Contextualization

Geometry is one of the oldest areas of mathematics, originating from the land measurement practices and astronomical calculations of the ancient Egyptians and Babylonians. Over the centuries, its application and study have expanded vastly, being used today in numerous scientific, technological, and artistic fields, from engineering to graphic design.

Translations and symmetries play a fundamental role in geometry and have practical applications in many fields of human knowledge. You can find translations in tile patterns, floors, wallpapers, and even in logo design. A good understanding of the concept of translations will allow you to appreciate and create geometric patterns, as well as enhance your spatial reasoning skills.

Theoretical Introduction

The term 'translation' comes from the Latin 'translatio', which means 'to carry across'. In mathematics, a translation is a geometric transformation that moves each point of a figure or space the same distance in a specific direction. It is important to emphasize that the figure does not change, it just moves to a different position in space.

Translation preserves the size and shape of the figure, as well as the respective distances and angles. This means that the original figure and the figure after the translation are congruent. You can visualize a translation as a 'slide' of a figure along a line.

In terms of coordinates, a translation moves each point a constant distance in the direction of the x-axis (horizontal) and the y-axis (vertical). For example, if a point (x, y) is translated 3 units to the right and 2 units up, its new coordinates will be (x + 3, y + 2).

Practical Activity: The World of Translation Through Art and Architecture

Project Objective

This project aims to apply the acquired knowledge about translation in creating an artistic geometric pattern that represents a work of art or architectural element. Students should explain the theory behind the translation process and its results, as well as the socio-emotional skills acquired throughout the project.

Detailed Project Description

Students will work in groups of 3 to 5, where each group will choose a known work of art or architectural element that contains geometric patterns that can be represented by translations. They will have to reproduce it, highlighting and explaining the translation patterns present in the choice.

Required Materials

• Pencil and eraser
• Ruler and Compass
• Graph paper
• Computer with Internet access
• Dynamic geometry software (such as GeoGebra)

Activity Steps

1. Research and Selection of the Artwork: Start the project by researching and selecting a work of art or architectural element that contains geometric patterns that can be represented by translations. Justify the choice.

2. Identification of Patterns: Identify and explain the translation patterns present in the chosen artwork.

3. Reproduction of the Artwork: Use graph paper and dynamic geometry software to reproduce the chosen artwork or architectural element, highlighting the translation patterns.

4. Discussion and Presentation: Discuss the observations and reactions of the group members regarding the project. Prepare a presentation for the class, including photos, diagrams, and drawings, explaining the detected translation patterns and how they were reproduced.

5. Report Development: Prepare a written report to accompany the presentation, explaining the work done, the applied translation theory, and the conclusions drawn. The report should follow the structure indicated earlier: Introduction, Development, Conclusions, and Bibliography.

Project Deliverables

• Presentation: Each group must prepare and deliver a visual presentation (PowerPoint, Google Slides, or similar) explaining the translation patterns in the chosen artwork or architectural element, including images, diagrams, and graphics.

• Graphic Reproduction: Each group must deliver the graphic reproduction of the chosen artwork or architectural element, clearly showing the translation patterns.

• Written Report: Each group must write a well-founded report explaining the work done, the applied translation theory, the detected patterns, the graphic reproduction, and the conclusions drawn. The introduction should contextualize the selected artwork or element and explain the importance of translations in mathematics and art or architecture. The development should explain in detail the process of identifying and reproducing the translation patterns. The conclusion should summarize the main points, explain the learnings obtained, and the conclusions drawn about the project. The bibliography should indicate all sources used in the project.