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.

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.

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 and period (and range) in the debrief.

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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.

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