project.






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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 greatgrandparents, and the female bee has 5 greatgrandparents. The male bee has 5 greatgreatgrandparents, and the female bee has 8 greatgreatgrandparents. The male bee has 8 greatgreatgreatgrandparents, and the female bee has 13 greatgreatgreatgrandparents.)







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You can evaluate students using the following threepoint 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






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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.)






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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.






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algorithm
Definition:A stepbystep 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.






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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 K12 Education: 2nd Edition and have been provided courtesy of theMidcontinent Research for Education and Learningin Aurora, Colorado.
Grade level:912
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:912
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 realworld problems (e.g., home mortgages or annuities).
Grade level:912
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.






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Chuck Crabtree, freelance curriculum writer. 





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