Objectives (5 - 7 minutes)

1. Understand the Concept of a Function: The teacher will introduce the concept of a function, emphasizing that it is a relationship between an input and an output where each input value is associated with exactly one output value. The students will learn that a function can be represented using a set of ordered pairs, a table, a mapping diagram, or a graph.

2. Recognize the Components of a Function: The teacher will explain the key components of a function - the input, the output, the rule, and the relationship. The students will understand that the input (or the independent variable) is the value that is put into the function, the output (or the dependent variable) is the value that comes out, the rule defines how the input is transformed into the output, and the relationship shows how the input and output are related.

3. Learn to Graph Functions: The teacher will introduce the concept of graphing functions, highlighting that a graph is a visual representation of a function. The students will be taught how to plot points and draw lines or curves on a graph to represent a function.

Secondary Objectives:

1. Develop Problem-Solving Skills: Through the lesson, the students will improve their ability to understand and solve problems related to functions and graphs. This objective will be achieved through various activities and exercises that require them to apply their knowledge.

2. Enhance Critical Thinking: The students will be encouraged to think critically about the connection between the input and output of a function, the rule that defines the function, and the relationship between the input and the output. This objective will be achieved through discussions and analysis of examples and problems.

3. Cultivate Collaborative Learning Skills: The students will be given opportunities to work together in pairs or groups during the lesson, fostering their collaborative learning skills. This objective will be achieved through group activities and discussions.

Introduction (8 - 10 minutes)

1. Recalling Prior Knowledge: The teacher will begin the lesson by reminding students of the concepts they have already learned that are essential for understanding functions and graphs. This will include the understanding of variables, coordinates, and basic mathematical operations like addition, subtraction, multiplication, and division. The teacher will also ask a few review questions to ensure that the students have a basic understanding of these concepts.

2. Problem Situations as Starters: The teacher will then present two problem situations that will serve as starters for the development of the theory. The first problem could be about a car's speed over time, where the students will be asked to predict the distance traveled at different times. The second problem could be about a fruit stall, where the students will be asked to determine the cost of a certain number of apples.

3. Real-World Applications: The teacher will then contextualize the importance of understanding functions and graphs by relating them to real-world applications. The teacher can explain how functions and graphs are used in various fields like physics, economics, computer science, and even in everyday life situations like planning a trip or managing finances. To make it more engaging, the teacher can share a couple of interesting facts related to the topic. For instance, how the concept of functions and graphs is used in video games to determine the movement of characters or in music to create different sounds.

4. Topic Introduction: The teacher will finally introduce the topic of the day - "Functions: Graph". The teacher will pique the students' interest by sharing that understanding functions and graphs can help them predict and understand patterns, make decisions based on data, and even create their own mathematical models. The teacher will also assure the students that by the end of the lesson, they will be able to graph functions confidently and use them to solve problems.

Development (20 - 25 minutes)

1. Theory and Explanation (7 - 10 minutes)

   1.1. The Teacher's Role: The teacher will provide a clear and concise definition of a function, emphasizing the concept that it is a rule that relates each input value to one and only one output value. The teacher will make sure to explain that functions are not limited to mathematical contexts but can also describe real-world situations like the speed of a car or the cost of buying a certain number of items. They will then explain the different ways a function can be represented, including as a set of ordered pairs, a table, a mapping diagram, or a graph.

   1.2. Graphical Representation of a Function: The teacher will then explain how to graph a function, using the example of a simple linear function (y = x). They will show how to create a table of ordered pairs, plot those points on a Cartesian plane, and connect them to form a line. The teacher will also highlight the importance of labeling the x-axis and y-axis and the units used.

   1.3. Components of a Graph: The teacher will explain the components of a graph, including the x-axis, y-axis, coordinates, and the line or curve itself. They will also introduce the concept of slope and how it is represented on a graph.

   1.4. Graphing Non-Linear Functions: The teacher will then explain how to graph a non-linear function, using the example of a quadratic function (y = x^2). They will show how to create a table of ordered pairs, plot those points on a Cartesian plane, and connect them to form a curve. The teacher will also demonstrate the difference between a line (representing a linear function) and a curve (representing a non-linear function).

   1.5. Interpreting a Graph: Lastly, the teacher will explain how to interpret a graph, emphasizing that it can be used to understand the function's rule, predict its behavior, and find the relationship between the input and output values.

2. Class Activity: Group Graphing (8 - 10 minutes)

   2.1. Problem Presentation: The teacher will provide each group with a set of ordered pairs that represent a function and a blank graph. The task is to plot the points and correctly represent the function on the graph.

   2.2. Group Work: Students will work together in their groups to solve the problem. The teacher will circulate the classroom, monitoring the groups, and providing assistance as needed.

   2.3. Discussion: After the groups complete their tasks, the teacher will facilitate a class-wide discussion. Each group will explain their approach and the decisions they made when graphing their function. The teacher will provide feedback and clarify any misconceptions.

