Contextualization

In our daily lives, we often encounter repeating patterns, whether in nature, art, the objects we use, and even in our bodies! These repeating patterns are called symmetries.

Symmetry is a mathematical concept that studies the harmony and balance of shapes. It is based on the existence of an axis, a center, or a plane that divides a figure into equal parts. For example, if you think of a heart, you will notice that it is symmetrical, meaning we can fold it in half and the two parts will perfectly coincide.

Symmetry is present in many areas of knowledge, such as Biology (symmetries in living beings), Physics (symmetries in natural laws), Music (symmetries in rhythm and melody), and of course, Mathematics (symmetries in geometric figures). And it is precisely in Mathematics that we will focus our study on symmetry!

Introduction

Symmetry in the Cartesian plane is an essential concept for understanding how figures behave when rotated or reflected. In the Cartesian plane, we have two perpendicular axes, the X-axis and the Y-axis, which divide the plane into four quadrants. Each point in the plane is represented by an ordered pair (x, y), where x is the coordinate on the X-axis and y is the coordinate on the Y-axis.

Symmetries in the Cartesian plane can be classified into two types: symmetry with respect to the X-axis and symmetry with respect to the Y-axis. Symmetry with respect to the X-axis occurs when, upon reflecting a figure with respect to this axis, the resulting figure coincides with the original figure. Similarly, symmetry with respect to the Y-axis occurs, but with the reflection happening with respect to the Y-axis.

Symmetry in the Cartesian plane has important applications in various areas, such as Geometry, Physics, and even Computing! Therefore, understanding this concept is fundamental for the development of logical reasoning and for the comprehension of other mathematical topics. In our project, we will explore and apply these concepts in a practical and fun way!

Practical Activity

Activity Title: "Creation of a Symmetrical Mural"

Project Objective:

• Understand the concepts of symmetry in the Cartesian plane.
• Apply the concepts of symmetry in practice, through the creation of a symmetrical mural.
• Develop the ability to work in a group, solve problems, and think creatively.

Detailed Project Description:

In this activity, students will work in groups to create a symmetrical mural. Each group will receive a large cardboard paper and a variety of art materials. The task will be divided into two stages. In the first stage, students will plan the design of the mural, creating a symmetrical image on the paper. In the second stage, they will color and decorate the mural following the planned design.

Required Materials:

• Large cardboard paper (one per group)
• Pencils
• Ruler
• Eraser
• Colored pencils
• Markers
• Glue
• Scissors
• Magazines (for cutting and pasting, if desired)

Detailed Step-by-Step:

First Stage: Mural Planning (2 days)

1. Group Discussion: On the first day, the group should discuss and decide on a theme for the mural. It can be something abstract (like "the universe") or something more specific (like "a garden of flowers").

2. Drawing the Plan: After deciding on the theme, students should draw a plan on the cardboard paper. The plan should show how the mural will be divided into two symmetrical parts (with respect to an axis or the center).

3. Drawing the Image: Within each part, students should draw an image that represents the chosen theme. Remember, the two images should be symmetrical.

4. Review and Adjustments: After finishing the drawing, students should review and make adjustments if necessary. Make sure that both parts of the mural are symmetrical!

Second Stage: Mural Creation (3 days)

5. Outline and Color: On the second day, the group should outline the images with black markers or colored pencils. Then, they can color the mural using colored pencils, markers, and other art materials.

6. Decoration: Finally, students can decorate the mural as they wish, using magazine cutouts, glitter, stickers, etc. Remember, the decoration should also be symmetrical!

7. Final Presentation: At the end of the third day, each group will present their mural to the class. During the presentation, students should explain how they used the concepts of symmetry in the planning and creation of the mural.

Delivery Format:

Each group must deliver the complete mural, which will be displayed in the classroom. Additionally, students should prepare a brief presentation (5 to 10 minutes) about the mural creation process, explaining how they used the concepts of symmetry and what challenges they faced.

This project will be of great importance, as it will allow students to apply, in practice, the concepts of symmetry in the Cartesian plane, while developing skills such as teamwork, problem-solving, and creative thinking. Good luck!