Illuminations: Golden Ratio

# Golden Ratio

 Students explore the Fibonacci sequence, examine how the ratio of two consecutive Fibonnacci numbers creates the Golden Ratio, and identify real-life examples of the Golden Ratio.

### Learning Objectives

 Students will: Discover the Fibonacci Sequence with the rabbit problem Use a spreadsheet to examine the Fibonacci Sequence Graph the Fibonacci numbers in a scatterplot Examine similar rectangles Recognize the Golden Ratio in nature, architecture, and art

### Materials

 Fibonacci Rabbits Activity Sheet Fibonacci in Nature Overhead Computer Spreadsheet Program, such as Microsoft Excel® Dynamic Geometry Software, such as Geometers Sketchpad® Measuring tape or rulers Calculators Apple, lemon (optional)

### Questions for Students

 How do we create the Fibonacci Sequence? [Start with 1, then 1, and then add the two previous numbers to get the next number in the sequence.] How is a spreadsheet used to create the Fibonacci Sequence? [The program extends the pattern using the fill down feature.] What happens when you divide one Fibonacci Number by the previous or next Fibonacci Number? [You get the Golden Ratio or its reciprocal.] How are golden rectangles related? [They share the same ratio of length and width.] Why was the Golden Ratio used in buildings? [It was the most visually appealing to the eye.]

### Assessment Options

 As students are creating their spreadsheets and scatterplots, the teacher should be assessing student work by circulating throughout the classroom. As students identify examples of the Golden Ratio in their world, the teacher should listen to student responses to get a sense of their understanding of what a Golden Ratio is. If so desired, students could write their responses to the Questions for Students (above) and the teacher could collect the written responses as another form of assessment.

### Extensions

 Examine the Golden Spiral. Each stage will show the square and the rectangles referred to in the Sketchpad activtiy. On a sheet of graph paper, have students draw a square in the center with side length 1, the first Fibonacci Number. Next to it, draw another with side length 1, the second Fibonacci number. Above these two, draw a square with side length 2. Notice you can now draw a square with side length 3 next to it, each time, adding a square with side length the next Fibonacci number. Keep this going until you run out of paper. The rectangles themselves are intriguing, as they also show the sums of the previous two numbers. But if you draw in the diagonal in the first square, and then keep it going into the next and next again, you will get the spiral, known as the golden spiral, that resembles a nautilus shell! Students can view the following spiral demonstration. Students can find examples of spirals in pine cones, sunflowers, and pineapples.

### Teacher Reflection

 As students were creating their spreadsheet file, did they demonstrate an understanding of why the file was being set up a certain way (e.g. how the formulas worked)? Did you have to adjust your teaching to increase student understanding of this application? Did the problems posed in this lesson (e.g. the Rabbits problem, finding examples of the Golden Ratio in the world around us) provide sufficient motivation for the students, or did you have to adjust the lesson to engage all students? How did this lesson address auditory, tactile, and visual learning styles?

### NCTM Standards and Expectations

 Algebra 6-8Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules. Model and solve contextualized problems using various representations, such as graphs, tables, and equations.

### References

 Naylor, Rhonda. "NTTI Lesson: I Am Golden." http://www.thirteen.org/edonline/nttidb/lessons/dn/golddn.html (accessed May 8, 2006).
 This lesson prepared by Rhonda Naylor.

1 period

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