Contextualization

Rotations are fundamental geometric transformations that allow the change of position of an object without altering its size or shape. This concept is used in a variety of contexts, from graphic design to the locomotion of autonomous robots. This project will focus on 2-dimensional rotations, which is the most common type of rotation studied in school.

In mathematics, a rotation is a movement of an object, in the plane or in space, around a fixed point called the center of rotation. This concept is essential to understand many aspects of geometric elements and their transformations. Rotations can be presented through shapes and patterns, which, when rotated, maintain their proportions and only change their positions.

Rotations are a fundamental part of everyday life. When a basketball player shoots a ball, they use rotation. When a door opens, a rotation occurs around the hinges. When a car makes a turn, it is also in rotation. This concept is present from the way planets move around the Sun to the way a key turns inside a lock.

Rotation is one of the axes in understanding the three-dimensional world and graphic interpretation. It is present in the arts, product design, architecture, among other areas. Furthermore, understanding rotations can improve the ability to visualize in 3D, a critical skill in many fields of study and work.

To enhance understanding of this topic, it is recommended to read the chapter on 'Rotations' in the book 'Dynamic Geometry' by Dorival Antonio de Moraes. In addition, students can watch educational videos on rotations available on the YouTube platform, such as the channel 'Matemática Rio with Prof. Rafael Procopio'. They can also access the 'Geogebra' platform, which has a section dedicated to interactive exploration of 2D rotations.

Practical Activity

Activity Title: Experimenting 2D Rotations

Project Objective

This project aims to deepen students' understanding of 2-dimensional rotations by experimenting with rotations on objects of different shapes, recording their findings, and preparing a presentation for their classmates.

Detailed Project Description

Students should form groups of 3 to 5 people. Each group must choose a geometric shape and perform rotations on this object from different angles (90°, 180°, 270°, and 360°) and observe what happens. The step-by-step below suggests a sequence to carry out the project.

Required Materials

• Graph paper.
• Geogebra or other dynamic geometry software.
• Ruler, compass, and pencil.
• Camera or smartphone to document the process.

Step by Step

1. Each group must choose a type of geometric figure (square, triangle, hexagon, etc.).
2. Draw the chosen figure on graph paper.
3. Use the compass and ruler to perform a rotation of 90°, 180°, 270°, and 360° on the figure and draw the result.
4. Document each rotation with images and notes.
5. Repeat steps 2 to 4 using the Geogebra software.
6. Compare the results obtained on paper and on the software.
7. Prepare a presentation of up to 10 minutes explaining your findings. The presentation should include a live demonstration of a rotation using the software.
8. Write a report based on the guidelines below, including all the images and notes you made during the project.

Project Deliverables

Students must submit the presentation and report at the end of the project. The report should contain:

• Introduction: Contextualization of the rotation concept and its relevance in mathematics and daily life.
• Development: Detailed description of the practical activity, including the theory of rotations, the methodology used, the results obtained, and the discussions that arose during the process. Images of the rotations and figures obtained should be included.
• Conclusions: Review of the main points of the report, what was learned during the project, and how it contributed to the understanding of the rotation concept.
• Bibliography: References to all resources (books, videos, websites) used during the project.

The report should be in Word or PDF format and the presentation in PowerPoint or Prezi. Both should be sent by email to the teacher within one week after the start of the project.