Objectives (5 - 10 minutes)

1. Introduction to Functions:

   • Understanding what a function is and its relevance in Mathematics and various areas of everyday life.

2. Interpretation of Function Representations:

   • Developing skills to interpret different forms of function representation, such as graphs, tables, and algebraic expressions.

3. Practical Applications:

   • Understanding and applying the concept of function in real-life situations, such as in studies of natural phenomena, economics, sciences, and engineering.

Secondary Objectives (optional):

1. Connection with Previous Content:

   • Reviewing previous mathematical concepts that are fundamental for understanding functions, such as Cartesian coordinates and linear equations.

2. Development of Problem-Solving Skills:

   • Promoting the ability to analyze and solve complex mathematical problems through the use of functions.

Introduction (10 - 15 minutes)

1. Review of Previous Content:

   • The teacher should start the lesson by briefly reviewing the concepts of Cartesian coordinates, linear equations, and the use of graphs to represent relationships between variables. These are fundamental concepts for understanding functions and should be reviewed so that all students are on the same page.

2. Problem Situation 1:

   • After the review, the teacher should propose the following situation: 'Imagine we are measuring the number of study hours of a student and the grade he gets on a test. How can we mathematically represent this relationship?'. This question serves to introduce the idea of a function as a tool to describe relationships between variables.

3. Problem Situation 2:

   • Next, the teacher can present another situation: 'Now, consider that we have a car moving at a constant speed of 60 km/h. How can we describe the distance traveled by the car as a function of time?'. This situation aims to show students how functions can be used to model real-life phenomena.

4. Contextualization of the Subject's Importance:

   • The teacher should then explain that functions are widely used in many areas of science, engineering, economics, and even in everyday life. Examples may include predicting population growth, modeling natural phenomena such as a comet's trajectory, or determining economic trends.

5. Introduction to the Topic:

   • Finally, the teacher should formally introduce the topic of the lesson: Functions: Representations and Applications. It should be explained that students will learn to interpret different forms of function representation, such as graphs, tables, and algebraic expressions, and how to apply the concept of function in real-life situations.

This stage of the lesson is crucial to capture students' attention, show the relevance of the subject, and prepare them for the content that will be covered.

Development (20 - 25 minutes)

1. Activity 1: Building the Function Graph (10 - 12 minutes):

   • Description: In this activity, students will build the graph of a linear function from a table of values. They will be divided into groups of 3-4 people, and each group will receive a table of values representing a linear function. They will have to build the corresponding graph on graph paper.
   • Step by step:
     1. The teacher should distribute the tables of values and graph paper to each group.
     2. The students, in their groups, should discuss how to use the values from the table to build the function's graph.
     3. They then start drawing the graph on the graph paper, making sure all points from the table are represented.
     4. After completing the graph, they should verify if it correctly represents the linear function from the table.
     5. The teacher should circulate around the room, providing guidance and clarifying doubts as needed.

2. Activity 2: Functions in Real Life (10 - 12 minutes):

   • Description: In this activity, students will explore how functions are used to model real-life phenomena. The teacher will present several problem situations that can be modeled by functions, and the students, in their groups, will have to identify the variables involved, write the corresponding function, and draw the graph.
   • Step by step:
     1. The teacher should present a series of problem situations. For example, the speed of a car as a function of time, the temperature as a function of soil depth, the height of an object thrown as a function of time, etc.
     2. The students, in their groups, should identify the variables involved in each situation and discuss how to write the corresponding function.
     3. They should then draw the function's graph on the graph paper.
     4. After completing all situations, each group should present their answers to the class, explaining how they arrived at the function and what the graph represents.
     5. The teacher should provide feedback and clarify any doubts that may arise.

3. Activity 3: Group Discussion (5 minutes):

   • Description: After the completion of the previous activities, students will have the opportunity to discuss in their groups what they have learned. The teacher will propose some questions to guide the discussion, such as 'What was the most challenging activity?' and 'How can functions be useful in your lives outside the classroom?'.
   • Step by step:
     1. The teacher should propose the questions for discussion.
     2. The students, in their groups, should briefly discuss their answers.
     3. Some ideas from each group should be shared with the class.
     4. The teacher should end the activity, highlighting the main ideas discussed and reinforcing the importance of functions in everyday life.

