Contextualization

Introduction

Mathematics is much more than just numbers and formulas; it is the foundation for understanding and describing the world we live in. Within the mathematical universe, we have Geometry, an area that studies shapes and their properties. One of the most common shapes we encounter in our daily lives is the cylinder. This, in turn, is a three-dimensional figure with two parallel and equal bases and a lateral surface. Calculating the area of a cylinder is a fundamental mathematical skill that has a wide range of applications in various disciplines beyond mathematics, such as physics, engineering, architecture, among others.

The area of a cylinder is calculated by adding the area of the lateral surface (twice the product of the radius by the height times pi) with the area of the two bases (twice the product of the radius squared times pi). This concept, although simple, requires a deep understanding of geometry and area calculation.

Contextualization

In everyday life, we come across many objects that have the shape of a cylinder or parts of a cylinder. Some examples are trash cans, soda cans, test tubes, barrels, silos, poles, among many others. Calculating the area of a cylinder can be vital to determine the amount of material needed to cover or build these objects, which has direct implications on aspects such as cost and sustainability. Furthermore, in engineering, the ability to calculate the area of a cylinder is essential for the design and construction of a variety of structures, from simple aluminum cans to complex airplane engines.

Applications of calculating the area of a cylinder are not limited to the tangible world. It also plays a crucial role in fields such as computer graphics and particle physics, where the concept of a three-dimensional space is essential.

Mathematics, and the calculation of the area of a cylinder in particular, are therefore essential skills to understand and shape the world around us.

Practical Activity

Activity Title

"Building and Calculating Cylinders: A Practical and Playful Approach"

Project Objective

The objective of this project is for students to be able to calculate the lateral surface area and the total surface area of a cylinder, based on the physical construction of a model and solving problems involving these calculations.

Detailed Project Description

In this project, students will work in groups of 3 to 5 people to build a physical model of a cylinder. From this model, they will calculate the lateral surface area and the total surface area of the cylinder. Additionally, based on a proposed problem, they will perform calculations involving the lateral and total areas of the cylinder.

Required Materials

For this activity, the following materials will be needed:

1. Cardboard or cardstock.
2. Ruler.
3. Compass.
4. Scissors.
5. Glue.
6. Pens or markers for notes.
7. Calculator.

Detailed Step-by-Step

1. Each group should choose the dimensions of the cylinder they will build (height and radius).
2. With the help of the compass and ruler, draw on the cardboard or cardstock two circles with a radius equal to the chosen one and a rectangle with a height equal to the chosen height and a length equal to the circumference of the cylinder base (2piradius).
3. Cut out the drawn shapes and assemble the cylinder, gluing the parts correctly.
4. Calculate the lateral surface area of the cylinder using the formula: 2piradius*height.
5. Calculate the total surface area of the cylinder by adding the area of the two bases (2piradius²) to the lateral area.
6. Imagine that this cylinder will be used as a silo to store grains and you need to buy paint to completely paint it, both on the inside and outside. The paint cans available on the market cover an area of 3m² per liter. How many liters of paint would you need to buy? (Remember that, in this case, the internal area of the silo must also be painted, that is, it is as if we had two cylinders, one internal and one external).
7. Record all steps taken, calculations, decisions made, and answers found to write the final report.

Project Deliverables

Each group must deliver:

1. The built cylinder model.

2. A written report containing:

   • Introduction: Contextualization of the calculation of cylinder areas in the real world, relevance and application of this learning, and description of the project's objective.
   • Development: Detailed description of the activity performed, theoretical explanation of the cylinder area calculation, methodology used, discussion of the results obtained, and resolution of the proposed problem.
   • Conclusions: Recap of the main points of the project, lessons learned, and conclusions drawn about the project.
   • Bibliography: Indicate all sources used in the project's elaboration.

The report should be written clearly and cohesively, in a way that provides anyone who reads it with a clear understanding of what was done and learned during the project.