Lesson Plan | Active Learning | Geometric Constructions

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 crucial for establishing a clear and directed focus for the lesson. By defining specific objectives, the teacher guides students on what is expected of them to learn and apply. This section also serves to align expectations and ensure that all participants are engaged in the learning process. The main objectives are outlined based on the skills students need to develop and the competencies that will be enhanced during the lesson.

Main Objectives:

1. Empower students to understand and apply algorithms and flowcharts for constructing medians and angle bisectors.

2. Develop skills for constructing specific angles (30º, 45º, 60º) and regular polygons using geometric methods.

Side Objectives:

1. Foster critical thinking and problem-solving through the practical application of geometric concepts.
2. Encourage collaboration and communication among students during practical activities.

Introduction

Duration: (15 - 20 minutes)

The introduction serves to engage students with the content they studied previously, using problem situations that prompt them to think critically and apply their knowledge practically. Furthermore, the contextualization aims to illustrate the relevance of the topic in daily life, enhancing interest and the perceived importance of studying geometric constructions.

Problem-Based Situations

1. Imagine you are an architect and need to design a park with paths that intersect at specific angles of 30º, 45º, and 60º. How would you use geometric constructions to ensure that these angles are exact?

2. An interior designer wants to create a new room with a ceiling formed by a regular polygon. How could you ensure that all sides of the polygon are equal and that the angles are precisely equal?

Contextualization

Geometric constructions are not just theoretical tools; they have practical applications in various fields such as architecture, engineering, and design. For example, using specific angles is crucial in architecture to ensure that structures are stable and visually pleasing. Additionally, understanding how to construct regular polygons can aid in the design of objects that require symmetry, such as logos and building embellishments.

Development

Duration: (70 - 75 minutes)

The Development stage is designed to allow students to practically and creatively apply the concepts of geometric constructions they studied previously. By working in groups, they not only solidify their theoretical understanding but also develop collaboration and communication skills. The proposed activities are contextualized and challenging, encouraging students to think critically and solve problems innovatively.

Activity Suggestions

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

Activity 1 - The Lake of Angles and Polygons

> Duration: (60 - 70 minutes)

- Objective: Apply knowledge of geometric constructions to solve a practical park design problem, developing skills in precision and creativity.

- Description: In this activity, students will be challenged to design an artificial lake in a park, considering the creation of islands in the shapes of regular polygons and paths that intersect at specific angles of 30º, 45º, and 60º. They will use techniques of geometric constructions to draw the layout of the lake, ensuring that the angles and shapes are precise.

- Instructions:

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

• Each group will receive a blueprint of the area where the lake will be built.

• Students should use protractors, rulers, and compasses to draw the medians and angle bisectors needed for the desired angles and shapes.

• After constructing the medians and angle bisectors, students should use techniques for constructing regular polygons to draw the islands of the lake.

• The intersecting paths must form the specified angles, and students must ensure that these are symmetrical and precise.

Activity 2 - The Math Magical City

> Duration: (60 - 70 minutes)

- Objective: Develop technical drawing skills and the application of geometric concepts in an urban design context, fostering collaboration and critical thinking.

- Description: Students will design a small city using concepts of geometric constructions to create buildings with symmetrical facades and specific angles. Each group will be required to draw blueprints of the buildings, incorporating angles of 30º, 45º, and 60º and shapes of regular polygons.

- Instructions:

• Form groups of up to 5 students and distribute the necessary materials (paper, pencil, ruler, compass, protractor).

• Each group chooses a type of building (residential, commercial, cultural) and must draw the blueprint of the building, incorporating the requested angles and geometric shapes.

• Students must use techniques for constructing medians and angle bisectors to ensure that the internal angles and facades are precise.

• Present the completed blueprints in a digital or physical model, highlighting the application of geometric concepts.

Activity 3 - The Challenge of Magic Polygons

> Duration: (60 - 70 minutes)

- Objective: Promote understanding of the properties of regular polygons and develop manipulation and creativity skills in constructing geometric figures.

- Description: In this challenge, students will create a set of figures using exclusively regular polygons that fit perfectly without leftovers or empty spaces. They will need to use their skills in geometric construction to cut and fit the shapes so that they occupy the entire available space.

- Instructions:

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

• Provide each group with a set of regular polygons (triangles, squares, pentagons), all with equal length sides.

• Students should use scissors and glue to cut and manipulate the polygons to form figures that fit perfectly, leaving no empty spaces.

• The groups must present their figures, explaining the construction process and the geometric properties involved.

Feedback

Duration: (15 - 20 minutes)

The purpose of this stage is to consolidate learning, allowing students to articulate acquired knowledge and share their experiences and discoveries with peers. The group discussion helps develop communication and argumentation skills, as well as providing additional insights into the practical application of geometric construction concepts. This collective reflection also serves to assess students' understanding and identify any areas that may need review or additional explanation.

Group Discussion

To start the group discussion, the teacher should ask each group to briefly present what they constructed and explain the thinking process behind their creations. The discussion questions can be directed for students to reflect on the challenges encountered, the creative solutions they applied, and how the concepts of geometric constructions can be utilized in real situations. Additionally, it is important for students to discuss the differences and similarities between the projects of the different groups.

Key Questions

1. What were the biggest challenges in applying the concepts of medians, angle bisectors, and the construction of angles and polygons in the activity?

2. How did precision and attention to detail influence the final result of your project?

3. In what ways can the concepts of geometric constructions be used in other subjects or everyday situations?

Conclusion

Duration: (10 - 15 minutes)

The purpose of the Conclusion is to solidify the knowledge acquired by students, linking the theory studied with the practice carried out during the lesson. Additionally, it aims to emphasize the applicability and importance of geometric constructions in practical and theoretical contexts, reinforcing students' perception of the usefulness of what they have learned. This stage also serves to assess students' understanding and encourage them to continue exploring and applying mathematical concepts in their lives.

Summary

In this final stage, the teacher should summarize the main topics addressed regarding geometric constructions, reaffirming the construction techniques for medians, angle bisectors, specific angles (30º, 45º, 60º), and regular polygons. It is crucial to recap how these concepts were applied in practical activities, such as the creation of a park with a lake and paths, a city with symmetrical buildings, and a composition of regular polygons.

Theory Connection

Today's lesson was structured to effectively connect theory and practice. Through practical activities and discussions, students were able to directly apply the geometric concepts studied, using drawing and construction tools to solve real problems and design creative solutions. This approach not only solidified theoretical learning but also demonstrated the importance and utility of geometric constructions in practical and professional contexts.

Closing

Finally, it is important to highlight the relevance of geometric constructions in everyday life. The ability to think and act geometrically is essential in various fields, from product design and architecture to engineering and technology. Understanding and applying these concepts not only enriches students' mathematical knowledge but also prepares them to face real challenges in their future careers and personal lives.