The students in my precalculus class have studied trigonometric functions and the unit circle in Algebra 2, so they come to me with some background information about these concepts. However, I don't expect them to be able to recall trigonometry concepts at the drop of a hat. The nice thing about this lesson is that it is still rich and challenging for students who already know trigonometry and for those who have never seen it before.
To launch this task, give students the task worksheet and ask them to solve the problems in their groups. I don’t want them to tell them that we are working with trigonometric functions, so I would not give them any background information about these problems.
The scenarios are pretty straight forward so they should not require much introduction on your part. For question #1, most students will sketch the graph that has linear “pieces” since the question specifies that the lights move at a constant rate. In the Ferris wheel question, students may do the same thing and sketch a graph that has linear “pieces” instead of being sinusoidal. If this occurs, you can ask students how they know that the distance from the ground is increasing at a constant rate when that information is not given. Get them thinking about this and it will definitely get addressed in the class discussion.
Even if students draw the Ferris wheel graph incorrectly, they will still be able to find the period and amplitude. I want them to think about if the graphs are functions because the path of the Ferris wheel does not pass the vertical line test, so some students may mistakenly say that it is not a function.
During the class discussion, choose a student who completed the Power Tower graph correctly and have them share their work. Talk about the period and amplitude of the function and ask students how they figured it out from the context of the problem.
For the Ferris wheel problem, start with a student who drew the graph with linear pieces and ask them to explain their work. (If no students drew the zigzag graph for the Ferris wheel, draw it yourself.) See if everyone agrees with the student. If no one speaks up, ask a student who drew a sinusoidal graph to explain why they thought the distance as a function of time would be curved and not linear. If everyone drew linear graphs, tell them that students in a different hour were drawing sinusoidal graphs and have them discuss which is correct. In the video below I have some suggestions if students are insistent that the graph should be linear. Here is another image that shows why the height off the ground will differ depending on the start and end points on the Ferris wheel (and why the function is not linear).
After that misconception is addressed, talk about the fact that these functions are periodic. Discuss what that means and gather responses about the characteristics of those graphs. Students may bring up sine and cosine graphs. If they do not, leave it until tomorrow. It is always better to give students the opportunity to make the connection rather than just telling them about it.
If you have a CBR data collection device for your Texas Instruments graphing calculator, you can use it to demonstrate the Ferris wheel problem. The CBR will measure the distance it is from a wall, for example, and you can have a student move it in a circle like a Ferris wheel. The graph will show up as the distance from the wall as a function of time, and students will be able to see that it is indeed sinusoidal. Note: I would not demonstrate the Power Tower example by this method as it may look sinusoidal unless your rate of movement is exactly constant, which is almost impossible to do with the device.
The assignment below uses the Ferris wheel metaphor again, but this time the unit circle is acting like our Ferris wheel. If students complete the graph correctly, they will have graphed y = sin x. If a student has never studied trigonometry before, this assignment may be too much of a jump, so you might want to graph some other Ferris wheel examples before you move directly to this assignment. Since my students have graphed sinusoidal functions in the past, I am giving this assignment to see if it will spark their memory. We will discuss this assignment in depth tomorrow and will use it as the main focus of the lesson. In the video below I give some information about how you could use this worksheet as an introduction to many, many trigonometry topics.