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.
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.
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.