These activities are designed to engage students actively and practically, allowing them to experience building function graphs and modeling real-life phenomena. Additionally, group discussion promotes the exchange of ideas and reflection on what has been learned.

Return (10 - 15 minutes)

1. Group Discussion (5 - 7 minutes):

   • Description: At this stage, the teacher should gather all students and promote a group discussion on the solutions or conclusions found by each team during the activities. Each group will have up to 3 minutes to share their findings, and the teacher will guide the discussion by asking questions to deepen students' understanding of the topic.
   • Step by step:
     1. The teacher should get everyone's attention and ask each group to share their solutions or conclusions.
     2. While the groups share, the teacher should ask questions to check students' understanding and promote meaningful discussion.
     3. After each group presents, the teacher should summarize the main ideas and clarify any misunderstandings that may have arisen.
     4. The teacher should end the discussion by emphasizing the importance of teamwork and clear communication when solving mathematical problems.

2. Connection with Theory (3 - 5 minutes):

   • Description: At this stage, the teacher should connect the practical activities carried out by students with the theory presented at the beginning of the lesson. The goal is to reinforce the concepts learned and show how they apply in practice.
   • Step by step:
     1. The teacher should briefly summarize the main theoretical concepts presented at the beginning of the lesson.
     2. Then, the teacher should show how these concepts were applied during the practical activities.
     3. The teacher should highlight the importance of being able to interpret different forms of function representation and apply the concept of function in real-life situations.
     4. The teacher should end this stage by emphasizing that theory and practice are complementary and that it is important to understand both to have a good grasp of the subject.

3. Individual Reflection (2 - 3 minutes):

   • Description: Finally, the teacher should propose that students reflect individually on what they have learned during the lesson. They should think about the answers to the questions: 'What was the most important concept you learned today?' and 'What questions have not been answered yet?'.
   • Step by step:
     1. The teacher should propose the reflection questions and ask students to think about them for a minute.
     2. After the reflection time, the teacher can ask some students to share their answers with the class.
     3. The teacher should end the lesson by reinforcing key concepts and informing that any remaining doubts will be addressed in the next lesson.

This Return stage is essential to consolidate students' learning, promote reflection on what has been learned, and identify possible gaps in understanding that need to be addressed. Additionally, group discussion and connection with theory help make learning more meaningful and in-depth.

Conclusion (5 - 7 minutes)

1. Lesson Summary (2 - 3 minutes):

   • The teacher should start the Conclusion of the lesson by giving a brief summary of the main points covered. This includes the definition of function, the different forms of representation (graphical, tabular, and algebraic), and how to apply the concept of function in real-life situations.
   • The teacher can ask for student participation to help recap the concepts and ensure everyone is aligned.

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

   • Next, the teacher should highlight how the lesson managed to articulate theory with practice and applications.
   • It should be emphasized how, through practical activities, students were able to experience building function graphs and modeling real-life phenomena, which reinforced the presented theory.
   • The teacher can mention again the problem situations presented at the beginning of the lesson and how students were able to apply the concept of function to solve them.

3. Additional Materials (1 minute):

   • The teacher should suggest some complementary study materials for students. This may include reference books, math websites, educational videos, and online exercises.
   • The teacher may also recommend that students practice building function graphs and solving problems involving functions at home.

4. Importance of the Subject (1 - 2 minutes):

   • Finally, the teacher should reinforce the importance of the subject covered for daily life and other disciplines.
   • It should be emphasized how the ability to interpret and use functions is crucial in many fields, such as engineering, sciences, economics, and even in everyday life.
   • The teacher can also mention how understanding functions can facilitate the study of other mathematical topics, such as calculus and algebra.

At the end of the lesson, students should have a clear understanding of what functions are, how to interpret their different representations, and how to apply them in real-life situations. Additionally, they should feel motivated to continue exploring the subject and practicing the learned skills.