Contextualization

Mathematics is a discipline that permeates the world around us, even if we often do not realize it. It is in everything, from the way planets rotate and revolve, to the design of the simplest objects in our daily lives, like a simple soda can. In the third year of high school, we further explore these connections, and one of the fundamental topics we study is rotation.

Rotation is a mathematical operation that moves an object in a circle around an axis. This axis can be centered at any point of the object, or even outside of it, and is fundamental in various fields of exact sciences, such as physics, engineering, and of course, mathematics.

In this project, we will explore rotations in a playful and fun way, using mathematics to create beautiful geometric mosaics, similar to the patterns you may have seen on tile floors and walls or on fabric prints. Through this exercise, we will understand and apply the rules of rotation, and see how this concept is important in various parts of our daily lives, from art to engineering.

Rotations, like other isometric transformations (translation and reflection), play a central role in various areas of exact sciences. In civil engineering, for example, these transformations are used in the design of complex structures, and also in the creation of fractals, which are incredible figures that seem to repeat infinitely.

Practical Activity: Unraveling Rotations through Geometric Mosaics

Project Objective

The objective of this project is to explore the operation of rotation in geometry in a playful and visual way, using the construction of geometric mosaics to understand how rotation works and what it is used for. We want students to be able to apply the concepts of rotating figures and isometric transformations they have learned in the classroom, and also develop teamwork, time management, and creative thinking skills.

Project Description

Each group of students (3-5 students per group) will be responsible for creating a mosaic using rotation concepts. The mosaic will consist of a basic geometric figure chosen by the students (triangle, square, pentagon, etc.), which will be repeated and rotated to form an interesting and visually appealing geometric pattern.

Necessary Materials

• Large grid paper
• Colored pencils
• Compass
• Ruler
• Protractor

Step by Step

1. Planning: The group should start by discussing and planning the type of mosaic they want to create. They should think about which geometric figure to choose and how they will rotate it to create an interesting pattern. The planning should include a small sketch of the mosaic they plan to create.

2. Construction: Using the grid paper, the group should start drawing their mosaic. They should use the compass, ruler, and protractor to draw and rotate their figures accurately.

3. Coloring: Once all the figures have been drawn and rotated, the group should use colored pencils to fill in the mosaic, creating a vibrant and colorful work of art.

4. Analysis: Groups should record the number of times each figure was rotated, the angle of each rotation, and any other interesting or relevant data.

5. Report: Once the mosaic is completed, each group should prepare a written report detailing the process of creating the mosaic, the decisions made along the way, and the analysis of the collected data. The report should follow the format of 'Introduction, Development, Conclusions, and Bibliography used'.

• In the Introduction, students should contextualize the theme, its relevance and real-world application, as well as the objective of the created mosaic.
• In the Development, students should explain the concept of rotation, describing the step-by-step of the activity carried out and present the analysis of the collected data, explaining how rotation was applied in the creation of the mosaic.
• In the Conclusions, students should summarize the main points of the work, considering what was learned during the process and what were the difficulties and facilities encountered.
• In the Bibliography, students should indicate the sources consulted for the project.

Students will have one month to complete the project, which should require between five to ten hours of work per student.