Contextualization

Mathematics, as we know it, goes far beyond numbers and calculations. It is also the science that studies shapes, the relationships between shapes, and space. Among the numerous concepts that mathematics provides, the theme of this project emerges: 'Reflections'. Reflection, in the mathematical context, is a type of geometric transformation that produces the mirrored image of a figure. Thus, when applied, reflection produces a mirroring of the figure in relation to a line, which we call the axis of reflection.

The study of reflections is extremely relevant for the understanding of various natural phenomena, as well as for the most varied practical applications. In physics, for example, the study of reflections is essential to understand how mirrors and lenses work. In art, reflections are used to create symmetries and patterns. In graphic design, reflections are also widely used.

Reflections, in mathematics, also have great importance for their applications in the real world. The idea of reflection is used, for example, in civil engineering, for the construction of flat and curved mirrors, used in many optical devices, such as telescopes and microscopes. It is also used in art and design, where symmetry and reflection are key principles. Furthermore, in computer science, the concepts of reflection are used in computer graphics and image processing.

For this project, I suggest that you use the following resources to delve deeper into the subject:

1. Reflection: What it is, characteristics and examples - Mundo Educação
2. Transformations in the plane: reflection - Khan Academy

These resources contain detailed information and visual examples about the concept of reflection. Additionally, they offer a wide variety of practical exercises and solutions that will help you understand this topic more deeply.

Practical Activity

1. Activity Title: Mirror, Mirror on the Wall

2. Project Objectives:

The 7th-grade students will form groups of 3 to 5 members with the objective of carrying out a playful project to deepen their knowledge about the mathematical concept of reflections. They will build figures obtained by reflection symmetries using simple and easily accessible materials. By the end of the project, they should be able to:

• Identify the axis of reflection in a figure.
• Build the mirrored image of a figure through the axis of reflection.
• Apply the concept of reflection to create works of art.

3. Detailed Project Description:

The students will be encouraged to explore geometric reflections through art, using reflective symmetry to create drawings. To achieve this goal, they should use simple materials such as paper, pencils, and mirrors to create symmetrical patterns with reflections.

4. Necessary Materials:

To develop this project, each group will need the following materials:

• Paper (can be sulfite, cardboard, cardstock, or any other type of paper suitable for drawing/painting).
• Colored pencils or markers.
• Ruler.
• Small mirror (preferably the same width as the paper).

5. Detailed Step-by-Step:

1. With the paper horizontally, the students should draw a straight line in the center of the paper, which will be the axis of reflection.
2. Next, at the top of the axis, they should create a drawing of their choice. Important: The drawing should be created in such a way that, when placing the mirror on the axis of reflection, the reflection of the drawing on the mirror fits on the paper at the bottom of the axis.
3. After drawing, they should place the mirror on the axis of reflection and 'transfer' the reflection to the bottom of the axis, drawing freehand.
4. Finally, the students should color the drawing with colored pencils or markers, noting that the colors should be the same in the original drawing and in the reflected image.

Note: Students should dedicate two to four hours to complete the project, remembering that the work should be done as a team, promoting a collaborative and cooperative experience.

At the end of this project, the students should:

• Produce a written report containing: Introduction, Development, Conclusions, and Bibliography used.

In the Introduction, students should contextualize the concept of 'reflections' in mathematics, its relevance, and application in the real world. In the Development, they should explain the theory behind reflections, describe the practical activity carried out, the methodology used, and, finally, present and discuss the results obtained with images of the finished project. In the Conclusion, they should detail the learnings acquired, the difficulties encountered, the solutions, and the conclusions about the project. Finally, in the Bibliography, students should cite the sources used for theoretical foundation during the project.