Contextualization

Mathematics, far from being limited to abstract theory, plays a crucial role in the analysis and interpretation of the world we live in. This project will focus on one of these mathematical tools: Translations.

Translations, one of the isometric transformations, have a tangible practical application in our three-dimensional world. They describe the process of moving an object from one place to another in a two-dimensional or three-dimensional space without altering its orientation. The object maintains its shape and size, being only 'pushed' or 'pulled' to a new position.

It is a mathematical concept present daily in our lives. For example, when we move a piece of furniture from one room to another, we are performing a translation. You may not realize it, but when you open and close a drawer, you are executing a translation.

The study of Translations has substantial practical relevance. In the field of computer science, for example, they are intensively used in computer graphics to move and position objects on the screen. In physics, translations are the basis for concepts such as speed and acceleration. Furthermore, translations are also used in engineering and architecture to design and construct buildings and structures.

It is evident, therefore, the importance of this concept for various areas of knowledge. Studying and understanding it allows not only a deeper understanding of mathematics but also a clearer view of how our world is structured and functions.

Several resources can be used to study and better understand translations. Some of them are:

• Translations - KhanAcademy
• Translation - Mundo Educação
• Flat Geometry - Translations. Pure and Simple Mathematics - YouTube

Remember: mathematics is not just an abstraction, but a language to understand and shape the world around us. This is no less true for the concept of translations.

Practical Activity

Activity Title: 'Translating the World: The Mathematics of Translations'

Project Objective:

This project aims to provide a practical understanding of the application of mathematical translation in the real world. Students will apply the theory of translations to 'translate' or move an object on a two-dimensional grid. By applying this theory in practice, students' understanding of translations will deepen, as well as develop teamwork, time management, communication, problem-solving, and creative thinking skills.

Detailed Project Description:

Students will work in groups of 3 to 5 people. Each group will receive a sheet of graph paper, colored pens, and a numerical object (a square or circle made of cardboard that fits the grid). The work will be divided into two parts:

1. Moving the World: Students will be challenged to move the numerical object from an initial point to a final point on the grid, only through translations. They will have to record each translation performed, indicating the direction and distance of the translation.

2. Describing the Movement: Students will describe each movement on paper and the total trajectory of the object. They will also be encouraged to discuss how these translations relate to the movement of the object in real life.

Required Materials:

• Sheets of graph paper
• Colored pens
• Numerical objects (cardboard squares or circles)

Detailed Step-by-Step:

1. Group Division: Organize students into groups of 3 to 5 people.

2. Distribution of Materials: Provide each group with a sheet of graph paper, colored pens, and a numerical object.

3. Definition of Initial and Final Points: Mark an initial and final point on the grid for the object.

4. Route Planning: Ask students to plan the route that the object will follow from the initial point to the final point only through translations.

5. Translation Recording: As students begin to move the object, they should record each translation, noting the direction and distance of the translation.

6. Documentation of Trajectory: After the object reaches the final point, students should describe the total trajectory of the object, addressing the translations performed and how the concepts worked apply in the real world.

7. Document Writing: Finally, each group should write a document reporting their findings and reflections.

Project Deliverables and Connection to the Activity:

After completing the practical activity, students should write a document reporting their findings and reflections. This document should be divided into four parts: Introduction, Development, Conclusions, and Bibliography.

In the Introduction, students should briefly contextualize the concept of translation and its applied relevance in the real world. The planning of the object's route and the justification for it should be included here.

In the Development section, students should detail the methodology used to move the numerical object, a detailed description of the object's trajectory, a record of the translations performed, and a discussion of the results.

In the Conclusions section, groups should emphasize the skills learned, the difficulties encountered, how these were overcome, and what could be done differently. Additionally, they should discuss the overall meaning of their findings and the relevance of translations in everyday life.

Under Bibliography, all sources that the groups consulted throughout the project to understand the concept of translations and to write their document should be listed.

This project will, therefore, not only solidify students' knowledge of the concept of translations but also enhance their teamwork and communication skills, as well as apply the theoretical concepts covered in class in practice.