Tangent on the Unit Circle and Relationships between Trig Functions
Lesson 7 of 9
Objective: SWBAT use problems they've been working on to see a unit circle definition of the tangent function. They will also continue to make connections between sine, cosine, and tangent.
As today's class begins, I project the lesson opener leading with the question:
What does it mean to demonstrate mastery on these Student Learning Targets?
I circulate and have informal conversations with students about the learning targets.
I also remind students that Problem Set 14 is due at the end of class today, and that the purpose of today's class is to finish the assignment. Everyone moves straight into work time, with our visions of mastery in mind.
The ideal for today's class is that all students have completed the first six problems on Trig Problem Set 14, and are ready to complete problems 7 and 8 today. In my experience, this will be the case for about half of my students. I am prepared to work with students on any probelms they need, but for the purpose of this narrative, I'll focus on those last two problems. My notes on probelms 1 through 6 are included in the previous lesson.
I love this problem because it gives students a chance to see and make use of structure on their own (MP7). It's a rich problem that asks a lot of students. To begin, they have to understand the notation that is used to introduce the problem. Some students are confident with the notation, others are confused. If they're confused, I tell them simply to pick values for theta and alpha and to test if their values fulfill all constraints: are both numbers between 0 and 360? Is alpha greater than theta? Is the difference between them 180? It rarely takes more than one round of "guess and check" for students to understand what the notation is asking for. Of course, I want all of my students to be fully literate in the language of mathematics, and I consider problems like this an opportunity to build that literacy.
Next, the chart assumes some explanation that's not written, especially the last column. Rather than bogging down the document with an explanation of what points A and B represent, I give students the space to assume that point A is the point on the unit circle at the terminal side of angle theta. If they don't (correctly) assume this, I'm happy to explain it.
Question 7b asks students to make generalizations about points A and B. They may have some ideas after completing the chart, but whether they do or not, I expect them to have more to say after completing the graph in 7c.
On their graph, the straight line connecting A and B should go through the origin, and these angles are 180 degrees apart, and therefore we’ll have a diameter. As they complete their sketches, I ask students what angle their line makes with the x-axis. This will be the reference angle.
Then we get to the fun part: what is the slope of this line. There are several ways to approach this question, and I try to leave room for students to figure it out. If they are stuck, I point out that one point on the line in (0,0) and another has the coordinates of the cosine and sine of the chosen angle. Some students will realize, without any calculation, that the slope of this line is sin/cos, and they will recognize this as tangent. Others will divide the decimal values of sine and cosine on their calculators, and they will be amazed to see that the quotient matches the tangent they've recorded in their table. Either way, the idea is for students to see for themselves that the tangent of an angle is equivalent to the slope its diameter on the unit circle.
In small conversations, I direct students back to probelms 2 through 4. What angle does the line y=x make with the x-axis? What was the tangent of 45 degrees? It's a joy to watch all levels of students be amazed at this new aspect of slope.
This is more of an open-ended task than a problem, and it's a chance for students to illustrate any relationships they've found between the sine, cosine and tangent. This is also an opportunity for students to practice their abstract and quantitative reasoning skills (MP2) as they try to generalize these relationships in ways that makes sense. It's fascinating to see what students come up with here. It's also easy for them to ignore this problem, so I pay attention to their engagement here, and I make suggestions about some of the relationships they have noticed. I really like asking students to "use a combination of words and diagrams" to explain something, because it gives them space to express themselves however they're comfortable. With this task, I'm simply trying to create a space where students can synthesize what they've learned about the trig functions over the last few classes. In tonight's homework, and in our next class, we'll move back toward formal exercise as we try to turn this knowledge into a set of content skills. For now, it's great to give students room to make their own sense of what they have learned.
On the back of Problem Set 14 is a space for students to assess their mastery of the four SLTs on this assignment. With 15 minutes left in the class, I tell students how important I consider this part of the assignment to be.
I return the index card essays that served as exit slips for the previous class, and I say that today the task is to write four such essays. "For each of the learning targets on this assignment," I say, "I expect a topic sentence in which you tell me what your grade should be, followed by at least two supporting sentences that include specific details about how you demonstrated mastery of this learning target on this assignment."
What's really happening here, even though I don't explicitly say this to kids, is that students are constructing arguments (MP3) about what they understand, and they're engaging in self-critique. When I grade this problem set, of course I am very interested in what students were able to accomplish in their problem solving. I am equally interested in how well they're able to explain how their work translates to mastery of these learning targets, because this reveals that they really understand what they've learned and what it means.