Contextualization

Mathematics, at its core, is a tool that helps us understand the world around us. It provides us with a universal language that can be used to describe phenomena of various natures. One of these phenomena that mathematics can describe quite adequately are exponential processes. These processes are found in various everyday situations and in various areas of knowledge, such as in medicine, physics, economics, among others.

Exponential equations are equations in which the variable appears as an exponent. This makes these equations have characteristics quite different from what we are used to seeing in first or second-degree equations. One of the main characteristics of exponential equations is the fact that they model situations where there is non-linear growth (or decay). This means that, depending on the context, small changes in the variables involved can cause significant impacts on the results.

Introduction

In our daily lives, we encounter several examples of phenomena that can be described by exponential equations. For example, the growth of a population of bacteria, radioactive decay, or the growth of an investment with compound interest. These are just a few examples of how exponential equations can be used to describe real situations.

Regarding exponential equations, it is important to understand two main concepts: the base and the exponent. The base is the number that is being multiplied by itself, and the exponent indicates how many times this number is being multiplied.

Solving an exponential equation, therefore, involves finding the value of the variable that makes the equation true. This process may seem simple, but it is important to emphasize that it requires a good understanding of logarithms and the properties of exponential equations.

Practical Activity

Activity Title: "Unraveling the Mysteries of Exponential Equations"

Project Objective

The objective of this project is to deepen students' understanding of exponential equations and their practical applications. Additionally, it aims to foster socio-emotional skills such as teamwork, time management, and problem-solving.

Detailed Project Description

Each group of students (3-5 people) will have the mission of deciphering a mystery, solving mathematical problems and challenges based on exponential equations. The mystery will be represented by a fictional story about a hidden treasure, which can only be found by solving exponential equations.

Another aspect of the project will be the preparation of a presentation on a specific topic related to exponential equations, such as population growth, financial problems with compound interest, or radioactive decay and its resolution.

Required Materials

1. Paper and pen for sketches and initial calculations.
2. Internet access for research.
3. Presentation resources such as PowerPoint, Google Slides, or Prezi.
4. Scientific calculator or similar software to facilitate mathematical calculations.

Activity Step by Step

1. Divide into groups. Each group should have between 3-5 members.

2. Research exponential equations and their practical applications.

3. Each group will receive a mystery represented by a series of mathematical problems. Solve these problems using the knowledge acquired about exponential equations.

4. In addition to solving the problems, each group must prepare a presentation on a specific topic related to exponential equations.

5. The presentations should detail the chosen topic, explaining how exponential equations are applied in it.

6. Finally, prepare a final report in the form of a scientific article.

Project Deliverables

1. Final report in the form of a scientific article, with the following topics: Introduction, Development, Conclusions, and Bibliography.

   • Introduction: Provide context for the theme and the objective of this project. Explain the relevance of exponential equations and their applications in the real world.

   • Development: Explain the theory of exponential equations and discussions on the chosen topic. Detailed description of the methodology used and the resolution of the proposed problems.

   • Conclusions: Summarize the main points of the work, the lessons learned, and the conclusions about the project.

   • Bibliography: Indicate the study sources and inspiration for the project.

2. Presentation on the chosen topic, with the resolution of the proposed problems.

The estimated time for the project is one month. It is recommended that students invest between 5 to 10 hours per week to complete all stages of the project, including the preparation of the report and presentations.