Experience First:

Today we have another hands-on lesson where students create the graphs of sine and cosine using the unit circle and uncooked spaghetti. In addition to the spaghetti, students will need either glue or tape to secure the spaghetti lengths onto their graphs.

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In this activity, students will learn that the cosine and sine values found on the unit circle can be plotted as outputs of a function where the input is the angle. They will break off spaghetti according to the appropriate lengths on the given unit circle and these spaghetti pieces will represent the y-values on the graph.

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During the activity portion, students think about low points and high points and how long it takes for the graph to start repeating before being introduced to the formal vocabulary of amplitude, period, and range in the debrief.

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Monitoring Questions:

• What does this spaghetti length represent?

• How could we find outputs of the sine function for angles that aren’t marked on the unit circle?

• Why does the sine function have negative values on the interval [π, 2π]?

• When is the sine function increasing? How can you tell from the graph? How can you tell from the unit circle?

• What is the y-intercept of y=cos(theta)? Why?

• Why do you think the sine and cosine function graphs look so similar?

• Why is the range of cosine [-1,1]?

• What do you notice about the symmetry of the graph of y=sin(theta)?

• What do you notice about the symmetry of the graph of y=cos(theta)?

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Formalize Later:

The trickiest cognitive challenge in this lesson is the idea that the input and x-axis variable is the angle and the output or y-axis variable is the cosine or sine ratio. Students that rely heavily on the “x is cosine”, “y is sine” shortcut may struggle to graph the cosine as an output. For this reason we have emphasized in all previous lessons that sine and cosine represent ratios of sides, and that ratio is easiest to see when the hypotenuse is 1, in which case the legs of the triangles themselves represent the sine and cosine values. Although it might seem tedious, be precise in your language around sine and cosine. Clarify that on the unit circle the x-coordinate represents the cosine and the y-coordinate represents the sine, and when the hypotenuse is 1, the adjacent side represents the cosine and the opposite side represents the sine.

 

Today’s lesson is all about the parent functions sine and cosine. Students should be able to explain why the domain, range, period, and amplitude is what it is based on the unit circle. The Check Your Understanding questions help students further classify the cosine function as an even function and the sine function as an odd function.

 

Best of all, the cyclical nature of a circle is seen in the periodic behavior of the sine and cosine graphs!

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