Objectives (5 - 7 minutes)

1. Understanding the Concept of Negative Bases: Students should grasp the fundamental concept of negative bases and how they differ from positive bases in relation to exponents.

2. Exploring Exponent Rules with Negative Bases: Students will learn the rules and properties of exponents when applied to negative bases. They will understand the implications of even and odd exponents on the sign of the result.

3. Applying the Concept in Problem Solving: Students will utilize their understanding of negative bases and exponents in solving mathematical problems. They will demonstrate their learning by successfully working through problems that require the application of these concepts.

Secondary Objectives:

• Promoting Collaborative Learning: As part of the hands-on methodology, students will work together in pairs or small groups, promoting the development of communication, teamwork, and peer-based learning.
• Encouraging Critical Thinking: Through problem-solving activities, students will enhance their critical thinking and reasoning skills. They will learn to analyze problems, apply appropriate concepts and rules, and evaluate their solutions.
• Boosting Confidence in Handling Negative Numbers: By mastering the concept of negative bases raised to a power, students will improve their overall confidence in dealing with negative numbers in different mathematical contexts.

Introduction (10 - 12 minutes)

1. Recalling Prior Knowledge: The teacher begins the lesson by prompting students to recall their previous knowledge about exponents with positive bases. This should include the law of exponents and how they are used in mathematical operations. A few quick problems involving positive bases and exponents can be solved collectively to jog their memory.

2. Problem Situations: The teacher then presents two problem situations to the students:

   a) "Imagine you owe your friend $5. He decides to forgive your debt every other day, but on the days he doesn't forgive your debt, he doubles what you owe. How would you represent this situation using exponents and negative numbers?"

   b) "If you have a temperature of -3 degrees Celsius and it doubles every hour, what will be the temperature after 2 hours, and after 3 hours?"

3. Real-World Relevance: The teacher contextualizes the importance of the subject by highlighting its real-world applications. For instance, they can explain how understanding exponents with negative bases helps in calculating financial interests, solving scientific problems involving extremely small or large numbers, and understanding patterns in nature and technology.

4. Topic Introduction: The teacher introduces the topic of exponents with negative bases by posing a few intriguing questions: "What happens when the base of an exponent is a negative number? Does it follow the same rules as a positive base? How does the power affect the sign of the outcome?"

5. Curiosities and Stories: To grab students' attention and stimulate their interest in the topic, the teacher shares a couple of curiosities or stories related to the topic:

   a) Curiosity: "Did you know that the concept of negative numbers was not accepted by European mathematicians until the 17th century? They found it hard to comprehend how a number could be less than nothing."

   b) Story: "In the world of computer science, negative bases are used in the representation of negative binary numbers. This concept is fundamental in the creation and functioning of the digital devices that we use every day, such as smartphones and computers."

6. Related Applications: The teacher can mention how understanding the concept of exponents with negative bases is crucial in fields like physics, engineering, and computer science. For example, explaining that electrical engineers use exponents with negative bases to calculate resistance, capacitance, and inductance in alternating current circuits. Or how in physics, it helps in understanding phenomena like radioactive decay, and in computer science, it's used in algorithms and data structures.

By the end of the introduction, students should have a firm grasp of what they will be learning, why it's important, and how it applies to the world around them. They should be ready to dive into the main lesson with interest and curiosity.

Development (15 - 20 minutes)

Before delving into the activities, the teacher provides a brief review of exponents with negative bases, highlighting the difference in outcomes when the exponent is odd or even. Then, they proceed with the hands-on activities.

Activity 1: The Exponent Game (5 - 7 minutes)

1. Setup: The teacher prepares a deck of flashcards, half of them having negative bases (-2, -3, -4, etc.) and a variety of exponents (2, 3, 4, etc.) Each card will therefore represent a mathematical problem, e.g., (-2)³, (-3)², (-4)⁴, and so on. In addition, the teacher prepares a large one-page table with rows and columns–rows representing different negative bases and columns representing different exponent values.

2. Grouping: The teacher divides the class into pairs

3. The Game: Each pair picks a flashcard from the deck, tries to solve it, and writes down the result in the matching row and column on the table. This continues until the table is entirely filled, or time runs out.

4. Reflection: Students will observe the pattern that results appear positive when the exponent is even, and negative when the exponent is odd. The teacher emphasizes this point by explicitly showing some examples on the board and asking the students why they think this pattern occurs.

Activity 2: Exponent Battleship (8 - 10 minutes)

1. Setup: The teacher prepares a grid (like Battleship) labeled along one axis by negative base numbers and on the other by exponents. The students are given mini pin flags (or something similar) with the results written on them.

2. Grouping: Separate the class into two teams.

3. Playing the Game: The teams alternately call out a base and an exponent (for example, '-2 raised to the power of 3') and then place the appropriate pin flag in that grid square. The team with the most correct placements at the end is the winner.

