Contextualization

In this project, we will explore the concept of a cylinder, which is a 3D shape composed of two parallel bases that are equal circles and a curved surface connecting the two bases. A cylinder is considered a geometric solid and has a long history of use in geometry, engineering, and everyday life. Volume, which is the three-dimensional space that an object occupies, is a fundamental property of cylinders and has practical implications in a variety of fields.

The volume of a cylinder can be calculated by the formula V = πr²h, where "r" is the radius of the cylinder's base and "h" is the height of the cylinder. This formula is derived from the principle that the volume of a cylinder is equivalent to the product of the base area by the height. That is, if we could stretch the curved surface of the cylinder to form a rectangle, the area of that rectangle (base x height) would be the same as the volume of the original cylinder.

The concept of cylinder volume plays an important role in many areas of everyday life and science. For example, it is useful for calculating the capacity of containers, the volume of water in a cylindrical tank, the amount of concrete needed to build a column, among others. Furthermore, a solid understanding of cylinder volumes is essential for careers involving space and capacity calculations, such as engineering, architecture, and logistics.

To acquire in-depth knowledge about the concepts related to cylinders, we recommend the textbook "Fundamentos de Matemática Elementar" by Gelson Iezzi, which contains a detailed section on the calculation of areas and volumes. Additionally, websites like the OBMEP Mathematics Portal, Khan Academy, and the YouTube channel Matemática Rio offer a variety of free online resources, including video explanations, concept summaries, and practical problems.

Practical Activity: "Building Cylinders and Calculating Volumes"

Objective

This practical activity aims to apply the theoretical concept of calculating the volume of cylinders in a concrete and tangible scenario. Each group of students will be tasked with building their own cylinders using readily available materials and then calculating the volume of these cylinders.

Description

The project will be carried out in groups of three to five students and is expected to last two to four hours per student, but this may vary depending on each group.

After building their cylinders, the groups will be invited to fill the cylinders with water or sand (depending on the available material) to verify the practical application of the volume concept. Students will verify the accuracy of their calculations by comparing the theoretical results with those obtained experimentally.

Required Materials

1. Cardboard, cardstock, or other sturdy paper for constructing the cylinder.
2. Adhesive tape.
3. Ruler or measuring tape.
4. Pen or pencil for markings.
5. Container to accommodate the material (water or sand) used to fill the cylinder.
6. Scale to weigh the filled material, if available.

Step by Step

1. Each group should create one or two cylinders using cardboard or cardstock and adhesive tape. The group is free to choose the radius and height of the cylinder, as long as both are greater than zero.

2. After building the cylinder, students should measure and record the radius and height of the cylinder.

3. Using the cylinder volume formula (V = πr²h), students should calculate the theoretical volume of the cylinder they built and record this value.

4. Next, students will fill the cylinder with the chosen material (water or sand) and try to determine the experimental volume of the cylinder. If students are using water, they can measure the volume directly. If they are using sand, they can weigh the sand and use the density of the sand to find the volume.

5. Finally, students will compare the theoretical volume with the experimental volume, discussing possible reasons for any discrepancies between the two values.

Project Deliverables

The group must produce a detailed report of the entire project. This report should include:

1. Introduction: In this section, students should contextualize the theme, explaining the relevance and application of calculating cylinder volumes in the real world. Additionally, they should present the project's objectives.

2. Development: This should be the most extensive section of the report. Here, students should explain the theory they used to carry out the project, describe in detail the practical activity (cylinder construction and volume calculation), outline the methodology used, and present the results obtained, comparing the theoretical volume and the experimental volume, and discussing the reasons for any discrepancies.

3. Conclusions: In this final section, students should recap the main points covered, elucidate the learnings obtained, and draw conclusions about the project.

4. Bibliography: Students should list all sources consulted during the project, including books, web pages, videos, etc.

This project will allow students to apply a theoretical concept in a practical way, making the learning of the subject more meaningful and lasting. Furthermore, teamwork, time management, communication, problem-solving, and creative thinking will be valued and developed throughout the project.