Contextualization

The sphere is a three-dimensional geometric figure present in many aspects of our daily lives. Balls, planets, soap bubbles, and even the Earth itself are some examples of spheres that we encounter in our daily lives. Furthermore, the notion of a sphere is fundamental in numerous areas of knowledge, from physics and engineering to medicine and art.

The surface area of a sphere is a measure that tells us how much material would be needed to completely cover the surface of the sphere. For example, if we want to paint a ball, the amount of paint we need is directly proportional to the surface area of the ball. In medicine, the amount of skin needed for a graft can be calculated by the surface area of the affected organ. Therefore, calculating the surface area of a sphere has very real practical applications.

Introduction

It is important to note that a sphere is not a circle; they are two distinct geometric figures. The circle is a two-dimensional figure (has only two dimensions: width and height), while the sphere is a three-dimensional figure (has three dimensions: width, height, and depth). Thus, while the area of a circle is given by the formula πr², where r is the radius of the circle, the surface area of a sphere is given by a different formula: 4πr².

The formula for the surface area of a sphere is derived from integral calculus, an advanced area of mathematics. However, you do not need to understand integral calculus to use the formula. Just understand that the radius of a sphere is the distance from the center to any point on its surface, and that π is a mathematical constant whose approximate value is 3.14159.

For this project, you will delve into understanding the formula for the surface area of a sphere, explore its practical applications, and develop collaboration and critical thinking skills.

Practical Activity

Activity Title: "Building and Exploring Spheres"

Project Objective:

The objective of this project is to apply and understand in practice the formula for calculating the area of a sphere, explore its real-world applications, and develop socio-emotional skills such as teamwork, effective communication, and problem-solving.

Detailed Project Description:

In this project, students will be divided into groups of 3 to 5 members. Each group will build spheres using recyclable materials and calculate the surface area of these objects. Subsequently, they will contextualize the importance of such calculation in the real world by creating a presentation on the importance and practical applications of the acquired knowledge.

Required Materials:

• Styrofoam balls of different sizes or papier-mâché for sphere construction.
• Tape measure or ruler.
• Paper, pens, pencils, eraser.
• Decoration materials (paints, markers, etc., if desired)
• Research material (books, internet access, etc.)

Detailed Step-by-Step for Activity Execution:

1. Building the Spheres: Each group must build or obtain Styrofoam balls of different sizes. The construction can be done through papier-mâché modeling. After construction, students must measure the radius of each sphere using the tape measure or ruler.

2. Calculating the Surface Area: Using the formula 4πr², each group must calculate the surface area of each sphere.

3. Practical Verification: To confirm the validity of the calculation, groups must cover one of the spheres with squares of paper of the same size, counting how many squares were needed and comparing this measure with the calculated value.

4. Contextualization: Groups must research the practical applications of the surface area of a sphere in the real world and prepare a presentation on the topic.

5. Report Elaboration: Students must produce a detailed report on the project, including information about the sphere construction process, the calculations performed, the practical verification, the contextualization, and their presentation.

Deliverables:

• Spheres created with recyclable materials.
• Written and justified calculations of the surface area of the spheres.
• Presentation on the practical application of calculating the surface area of a sphere.
• Final report, including:
  • Introduction: The student must contextualize the theme, its relevance and application in the real world as well as the objective of this project.
  • Development: The student must explain the theory behind the calculation of the surface area of a sphere and describe in detail the activities carried out, indicating the methodology used. It is important to present the results obtained and discuss the practical verification.
  • Conclusion: The student must conclude the work by summarizing its main points, explaining the learnings obtained, and drawing conclusions about the project.
  • Bibliography: The student must indicate the sources they relied on to work on the project such as books, internet pages, videos, etc.