Lesson Plan | Active Learning | Exponential Equation

Assumptions: This Active Lesson Plan assumes: a 100-minute class, prior student study with both the Book and the start of Project development, and that only one activity (among the three suggested) will be chosen to be conducted during the class, as each activity is designed to take up a significant portion of the available time.

Objectives

Duration: (5 - 10 minutes)

The Objectives phase aims to clearly establish what is expected for students to learn and be able to do by the end of the lesson. By defining specific and measurable objectives, the lesson plan guides both student preparation and the conduct of classroom activities. This section serves as a guide to ensure that all important aspects of solving exponential equations are addressed and understood by the students.

Main Objectives:

1. Empower students to solve simple and compound exponential equations, such as the equation 2^x = 4, applying properties of logarithms and exponentials.

2. Enable students to apply the concept of exponential equations in solving practical problems from everyday life and in other subjects.

Side Objectives:

1. Develop logical and critical reasoning skills through the manipulation of exponential expressions.
2. Encourage collaboration and debate among students during practical activities.

Introduction

Duration: (15 - 20 minutes)

The Introduction phase serves to engage students and connect the content they have studied previously with practical and real-world applications. By presenting problem situations, reasoning and curiosity are stimulated, laying the groundwork for the practical application of knowledge in class. The contextualization aims to show the relevance of the topic in daily life, increasing interest and awareness of the importance of studying exponential equations.

Problem-Based Situations

1. Imagine that you are responsible for calculating the depreciation of an asset over time, where the depreciation rate is an exponential function. How would you use the exponential equation to predict the value of the asset in a specific year?

2. Think of a situation where a person is investing in an investment fund that promises to double the initial invested amount every 5 years. How could you use exponential equations to determine the time necessary for the investment to double its initial value?

Contextualization

Exponential equations are fundamental for modeling phenomena that grow or decay rapidly, common in areas such as economics, biology, and physics. For example, the population growth rate of a city or the radioactive decay of an element are modeled by exponential equations. Furthermore, understanding these equations helps make informed decisions in financial matters, such as investments that promise exponential returns.

Development

Duration: (75 - 80 minutes)

The Development phase is designed to allow students to apply and deepen their acquired knowledge of exponential equations in a practical and engaging context. Through playful and challenging activities, they will work in teams to solve complex problems that simulate real-world situations, thus promoting collaborative learning and problem-solving through mathematical methods.

Activity Suggestions

It is recommended to carry out only one of the suggested activities

Activity 1 - The Expansion of the Kingdom of Exponents

> Duration: (60 - 70 minutes)

- Objective: Apply the concept of exponential equations to solve a practical population growth problem.

- Description: In this activity, students are challenged to solve a mathematical riddle involving the expansion of a magical kingdom. They must use their skills in exponential equations to help a king calculate how many days he needs to double his army, considering an exponential birth rate.

- Instructions:

• Students should form groups of up to 5 people.

• Each group will receive a set of data describing the current population, daily growth rate, and the goal of doubling the population.

• Using the exponential equation y = a * (1 + r)^x, where a is the initial population, r is the growth rate (in decimal), and x is the number of days, students must calculate the value of x for double the initial population.

• After calculating, each group should present their solution and the method used to arrive at it, providing a brief explanation of how the exponential equation works.

Activity 2 - Mission Impossible: The Radioactive Decay

> Duration: (60 - 70 minutes)

- Objective: Use exponential equations to solve a safety problem with real applications of radioactive decay.

- Description: Students are secret agents on a mission to deactivate a bomb that uses radioactive material. They must calculate the time required for the amount of radioactive material to fall to a safe level, using exponential decay equations.

- Instructions:

• Form groups of up to 5 students.

• Each group receives data about the initial amount of radioactive material and the decay rate.

• Students must use the equation N = N0 * e^(-kt), where N is the final amount, N0 is the initial amount, k is the decay coefficient, and t is the time, to calculate the time required for the amount of material to be safe.

• Prepare a presentation where they explain the calculation process and how radioactive decay is modeled by the exponential equation.

Activity 3 - The Powers Puzzle

> Duration: (60 - 70 minutes)

- Objective: Develop understanding and application of logarithms in solving exponential equations, in addition to reinforcing the concept of properties of exponentials.

- Description: In this challenge, students must solve a mathematical puzzle where the goal is to find the value of x in the equation 2^x = 128, using properties of logarithms and exponentials.

- Instructions:

• Divide the class into groups of no more than 5 students.

• Give each group a set of numbered cards with successive powers of 2 (2^1, 2^2, 2^3, etc.) up to 2^10.

• Students must use the concept of logarithms to determine the value of x in the equation 2^x = 128.

• After solving, each group must explain the reasoning used and how the concepts of exponential equations and logarithms apply to solving the problem.

Feedback

Duration: (15 - 20 minutes)

The purpose of this phase of the lesson plan is to consolidate students' learning, allowing them to reflect on the practical applications of exponential equations and share their experiences and discoveries with their peers. This group discussion helps reinforce understanding of the content, promotes the verbalization of mathematical thinking, and stimulates learning through the exchange of ideas and perspectives.

Group Discussion

Start the group discussion with a brief introduction, highlighting the importance of sharing discoveries and strategies used. Ask each group to present a summary of their results and the reasoning process followed to reach the solutions. Encourage students to ask each other questions about the approaches used and the rationale behind them.

Key Questions

1. What were the main challenges in applying exponential equations in the activities and how did you overcome them?

2. How can knowledge about exponential equations apply to other areas outside of mathematics?

3. Is there any real-life situation where you could apply the concept of exponential equations after these activities?

Conclusion

Duration: (5 - 10 minutes)

The purpose of the Conclusion phase is to ensure that students have a clear and consolidated understanding of the topics discussed during the lesson. This section allows students to connect the theory studied with the practical activities carried out, as well as understand the importance and applicability of exponential equations in real contexts. The summary helps reinforce learning, while the discussion about the link between theory and practice and practical applications reinforces the relevance of the content for students' lives.

Summary

In the final phase of the lesson, the teacher should summarize and recapitulate the main concepts covered regarding exponential equations, emphasizing essential properties such as base and exponent, as well as practical applications in real and theoretical situations. This summary serves to consolidate learning and ensure that students have understood and memorized the techniques discussed during the lesson.

Theory Connection

During the lesson, the connection between theory and practice was established through interactive activities and contextualized problems, such as the expansion of the magical kingdom and radioactive decay. These activities illustrated how exponential equations, besides being fundamental for mathematics, have direct applications in areas such as science, economics, and engineering, reinforcing the importance of theoretical content for solving practical and everyday problems.

Closing

Finally, it is crucial to highlight the relevance of studying exponential equations in students' daily lives. Understanding these equations helps not only in solving mathematical problems but also in making informed decisions in various practical situations, such as financial investments and modeling natural phenomena. This understanding enhances students' skills to apply mathematical knowledge in varied contexts, contributing to a more comprehensive and prepared education.