Objectives (5 - 7 minutes)

1. Understand the definition and function of the graph of a quadratic function: The teacher should explain the definition of a quadratic function and how it relates to its graphical representation. Students should be able to identify the most important characteristics of a graph, such as the vertex, the axis of symmetry, the roots, and the concavity.

2. Learn to interpret the graph of a quadratic function: After understanding the definition of the graph, students should learn to interpret the information present in it. They should be able to determine the domain and range of the function, as well as identify if the function is increasing or decreasing in different intervals.

3. Construct the graph of a quadratic function from a table: Finally, students should learn to construct the graph of a quadratic function from a table of values. They should understand how the points in the table relate to the graph and how this graphical representation can be useful for interpretation and problem-solving.

Secondary Objectives:

• Develop problem-solving skills: Through the study of quadratic functions and their graphs, students will develop problem-solving skills, which are essential for the study of mathematics and other disciplines.

• Stimulate critical thinking and logic: The study of quadratic functions and their graphs also helps develop students' critical thinking and logic. They will have to analyze the information present in the graph, make predictions, and justify their answers, contributing to the development of critical and logical thinking skills.

Introduction (10 - 15 minutes)

1. Review of previous content: The teacher should start the lesson by reviewing the concepts of function and linear function, as the quadratic function is an extension of these. It is important that students are familiar with the idea of variables and how they relate in a function. (3 - 5 minutes)

2. Problem situation 1 - The athlete's jump: The teacher should present the following situation: 'An athlete jumps from a trampoline. His height over time can be represented by a quadratic function. How can we use the graph of this function to analyze the athlete's jump?' This situation will introduce the practical application of the content to be studied. (3 - 5 minutes)

3. Problem situation 2 - The movement of a projectile: The teacher should present the following situation: 'A projectile is launched into the air. The height of the projectile over time can be represented by a quadratic function. How can we use the graph of this function to analyze the movement of the projectile?' This situation will reinforce the importance of studying quadratic functions and their graphs. (3 - 5 minutes)

4. Contextualization: The teacher should explain that quadratic functions and their graphs are widely used in various areas, such as physics, engineering, economics, among others. He should emphasize that the ability to interpret and construct graphs of quadratic functions is a very useful and valuable tool. (2 - 3 minutes)

5. Introduction to the topic: To introduce the topic of the lesson, the teacher can present some interesting curiosities and applications of quadratic functions. For example, he can mention that the trajectory of a projectile in a uniform gravitational field (such as that of a ball kicked upwards or a missile) is a quadratic function. In addition, he can mention that the famous equation by Einstein, E=mc², is a quadratic function, where E represents energy, m represents mass, and c represents the speed of light in a vacuum. (2 - 3 minutes)

Development (20 - 25 minutes)

1. Activity 1 - 'Discovering the graph' (10 - 12 minutes):

   • Description: The teacher should divide the class into groups of 3 or 4 students. Each group will receive a large sheet of paper and a series of cards. Each card will have a quadratic expression, the value of the coefficient a (which determines the concavity of the parabola) varying from card to card, and the value of the coefficient c (which determines the value of y when x=0). The value of the coefficient b (which determines the vertex of the parabola) will always be 1.

   • Objective: The objective of the activity is to construct the graph of each quadratic expression on the large paper, using the cards as a guide.

   • Step by step: Students should start with the card with the smallest value of a and proceed in ascending order. For each quadratic expression, they should identify the vertex, the concavity, the value of y when x=0, and draw the parabola on the large paper. At the end of the activity, each group will have a 'mural' with several graphs of quadratic functions.

2. Activity 2 - 'Connecting the dots' (10 - 12 minutes):

   • Description: Still in groups, students will receive a set of tables with x and y values corresponding to a quadratic function. They will be challenged to create the corresponding graph on the large paper, connecting the points from the table.

   • Objective: The objective of this activity is for students to practice the skill of constructing the graph of a quadratic function from a table of values.

   • Step by step: Students should start by identifying the value of a, b, and c in the quadratic function. Then, they should plot the points from the table on the large paper and, finally, connect the points to form the parabola.

