Objectives (5 - 7 minutes)

1. Understanding the concept of factoring by grouping and evidence: Students should be able to understand what factoring by grouping and evidence is, recognizing situations where these techniques can be applied.

2. Identification of problem situations for the application of factoring by grouping and evidence: After understanding the concept, students should be able to identify situations in mathematical problems where factoring by grouping and evidence is the appropriate technique to be applied.

3. Development of skills for problem solving using factoring by grouping and evidence: Finally, students should be able to solve problems using the techniques of factoring by grouping and evidence, applying the concepts learned effectively and accurately.

Secondary objectives:

• Encourage active student participation in the learning process: Through the use of active methodologies, the lesson plan seeks to encourage student participation and engagement in the learning process, promoting a collaborative and meaningful learning environment.

• Develop critical thinking and problem-solving skills: In addition to the specific content of factoring by grouping and evidence, the lesson plan aims to develop critical thinking and problem-solving skills, which are essential for students' academic and professional success.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher starts the lesson by reviewing the concepts of factoring and algebraic expressions, which were covered in previous classes. He may propose some quick questions to the students to verify if they remember these concepts and are prepared for the new stage of learning. For example, "What is factoring?" or "What are the steps to factorize an algebraic expression?".

2. Problem situation 1: Next, the teacher presents the students with a contextualized problem that can be solved using the grouping factoring technique. For example, "Imagine you have a rectangle with an area of 20x^2 + 2x - 15. Can you divide this expression into two smaller rectangles of equal areas? How would you do that?".

3. Problem situation 2: The teacher then proposes a second problem, this time using the evidence factoring technique. For example, "The expression 3x^2 + 10x + 8 represents the area of a rectangle. Can you find the dimensions of this rectangle?".

4. Contextualization: The teacher explains that factoring by grouping and evidence are very useful techniques in Mathematics and in various other areas of knowledge. They can be applied, for example, in solving Engineering problems, analyzing economic data, and even in encrypting information.

5. Introduction to the topic: To spark students' interest, the teacher can share curiosities or stories related to factoring. For example, he can talk about how factoring is used in cryptography to protect confidential information, or about how factoring large numbers is a challenging and very important problem in the field of Computing.

6. Presentation of the lesson's objective: Finally, the teacher presents the lesson's Objectives, which are to understand the concept of factoring by grouping and evidence, identify problem situations for the application of these techniques, and develop skills for problem solving using factoring by grouping and evidence.

Development (20 - 25 minutes)

1. Activity 1 - The Mystery of Disordered Expressions (10 - 12 minutes)

   • Description: The teacher proposes the following situation: "You are famous mathematicians who have been hired to solve a mystery. One day, on a mysterious island, you find an enigma written on a rock: (3x + 5)(2x - 7) - 4(3x + 5). What could this mean? How can you unravel this mystery?".

   • Development: Students, in groups of 4 or 5, are challenged to solve the mystery. They must start by identifying the expression as a factoring question. Then, they should apply the grouping technique to factor the expression.

   • Step by step:

     1. Identify the expression as a factoring question.
     2. Apply the grouping technique: (3x + 5) and -4 are common factors, as well as (2x - 7) and -4.
     3. Factor the expression: (3x + 5)(2x - 7) - 4(3x + 5) = (3x + 5)(2x - 7 - 4) = (3x + 5)(2x - 11).

   • Objective: Through this activity, students will have the opportunity to apply the grouping factoring technique in a playful and challenging context, which will certainly increase their understanding and interest in the subject.

2. Activity 2 - Factoring of Colored Rectangles (10 - 12 minutes)

   • Description: For this activity, the teacher prepares colored cardboard rectangles with algebraic expressions that can be factored by evidence. Each cardboard should contain the expression in one color and the factored answer in another color.

   • Development: The class is divided into smaller groups and each group receives a set of cardboard rectangles. They must identify which expressions can be factored by evidence and then perform the factoring. The group that manages to factor all the expressions first wins the activity.

   • Step by step:

     1. Identify the expressions that can be factored by evidence.
     2. Factor the expressions.
     3. Verify if the factoring is correct by comparing with the answer on the cardboard.

   • Objective: With this activity, students will have the opportunity to apply the evidence factoring technique in a playful and practical way, which will certainly facilitate the understanding and internalization of the concept.

3. Group Discussion and Sharing of Solutions (5 - 7 minutes)

   • Description: After the activities, the teacher promotes a group discussion so that students can share the solutions found and the strategies used to solve the problems. This is an important moment for students to clarify doubts, consolidate learning, and develop communication and argumentation skills.

   • Development: Each group will have the opportunity to present their solutions for the activities carried out. The teacher should encourage other groups to ask questions and express their opinions. At the end of the presentations, the teacher should summarize the discussions and clarify any doubts that may have arisen.

