Illuminations: Graphs from the Unit Circle

# Graphs from the Unit Circle

 In this lesson, students use uncooked spaghetti to transfer lengths from the unit circle to a function graph on large butcher paper. In the process, they discover the key features of sine and cosine graphs. The activity is presented for students working in degrees, but another version of the handouts is provided for students working in radians.

### Learning Objectives

 Students will: Explore the properties of the unit circle Discover the graphs of sine and cosine curves Discover the domain, range, maxima, minima, and intercepts of sine and cosine graphs Compare the properties of sine and cosine curves

### Materials

 Butcher or art paper (about 8 ft per group) Uncooked spaghetti Masking tape Protractors Meter sticks Colored markers or pencils Twine, rope, or yarn (about 7 ft per group) Compass for chalk board or white board (optional) Pre-Assessment Overhead Questions from the Unit Circle Answer Key If working in degrees: If working in radians:

### Questions for Students

 What happens to the sine value on the unit circle at 180°? How will you show that on the function graph? [The circle intersects the x-axis, so the sine is 0. Therefore, it should be a zero of the function or an x-intercept of the graph.] What happens to the cosine value on the unit circle at 180°? How will you show that on the function graph? [Cosine is the x-value in the unit circle, so that would be –1. At 180° on the function graph, the y-coordinate should be –1, so this is a minimum point on the function graph.] What happens to the cosine value on the unit circle at 90°? How will you show that on the function graph? [The x-value on the unit circle for a 90° angle in standard position is 0, so the cosine is 0 at 90°. It should be a 0 of the function or an x-intercept of the graph.]

### Assessment Options

 Use the check points to facilitate ongoing assessment. Later, determine some number of points for each section in order to assign a grade for the assignment. Show students graphs created by other students or graphs you've created with mistakes and ask them to correct them. For example, flip the sine graph vertically across the x-axis and ask students how they know it is wrong. Assign a journal-writing task in which students explain the connection between the unit circle graph and the function graph.

### Extensions

 Similar activities can be done for tangent, cotangent, secant, and cosecant. You can introduce the other trigonometric functions using the same activity, or have students create the graphs on their own on a smaller scale.

### Teacher Reflection

 Was students’ level of enthusiasm/involvement high or low? Explain why. Did you challenge the achievers to go beyond quick answers and pursue deeper explanations and understanding? How? Did all members of each group contribute to the project and remain fully engaged? How do you know? How could you improve that next time? Did you provide appropriate support for students who struggle with graphing by hand? How? How did students demonstrate understanding of the relationship between the unit circle and the function graph? Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were the effective?

### NCTM Standards and Expectations

 Algebra 9-12Use symbolic algebra to represent and explain mathematical relationships. Understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions. Use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts.

### References

 Peterson, Blake E., Patrick Averbeck, and Lynanna Baker. 1998. Sine curves and spaghetti. Mathematics Teacher 91: 564–67.
 This lesson was prepared by Margie Coleman as part of the Illuminations Summer Institute.

3 periods

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