Contextualization

The sphere is a spatial geometric figure quite present in our day to day lives and is easily recognized by its shape. It is defined as the geometric locus of the points in space that are the same distance from a fixed point called the center.

Understanding the metric relations in spheres is fundamental to solving a wide variety of problems. From these relations, we can find the volume, surface area, and the intersection between a sphere and a plane. The metric relations in the sphere have several practical applications, from physics and engineering to art and design.

The importance of this topic is broad and comprehensive. In physics, the sphere is used to model celestial bodies, such as stars and planets. In engineering, it is used for the development of rolling spheres, ball bearings, and even in the creation of rolling robots, such as Star Wars' BB-8! In art and design, the sphere can be used to create visual illusions, models, and 3D modeling.

The challenge in this project is that students can deepen their understanding of these metric relations, understand their applicability, and develop teamwork and problem-solving skills. The theme is complex and requires a good dose of dedication, but the results can be surprising and rewarding.

To delve into the subject, we recommend the following resources:

• Book "Fundamentals of Elementary Mathematics - Vol 8 - Spatial Geometry, Relative Positions, Polyhedrons" by Gelson Iezzi, Carlos Murakami.
• Explanatory videos from Khan Academy about spheres.
• Brasil Escola website with articles on sphere.

Practical Activity: "Traveling through Space"

Project Objective

This project aims to engage students in understanding the metric relations in the sphere. Students will create a scale model of the solar system using spheres of different sizes to represent the planets. They will also solve problems related to the positioning of the planets around the sun and the distances between them. In this process, they will apply mathematical concepts involving metric relations in the sphere, working in teams to carry out the activities and overcome the challenges.

Detailed Project Description

The students, divided into groups of 3 to 5 people, will start by creating a scale model of the solar system. They will use spheres of different sizes to represent the planets and the sun. The spheres will be hung according to the arrangement of the planets in the solar system.

Students will have to calculate the scale of the model, taking Earth as a reference and defining its size in relation to the real size of Earth. The other planets and the sun must be in scale relative to the Earth.

From there, students will solve several problems related to the intersection of a plane with a sphere, such as calculating the area of the section of a planet caused by sunlight at different positions of the planet in its orbit around the sun.

Finally, students should develop and present a detailed report describing the entire process of creating and implementing the project, the calculations performed, the difficulties encountered and how they dealt with them as a team, and the conclusions drawn from the results.

Required Materials

• Styrofoam spheres of different sizes (these will represent the planets and the sun)
• Acrylic paint in various colors
• Paintbrushes
• Fishing line or nylon thread
• Ruler
• Pencil and paper
• Calculator
• Computers with internet access for research

Detailed Step by Step

1. Research the size of the planets and sun and, based on Earth's size, determine the scale of your model.
2. Paint the styrofoam spheres to represent the planets and the sun. Remember to use the defined scale to choose spheres of the correct size.
3. Hang the spheres in the correct order to represent the solar system.
4. Calculate the area of the section of a planet caused by sunlight at different positions of the planet in its orbit around the sun.
5. Record all calculations and steps used to solve them.
6. Write a detailed report describing the entire project, from start to finish.

Project Deliverables

After completing the practical activities, students will write a document in the form of a report that must contain the following topics:

1. Introduction: Description of the topic, importance, and applicability in the real world, and the objective of this project.
2. Development: Theoretical explanation of the theme(s), detailed description of the activity in detail, explanation of the methodology used, and presentation and discussion of the results obtained.
3. Conclusion: Demonstration of the main points and learning obtained during the project and the conclusions drawn.
4. Bibliography: Indication of all sources consulted during the project.

Students should strive to make the calculations and descriptions as accurate as possible, highlighting the teamwork and knowledge exchange during the development of the project.