Contextualization

Geometric constructions are an important part of mathematics, with roots dating back to great ancient Greek mathematicians like Euclid. He used geometric constructions to aid in the understanding of mathematical concepts, and this is still relevant today. Geometric constructions involve creating precise figures and shapes using only a ruler and a compass, and they form the basis for understanding many concepts in geometry and even in other areas of mathematics.

Geometry plays a crucial role in many aspects of our daily lives and in various professions. For instance, architects, artists, game designers, and civil engineers use geometric principles in their daily work. It is also useful in teaching valuable skills such as problem-solving and critical thinking. Furthermore, one could argue that it helps develop an appreciation for the beauty and harmony found in mathematics.

Geometric constructions allow us to explore and understand mathematical concepts in a more visual and concrete way. For example, by constructing an equilateral triangle with a ruler and compass, we are not only creating a shape but also exploring important mathematical ideas, such as the concept of equality and the properties of triangles.

Geometric construction is a fascinating field of mathematics that takes us back to antiquity but remains relevant today. Understanding and being able to perform geometric constructions opens the doors to a better understanding of more advanced mathematical concepts and allows us to appreciate the intrinsic beauty of mathematics.

The sources that can be used to delve deeper into the subject and assist in project implementation are:

1. Book "Elements", by Euclid. Available online and free at: Euclid - Elements
2. YouTube channel "Matemática Rio with Prof. Rafael Procopio" has a list of videos explaining geometric constructions: Matemática Rio
3. Article "The Use of Geometric Constructions in Mathematics Education", available at: Article
4. Website "Prof. Luiz Antonio" has examples of constructions and activities for practice: Prof. Luiz Antonio
5. Website "Mundo Educação" has a section dedicated to geometric constructions: Mundo Educação

Practical Activity: "Shaping the World: Interactive Geometric Constructions"

Activity Title:

"Building a Geometric City"

Project Objective:

The objective of this project is for students to build a "Geometric City" from geometric constructions using their knowledge and skills in mathematics and geometry.

Detailed Project Description:

Students will be divided into groups of 3 to 5 people. Each group will be tasked with designing and building a "Geometric City" using paper, pencils, ruler, compass, and, if preferred, dynamic geometry software like GeoGebra.

The "Geometric City" should include a variety of elements, such as buildings (in the form of regular polygons), streets (mediatrices), avenues (bisectors), squares (circles), among other options that the group chooses. It is essential that each chosen element be constructed from studied geometric concepts, such as angles of 90°, 60°, 45°, and 30°, mediatrices, bisectors, and regular polygons.

All decisions and project development must be documented, going through the rationale for choosing the elements, the construction process, challenges encountered, solutions found, and lessons learned.

Necessary Materials:

• Paper;
• Pencils;
• Ruler;
• Compass;
• Dynamic Geometry Software (optional);
• Internet for research (optional).

Detailed Step-by-Step for Activity Execution:

1. Group Formation and Initial Planning: Students must form their groups and, together, discuss and plan how their "Geometric City" will be. Remembering that teamwork is an essential skill to be developed.

2. Choice of Elements and Their Constructions: Each group must decide which geometric elements they will use to create the different components of the city.

3. Construction of the 'Geometric City': Based on the decisions made in the previous step, the groups must start building the elements, either on paper using ruler and compass, or using dynamic geometry software.

4. Documentation: Throughout the process, groups must document from the initial conception, choice of elements, rationale used, challenges encountered, solutions adopted, to final conclusions.

5. Presentation: At the conclusion of the project, each group must present their "Geometric City" to the class, explaining the construction process and the geometric concepts used.

The deliverables of this project will be:

• The built "Geometric City". If it is a virtual construction, the files must be submitted. If it is a physical construction, photos or videos can be used for documentation.
• The written document, with the Introduction explaining the project and its relevance, the Development detailing the theory, methodology, activity, and result, and the Conclusion, synthesizing the learning and appreciation of the experience. The Bibliography used should also be included.