Today my students will use The Ambiguous Case worksheet from yesterday to discover methods for solving triangles in the ambiguous case, when there are two possible solutions for the measures of a triangle.
The bell work I have planned is similar to the Questions 4-6 on the worksheet. Today I have the students focus on the side measures and the measurement of the angle given in the activity. As students look at the information, it is important for them to understand that the measure of the side opposite the known angle has an important relationship to the measurement of the angle.
The goal of the Question 1 is for students to realize that they will always have 1 solution if the side opposite the angle is longer than the other given side. I sometimes need to coax my students to write out the three situations that resulted in a single solution. When the information is organized, it is much easier to compare than when you are looking at 3 tables that are on the worksheet.
The second question focuses on when a solution is not possible. For students to connect information from geometry I ask:
The final questions allows students to see that the only time they will have two solutions is when the given angle is acute. Students also notice that the side opposite the given angle is smaller than the other side and this is the same as when they could have 2 solutions. I ask the students to try and see if they can find a way to determine how small the opposite side has to be before it is too small to produce a triangle.
Of course there is a way to determine this by comparing the opposite side to what would be the altitude of a triangle. I usually allow students to say that will have 2 or no solution if they are just identifying the number of solutions.
I now give the students time to discuss what we have just minutes working with their groups to develop a process for determining the number of solutions they could have when given 2 sides and a non-included angle. After 5 minutes the groups share out what they have developed and critique/improve on the process.
If the only goal was to determine the number of triangles, I would have students be more exact when the side opposite the given angle is smaller than the other side. My goal is for students to solve problems with the Law of Sines. I have students state there are 2 or 0 triangles when they have an acute angle and the opposite side is smaller. When a student has reasoned that they may have 2 or no solutions, the student knows there will be 2 answers if when solving they get an answer for the angle.
Since real problems usually do not deal with special angles (30,45,60, etc.), students will be using calculators so the student will be able to determine when the situation does not have a solution.
I put a problem on the board that has no solution so they can see what the calculator states. On the TI-84 it says domain error. I do not go into more detail at this point. We will discuss what is meant by domain error more when we develop the inverse trigonometric functions.
I now give students time to discuss the bell work. Students work in groups to determine what needs to be considered when working with information about a triangle. After a few minutes (4-5 minutes) I have students share what was discussed. We then try to develop some questions to ask as we solving triangles. Some questions that I want students to consider include:
Once we have a set of questions I suggest students put these in their notes for use when solving triangles.
The students are given 10 problems from the Larson Precalculus with Limits, 2nd ed, page 434 problems 25-34. The selected problems ask students to determine the number of possible triangles based on the given information provided.
Students work on the problems in groups. After about 5 minutes groups share some of the answers with explanation to the class. The students answer each others questions as the problems are shared.
When I began teaching trigonometry I was a teacher that just went over the rules in the book with no explanation. I never develop why the new triangle had an angle that was supplementary to the angle found in the first equation. I sometimes just told the students anytime you have an acute angle you may have 2 solutions so you need to check.
As I develop my understanding and teaching approaches I realized that having a deeper understanding of the ambiguous case helps the students in all areas of solving triangles. Students need to explore to see the relationships among the sides and angles.
For me to develop the activities in these last 2 lessons I have had to look at the structure of the book explanation and then see how I could make an activity that would help students understand the structure. I am still working on improving the activity for this topic as well as how I facilitate students learning.
I may never have the perfect lesson but I know this lesson is far above what I did 20+years ago.
I will ask my students to complete and turn in this exit slip as they leave class today. I will use this to assess which students need more assistance in understanding the ambiguous case.