Law of Sines Introduction

Today's class starts with students solving a right triangle and an oblique triangle. Students work to solve the problems while I walk move around the room checking over student work and talking to students who had errors on yesterday's exit slip. I expect my students to struggle with the oblique triangle. I may ask a questions such as, "Can we use what we did yesterday?" I want my students to try some different techniques that we can discuss before beginning to develop the Law of Sines (MP1).

After students have worked for about 5 minutes, I will ask a student to share his/her process for solving the first problem. After we discuss one student's work, I plan to ask, "Did anyone use a different method?" After different methods are put on the board, we compare the methods. I want students to approach today's lesson thinking about alternatives. Some alternatives that I hope will be shared are using different trigonometric functions and the use of the Pythagorean Theorem. When using trigonometry, I think it is especially important for students to learn how to approach a problem in different ways. 

When we begin to discuss solutions for the second problem, I expect that many students started the problem, but were unsure how to complete it. Typically, some students consider drawing an altitude as in the diagram to make a right triangle. However, students are not always sure how to proceed from this idea to a solution. I am prepared to work as a class to solve the second problem. I plan to guide students in solving the problem with questions like:

• If we use the small right triangle with the side of 20 cm what part of the larger triangle can we find?
  
• What do we need to find to help find the rest of the sides for the large triangle?
• If we have 2 angles equal in a triangle what do we know about the sides?
• How can we find AB?

Once we are done with the bell work I plan to make a comment like, "Boy that took a lot of work?" unless a student beats me to it. The important point that I want students to consider is whether the process that we used can be reused. I will make a comment to suggest that sometimes mathematicians discover new processes by repeating methods, and observing their structure (MP7, MP8). With this opaque guidance, I will distribute the Law of Sines activity.

I tell my students that I would like them to work on this activity in groups. The activity guides students through a proof for the Law of Sines. As students work I answer any questions groups have about the proof.  I expect that my students may have questions about the prompts on the worksheet, but they will not have difficulty with the process.

• Some students will need a reminder of the properties of an altitude
• Step 4 can confound students. I have them write one equation from step 3 and say "now how can you isolate k?" 
• In Step 6, when it says to group sin A with a and sin C with c, I often rephrase the step by saying you want all the quantities related to the measurement of angle A and side a on one side of the equation and measurements for C and c on the other side of the equation.

I want my students to sustain their momentum, so I assign problems 7, 11, and 12 from page 434 from Larson Precalculus with Limits 2nd ed. Each of these problems will only have one solution. With just a few minutes left of class I ask "When do we need to use the Law of Sines?" I want my students to understand that we can use the Law of Sines with right triangles, but right triangles are a special case because sin (90 degrees) = 1.

Before leaving for the day, I ask my students to write out the Law of Sines in their notes, including the information that is needed to use the Law. I want students to be aware that they need at least one side and three pieces of information to solve.

We will discuss the ambiguous case when there is more than one possible solution tomorrow.