Objectives (5 - 7 minutes)

1. Understand the concept of limits in calculus: The students should be able to explain what limits are in the context of calculus. They should know that a limit is a value that a function or sequence "approaches" as the input or index approaches some value.

2. Learn how to calculate limits: The students will learn various methods to calculate limits, including direct substitution, factoring, and rationalizing techniques. They should be able to solve simple limit problems by the end of the lesson.

3. Comprehend continuity in calculus: The students should be able to define continuity in terms of limits. They should understand that a function is continuous at a point if its limit at that point exists and is equal to the value of the function at that point.

   Secondary Objectives:

   • Develop problem-solving skills: The lesson will involve several hands-on activities and problem-solving exercises. This will help the students to develop their critical thinking and problem-solving skills.
   • Enhance collaborative learning: The activities will be conducted in groups, encouraging students to work together and learn from each other. This will promote collaborative learning and improve their communication and teamwork skills.

Introduction (10 - 12 minutes)

1. Review of previous knowledge (3 - 4 minutes): The teacher starts the class by recalling the concepts of functions, sequences, and their properties. This includes discussing what a function is, how to find the value of a function at a given point, and the concept of a sequence. The teacher also reminds the students about the importance of these concepts in calculus.

2. Problem situations (3 - 4 minutes): The teacher presents two problem situations to the students. The first problem is about finding the height of a ball thrown in the air at a specific time. The second problem is about finding the speed of a car at a particular moment during a journey. The teacher asks the students how they would approach these problems using the concepts of functions and sequences.

3. Contextualization (2 - 3 minutes): The teacher explains the importance of the subject by showing its real-world applications. The teacher can mention that calculus is widely used in physics, engineering, economics, and computer science. They can also give examples such as how limits are used in designing computer algorithms, or how continuity is used in predicting the weather.

4. Introduction to the topic (2 - 3 minutes): The teacher introduces the topic of calculus, specifically focusing on limits and continuity. They explain to the students that limits are used to define the derivative and integral, which are the two central concepts of calculus. The teacher also emphasizes that continuity is a fundamental property of functions, and understanding it is crucial in calculus.

5. Curiosities and applications (2 - 3 minutes): To spark the students' interest, the teacher can share some interesting facts or stories related to the topic. For instance, they can share the story of how Sir Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus in the 17th century.

6. Visual aids and technology (1 - 2 minutes): The teacher uses a whiteboard or a projector to draw graphs and diagrams that help visualize the concepts of limits and continuity. They can also use interactive online tools or apps to demonstrate how to calculate limits and check the continuity of a function.

Development (20 - 25 minutes)

Activity 1: "The Great Limit Race" (10 - 12 minutes)

1. The teacher divides the class into five groups and provides each group with a set of cards. The cards contain expressions of functions with undefined values at some points. For example, one card could have the function (x^2 - 1) / (x - 1), where the limit is not defined at x = 1.
2. Each group arranges their cards in the order they believe the limits of the functions are, from left (the function has the smallest limit) to right (the function has the largest limit).
3. Once all groups have ordered their cards, the teacher verifies the correct order and explains the concept of limits using these examples.
4. The teacher then announces that it's time for "The Great Limit Race", where the groups have to solve the limit problems on their cards using the techniques they have learned.
5. The first group to correctly solve all their limit problems wins. However, if any of their solutions are incorrect, they have to go back and rework their limits.

Activity 2: "The Mystery of Continuity" (10 - 12 minutes)

1. The teacher introduces a mystery function that is continuous at some points but not at others. For example, the piecewise function f(x) = x^2 if x is rational, and 0 if x is irrational.
2. The teacher then assigns each group a different interval on the function's domain, without telling them whether the function is continuous in that interval or not.
3. The groups must analyze the given interval and use the definition of continuity to determine whether the mystery function is continuous within the interval or not.
4. Once all groups have made their decisions, they present their findings, and the teacher clarifies whether the function is continuous in the given interval or not.
5. The teacher then explains the concept of continuity more in-depth, using the mystery function and the intervals chosen by the students as examples.
6. To conclude the activity, the teacher asks the students to reflect on what they have learned about limits and continuity from these activities.

Secondary Development Activities:

1. Problem Sets (5 - 7 minutes): After the group activities, the teacher gives each student a set of problem sets, which focuses on the concept of limits and continuity. The students work on these problems individually, applying what they have learned from the group activities.

2. Class Discussion (5 - 7 minutes): After the students have worked on the problem sets, the teacher facilitates a class discussion where the students can share their solutions and approaches. The teacher can also provide additional insights and explanations as necessary.

Feedback (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes): The teacher asks each group to share their solutions or conclusions from the activities. This discussion should focus on how they arrived at their answers, what strategies they used, and any challenges they faced. The teacher should guide the conversation to ensure that the students are linking their group work to the theoretical concepts of limits and continuity.

2. Connecting Theory and Practice (2 - 3 minutes): The teacher then summarizes the key learning points from the group activities, connecting them to the theoretical concepts of limits and continuity. They should highlight how the activities illustrated the practical application of these concepts and helped the students to understand them better.

3. Individual Reflection (2 - 3 minutes): The teacher asks the students to take a moment to reflect on what they have learned in the lesson. They should think about the answers to the following questions:

   • What was the most important concept learned today?
   • What questions have not yet been answered?

4. Question and Answer Session (1 - 2 minutes): After the students have had time to reflect, the teacher opens the floor for a question and answer session. The students can ask any clarifying questions or seek further explanation on any concepts they found challenging. The teacher should answer these questions, providing additional examples or demonstrations as necessary.

5. Summary (1 minute): To conclude the feedback session, the teacher summarizes the main points of the lesson, reiterating the importance of understanding limits and continuity in calculus. They also remind the students to practice these concepts on their own and seek help if they encounter any difficulties.

6. Homework Assignment (1 minute): The teacher assigns homework that involves solving a variety of limit and continuity problems. This will provide the students with an opportunity to practice the concepts learned in the lesson and prepare them for future, more complex calculus topics.

Conclusion (5 - 7 minutes)

1. Recap of the Lesson (2 - 3 minutes): The teacher begins by summarizing the main points of the lesson. They remind the students that limits are values that a function or sequence approaches as the input or index approaches some value. The teacher also recaps the methods of calculating limits, such as direct substitution and factoring, and the concept of continuity, which states that a function is continuous at a point if its limit at that point exists and is equal to the value of the function at that point. The teacher emphasizes the importance of understanding these concepts as they form the basis of calculus.

2. Connecting Theory, Practice, and Applications (1 - 2 minutes): The teacher then explains how the lesson connected theory, practice, and applications. They mention how the group activities, such as "The Great Limit Race" and "The Mystery of Continuity", allowed the students to apply the theoretical concepts of limits and continuity in a fun and engaging way. The teacher also highlights the real-world applications of these concepts, such as in physics, engineering, economics, and computer science.

3. Additional Materials (1 minute): The teacher suggests additional materials for the students to further their understanding of the lesson's topics. This can include textbooks, online resources, and video lectures. They can recommend specific sections of a textbook that provide more detailed explanations and examples of limits and continuity. They can also suggest interactive online resources and video lectures that demonstrate the calculation of limits and the concept of continuity.

4. Importance of the Topic (1 - 2 minutes): Finally, the teacher explains the importance of the topic for everyday life. They remind the students that calculus, including the concepts of limits and continuity, is used in various fields, from designing computer algorithms to predicting the weather. They also stress that understanding these concepts can help the students to think critically, solve complex problems, and make informed decisions in their future academic and professional pursuits. The teacher concludes the lesson by encouraging the students to continue exploring and learning about calculus.