Contextualization

Translation is one of the most fundamental and common geometric transformations in mathematics and related disciplines. Basically, a translation is a displacement of an object from one place to another, maintaining its orientation and shape. For example, if you move a book from one table to another, you have performed a translation of the book.

In mathematics, we study translations because they help us understand how different parts of a system are connected. Translation is a powerful tool that allows us to simplify complex problems, as we can 'move' parts of the problem to a more convenient space for analysis.

Importance of Translation

Translation plays an important role in various areas such as Physics, Engineering, Computer Graphics, among others. The movement of objects without altering their shape or orientation is a fundamental concept in physics, for example. In computer graphics, translation is used to create animations and games, moving objects on the screen in a way that makes them appear to be moving in space.

Furthermore, the ability to see and understand geometric transformations, such as translation, allows us to have a better sense of space and be able to locate ourselves better in the world. This is useful in various contexts, from knowing where the bathroom is in a new building to being able to navigate in a new city or country.

Study Sources

To deepen your knowledge on the subject, we recommend the following sources:

• Book: 'Descriptive Geometry' by José Luís Santos. This book is a complete reference on the subject, covering translation and other geometric transformations.

• YouTube Video: 'Geometric Transformations - Matemática Rio' link. In this video, the teacher explains dynamically and in an easy-to-understand way how geometric transformations work.

• Web Page: 'Transformations in the plane' from the website Só Matemática link. This site provides a detailed and visual explanation of translations, as well as other types of transformations.

Remember: reading and studying the subject are essential for good performance in practical activities!

Practical Activity

Activity Title: 'Exploring Translations in the World Around Us and in the Universe of Video Games.'

Project Objective

The objective of this activity is to explore the concept of translation and other geometric transformations, their practical applications, and their importance in problem-solving. Additionally, this activity will allow students to develop skills in teamwork, critical and creative thinking, time management, communication, and proactivity.

Detailed Project Description

For the completion of this project, students must form groups of 3 to 5 members. They will investigate and document examples of translation and other geometric transformations in two distinct contexts: in the world around them (Civil Engineering, Architecture, Art, Nature, etc.) and in the universe of video games. Additionally, each group must program a small game simulation involving the use of translations and discuss their implications.

For the first part of the project, students will investigate, photograph (or draw), and document at least 5 examples of translations and other geometric transformations that can be found in their school or home, city, and even in works of art or nature. They must explain and justify why each example represents a translation or another transformation, and what effects or implications such transformations have on the objects or environments involved.

In the second part of the project, students must choose a well-known video game (online, console, mobile, etc.) and identify at least 5 examples of translations and other geometric transformations present in the game. This may include character movements, objects moving through the game environment, or even elements of the user interface. Students must document these examples with screenshots and a description of the identified transformations.

Finally, in the third part of the project, each group must program a small simulation of a mini-game involving the use of translations. They must explain how translation is used in the simulation and discuss the implications of their choices. The simulation can be written in any programming language that the students are comfortable with, such as Python, JavaScript, Processing, among others.

Required Materials

• Camera (or cell phone) for taking photos
• Computer with Internet access
• Text editing software (e.g., Microsoft Word, Google Docs)
• Software for programming the game simulation (e.g., Pygame for Python, p5.js for JavaScript)
• Paper and pen for notes

Step by Step

Step 1: Formation of groups and planning of activities. Define deadlines and task distribution among group members.

Step 2: Research and documentation of examples of translations and other geometric transformations in the real world. Each group must search for and document at least 5 examples.

Step 3: Research and documentation of examples of translations and other geometric transformations in video games. Each group must choose a video game and identify and document at least 5 examples.

Step 4: Development of the simulation of a mini-game involving the use of translations. Each group must program the simulation and prepare an explanation of the code and the choices made.

Step 5: Writing the final report, which must follow the following structure:

1. Introduction: contextualize the theme, its relevance and application in the real world and in the universe of video games, as well as the objective of this project.
2. Development: explain the theory behind translations and other geometric transformations, detail the activity, indicate the methodology used, and finally present and discuss the results obtained.
3. Conclusion: conclude the work by summarizing its main points, explaining the learnings obtained, and drawing conclusions about the project.
4. Bibliography: indicate the sources on which they relied to work on the project such as books, web pages, videos, etc.

Step 6: Each group presents their report and the game simulation to the class.

Project Deliverables

Each group must deliver:

• Photos or drawings and descriptions of examples of translations and other geometric transformations found in the real world.
• Screenshots and descriptions of examples of translations and other geometric transformations found in video games.
• Source code and documentation of the mini-game simulation.
• Final report printed and digital, following the proposed structure.