To start today's class I have students work on a problem. After a few minutes a student shares a process for solving the problem. After the process is on the board I ask:
The last questions is usually answered "yes" by most students. Some always think I am trying to trick them, so they are hesitant to say yes.
I give the student a new problem. Students begin working and some begin to realize that they cannot solve it. I say "Is this an ambiguous case? Usually we get a single answer if it was not the ambiguous case. How many solutions does this problem have? Why?"
I expect one or more students will realize that you need to have at least one side to solve. Once students broach this idea I'll say, "Oh, you need to know the measure of a side or you cannot determine the measurements for a triangle. Why is that?" This is a good opening for us to review the students' knowledge of similar triangles.
Once we have seen discussed the importance of knowing the measure of a side in solving, I will put another problem on the board. As we begin to work on the problem, I want to ask students, "How many possible solutions are there? How do you know?" I ask myself these questions, so I want to my students to consider this approach as a model. Most students will come to the conclusion that there are either 2 or 0 solutions. The students are given a minute or 2 to determine if there is a solution. Once students know there are 2 solutions, we will work together to find the first solution, putting this information on the board.
After the first solution, I make a big deal of asking the students, "Are we done?" I want students to realize that there is another solution. Then I ask, "How are we going to find the second solution?"
If students cannot offer an explanation, I refer the students back to the The Ambiguous Case from The Ambiguous Case Day 1 of 2. As a class we draw a triangle to represent the first solution. I tell students we know angle A and sides a and c. I then draw another side to represent side a swinging in to produce another triangle. I explain to the students that we label the new angle C as C' (C-prime). and the new sides with primes. I then ask students to tell me what they can determine from the diagram. We discuss how there is an isosceles triangle so the base angles are equal. I then ask "If we know the base angles are C how can we find C'?" Students realize that the C' is the supplement of angle C.
Once this is determined we return to the problem. We find B' together. I then ask: "Besides angle B what else will change?" The students then find the rest of the second solution and share the results on the board.
For practice with problem solving Students are given problems 25, 28, and 34 from page 434 in Precalulus with Limits by Larson. These problems are chosen so that students can practice determining the number of possible solutions and then solving. The problems are not in context at this point because I want students to focus on the structure of the triangles and how to use the structure in solving. As we progress through this unit I will give students more contextual problems.
As class ends I put a question on the board for students to discuss in their groups. Each group is given sticky notes. Students will put a sticky note by the number that can be solved using Law of Sines. The sticky notes will include the names of the group members and any question they have about the problem. I also have a "Parking Lot" by the board for student questions about the Law of Sines.
This activity is foreshadowing the next topic which is the Law of Cosines. At this point students can solve right triangles and some oblique triangles after the next topic they will have all the tools necessary to solve any triangle.