##
* *Reflection: Developing a Conceptual Understanding
Ambiguous Case Day 1 of 2 - Section 2: Developing the Ambiguous Case

Here is an example of what students determined during the activity. On the second page students wanted to be very brief in answering the questions. This is obvious in question 6. As I question why the student said one is acute and one is obtuse I realize the student did not look at the sides. This student misunderstood the question. I need to question the student more to know what the student is thinking. I would ask questions such as:

- Why did you say one is acute and one is obtuse?
- What do you know notice about the known sides?

My first idea about the answer is that the student was looking at the measure of the angle C since when we found the 2 triangles in question 1 one of the angles was acute and the other was obtuse. My goal was for the students to notice that the opposite side is smaller than the other given angle and the given angle is acute for us to have a possibility of 2 solutions.

*Developing a Conceptual Understanding: Student Example of Activity*

# Ambiguous Case Day 1 of 2

Lesson 4 of 13

## Objective: SWBAT will solve the ambiguous case for oblique triangles.

#### Bell Work

*10 min*

As my students come into class today, a contextual problem is on the board. My plan for this problem is that after about 3-4 minutes I will have a student share a diagram for the problem. Then, I will give the students a few more minutes to work on a solution.

As students work I move around the room and answer questions. Many students will try to solve by using right triangles. When I observe this happening, I will ask students to explain their reasoning. I ask questions like "Is that ratio correct?" and "What type of triangle are you given in this problem?"

Once a good number of students have a solution, I will ask a student (or two) to put his/her solution on the board. Since I expect my students to have struggled with the problem, I will give the class time to ask questions about the presented solution. If necessary, I will help the discussion along by making sure students hear how the Law of Sines organizes the measurements from the triangle using ratios. My goal is to both identify and remedy lingering confusion about the structure and application of the Law of Sines.

Before leaving the problem, I will give students a chance to ask questions about the problem description. Sometimes, students find these types of problems confusing to model. I will do this after the problem, because I want students to persevere in all stages of solving a problem. So far today, some may have spent most of their time reading and interpreting the problem.

#### Resources

*expand content*

*This activity requires spaghetti, toothpicks, or some rigid material the students can cut or break easily, rulers and protractors.*

Students are given The Ambiguous Case worksheet. The activity has students look at the number of triangles that can be made when you know 2 sides and a non-included angle. The worksheet has one side and one angle fixed. Students are given different measures for the opposite side and ask to determine how many triangles can be formed. I remind students that the side opposite can move in and out to make a triangle. The students will draw any triangles possible and measure angle C for each triangle.

Students are given brief directions about the activity. As students work on the activity I go to the different groups to make sure students are on task and understand the process. If a student has not found 2 triangles with BC=4 cm (in the first question) I will ask them if they have checked for more than one triangle. Many students will only draw the triangle with angle C being acute. I ask if they can take the 4 cm and move closer to angle A to get another triangle.

After exploring the triangles students will analyze the results and try to answer 4-6 on the worksheet. I will ask questions such as:

- Look at all the answers with one solution, what do you notice about BC and AB?
- Why did you either get no solution or 1 solution when angle A is obtuse or right?

These are questions students should ask themselves as they work with the Law of Sines.

#### Resources

*expand content*

#### Closure

*5 min*

My plan is to have my students work on the activity until there are about 5 minutes left in class. We will need some time today to put the materials put away. As we clean up, I will ask students to continue to think about Questions 4-6.

Once the cleanup is complete, I will ask my students, "What type of angle (acute, right, or obtuse) will you be given to have at most 1 solution? Why?" I want them to record their answer on a piece of paper and turn it in as an exit slip for today's lesson.

*expand content*

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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