Contextualization

The world is constantly changing, and Mathematics helps us better understand these changes. In the field of Mathematics, we will study the theme of 'Transformations', essential to understand changes in figures and objects, with broad applications in areas such as engineering, construction, graphic design, and even video games.

Geometric transformations are operations that alter the position, size, and shape of figures in space. There are four main types of transformations: translations, reflections, rotations, and dilations. In this project, we will focus on studying reflections and dilations, particularly in the context of the Cartesian plane.

Mathematics, besides being a discipline in itself, is also a valuable tool that we use in many aspects of everyday life. Transformations have a wide application in many practical fields. For example, when designing a house, an engineer will use transformations to create a 3D model on a different scale. When we play video games, transformations are used to move and rotate characters. Artists use transformations to create illusions of symmetry and perspective. By knowing transformations, you will be able to understand and apply these concepts in a variety of situations.

To delve deeper into the subject, I suggest the following reliable resources in Portuguese:

1. Book: 'Geometria Plana e Espacial' (Flat and Spatial Geometry), Moderna Publisher, by Antônio Alfredo Nicchio. This book has an entire chapter dedicated to transformations.

2. Khan Academy Website: The free online course Transformações no Plano is an excellent way to familiarize yourself with the subject.

Therefore, we hope you become interested and engaged in this project, where we will have the opportunity to explore a dynamic and interesting theme.

Practical Activity: Building a Geometric Amusement Park

Project Objective

Students, in groups of 3 to 5, must create an amusement park on the Cartesian plane, using and applying geometric transformations (reflections and dilations), demonstrating their collaboration, creativity, time management skills, and problem-solving abilities.

Project Description

Students will design an amusement park, representing each ride as a polygon on the Cartesian plane. They will then apply geometric transformations (reflections and dilations) to change the position, size, and shape of the rides.

The connection with another discipline will be Geography, as students will need to plan the layout of the rides in the park, taking into account the available space, expected population density, and accessibility of each ride.

Necessary Materials

• Graph paper.
• Pencils and erasers.
• Rulers.
• Colored Pencils.

Procedure

1. Gather with the group and discuss ideas for the park. Each member should propose at least one ride to be included in the park.

2. Draw each ride as a polygon on the graph paper, using your knowledge of geometry. Make sure to record the coordinates of each vertex.

3. Decide and apply appropriate geometric transformations (reflections and dilations) to the rides. Take notes on the transformations applied and the resulting effect on each ride.

4. Plan the layout of the park, considering the amount of available space, expected population density, and accessibility of the rides.

5. Finalize and color the drawing. Note down the final coordinates of each ride after applying the transformations.

6. Prepare a presentation of your park for the class, explaining the transformations performed and your justifications for the park layout.

Project Delivery

After completing the practical part, students must write a report containing:

1. Introduction: Provide context for the project, explaining the importance and application of geometric transformations. Describe the project's objective of creating a geometric amusement park.

2. Development: Explain in detail about the park, the rides (polygons) created, the transformations applied (reflections and dilations), and how these transformations affected the rides. Describe the methodology used for planning the park layout, connecting it with the Geography discipline.

3. Conclusion: Discuss the results obtained, what you learned about geometric transformations and spatial planning. Reflect on the experience of teamwork.

4. Bibliography: Indicate all sources of information (books, websites, videos, etc.) used to carry out the project.

The report should complement the practical work of the project in order to highlight what was learned in practice. Each transformation applied to each ride should be highlighted and mathematically explained, and the concepts of the Geography discipline applied in the layout should be clearly described.

The project has a duration of approximately 15 hours, which can be divided into 5 sessions of 3 hours each.