Lesson Plan | Active Learning | Sum of the Interior Angles of a Triangle

Assumptions: This Active Lesson Plan assumes: a 100-minute class, prior student study with both the Book and the start of Project development, and that only one activity (among the three suggested) will be chosen to be conducted during the class, as each activity is designed to take up a significant portion of the available time.

Objectives

Duration: (5 - 10 minutes)

The Objectives stage is designed to clearly establish what students should be able to do by the end of the lesson. By focusing on identifying and applying the fundamental property of the sum of the interior angles of a triangle, and on calculation skills involving this property, students will gain a practical and theoretical understanding of the topic. This segmentation helps guide both the teacher in preparing activities and the students in executing and understanding the proposed exercises.

Main Objectives:

1. Empower students to recognize and apply the mathematical property that states the sum of the interior angles of a triangle is always 180°.

2. Develop skills to calculate the third angle of a triangle, given that the other two angles are known.

Side Objectives:

1. Encourage collaboration and group discussion for solving mathematical problems.
2. Promote the use of logical reasoning and argumentation in justifying solutions.

Introduction

Duration: (20 - 25 minutes)

The Introduction serves to engage students with the content they have previously studied, using problem situations that stimulate critical thinking and the practical application of knowledge about the sum of angles in a triangle. The contextualization, in turn, seeks to connect mathematical content with real situations and curiosities, increasing student interest and demonstrating the relevance of the topic in their lives and future professional applications.

Problem-Based Situations

1. Consider a triangle with two angles measuring 60° and 40°. How could you calculate the value of the third angle without measuring it directly?

2. If in a triangle, one angle measures 90° and the second angle measures 30°, what is the value of the third angle and why?

Contextualization

Understanding the sum of the interior angles of a triangle is essential not only for solving mathematical problems but also has practical applications in various fields, such as engineering, architecture, and geography. For example, engineers use this property to calculate load distribution in triangular structures, while architects may apply the concept to create more efficient shapes and structures. This basic knowledge of geometry is a fundamental tool for solving everyday and professional problems.

Development

Duration: (75 - 80 minutes)

The Development stage is designed for students to apply the knowledge acquired about the sum of the interior angles of a triangle in a practical and collaborative way. Through playful and contextualized activities, students have the opportunity to explore the concept in varied scenarios, promoting a deeper and more enduring understanding of the theme. This approach not only reinforces learning but also develops teamwork skills, critical thinking, and the application of mathematical concepts in real-world contexts.

Activity Suggestions

It is recommended to carry out only one of the suggested activities

Activity 1 - Angle Detectives

> Duration: (60 - 70 minutes)

- Objective: Apply knowledge about the sum of the interior angles of a triangle through a practical and collaborative problem.

- Description: In this activity, students will be divided into groups of up to 5 people and take on the role of detectives who need to uncover a mathematical mystery. Each group will receive a set of evidence consisting of partial measurements of two interior angles of a triangle and must use their knowledge of the sum of interior angles to determine the third angle and thus solve the case.

- Instructions:

• Divide the class into groups of up to 5 students.

• Distribute to each group an envelope containing the evidence which are the measurements of two angles of a triangle.

• Ask each group to calculate the third angle using the property of the sum of the interior angles of a triangle.

• Each group must present its solution and the method used to arrive at it.

• Conduct a class discussion to compare the different methods and solutions found by the groups.

Activity 2 - City Builders

> Duration: (60 - 70 minutes)

- Objective: Promote practical understanding of the sum of angles in a playful and educational urban planning context.

- Description: Students, organized in groups, will design a fictional city on graph paper, where each building must be represented by a triangle. Using the properties of angles, groups must ensure that the triangles represented by the buildings comply with the rule that the sum of the interior angles equals 180°.

- Instructions:

• Divide the students into groups of up to 5 people.

• Provide each group with a large sheet of graph paper and colored markers.

• Explain that each group must design a part of the city, where each building is represented by a triangle.

• Students must use the properties of angles to ensure that the triangles of the buildings have their interior angles sum to 180°.

• At the end, each group presents their part of the city and explains how they applied the properties of angles in the design.

Activity 3 - Music and Mathematics Festival

> Duration: (60 - 70 minutes)

- Objective: Utilize the property of the interior angles of triangles to plan an event, integrating mathematics and practical situations.

- Description: In this activity, students will plan a music festival where the stage, equipment, and audience are positioned to create triangles. Each group must calculate the angles of the triangles to ensure the festival is visually appealing and safe.

- Instructions:

• Form groups of up to 5 students.

• Present the scenario of a music festival and explain that the stage, equipment, and audience can be positioned to create triangles.

• Students must calculate the angles of the created triangles to ensure the festival is visually appealing and safe (for example, avoiding very acute angles that could pose risks).

• Each group presents their plan for the festival, explaining how they calculated the angles and justifying their choices.

• Encourage a discussion about the different solutions found and the criteria used to position the elements.

Feedback

Duration: (10 - 15 minutes)

The Return stage is crucial to consolidate the learning acquired during the practical activities. In discussing in groups, students have the opportunity to verbalize and reflect on the mathematical processes used, which strengthens their understanding of the topic. This discussion also allows the teacher to individually assess students' understanding, correct any misconceptions, and reinforce key concepts. Additionally, the exchange of ideas between groups promotes a diversified and enriching view of the subject.

Group Discussion

Initiate the group discussion by inviting each group to share their discoveries and solutions. Suggest they start by describing the problem they received, the reasoning process they used to solve it, and finally, the solution found. Encourage students to justify their answers based on the mathematical properties discussed. This is a moment of exchange where each group can learn from the strategies of others and discuss the effectiveness of the methods employed.

Key Questions

1. What were the greatest challenges your group faced when trying to calculate the third angle and how did you overcome them?

2. Was there any situation where different groups arrived at different solutions for the same problem? How do you believe that happened?

3. How can the property of the sum of the interior angles of a triangle be applied in everyday or professional situations?

Conclusion

Duration: (5 - 10 minutes)

The conclusion of the lesson aims to reinforce learning, consolidate the discussed concepts, and demonstrate the importance of applying mathematics in everyday life. This stage allows students to reflect on what they have learned and helps to highlight the relevance of the content covered in the class for their lives and future careers, ensuring a holistic and lasting understanding of the theme.

Summary

To conclude, the teacher should summarize the main points addressed in the lesson, recalling the property of the sum of the interior angles of a triangle, which is always equal to 180°, and how to calculate the third angle from the two known angles. It is important to emphasize the significance of this property in solving mathematical problems and in practical applications.

Theory Connection

During the lesson, students were able to experience the connection between mathematical theory and practice through interactive and contextualized activities. Through problems based on real situations and the use of active methodologies, such as the flipped classroom, students applied theoretical knowledge in scenarios that simulate the applicability of the concept in the real world.

Closing

Finally, it is important to highlight that understanding the sum of the interior angles of a triangle is fundamental not only for Mathematics but also for various fields of knowledge and everyday applications. This concept aids, for example, architects in constructing stable structures and engineers in calculating forces and supports, demonstrating its relevance and practical utility.