##
* *Reflection: Relevance
Ambiguous Case Day 2 of 2 - Section 1: Bell Work

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.

*Why let students state 2 or no solutions*

*Relevance: Why let students state 2 or no solutions*

# Ambiguous Case Day 2 of 2

Lesson 5 of 13

## Objective: SWBAT determine the number of solutions possible from given information.

## Big Idea: By analyzing the information given a student can determine the number of possible solutions and find the solutions.

*45 minutes*

#### Bell Work

*15 min*

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:

- Why is this true when you know the angle is obtuse or right?
- In a triangle the largest side is always opposite what angle?
- The smallest side is always opposite what angle?

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.

*expand content*

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:

- Do I have enough information to find a solution?
- Do I have the ambiguous case (SSA)?
- What type of angle was given?
- How does the opposite side compare to the other side?
- How do I set the problem up to solve?

Once we have a set of questions I suggest students put these in their notes for use when solving triangles.

*expand content*

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.

#### Resources

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#### Resources

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- UNIT 1: Introduction to Learning Mathematics
- UNIT 2: Functions and Piecewise Functions
- UNIT 3: Exponential and Logarithmic functions
- UNIT 4: Matrices
- UNIT 5: Conics
- UNIT 6: Solving Problems Involving Triangles
- UNIT 7: Trigonometry as a Real-Valued Functions
- UNIT 8: Graphing Trigonometric Functions
- UNIT 9: Trigonometric Identities
- UNIT 10: Solving Equations
- UNIT 11: Vectors and Complex Numbers
- UNIT 12: Parametric and Polar graphs and equations

- LESSON 1: Review of Angles
- LESSON 2: Solving Right Triangles
- LESSON 3: Law of Sines Introduction
- LESSON 4: Ambiguous Case Day 1 of 2
- LESSON 5: Ambiguous Case Day 2 of 2
- LESSON 6: Finding the Second Solution
- LESSON 7: Problem Solving with Triangles
- LESSON 8: Law of Cosines Day 1 of 2
- LESSON 9: Law of Cosines Day 2 of 2
- LESSON 10: Area of Triangles
- LESSON 11: Review of Solving Triangles Day 1 of 2
- LESSON 12: Review of Solving Triangles Day 2 of 2
- LESSON 13: Solving Triangles Assessment