Lesson Plan | Active Learning | Regular Polygons: Angles and Diagonals

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 minutes)

The objectives stage is crucial to guide both the teacher and the students on what is expected to be achieved by the end of the lesson. Clearly establishing objectives helps focus activities and discussions in the classroom, ensuring that time is well utilized and that students can consolidate and apply the prior knowledge acquired about regular polygons. Furthermore, this clarity of goals facilitates the assessment of learning and the progress of students during the lesson.

Main Objectives:

1. Empower students to recognize and identify regular polygons, understanding their characteristics and properties.

2. Develop the ability to calculate the number of diagonals that pass through the center of a regular polygon and the internal and external angles of these polygons.

Introduction

Duration: (15 minutes)

The introduction serves to engage students with the content they have studied previously in an active way, using problem-based situations that stimulate reflection and the application of theoretical knowledge in practical, everyday contexts. Furthermore, by contextualizing the importance of regular polygons in real situations and history, students can see the relevance of the subject studied and feel motivated to explore mathematical applications more deeply.

Problem-Based Situations

1. Imagine you are planning a winter garden and want to use tiles to create a floor pattern with regular polygons. How would you decide on the quantity of tiles needed to form each pattern and which angles would be most suitable to create a visually pleasing design?

2. An architect is designing the facade of a new building and is considering using regular polygons in some decorative elements. He wants to know how many connection points there will be if he uses each vertex as a central point for the construction of new structures. How could he calculate this for different polygons?

Contextualization

Regular polygons are fundamental not only in mathematics but also have practical and aesthetic applications in design, architecture, and art. For example, the use of tile patterns in Islamic mosques is a classic example of how regular polygons are used to create complex and symmetrical designs. Additionally, understanding the properties of regular polygons assists in solving practical problems, such as calculating the amount of material needed to cover a surface or determining the stability of a structure based on its symmetry and force distribution.

Development

Duration: (75 - 80 minutes)

The development stage is designed to allow students to apply theoretical knowledge about regular polygons in a practical and interactive way. By working in groups, they not only reinforce individual learning but also develop social and communication skills. Each proposed activity is structured to challenge students to think critically and solve problems creatively, utilizing the mathematical concepts studied. The choice of a single activity allows for a focus on the depth of learning, ensuring that students can explore the topics in a meaningful way.

Activity Suggestions

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

Activity 1 - Polygon Olympics

> Duration: (70 - 75 minutes)

- Objective: Apply knowledge about regular polygons to solve practical and theoretical problems, developing calculation and teamwork skills.

- Description: In this activity, students will be divided into groups of up to five people and will participate in a competition to solve problems related to regular polygons. Each group will receive a set of cards with different polygons and must calculate the number of diagonals that pass through the center of each one, as well as the internal and external angles. The cards will also include practical challenges, such as calculating the area of a garden with a polygonal tile pattern.

- Instructions:

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

• Distribute sets of cards with different polygons and instructions.

• Each group must solve the proposed questions on the cards, noting their answers and the calculations used.

• After completion, each group will present their findings to the class, explaining the methods used and the solutions found.

• Facilitate a class discussion to compare the methods used by the different groups and clarify any doubts.

Activity 2 - Polygon Builders: The Art of Geometry

> Duration: (70 - 75 minutes)

- Objective: Promote practical understanding of the properties of regular polygons and stimulate creativity and teamwork.

- Description: Students, in groups, will receive construction materials (sticks, modeling clay, ruler) to build models of regular polygons. They must calculate and represent the diagonals that pass through the center of each polygon, as well as measure and calculate the internal and external angles. The final challenge will be to create a mural with the constructed polygons, applying concepts of symmetry and aesthetics.

- Instructions:

• Organize students into small groups.

• Provide each group with the necessary materials to build the polygons.

• Students should build the polygons, calculate the diagonals, and measure the angles.

• Each group presents its polygon and calculations to the class.

• Together, the groups will create a mural with the polygons, exploring different arrangements and symmetries.

Activity 3 - Detectives of the Missing Diagonals

> Duration: (70 - 75 minutes)

- Objective: Develop logical reasoning skills and application of mathematical concepts to solve complex geometry problems.

- Description: In this investigative game, students will have to unravel a 'mathematical mystery' involving locating 'missing' diagonals in different regular polygons. They will use clues based on angle calculations and symmetry to determine the possible missing diagonals and justify their answers.

- Instructions:

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

• Give each group a set of 'files' containing partial information about the polygons and the diagonals that may be missing.

• Students must use the information to calculate where the missing diagonals could be and justify their decisions based on calculations and logical reasoning.

• Each group presents their conclusions and reasoning process to the class.

• Discuss the different approaches and solutions found by the groups.

Feedback

Duration: (15 - 20 minutes)

This feedback stage is essential for consolidating students' learning, allowing them to reflect on the practical and theoretical activities performed. Group discussion helps to identify points of confusion or misunderstandings, which can be clarified by the teacher or by fellow students. Furthermore, by sharing their experiences, students can learn from each other and see different perspectives and approaches to solving the same problems, enriching their learning and promoting a deeper understanding of the mathematical concepts covered.

Group Discussion

Start the discussion with a quick review of the main concepts covered in the activities, asking each group to share their findings and challenges overcome. Encourage students to discuss the different approaches used and how they varied among groups. A friendly competition can also be introduced to see which group came closest to the correct answers, encouraging the exchange of learnings and strategies.

Key Questions

1. What were the biggest challenges in calculating the diagonals and angles of the polygons during the activities?

2. How did the symmetry of the polygons help or complicate the calculation of the diagonals and angles?

3. Is there any practical situation where you can imagine applying the knowledge about regular polygons learned today?

Conclusion

Duration: (5 - 10 minutes)

The conclusion stage is crucial to reinforce learning, ensuring that students can consolidate the knowledge acquired throughout the lesson. Summarizing key points helps to solidify information, while discussing the practical application of concepts encourages students to view mathematics not just as an academic discipline but as a powerful and relevant tool in their lives.

Summary

To conclude, the teacher should summarize the main points addressed concerning regular polygons, recapping the calculation of the number of diagonals that pass through the center, as well as the internal and external angles. It is essential to reinforce how these concepts apply in practical and theoretical situations, highlighting the different activities carried out by the students.

Theory Connection

During the lesson, the connection between the mathematical theory of regular polygons and its practical application was established through interactive activities and real contextualizations. Students were able to see how the theory previously studied translated into practice through problem situations that mimic real contexts, such as planning a winter garden or the architecture of a building.

Closing

Finally, it is important to highlight the relevance of regular polygons in everyday life, whether in architecture, design, or other areas that involve concepts of symmetry and geometric calculation. Understanding these concepts not only broadens students' mathematical knowledge but also prepares them to apply these principles in their future careers and in solving everyday problems.