Objectives (5 - 7 minutes)

1. Understand the concept of factorization, identifying the elements involved in the operation and its importance in the study of mathematics.

2. Develop skills to perform grouping and highlighting in the factorization of algebraic expressions, applying these techniques in practical exercises.

3. Apply the acquired knowledge in real situations, recognizing the usefulness of factorization in everyday life and in other areas of knowledge.

Secondary objectives:

• To stimulate logical reasoning and the students' capacity for abstraction, favoring the development of general mathematical skills.

• To promote interaction and collaboration among students through solving problems in groups, strengthening cooperative learning.

Introduction (10 - 15 minutes)

1. Reviewing the necessary content: The teacher starts the class by reviewing the concepts of algebraic expressions, polynomials and common factors, since these are fundamental elements for understanding factorization by grouping and highlighting. The teacher can do this interactively, for example, by proposing small challenges or asking the students what they already know about the subject.

2. Presenting problem situations: The teacher presents two situations that require the use of factorization. The first one can be the factorization of an algebraic expression to simplify a complex calculation. The second one can be the factorization of a polynomial to solve an equation. The teacher asks the students how they would solve these situations, instigating critical thinking and curiosity.

3. Contextualizing the importance of the subject: The teacher contextualizes the importance of factorization, explaining that it is widely used in many areas of mathematics, physics, engineering and economics. In daily life, factorization is used, for example, to simplify calculations, solve equations, find roots of polynomials, among other applications.

4. Introducing the topic with curiosities and practical applications: To arouse the interest of the students, the teacher can share some curiosities about factorization. For example, factorization is one of the oldest and most important problems in mathematics, and the search for efficient factorization algorithms is one of the main challenges of modern cryptography. In addition, the teacher can present some practical applications of factorization, such as in the decomposition of numbers into prime factors, which is used in cryptography algorithms, or in the factorization of polynomials in the solution of differential equations, which is widely used in physics.

5. Introducing the topic with a story or anecdote: The teacher can tell the story of how factorization was discovered and how it has been used throughout history. For example, the teacher can describe how ancient Greek mathematicians used factorization to solve equations and how this technique was improved over time, culminating in the development of the Fundamental Theorem of Algebra which says that every polynomial with complex coefficients can be factored into linear polynomials.

Development (20 - 25 minutes)

1. Presenting the Theory (10 - 12 minutes): The teacher explains the theory of factorization by grouping and highlighting, detailing each step of the process and providing examples to facilitate understanding. During the explanation, the teacher should encourage the students to take notes and ask questions to clarify any doubts that may arise. The main topics to be addressed are:

   • Factorization by Grouping: The teacher explains that factorization by grouping is used when the algebraic expression has four or more terms. He demonstrates how to group the terms so that they can be factored in common. For example, considering the expression x² + 3x + 2 - 2x² - 4x - 2, the teacher shows how to group the terms so that they can be factored in common: (x² - 2x²) + (3x - 4x) + (2 - 2) = x²(1 - 2) + x(3 - 4) + (2 - 2) = -x² - x.

   • Factorization by Highlighting: The teacher explains that factorization by highlighting is used when the algebraic expression has three terms and the coefficient of the quadratic term is 1. He demonstrates how to identify the pairs of terms that can be factored and how to factor them. For example, considering the expression x² + 5x + 6, the teacher shows how to identify the pairs of terms that can be factored: x² + 2x + 3x + 6. Next, he shows how to factor each pair of terms: x(x + 2) + 3(x + 2) = (x + 3)(x + 2).

   • Differences between Grouping and Highlighting: The teacher highlights the differences between factorization by grouping and factorization by highlighting, explaining that factorization by grouping is used when the expression has four or more terms, while factorization by highlighting is used when the expression has three terms and the coefficient of the quadratic term is 1.

