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

This year I have the opportunity to see my lessons taught by an intern. The intern adjust the lessons to fit her style but the content is the same as are the activities.

While observing this lesson I was reminded of many issues I have faced when teaching the ambiguous case. The intern has great questioning techniques and has the ability to let students think and discuss ideas. This can be a challenge for a new teacher. During this lesson I saw a student get very frustrated because some students were not immediately seeing what he saw. Being able to facilitate a mathematical discourse can be a challenge especially if there are outside issues with the students.

It is hard to know when students have lost focus and need to be brought back to the question. It is important to consider the point you as the facilitator will stop a discussion. When 2 students are dominating the discussion I find is a time I will interrupt and ask someone in the class to explain what the 2 students are discussing and question the class on what we are trying to determine.

During the class discussion I noticed that students struggled when the given information was drawn in a different way than the students had explored. This tell me that I need to adjust the activity to do more exploration with the diagrams drawn in different ways.

For this lesson I found out after the lesson that the students have some old issues “drama” that distracted from the discourse. The outside issues caused a good class discourse to lose its effect. To recover from the loss of focus I intervened and brought the main concepts out by having very focused questions over the activity.

From this observation, I realized that I always have specific questions available if a discussion loses focus. I may or may not use the questions it depends on the class dynamics which I analyze on a daily basis.

*Difficulty for New Teachers*

*Developing a Conceptual Understanding: Difficulty for New Teachers*

# 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*

*Responding to Maria Martinez*I am glad you like the lesson. Let me know if you have suggestions to improve the lesson. I am always looking for ways to improve what I do in class. | 4 months ago | Reply

Dear Ms Katherine

Thank you for allowing me to try your lesson in my classroom. I was looking to modify the investigation on ambiguous case given in my textbook in a such a way that was understood by my students. I have second language learners and international students in my classroom. I am going to try your lesson today. I really like your lesson. I would like my students to come out with their own conclusions. :)

Maria Martinez

| 4 months ago | Reply*I am glad you found it useful. It is always great to know my activities work for others. | 7 months ago | Reply*

*This was such a great exercise. Thank you! It really helped the students visualize the ambiguous case. :) | 7 months ago | Reply*

*expand comments*

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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