Lesson Plan | Active Learning | Triangles: Cevians and Notable Points

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 objective definition stage is crucial for establishing a clear focus and directing classroom learning activities. By setting precise objectives, students can understand what is expected of them and what they need to achieve by the end of the lesson. This also aids in assessing progress and verifying that learning goals have been met.

Main Objectives:

1. Empower students to identify and understand cevians in a triangle, including altitude, median, and angle bisector, and their specific properties.

2. Develop understanding of the notable points of a triangle, such as orthocenter, incenter, and centroid, and how they relate to the cevians.

Side Objectives:

1. Promote practical application of the concepts of cevians and notable points in mathematical problems and everyday situations.

Introduction

Duration: (15 - 20 minutes)

The introduction serves to engage students, using problem situations that stimulate critical thinking and the practical application of prior knowledge about cevians and notable points in triangles. Furthermore, the contextualization seeks to connect the topic with real life, increasing students' interest and demonstrating the relevance of the subject in various areas of knowledge and practical applications.

Problem-Based Situations

1. Imagine you are planning the construction of a new triangular-shaped bridge. How can understanding cevians and notable points, such as the centroid, help distribute weight efficiently?

2. In a garden design competition, a participant proposes a garden with a triangular-shaped pond. Discuss how the height of the triangle can affect the depth of the pond and the aesthetics of the project.

Contextualization

Understanding cevians and notable points in triangles is fundamental not only in pure mathematics but also in practical applications such as architecture, engineering, and design. For example, the famous architect Santiago Calatrava employs mathematical principles to design structures with innovative and aesthetically pleasing shapes, where the use of cevians and notable points is essential. Additionally, understanding how these concepts apply to everyday problems, such as the distribution of forces in a construction, can highlight the importance of these concepts in daily life.

Development

Duration: (75 - 80 minutes)

The development stage is designed to enable students to apply and deepen their knowledge acquired about cevians and notable points in triangles in a practical and interactive manner. Through the proposed activities, students will be able to visually explore mathematical concepts, work in teams to solve complex problems, and apply their knowledge in practical and technological contexts, preparing them for real-world situations where these concepts are used. This stage also aims to strengthen communication and collaboration skills among students.

Activity Suggestions

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

Activity 1 - The Magic Triangle Challenge

> Duration: (60 - 70 minutes)

- Objective: Visually understand the cevians and notable points in a triangle and their geometric properties.

- Description: In this activity, students will be divided into groups of up to five people and will be challenged to construct a large triangle on the floor of the classroom using adhesive tape. Each group must then investigate and mark the cevians of the triangle (altitude, median, and angle bisector) and locate the notable points (orthocenter, incenter, and centroid) visually and practically.

- Instructions:

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

• Provide each group with adhesive tape, a large sheet of paper, a ruler, and a protractor.

• Instruct students to mark a large equilateral triangle on the classroom floor with the adhesive tape.

• Each group must then measure and mark the cevians (altitude, median, and angle bisector) of the triangle on paper.

• Students must also find and mark the notable points of the triangle (orthocenter, incenter, and centroid) on the paper.

• Each group will present their work to the class, explaining how they identified each cevian and notable point and discussing their findings.

Activity 2 - Triangle Detectives

> Duration: (60 - 70 minutes)

- Objective: Apply mathematical knowledge in a practical problem-solving context, reinforcing understanding of cevians and notable points.

- Description: Students, organized in small groups, will receive a series of cards with information about different triangles. Each card contains side lengths and angles, and students must use this information to identify the cevians and calculate the notable points of each triangle.

- Instructions:

• Organize students into groups of up to five people.

• Distribute a set of cards to each group, each representing a triangle with different side lengths and angles.

• Students must use the information from the cards to identify and draw the cevians (altitude, median, and angle bisector) on a separate paper.

• Next, they will calculate and mark the notable points (orthocenter, incenter, and centroid) of each triangle.

• Each group will present their results, discussing the challenges encountered and the strategies used to solve the problem.

Activity 3 - Triangle Builders

> Duration: (60 - 70 minutes)

- Objective: Utilize technology to visualize and apply mathematical concepts in an engineering project, emphasizing the importance of cevians and notable points in civil construction.

- Description: In this scenario, students, divided into groups, will take on the role of engineers who need to design a new triangular transmission tower. They will use 3D modeling software to create a representation of the tower and calculate the cevians and notable points, considering structural stability and aesthetics.

- Instructions:

• Divide students into groups of no more than five people.

• Introduce the 3D modeling software that will be used for the tower project.

• Students should design a triangular transmission tower in the software, considering factors such as height, materials used, and location of transmission equipment.

• After constructing the tower, each group must calculate and depict the cevians and notable points (orthocenter, incenter, and centroid) of the tower.

• Finally, each group will present their tower and explain the decisions made during the project, highlighting how the concepts of cevians and notable points were applied.

Feedback

Duration: (15 - 20 minutes)

The purpose of this feedback stage is to consolidate students' learning, allowing them to reflect on the practical activities and discuss the applications of the concepts of cevians and notable points. This discussion helps to identify gaps in understanding, promotes the exchange of ideas among students, and reinforces the importance of mathematical concepts in the real world. Additionally, this stage serves as an informal assessment of students' understanding, allowing the teacher to adjust future lessons based on the feedback received.

Group Discussion

At the end of the practical activities, organize all students in a large circle for a group discussion. Start the discussion with a brief introduction, highlighting the importance of teamwork and the application of the concepts learned. Next, ask each group to share their discoveries and the challenges encountered during the activities. Encourage students to discuss how the concepts of cevians and notable points can be applied in different situations, both theoretical and practical.

Key Questions

1. What were the most challenging points in identifying and calculating the cevians and notable points during the practical activities?

2. How can you apply the knowledge about cevians and notable points in other areas, such as design and engineering?

3. Was there any surprise or interesting discovery while working with triangles and their notable points?

Conclusion

Duration: (5 - 10 minutes)

The purpose of the conclusion is to consolidate the learning of the lesson, ensuring that students have grasped the main concepts and their applications. Additionally, it aims to reinforce the connection between theory and practice, highlighting the importance of triangles and their elements in real contexts. This stage also serves to prepare students for future applications of the concepts learned and for the continuation of the study of geometry.

Summary

To conclude, the teacher should summarize the main points addressed, reiterating the definition and properties of cevians (altitude, median, and angle bisector) and notable points (orthocenter, incenter, and centroid) in triangles. It is crucial that students have a clear vision of how these concepts intertwine and are applied in various geometric and practical situations.

Theory Connection

Today's lesson was structured to connect theory and practice, allowing students not only to understand mathematical concepts but also to apply them in real-world scenarios through practical activities and discussions. This approach not only solidifies learning but also demonstrates the relevance of triangles and their elements in everyday and professional applications.

Closing

It is essential for students to understand the importance of triangles and their geometric elements, as these are fundamental in various areas, from architecture to engineering. The ability to identify and work with cevians and notable points not only enriches mathematical knowledge but also prepares students to face real challenges in their future careers.