The more deeply that students think about and understand the first questions about symmetries on the Ferris Wheel Heights Warm-Up, the better prepared they will be to sketch graphs of the trigonometric functions. The big idea is: if students understand how to get information about one quadrant from another quadrant, they will more easily be able to create a graph. I think that seeing the symmetries on the circle first is a great way to make this leap.
There are a lot of different ways for students to think about the second problem. This is a good chance for students to think about multiple representations: they can use sketches of graphs, or start to set up data tables, or use a diagram of a Ferris wheel to identify the key information. At this point I hope that my students will start to look for generalizations—even though they don’t currently have any way to find a formula to fit the data—they can make some generalizations about how to use the four pieces of given information to find the maximum and minimum points. If they attempt this work, MP7 and MP8 will surely come into play.
Instructional Note: Some students may choose to spend their whole time trying to figure out this generalization—which is awesome, and it is great to note that they do not need to know anything about the trigonometric functions in order to make this generalization.
I include the third problem to make sure that students remember key information about how special right triangles work. My expectation is that my students will use their knowledge of the Pythagorean Theorem to set up equations, rather than memorizing the information about special right triangles. I inform students that the last problem is important; they will eventually apply their knowledge of these problems to the Ferris wheel context.
I think that a great question for them to think about during this lesson is, "How do these two types of problems connect with each other?"
Today's closing is a great time to ask students to make the generalizations. My students are already expressing informal observations, which is great. I will prompt them to make these more formal is by asking them:
If you were going to teach somebody some shortcuts about graphing Ferris wheel heights, what tricks could you show them? How did you figure these tricks out?
I expect that this question start the process of creating productive generalizations. Anthing that my students come up with today will be helpful once we start to write function rules more formally.
I will also using the closing activity of today's lesson to start a conversation about function rules:
Chances are, none of my students will have these ideas yet, because we really haven't talked about trigonometry at all, but hopefully with this question, more students will start to think about rules that may fit these situations. The more they learn from thinking and reasoning on their own, rather than waiting for us to tell them, the more belief they will have in their own reasoning and thinking abilities, which will empower them to do more reasoning and thinking!
Though it may seem inefficient to spend so much time looking at these graphs without using sinusoidal rules to describe them, the goal is for students to really want to find function rules for these graphs. The point of this whole week is that students can understand a lot about how these functions behave without even finding function rules to fit them.