4. Reflection: The game emphasizes the pattern noted in the first activity and provides students with ample practice in solving the problems. This game can also be digitally adapted using an online platform if schools are operating remotely.

Activity 3: Comic Strip Challenge (10 - 15 minutes)

1. Setup: Students are given drawing materials or can use digital tools if available.

2. Grouping: Students are divided into small groups.

3. Comic Creation: Their task is to create a comic strip where negative base numbers and exponents are characters. These characters go through various adventures that involve them interacting with each other (solving problems) according to the rules of negative bases and exponents.

The main challenge here lies in translating abstract mathematical concepts into an imaginative storytelling context. The comic strip needs to have at least three panels, showcase at least one specific example of a problem involving negative base and exponent, and depict the solution within the narrative of the strip.

4. Reflection: The students then present their comic strips to the class, explaining the mathematical problem and the solution involved in their story. This activity aids in deepening their understanding by requiring them to explain the problems and solutions in a fun, narrative, and accessible context, focusing on clarity over technical language.

Note: The teacher has to ensure the tables for the activities have been pre-checked for errors before giving them to the students.

At the end of these three activities, students must have hands-on practice in solving problems involving negative bases and their exponents. They should be familiar with the patterns involved and comfortable with the concept of raising a negative number to a power.

Feedback (8 - 10 minutes)

1. Group Discussions (3 - 4 minutes): The teacher encourages each group to share their solutions or conclusions from the activities. Every group gets a chance to explain their thought process and how they arrived at their solutions. This allows an exchange of ideas and promotes healthy discussions among the students.

2. Connecting Theory with Practice (2 - 3 minutes): The teacher then facilitates a discussion about how the group activities connect with the theory of exponents with negative bases. They highlight the importance of understanding the rules and patterns underlying the problems tackled in the activities. The teacher emphasizes that although the activities were designed to be fun and engaging, they serve a greater purpose of helping students grasp and apply the mathematical concepts in a practical context.

3. Reflective Questions (3 - 4 minutes): Finally, the teacher proposes that students take a minute to reflect on the following questions:

   a) Conceptual Understanding: "What was the most important concept you learned today?" This question encourages students to consolidate their understanding and identify the core concept they gained from the lesson.

   b) Unanswered Questions: "Which questions have not yet been answered?" This question aims to identify any gaps in understanding and provide guidance for future lessons. It also encourages students to take responsibility for their learning by identifying areas they need to work on.

4. Closing Remarks: The teacher concludes the session by summarizing the key points from the lesson and providing reassurance that any unanswered questions or areas of confusion will be addressed in the following lessons. They should commend the students for their active participation and encourage them to continue practicing the concepts learned.

5. Homework Assignment: To reinforce what was learned during the lesson, the teacher assigns homework that requires students to solve various problems involving negative bases and exponents. They should also encourage students to come up with their own real-world problems that can be represented using negative bases and exponents.

By the end of the feedback session, students should have a clear understanding of the day's lesson, a sense of accomplishment, and a direction for further learning and practice.

Conclusion (5 - 7 minutes)

1. Lesson Summary: The teacher wraps up the lesson by summarizing the main contents and key learning points. This includes recapping the concept of exponents with negative bases, the rules and patterns observed (e.g., negative bases raised to even exponents yield positive results, and to odd exponents yield negative results), and the importance of these concepts in mathematical problem solving.

2. Connecting Theory, Practice, and Applications: The teacher then explains how the lesson intertwined theory and practice. This includes the use of hands-on activities like the Exponent Game and Exponent Battleship, which provided practical examples for students to apply their knowledge about negative bases and exponents. The teacher also highlights the real-world relevance of these concepts, such as their application in financial calculations, scientific problems, and technological contexts.

3. Additional Learning Resources: The teacher suggests additional materials for further reading and practice. This could include textbooks, online resources, educational videos, and interactive math websites that cover the topic of exponents with negative bases. The teacher emphasizes that these resources will deepen the students' understanding and mastery of the subject.

4. Relevance to Everyday Life: Lastly, the teacher underscores the significance of the topic in everyday life and various professional fields. They can provide examples such as:

   a) Finance: Understanding the concept of negative bases and exponents is essential for calculating compound interest, evaluating investments, and making financial decisions.

   b) Physics: It helps in understanding phenomena like radioactive decay and calculations involving extremely small or large numbers.

   c) Computer Science: Negative bases and exponents are used in algorithms and data structures, which form the backbone of modern digital devices.

The teacher concludes by reinforcing the idea that the knowledge gained in this lesson is not just abstract, but highly practical and applicable in various real-world contexts. They encourage students to keep exploring the topic and to not hesitate to ask questions in the future.

By the end of the conclusion, students should feel confident in their understanding of exponents with negative bases, aware of the importance and applications of the topic, and motivated to continue learning and exploring the subject in depth.