3. Activity 3 - 'Interpreting the graph' (5 - 6 minutes):

   • Description: To conclude the group activities, students will be challenged to interpret the graphs they have constructed. The teacher will ask a series of questions, such as 'What happens to the function when x becomes very large?', 'What is the minimum/maximum value of the function?', 'Is the function increasing or decreasing in the interval x=?'. Students should discuss the answers in their groups and then share their conclusions with the class.

   • Objective: The objective of this activity is for students to practice the skill of interpreting the graph of a quadratic function.

   • Step by step: The teacher will ask the questions and give time for students to discuss in their groups. Then, a representative from each group will share their conclusions with the class.

Return (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes): The teacher should promote a group discussion with all students. Each group should share their conclusions from the activities carried out. The teacher should encourage students to explain how they arrived at their answers and to justify their conclusions. The teacher can ask questions to stimulate discussion and to ensure that all students are understanding the content. This is a moment to clarify doubts and deepen students' understanding of the topic.

2. Connection to Theory (2 - 3 minutes): After the discussion, the teacher should summarize the main ideas discussed and connect them to the theory presented at the beginning of the lesson. The goal is to show students how practice connects to theory and how the knowledge acquired can be applied in different situations. The teacher can use real or hypothetical examples to illustrate the application of the content.

3. Individual Reflection (1 - 2 minutes): The teacher should propose that students reflect individually on what they learned in the lesson. He can ask questions like: 'What was the most important concept you learned today?' and 'What questions have not been answered yet?'. Students should write down their answers on a piece of paper or in their notebooks. The goal of this activity is for students to consolidate what they have learned and identify any gaps in their understanding that may need further study or clarification.

4. Student Feedback (1 minute): Finally, the teacher should ask for quick feedback from students about the lesson. He can ask questions like: 'What did you think of today's lesson?' and 'Is there something you would like to learn more about on this topic?'. Student feedback is a valuable tool for the teacher to assess the effectiveness of his lesson and to make future adjustments, if necessary.

This Return is a crucial stage of the learning process, as it allows students to consolidate what they have learned, reflect on their learning, and identify any areas that may need further study. Additionally, it provides the teacher with a valuable opportunity to assess the effectiveness of his lesson and make adjustments, if necessary.

Conclusion (5 - 7 minutes)

1. Summary and Recapitulation (2 - 3 minutes): The teacher should start the Conclusion by summarizing the main points covered during the lesson. He should review the concepts of quadratic function, the definition and function of the graph of a quadratic function, how to interpret the graph of a quadratic function, and how to construct the graph of a quadratic function from a table. For example, the teacher can ask students to summarize what they learned in their own words.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes): Next, the teacher should explain how the lesson connected theory, practice, and applications. He should highlight how the theory presented at the beginning of the lesson was applied in the practical activities and how the skills developed are relevant for solving real-world problems. For example, the teacher can recall the problem situations presented in the Introduction and explain how the concepts and skills studied in the lesson can be used to analyze these situations.

3. Extra Materials (1 minute): The teacher should suggest some extra materials for students who wish to deepen their knowledge on the topic. He can recommend math books, explanatory videos online, educational websites, and math learning apps. For example, the teacher can suggest that students explore more about the subject on Khan Academy, an online platform that offers video lessons on math and other subjects.

4. Importance of the Subject (1 - 2 minutes): Finally, the teacher should emphasize the importance of the subject studied for daily life and for other disciplines. He should explain that quadratic functions and their graphs are powerful tools for understanding and describing many phenomena and processes in the real world, from the movement of a projectile to the variation of prices in an economy. For example, the teacher can suggest that students pay attention to everyday situations that can be described by quadratic functions, such as the trajectory of a ball that was kicked upwards or the temperature variation throughout the day.

The Conclusion is a crucial stage of the lesson plan, as it allows students to consolidate what they have learned, reflect on the application of the acquired knowledge, and identify any areas that may need further study. Additionally, it provides the teacher with a valuable opportunity to reinforce the main points of the lesson, suggest materials for additional study, and highlight the importance of the subject studied.