Introduction

Mathematics is a discipline that goes beyond numbers; it is present in many situations in our daily lives and is applied in different knowledge areas. Translations, specifically, are an indispensable concept of geometry that help us to understand structures, shapes, and movements.

We will begin our immersion into the world of translations by revisiting the concept of Cartesian coordinates, because this will be our starting point. Cartesian coordinates, or the Cartesian system, are a point of reference that helps us to locate a point on a plane (2D) or in space (3D) based on distances from a starting or reference point, generally indicated as (0,0) or (0,0,0). This concept is fundamental for us to understand the notion of translation.

A translation, in mathematics, is a type of geometric transformation involving the "sliding" or "movement" of a figure on a plane or in space. These movements are characterized by a direction and a distance that the figure is displaced. It is important to note that during a translation, the shape and size of the figure do not change, only its position.

We will also address the topic of translational symmetry. Translational symmetry is a type of symmetry in which a shape or pattern can be translated (or moved) a certain distance in a certain direction and still appear the same. This type of symmetry is very common in nature and in art, such as in the repetitive patterns on a carpet or the design of certain types of tiles.

Relevance of Translation in the Real World

The study of translations has a wide range of practical applications. In civil engineering and architecture, for example, it is necessary to understand the concept of translation in order to study the movement of structures. In physics, translation is used to describe the movement of particles. In the world of information technology, translation is used in computer graphics to move objects from one location to another, and in digital games to create the illusion of movement. And many other applications!

The fundamentals of translation are also used in the area of art. For example, in Escher's paintings, we can appreciate the translations of the figures that compose his works. Therefore, understanding translation can open up new doors to artistic appreciation and creativity.

Practical Activity

Activity Title: "Art of Translations: Creating Your Own Artwork Using Translational Symmetry"

Project Goal

The objective of this project is for students to deepen their understanding of Translations, Translational Symmetry, Cartesian Coordinates, and their practical applications in the creation of artwork, while developing their technical and socio-emotional skills, such as creativity, teamwork, time management, and communication.

Detailed Description of the Project

The students will have to create an artwork that exhibits translational symmetry, using their acquired knowledge of the concept of translation and Cartesian coordinates. This artwork can be a drawing, painting, collage, mosaic, digital work, or any other type, so long as it demonstrates the application of translational symmetry.

Materials needed

1. Paper for drawing or a canvas (if they choose to paint).
2. Pencils, colored pencils, pens, paints, or any other materials for drawing or painting that they want to use.
3. Ruler, compass, and protractor to assist in the creation of the figure and the translation.
4. Dynamic geometry software, such as GeoGebra (optional, and if they choose digital art).

Detailed step-by-step process for carrying out the activity

1. Form groups of 3 to 5 students.

2. Each member of the group should individually research translation, translational symmetry, and Cartesian coordinates in art, architecture, or nature. The references provided at the beginning of this project can be a good starting point.

3. The members of the group should come together to share and discuss their findings, and then jointly decide on a design or pattern that they would like to create that demonstrates translational symmetry.

4. The students should create a sketch of their design, marking the Cartesian coordinates and indicating the directions and distances of the translations that they will use to create the translational symmetry pattern.

5. Based on the sketch, the students should create the final artwork, making sure that each translation is correctly represented.

6. Each group should prepare a presentation explaining their artwork to the class, detailing the translations used, how the Cartesian coordinates were used, and what they learned by applying these mathematical concepts to art.

Project Deliverables and Document Preparation

Each group must submit the following documentation, in addition to the created artwork:

1. Introduction: A brief description of the chosen pattern or design, the relationship between the pattern and the concept of translation and translational symmetry, and the relevance of the study of translations.

2. Development: A detailed description of the process used to create the artwork, including how the sketch was created, how the Cartesian coordinates were used to plan the translations, and the directions and distances of the translations used. The students should also explain how they worked as a team, how they managed their time, and the challenges they faced.

3. Conclusions: A discussion of what they learned throughout the project, both about translations and Cartesian coordinates, and about teamwork, creativity, and time management.

4. Bibliography: A list of the sources consulted during the project. Each group should cite at least 4 sources, including books, web pages, videos, etc.

The groups should also present their artwork and share their findings and learning with the class.

I hope that this project deepens your understanding of translations and inspires your creativity, while fostering collaboration, communication, and time management.