Contextualization

Mathematics has a very interesting branch called geometry. Many of you may have heard of it or even studied it a bit. However, within geometry, we have a specific type called transformation geometry that studies the change in position of figures on the plane. One of these transformations is reflection.

Reflection in geometry is like looking at a figure through a mirror. The image you see is the reflection of the original figure. One of the fundamental concepts in reflection is the axis of symmetry, which is the "imaginary line" that you could draw through the figure where, if you folded the figure along this line, both parts would match exactly.

Reflection is not only an interesting topic but also a fundamental concept in mathematics. It is used in many fields of knowledge, such as physics, engineering, computer programming, and even in art. Furthermore, these reflection concepts are used in many aspects of everyday life, such as in building houses, bridges, in movie and videogame production, and even in the manufacturing of mirrors and lenses.

Understanding reflection and its properties can open many doors for you. Not only in terms of potential careers but also in terms of problem-solving skills and critical thinking. Additionally, since reflection is a visual concept, it is an excellent way to learn and better understand mathematics in a more concrete way.

To assist in your studies, you can use the following resources available in Portuguese:

1. Book: "Fundamentos de Matemática Elementar: Geometria Plana - volume 9", Gelson Iezzi and Carlos Murakami.
2. Website: Só Matemática This link provides a clear and comprehensive explanation of the concept of reflection in geometry.
3. Video: Geometric Reflection, channel "Matemática Rio with Prof. Rafael Procópio" on YouTube. This video presents an animated explanation of reflection with practical examples.

Practical Activity

Activity Title

"Reflections in Practice: An Interdisciplinary Approach"

Project Objective

The objective of this project is to explore, understand, and practically apply the concept of reflection in geometry. In addition to mathematics, it will involve aspects of art and technology, making it an interdisciplinary activity.

Detailed Project Description

Students, in groups of 3 to 5, will develop a mosaic representing an original work of art or emblem using graph paper and colored pens. This work must be symmetrical, using the concept of reflection.

The groups must then digitize their final work and, using dynamic geometry software (such as Geogebra), demonstrate the reflection and symmetry lines in their art.

As a final part of the project, students must present an analysis of a famous work of art, demonstrating the reflections and symmetries present. This analysis will be presented in the form of a written report and an oral presentation.

Required Materials

• Graph paper
• Colored pens
• Scanner or camera to digitize the art
• Computer with dynamic geometry software (Geogebra or similar)
• Research material (books, internet) for the analysis of the artwork

Detailed Step-by-Step for Activity Execution

1. Form groups of 3 to 5 students.
2. Choose a symmetrical figure to be the theme of your mosaic. This figure can be a work of art, an emblem, a logo, a complex geometric figure, etc.
3. Draw the chosen figure on graph paper using colored pens. Remember that the figure must be symmetrical.
4. Digitize the created image using a scanner or a camera.
5. Use the dynamic geometry software to import the digitized image and draw the reflection lines and highlight the symmetries present in the figure.
6. Select a famous work of art and research its symmetry and reflections present.
7. Write a report, following the suggested structure: Introduction, Development, Conclusions, and Bibliography used.
   • Introduction: Provide context for the chosen artwork, the relevance of symmetry in it and how it applies in the real world, as well as the objective of this project.
   • Development: Explain the theory of reflection and symmetry, describe the activity performed in detail, indicate the methodology used, and finally present and discuss the results obtained. Include screenshots from the geometry software to illustrate the points of reflection and symmetry.
   • Conclusions: Conclude the work, revisiting its main points, elaborating on the learnings obtained, the skills developed, and the conclusions drawn about the project.
   • Bibliography: List the sources you relied on to work on the project such as books, web pages, videos, etc.
8. Prepare an oral presentation of the work done and the findings.

Project Deliverables

• Mosaic drawn on graph paper.
• Digitized image of the mosaic and image of the mosaic with reflection and symmetry lines drawn using dynamic geometry software.
• Written report on the activity performed and analysis of the chosen artwork.
• Oral presentation of the project.

In their written report, students should describe the step-by-step of their practical activity, explaining how they drew the figure, how they digitized it, and how they used the geometry software to demonstrate the reflection and symmetry. Additionally, they should discuss the selected artwork, presenting their analysis of the reflections and symmetries present in it.