3. Theory Extension: Advanced Graphing (5 - 7 minutes)

   3.1. Introduction to Different Function Types: The teacher will introduce the concept of other function types like exponential, logarithmic, and trigonometric functions. They will briefly explain how these functions are different from linear and quadratic functions and how to graph them.

   3.2. Graphing Exponential, Logarithmic, and Trigonometric Functions: Using simple examples, the teacher will demonstrate how to graph these functions. They will also highlight the unique characteristics of each function type and how to interpret them on a graph.

By the end of this stage, students should have a good understanding of what a function is, how to graph different types of functions, and how to interpret a graph.

Feedback (5 - 7 minutes)

1. Assessment of Learning (2 - 3 minutes)

   1.1. The teacher will start by summarizing the main points of the lesson, emphasizing the definition of a function, the different ways to represent it, and how to graph different types of functions.

   1.2. The teacher will then ask a few questions to the class to assess their understanding. These could include: "Can someone give me an example of a function and its graph?" or "What is the difference between a linear and a non-linear function on a graph?"

   1.3. The teacher will also provide feedback on the group activity, highlighting the correct approaches and common mistakes made by the students. This will help students understand their areas of strength and areas that need improvement.

2. Reflection (2 - 3 minutes)

   2.1. The teacher will then encourage the students to reflect on what they have learned. The teacher can ask questions like:

   • "What was the most important concept you learned today?"
   • "Can you think of any real-world applications of functions and graphs?"
   • "What questions do you still have about functions and graphs?"

   2.2. The students will have a minute to think about these questions and then share their thoughts. The teacher will listen to their responses, provide clarifications, and address any remaining doubts or questions.

3. Connection to Real-World (1 - 2 minutes)

   3.1. Finally, the teacher will discuss the importance of the topic in everyday life. They will explain that understanding functions and graphs is not just about solving mathematical problems but also about understanding and predicting patterns, making decisions based on data, and creating mathematical models.

   3.2. The teacher can give examples of how functions and graphs are used in various real-world situations, such as in physics to describe the motion of objects, in economics to model supply and demand, in computer science to create algorithms, and even in everyday life situations like planning a trip or managing finances.

   3.3. The teacher will conclude the lesson by encouraging the students to continue exploring the topic and to apply what they have learned in their studies and daily life.

By the end of this stage, the students should have a clear understanding of the main concepts of the lesson, their application in real-world situations, and their own learning process. They should also feel motivated and confident to continue studying functions and graphs.

Conclusion (5 - 7 minutes)

1. Summarization of Content (2 - 3 minutes)

   1.1. The teacher will begin the conclusion by summarizing the main points of the lesson. They will reiterate the definition of a function, emphasizing that it is a rule that relates each input value to one and only one output value.

   1.2. The teacher will then recap the different ways a function can be represented - as a set of ordered pairs, a table, a mapping diagram, or a graph. They will also highlight the importance of the components of a graph, including the x-axis, y-axis, coordinates, and the line or curve itself.

   1.3. The teacher will also remind the students about the process of graphing a function, whether it is a linear or non-linear function. They will emphasize the importance of plotting points accurately and connecting them correctly to form a line or a curve.

2. Connection of Theory, Practice, and Applications (1 - 2 minutes)

   2.1. The teacher will then explain how the lesson connected theory, practice, and applications. They will remind the students that they started with the theoretical understanding of a function, then put that into practice by graphing different types of functions, and finally discussed real-world applications of functions and graphs.

   2.2. The teacher will also highlight the importance of the group activity, where students had the opportunity to apply their knowledge in a practical context and learn from their peers. They will encourage students to continue practicing graphing functions as it is a skill that will be used extensively in higher grades and in various fields.

3. Additional Materials (1 - 2 minutes)

   3.1. The teacher will suggest additional materials for students who want to explore the topic further. These could include textbooks, online resources, educational videos, and practice exercises.

   3.2. The teacher will also recommend specific topics for further study, such as more complex functions like exponential, logarithmic, and trigonometric functions, and their applications in different fields. They can also suggest advanced techniques for graphing functions and interpreting graphs.

4. Relevance to Everyday Life (1 minute)

   4.1. Lastly, the teacher will conclude by emphasizing the importance of understanding functions and graphs in everyday life. They will remind students that functions and graphs are not just abstract mathematical concepts but have practical applications in various fields and everyday life situations.

   4.2. The teacher can give a few final examples of how functions and graphs are used in real life, such as in predicting weather patterns, modeling the spread of diseases, designing buildings and bridges, creating computer games and animations, and even in personal finance planning.

By the end of this concluding stage, the students should have a clear and concise summary of the main points of the lesson, understand the connection between theory, practice, and applications, have additional resources for further study, and appreciate the relevance of the topic in everyday life.