2. Solving Exercises (10 - 13 minutes): After presenting the theory, the teacher suggests that the students solve some practical exercises on factorization by grouping and highlighting. The exercises should be chosen so as to address different difficulties and situations that the students may encounter. The teacher circulates through the room, helping the students who are struggling and clarifying questions. During the resolution of the exercises, the teacher should encourage the students to discuss their strategies and explain how they arrived at their answers.

3. Checking the Exercises and Discussing (5 - 7 minutes): After the students have solved the exercises, the teacher corrects them on the board, explaining step by step how to arrive at the correct answer. During the correction, the teacher should emphasize the main points to be observed in the factorization by grouping and highlighting and clarify any doubts that may arise. After the correction, the teacher promotes a discussion with the students about the exercises, asking them what they found most difficult and what they found easiest, and encouraging them to reflect on the factorization process and its applications.

Feedback (8 - 10 minutes)

1. Connecting with the Theory (3 - 4 minutes): The teacher begins this phase by connecting the practice performed by the students with the theory presented. He can do this by reviewing the main steps of the factorization by grouping and highlighting and by discussing the most common mistakes made by students during the resolution of the exercises. The teacher should emphasize the importance of understanding the concept behind the factorization techniques and of practicing a lot to gain fluency in its application. He should also point out that factorization is a powerful tool for simplifying calculations, solving equations and better understanding the structure of algebraic expressions.

2. Reflecting on the Learning (2 - 3 minutes): The teacher suggests that the students reflect for a minute on what they learned during the class. He can do this through questions like: "What was the most important concept you learned today?" and "What questions have not yet been answered?". After one minute of reflection, the teacher asks some students to share their answers with the class. He should listen attentively to the students' answers, valuing their contributions and clarifying any doubts that may arise.

3. Students' Feedback (2 - 3 minutes): The teacher asks for feedback from the students about the class. He can do this through questions like: "What did you find most useful in today's class?" and "What could be improved?". The teacher should note down the feedback from the students and use it to improve his future classes. He should also take this opportunity to reinforce the importance of feedback in the learning process and encourage students to give constructive and respectful feedback.

4. Preparing for the Next Class (1 minute): Finally, the teacher gives a brief overview of what will be taught in the next class, explaining how the new content connects to what was learned today. He can also suggest that students review the class contents at home and practice more factorization exercises to consolidate their learning.

Conclusion (5 - 7 minutes)

1. Summary of the Content (2 - 3 minutes): The teacher summarizes what was learned in the class, reinforcing the main concepts of factorization by grouping and highlighting. He points out the fundamental steps of each technique, as well as the situations in which they are applicable. The teacher can do this interactively, asking the students to recall the main points or to explain the techniques in their own words.

2. Connection between Theory, Practice and Applications (1 - 2 minutes): The teacher emphasizes how the class connected the theory of factorization with the practice of the exercises and the actual applications. He explains that factorization is not only a tool for solving mathematical problems, but also an important skill for understanding and simplifying algebraic expressions, solving equations and performing complex calculations. The teacher can also mention some of the practical applications of factorization that were discussed in class, reinforcing the relevance of the subject to different areas of knowledge.

3. Extra Materials (1 - 2 minutes): The teacher suggests some extra materials for students who wish to deepen their knowledge of factorization. These materials may include mathematics books, educational websites, explanatory videos and factorization games. For example, the teacher may recommend using a graphic calculator to visually explore polynomial factorizations. He can also suggest that students practice more factorization exercises at home to solidify what they learned in class.

4. Importance of the Subject (1 minute): Finally, the teacher reinforces the importance of factorization for everyday life and for other areas of knowledge. He explains that although factorization may seem like an abstract concept, it has many practical applications, from simplifying calculations to solving complex problems in areas such as physics, engineering and economics. The teacher also points out that factorization is a valuable skill that helps in developing logical thinking, abstract reasoning, patience and perseverance, which are useful skills not only in mathematics, but in many other aspects of life.