Introduction

Theoretical Introduction

The concept of volume is central to mathematics and physics, and is fundamental to understanding various natural phenomena and technological applications. The volume of an object is the amount of space it occupies, and is measured in cubic units.

Cylinders are a common shape in nature and technology. They are found in bottles, cans, trees, rivers, and even in the human body. The formula for the volume of a cylinder is based on the concept of a prism, and is given by V = πr²h, where r is the radius of the base of the cylinder and h is its height.

The surface area of a cylinder, in turn, is the sum of the areas of its two bases and its lateral surface area. Each base is a circle of radius r, so the area of each base is given by A = πr². The lateral surface area is a rectangle with width equal to the circumference of the base (2πr) and height equal to the height of the cylinder (h), so its area is 2πrh. Therefore, the total surface area of a cylinder is given by A = 2πr² + 2πrh.

Contextualization

The volume and surface area of a cylinder are concepts that we encounter in many real-life situations. If we are going to make a cake, for example, we need to know the volume of the pan to calculate the amount of batter we need to prepare. If we are going to paint a cylinder, we need to know its surface area to calculate the amount of paint that will be needed. In addition, industry and commerce constantly use these concepts to create and sell their products.

In the natural context, the volume and surface area of a cylinder are also very important. Tree trunks, for example, are approximately cylindrical, and their volume is used in the logging industry to calculate the amount of wood that can be obtained from a tree. Rivers can also be approximately modelled as cylinders, and their volume is important for controlling water supply and preventing flooding.

Practical Activity

Activity Title: Building the World with Cylinders

Project Objective

The project seeks to apply mathematical concepts of calculating the volume and surface area of cylinders in a practical and multidisciplinary context, exploring the interaction of mathematics with physics and art, and developing technical and socio-emotional skills in students.

Detailed Project Description

Students, in groups of 3 to 5, will create a model of a city using only cylinders. In their city, they must include at least 6 different types of buildings (buildings, factories, parks, etc.) and they must plan to use the cylinders in different ways (lampposts, trees, roads, etc.).

After building the model, the students will have to calculate the volume and surface area of at least three different cylinders used in the model. Additionally, they must calculate the amount of material (represented by the volume of the cylinder) and the amount of paint (represented by the surface area) needed to create each type of cylinder.

Students will use concepts of spatial geometry, especially the calculation of volume and surface area of cylinders, as well as notions of scale for the construction of the model. These skills will also be applied in planning the use of materials and paints, which involves logical reasoning and problem solving.

Necessary Materials

1. Cardboard of different colors and textures
2. Glue
3. Ruler
4. Compass
5. Scissors
6. Gouache paint
7. Brushes
8. Calculator

Detailed Step-by-Step Instructions for the Activity

1. Group formation: Students should organize themselves into groups of 3 to 5 members.
2. Model planning: The groups should plan their city, defining the types of buildings they will build and the layout of the city.
3. Construction of the cylinders: Each group should build the cylinders for their city, ensuring the diversity of their sizes and uses.
4. Model assembly: The groups should assemble their city using the constructed cylinders.
5. Volume and surface area calculation: Each group should select three different cylinders from their city and calculate the volume and surface area of each.
6. Material and paint usage calculation: Based on the volume and surface area calculations, the groups should estimate the amount of material and paint needed to build each type of cylinder.

Project Deliverables

Students will have to deliver the built model and a written report containing:

1. Introduction: With the theme of the project, the relevance of studying volumes and surface areas of cylindrical surfaces in everyday context and the objective of the work carried out.
2. Development: It should contain the theoretical description of the mathematical procedures for calculating the volume and surface area of a cylinder, the methodology used for constructing the model, the calculations performed and a discussion of the results, linking them to the real context (e.g.: "To build our project building, which is x cm high and y cm in radius, we would need z liters of material and w liters of paint").
3. Conclusion: It should revisit the main points of the work, the lessons learned and the conclusions drawn about the project. To this end, they can answer questions such as: What did we discover with this project? What difficulties did we face and how did we overcome them? How is mathematics applied in civil construction and architecture? How can understanding the calculation of volumes and areas be useful to us?
4. Bibliography: Indicate the sources consulted for the development of the project.