   • Objective: With this activity, students will have the opportunity to consolidate learning, develop communication and argumentation skills, and learn from their peers. Additionally, the teacher will be able to assess students' understanding of the content and identify possible difficulties that may need reinforcement in subsequent classes.

Return (8 - 10 minutes)

1. Group Discussion (3 - 5 minutes)

   • Description: The teacher promotes a group discussion so that students can share their experiences and conclusions about the activities carried out. This is a crucial moment for the teacher to assess students' understanding of the concept of factoring by grouping and evidence.

   • Development: The teacher asks each group to share their solutions and conclusions about the activities carried out. He should encourage students to explain the reasoning used to arrive at the answers and express their opinions on the strategies used. The teacher should ask questions to stimulate students' reflection and clarify any doubts that may have arisen during the discussions.

   • Objective: With this activity, the teacher will be able to assess students' understanding of the content and identify possible difficulties that may need reinforcement in subsequent classes. Additionally, students will have the opportunity to consolidate learning, develop communication and argumentation skills, and learn from their peers.

2. Connection with Theory (2 - 3 minutes)

   • Description: Based on the group discussions, the teacher makes the connection between the practice carried out and the theory presented in the Introduction of the lesson. He explains how the techniques of factoring by grouping and evidence were applied to solve the proposed problems and how they can be useful in various everyday situations.

   • Development: The teacher revisits the concepts of factoring by grouping and evidence, and briefly recaps the activities carried out. He shows, step by step, how the techniques were applied to solve the proposed problems, and explains again the benefits of using these techniques.

   • Objective: With this activity, the teacher aims to reinforce the lesson's content, clarify any doubts that may have arisen during the activities, and show students the relevance and applicability of what was learned.

3. Individual Reflection (2 - 3 minutes)

   • Description: To conclude the lesson, the teacher proposes that students reflect individually on what they have learned. He asks some questions to guide students' reflection, such as: "What was the most important concept you learned today?" and "What questions have not been answered yet?".

   • Development: The teacher asks students to think for a minute about the proposed questions. After this time, he can give the opportunity for some students to share their reflections with the class, if they wish.

   • Objective: With this activity, the teacher aims to stimulate students' metacognition, that is, their ability to reflect on their own learning process. Additionally, the teacher will be able to obtain valuable feedback on the effectiveness of the lesson and identify possible improvements for future classes.

Conclusion (5 - 7 minutes)

1. Summary and Recapitulation (2 - 3 minutes)

   • Description: The teacher should summarize the main points covered in the lesson, recapitulating the concepts of factoring by grouping and evidence, and highlighting the importance of these techniques in Mathematics and other areas of knowledge.

   • Development: The teacher can use visual resources, such as a whiteboard or a projector, to reinforce the concepts and show examples of how to apply them. For example, he can revisit the problem of the "Mystery of Disordered Expressions" and explain step by step how the grouping factoring was applied to solve it.

   • Objective: With this activity, the teacher aims to consolidate learning, reinforce key concepts, and ensure that students have understood the lesson's content.

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

   • Description: The teacher should highlight how the lesson connected theory, practice, and applications of the concept of factoring by grouping and evidence.

   • Development: For example, the teacher can recall the practical activities carried out and explain how they illustrate the application of the grouping and evidence factoring techniques. He can also revisit the problem situations proposed in the lesson's Introduction and show how the learned theory was useful to solve them.

   • Objective: With this activity, the teacher aims to reinforce the relevance and applicability of the content learned, and show students how theory and practice relate.

3. Additional Materials (1 - 2 minutes)

   • Description: The teacher should suggest additional materials for students who wish to deepen their knowledge on the subject. These materials may include books, websites, videos, and online exercises.

   • Development: For example, the teacher can recommend a Mathematics book that explains in detail the grouping and evidence factoring techniques, or a website with several factoring exercises for students to practice.

   • Objective: With this activity, the teacher aims to encourage students to study autonomously, deepening their knowledge on the subject. Additionally, the additional materials can be useful for students who have difficulties with the content and need additional explanation.

4. Importance of the Subject for Everyday Life (1 minute)

   • Description: Finally, the teacher should emphasize the relevance of the subject for students' daily lives, showing examples of how the techniques of factoring by grouping and evidence can be useful in practical situations.

   • Development: For example, the teacher can explain how factoring can be used to simplify complex mathematical expressions, facilitating problem solving. He can also mention that factoring is an essential tool in various areas of science and technology, such as Engineering and Computing.

   • Objective: With this activity, the teacher aims to show students that Mathematics is not something abstract and distant from reality, but a powerful tool that can be used to solve everyday problems and understand the world around us.