Contextualization

Translations are one of the fundamental concepts in the study of geometric mathematics. They are used to describe the movement of a figure on a plane, without altering its shape or size. When we perform a translation, we move a figure to a new position, without rotating it or changing its size. In the real world, this concept can be observed in various situations, such as in the planning of moving large objects, in graphic design, and even in the movements of tectonic plates!

The concept of translation is useful in various fields, such as physics, computer science, engineering, and life sciences. Understanding it well is essential for many studies and careers. Furthermore, understanding translations is also fundamental for the development of spatial skills and the comprehension of more complex concepts in geometry.

Introduction

Translation is a geometric transformation in which an object is moved, or translated, on a plane. In simple terms, if we take a figure and move it to a new location without rotating or changing its size, we perform a translation. An important distinction to make is that, during a translation, the figure does not rotate, change size, or change shape: it simply moves.

A translation is defined by the direction and the distance that the figure is moved. This is called the translation vector. The translation vector has both the direction and the magnitude (or length) of the translation. Every point in the figure is moved in the direction and by the distance defined by the translation vector.

Translations are an example of a rigid transformation, or isometry, which means they maintain the size and shape of the original figure. This means that any measurement you make on the original figure, such as side lengths, angles, areas, or perimeters, will be the same in the translated figure.

To deepen and base yourself for the project, I suggest the following resources:

1. Wikipedia Article on Translation (geometry)
2. Khan Academy Video on Translations
3. Só Matemática Article on Translations

Practical Activity

Title: "A Walk Through Translations"

Project Objective

The objective of this project is to apply the concept of translation in creating an artistic drawing that represents a story or theme chosen by each group. In this activity, students will practice the understanding and application of translations in mathematics, while also enhancing their teamwork, creativity, and communication skills.

Detailed Project Description

1. Planning Phase

Each group must choose a story or theme that will be represented through the drawing. The theme can be free, but it must allow for frequent use of translations to create the drawing.

2. Drawing Phase

Students must create a drawing applying translations to the objects that make up the scene. They must identify the translation vectors used and justify their choices.

3. Presentation Phase

Each group will present their drawing to the class, explaining the story or theme they chose and justifying the translations used.

Necessary Materials

• Drawing paper
• Pencils and erasers
• Ruler
• Compass
• Grid paper

Detailed Step-by-Step

1. Group Formation: Students should divide into groups of 3 to 5 people.
2. Theme Selection: Each group must choose a story or theme for their drawing.
3. Sketch Creation: Students must plan and sketch their drawing. At this stage, they must identify which objects in the drawing will be moved by translation and define the translation vectors for each one.
4. Final Drawing: Students must draw the final image on paper, applying the planned translations.
5. Presentation Preparation: Groups must prepare a brief explanation of their drawing and the translations used.
6. Presentation: Each group must present their drawing to the class, explaining the story or theme they chose and the translations used.

Project Deliverables

Students must deliver the finalized drawing and a written report.

The report should follow the following structure:

1. Introduction: Students must contextualize the chosen theme, explain its relevance and real-world application, and indicate the project's objective.
2. Development: Students must explain the theory of translations addressed, detail the activity carried out, describe the methodology used (including the translation vectors and the justification of their choice), and present and discuss the results obtained.
3. Conclusion: Students must conclude the work by summarizing its main points, discussing what they learned through the project, and the conclusions drawn. It is important that the learnings about translations and the socio-emotional skills exercised are highlighted.
4. Bibliography: Students must indicate the sources they used to work on the project, whether they are books, web pages, videos, etc.

Remembering that the main objective of the report is to complement the practical activity, therefore, it should explain in detail the translations used in the drawing, the chosen translation vectors, and how they were applied.