Experience First

In this final lesson of Unit 4, students are asked to apply their understanding of trigonometric functions to a real-world context. Students see two examples of scenarios that can be modeled with periodic behavior and ultimately we want students to understand that whenever the situation is cyclical or repeated, it is likely that sine and cosine will be used to model the situation.

Students begin my evaluating the function at three different times to make sense of the inputs and outputs. In question 2 they think about the max temperature. Note that they have not yet graphed the function, so we’re asking them to reason about where the maximum would occur based on the symmetry of the curve and the period (though informal reasoning is welcome here!) In question 3 they are asked to interpret the model. Though students’ answers may vary, some might make the argument that it is unlikely that the highest temperature would happen at noon, as the hottest temperature of the day tends to be about 3 PM (depending on where you live!).

Question 6 has students think about what would happen in the next 24 hour period. Since the period is 24 hours, students should see that the exact same temperatures would be reached as on day 1; temperatures would start at the minimum of 70˚, rise to 90˚ and then fall down to 70˚ again.

In question 7, students solve an equation using either a graphing calculator or analytical reasoning. Look for groups that use the symmetry of the graph to find the second time when the temperature reaches 78˚ instead of solving the equation twice.

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Formalize Later

The focus of this lesson is on interpreting concepts in context which is a key skill students need to develop for AP Calculus. The informal language of max and min to discuss range and “how long it takes for the graph to repeat” for period, are actually helpful in thinking about these topics in context. 

By the end of the lesson, students should be able to explain why certain scenarios are more likely to be modeled by trig functions and should be able to identify scenarios that model periodic behavior. A great assessment question would be to provide four scenarios and have students choose and explain which ones do or do not represent periodic behavior.