To open class, I provide students with the Unit Cirlce Entry Slip. This slip is designed to activate prior knowledge of the trig ratios for special angles on the unit circle. I allow students approximately 5 minutes to work on the activity. After they have finished, I ask for them write on a sticky note (I have these placed on their desks as they enter) what conclusions we can draw from the activity. It is important for the class flow and to engage all students to remind them to limit their thoughts to just one per sticky note, and no person is allowed to complete more than one.
After most of the sticky notes have made their way to the whiteboard, I ask 3 students to begin arranging them into piles of similar conclusions. As the students work, I tell them to provide commentary, "think alouds" for the class so that the everyone can offer input. The important thing is that I take a back seat and allow students to draw the conclusions. One generalization I hope surfaces is that the sine and cosine functions on the unit circle are never more than one, nor less than negative one since this information will be critical in the lesson. In fact I highlight this pile of responses as our launching pad for today’s lesson.
When the students come up with the initial claim, then the stage is set for me to assist them in taking their knowledge to the next level!
As we transition into the lesson, I begin by having students graph the values of theta as the horizontal axis. While leading them through this on the board, I tell them perhaps we should look at the results through a new lens to see if we can draw conclusions that do not show up in the sticky notes. The new lens, I tell them, is as simple as treating the sine and cosine as functions on a graph which I explain further in the Video Narrative.
After the lesson, I pose the following thought to the students:
So we have made some great conclusions so far, and discovered how the sine and cosine functions are periodic in nature. HOWEVER, there is one common misconception of which we must be careful... I am going to give you a hint. Think about the "dual" nature of y, if we assume as before that f(x) and y are the same thing on a graph.
To have the students respond to this, I run an online poll that allows the students to respond anonymously via iPad response. Using this method insures that everyone is comfortable in responding, and everyone has an equal opportunity... not just the "smart kid" who steals the show before the rest of the class has a chance to process the question. (Instead, the "smart kid" can eloquently explain is reasoning in writing to the class on the response.)
In response to the question, I hope that students can say that f(theta) and y represent different things... y, as we defined it on the unit circle, is the length of a side of a triangle formed in the unit circle. On the other hand, f(theta), does not mean the same thing! Changing the lens through which we are viewing the function must be preceded by an understanding of the variables that we are using to represent the events! Encouraging the students to be precise in their calculations (including the definition of variables) is an important Math Practice Standard (MP6).
To conclude the class period, I ask the class to graph 3 periods of the cosine and tangent function, just as we did for the sine function. I expect students to use the same process, and select points to complete this task. I ask students to complete this assignment in their notes, which is different from most of the homework assignments that I give them where I typically have then use separate paper and turn it in to me. I do this since in the next lesson, we will use their work to identify 5 significant points needed to plot the sine and cosine. We will also study how the tangent function differs in nature from the other two trig functions.