Introduction to Sine and Cosine
Lesson 6 of 11
Objective: Students will be able to solve problems by using the relationship between the sine and cosine ratios of complementary angles.
Model TTTT and Warm-Up
The Common Core requires students to be able to make sense of problems and persevere in solving them. Wordy context problems can pose challenges to students, which can lead to them blindly obeying “rules” for decoding what words mean in math problems or to them just giving up entirely. For this reason, I model for students how to Talk to the Text (TTTT), a literacy strategy that can help students to engage in a productive struggle when problem solving.
When I Talk to the Text for the first time with students, I do a think aloud, which is essentially me thinking out loud about the problem as I read it:
- I read the problem out loud
- I pause frequently and say what I am thinking
- I re-read as necessary
- I underline key pieces of information (visible on the document camera; see Talk to the Text Warm-Up)
- I circle words I am unsure of
- I make connections (and comparisons) to other problems (if possible; see TTTT Tangent Context Examples KEY)
- I summarize my understanding of the problem.
My goal is to model what good problem solvers do by sending the message that making sense of the problem itself is often the most challenging and time-consuming part of the process.
The ultimate goal for this part of the lesson is for students to really understand (1) the TTTT strategy and (2) my expectations for student work (evidence of TTTT in the problem itself, an accurate, well-labeled diagram that correctly matches the context, math work that matches their diagram). I ask for questions from the whole class before handing out the TTTT Tangent Context Examples, which will give students a chance to practice TTTT for themselves.
Over the last few lessons, through the context of the tangent ratio, we have developed an understanding that right triangle trigonometry is all about the relationship between sides and angles in right triangles. At this point in the unit, it is time to introduce the sine and cosine ratios. I tell students that given a reference angle, there are two other ratios we are interested in exploring: the sine ratio, which compares the opposite leg to the hypotenuse, and the cosine ratio, which compares the adjacent leg to the hypotenuse.
After this brief introduction, I give students time to practice analyzing a diagram, determining which trigonometric ratio they should, and solving for an indicated variable (see Sine and Cosine Notes).
Lastly, I end the notes by modeling angle of elevation and depression for the class. Since I am rather petite, I ask a tall student in the class to come up to the front of the classroom and look me in the eyes—for whatever reason, students tend to think the stark contrast in height is rather funny. I then explain the angle of elevation refers to the angle at which I look up at this student’s eyes, whereas the angle of depression is the angle at which he/she looks down at my eyes.
As I have taught right triangle trigonometry over the years, I have noticed that it is possible for students to identify the appropriate trigonometric ratio to use to correctly solve a problem without really understanding what they mean when they write “,” for example. While I want students to have procedural fluency, the Common Core also calls for students to be able to reason abstractly and quantitatively (MP2), which is why I want my students to be able to explain means in their own words and to be able to show their understanding by drawing several well-labeled triangles for which . If my students can say “given reference angle, , the opposite side is 2.5 times the length of the adjacent side,” then I would know that they understand the relationship between trigonometric ratios and triangle similarity and that they understand the meaning behind their algebraic/symbolic representation of a problem.
Because this task can be a bit challenging and because I want students to feel safe to make mistakes, express confusion, and to ask questions, I have students work in pairs on this activity. As a pair, students work together to make sense of what , , and really mean and how they can use this understanding to check if their answers are reasonable. When pairs finish, they call me over to check in with the class.
As is true with all pair or groupwork, students will finish at different rates. An extension I like to give is to pose questions to students about the relationship between the sine and cosine ratios for the same angle and to explain what they notice and why they think it is true.
Check for Understanding
Given the amount of notes and pair work for this lesson, it is essential that I assess individual student’s understanding, especially because the idea of the sine and cosine ratios is new. In this Check for Understanding, students can reason through the problem in several ways, as well as check their answers by using trigonometry or the Pythagorean Theorem. I ask students to include a diagram, show all work, and explain how they know to give me further insight into their thinking.
I need to give students an opportunity to practice tackling context problems by talking to the text on their own, which is why I give them this homework assignment. I make sure to tell students that they must be prepared to share out their work in the next lesson, where we will do a lengthy homework review exercise.