Introduction

Mathematics can often feel abstract and disconnected from the real world for some students. However, it is deeply embedded in many aspects of our everyday lives, and exponential inequalities are a prime example of this. Understanding their structure and how to solve them is fundamental, especially in fields involving growth and decay, such as economics, physics, and biology.

An exponential inequality is an inequality in which the unknown appears in the exponent. This adds an extra layer of complexity, as now we are dealing with growth that is not linear, but exponential. To tackle these types of growth and decay, we need to master the concept of logarithms, as they are the key to bringing the unknown from the exponent to the base of the equation.

In economics, for example, compound interest rates are calculated using exponential functions, where time is the exponential variable. In biology, the growth of a bacteria population or the progression of a disease can often be modelled using exponential functions. In physics, radioactive decay is another example of a phenomenon that can be described by an exponential inequality. Thus, mastering this topic is of utmost importance and has real-world applications.

Activity

Activity Title: Deciphering the Exponential Code

Project Goal

The goal of this project is to use the theory of exponential inequalities to solve a practical problem: following an exponentially coded map. Students will research exponential inequalities, learn about logarithms, and apply this knowledge to decode a code and, consequently, a map.

Detailed Project Description

Groups of 3-5 students will be given a coded map, where the challenges to progress from one point to another on the map are exponential inequalities. To solve the challenges, students will need to explore the properties of exponentials, learn how to manipulate logarithms, and how these are used to solve exponential inequalities. Solving the map will lead students to a "treasure," which can be an inspirational message, a new challenge, among other possibilities.

Materials

1. Exponentially coded map: This will be provided by the teacher and each group will receive a different map.

2. Calculator: For the computation of the inequalities.

3. Notebook and pens (or computer): For note-taking, computations, and writing the report.

Step-by-Step Detailed Instructions to Carry Out the Activity

1. The teacher should create the groups and distribute the coded maps, ensuring that each group receives a different map.

2. Students should start by researching exponential inequalities and logarithms using the resources suggested at the beginning of the assignment.

3. Students should then use their acquired knowledge to solve the exponential inequalities on the map, which will lead them through the path of the map.

4. Students should keep a logbook registering how they solved each inequality, the rationale used, and the difficulties encountered.

5. At the end of the map, students should write a report detailing the whole process of deciphering the map.

Project Deliverables

At the end of the project, groups of students should deliver:

1. The "logbook," where they registered the solutions of each inequality, questions, and discoveries during the process.

2. The final report, which should include:

   • Introduction: Contextualization of the topic, the relevance and real-world application of exponential inequalities and logarithms, as well as the goal of this project.

   • Development: Explanation of the theory of exponential inequalities and logarithms, detailing the activities carried out, with the methodology used to solve the inequalities, and finally presenting and discussing the results obtained.

   • Conclusions: Recapitulation of the main points, explicitation of the lessons learned, and the conclusions drawn about the project.

   • Bibliography: Indication of the sources used for the project.

This report should be written in a clear and precise manner, with appropriate language, in order to demonstrate all the knowledge acquired and applied during the realization of the project.