Contextualization

Symmetry in geometry is an important mathematical concept that helps us understand the world around us. Symmetrical shapes are appealing to the eyes and are everywhere, from nature to man-made constructions. Understanding and appreciating symmetry can enrich our appreciation for natural beauty and architecture.

In mathematics, symmetry involves the transformation of figures or areas so that they can still be considered the same object. These transformations can include reflection (as if it were a mirror), rotation, and translation (moving the figure without changing its orientation).

Introduction

For our project, we will focus on reflection or symmetry in relation to a line, which is the type of symmetry that creates a mirrored image. We will use the Cartesian Plane as our main tool, as it will allow us to visualize and manipulate our geometric figures clearly and precisely.

Reflecting a figure can be done in relation to the origin or any of the axes, producing a mirrored figure that is still the same figure but in a different position or orientation. By learning to reflect figures, we will explore important concepts of geometry and algebra, as well as develop spatial thinking skills.

We will use basic geometric figures such as triangles, squares, and rectangles, and learn how to calculate the vertices of their reflected images. This will be a useful skill for visualizing and solving more complex geometry and algebra problems in the future.

Practical Activity: Unraveling Flat Symmetry

Project Objective

The objective of this project is to enhance students' understanding of flat symmetry, in particular, the concept of reflection or symmetry in relation to a line on the Cartesian plane. Additionally, it aims to strengthen teamwork, time management, and communication skills.

Detailed Project Description

The work will be carried out in groups of 3 to 5 students. Each group will create a set of geometric figures: triangles, squares, and rectangles on the Cartesian plane and then find the symmetric of each figure in relation to the origin and the axes.

To challenge themselves, each group should calculate the sum of the new coordinates and the distance between the original vertex and the new one for each figure.

Required Materials

1. Graph paper to draw Cartesian planes.
2. Colored pencils to draw the figures and their reflections.
3. Rulers to create straight and precise lines.
4. Calculator for distance calculations.
5. Computers with internet access for research and report preparation.

Detailed Step-by-Step for Activity Execution

1. Divide into groups of 3 to 5 students.
2. Each group should draw a Cartesian plane on graph paper.
3. Next, the group should draw basic geometric figures (triangles, squares, and rectangles) on the Cartesian plane and note the coordinates of the vertices of each figure.
4. After drawing the figures, each group should calculate the vertices of the reflected figures in relation to the origin and the axes.
5. Draw the reflected figures on the Cartesian plane, using a different color for each original figure and its respective reflected image.
6. Then, calculate the sum of the new coordinates for each reflected figure and the distance between each original vertex and the new one.
7. Finally, students should discuss their findings among themselves and prepare a project report detailing their experiences and the concepts learned.

Project Delivery and Written Document

The project delivery will be in two formats:

1. Live Presentation: Each group will present their Cartesian plane, the original and reflected figures, explaining the calculations performed.

2. Written Document: Each group should prepare a project report. This report should include:

   • Introduction: In this section, students will contextualize the theme of flat symmetry, its relevance and real-world application, and the objective of this project.
   • Development: Here, students must explain the theory behind reflection or symmetry in relation to a line, explain the activity in detail, indicate the methodology used, and finally present and discuss the results obtained.
   • Conclusion: Students should conclude the work by summarizing their main points, explaining the learnings obtained, and drawing conclusions about the project.
   • Bibliography: Indication of the sources they relied on to work on the project, such as books, web pages, videos, etc.

This work will allow students to understand the theory of flat symmetry through practice and also enhance teamwork, time management, and communication skills.