Objectives (5 - 10 minutes)

1. Understand the concept of the remainder of a division of polynomials and its importance in solving mathematical problems.
2. Learn how to calculate the remainder of a division of polynomials using the Ruffini's rule, an effective and useful tool.
3. Develop skills to solve practical exercises involving the calculation of the remainder of a division of polynomials.

Secondary Objectives:

• Stimulate logical and critical thinking skills through solving mathematical problems.
• Promote the practice of mathematical calculations in an organized and systematic way.
• Encourage active participation of students during the class, through questions and discussions on the subject.

Introduction (10 - 15 minutes)

1. Content Review: The teacher should start the class by briefly reviewing the concepts of polynomials, polynomial division, and their properties. This review may involve brief questions to the students to assess their prior understanding of the topic. (3 - 5 minutes)

2. Problem Situation 1: The teacher can propose a practical problem involving polynomials, such as dividing a polynomial by an expression of the type (x-a) where 'a' is a constant. For example, "Let P(x) = x^3 - 4x^2 + 5x - 2 be a polynomial and we want to divide P(x) by (x-2), how can we do that? What is the meaning of the result obtained?" This problem situation serves to spark the students' interest in the topic and demonstrate the importance of the concept of the remainder of a division of polynomials. (2 - 3 minutes)

3. Contextualization: The teacher should then contextualize the importance of the subject, showing practical applications in real life, such as in engineering, computer science, and economics, where polynomial division and remainder calculation are used to solve complex problems. (2 - 3 minutes)

4. Problem Situation 2: The teacher can present a second, more complex problem, yet still applicable to real life. For example, "Suppose we have a set of data that follows a polynomial pattern, and we want to find the 'residue' that does not fit the pattern. How can we use polynomial division and remainder calculation to solve this?" This problem situation serves to show students how an apparently abstract concept, such as the calculation of the remainder of a division of polynomials, can have real and useful applications. (2 - 3 minutes)

5. Introduction to the Topic: Finally, the teacher should introduce the topic of the class - the calculation of the remainder of a division of polynomials. The teacher may mention that, although polynomial division can be a complex process, calculating the remainder is quite simple and direct, and that we will learn a useful tool for this - the Ruffini's rule. (2 - 3 minutes)

Development (20 - 25 minutes)

1. Theoretical Explanation (10 - 15 minutes): The teacher should explain the theory behind the calculation of the remainder of a division of polynomials. This includes presenting the Ruffini's rule and its step-by-step application. The teacher can use the whiteboard or a presentation software to illustrate the explanation. During the explanation, the teacher should:

   • Define the Concept of Remainder: Start by explaining what the remainder of a division is, and how it is represented in polynomial notation (usually as R).
   • Introduce Ruffini's Rule: Present Ruffini's rule, which is an efficient way to calculate the remainder of a division of polynomials. Explain that Ruffini's rule is based on the idea that, for a division to be exact, subtracting the product of the divisor by the quotient from the dividend should result in zero.
   • Apply Ruffini's Rule: Apply Ruffini's rule in a simple example, step by step, to illustrate how it works. The teacher can use an example like "Divide (x^3 - 3x^2 + 4x - 1) by (x-1) and calculate the remainder".
   • Discuss the Importance of the Remainder: Discuss the importance of the remainder in polynomial division, and how it can be used to determine if an expression is a factor of another.

2. Guided Practice (5 - 10 minutes): After the theoretical explanation, the teacher should conduct guided practice, where students solve problems under the teacher's guidance. The teacher should:

   • Provide Guidance: Give students clear instructions on how to apply Ruffini's rule to calculate the remainder of a division of polynomials. The teacher should emphasize the importance of following each step of Ruffini's rule correctly.
   • Solve Examples Together: Solve some examples in front of the class, allowing students to see the practical application of Ruffini's rule. The teacher should encourage students to ask questions and actively participate in the discussion.
   • Give Immediate Feedback: Provide immediate feedback to students on their calculations, correcting any errors and reinforcing the correct concepts. This helps ensure that students are understanding the material and correct any misunderstandings.
   • Gradually Increase Difficulty: As students gain confidence in using Ruffini's rule, the teacher should gradually increase the difficulty of the problems, to challenge students and help them develop their problem-solving skills.

3. Independent Practice (5 - 10 minutes): Finally, the teacher should give students the opportunity to practice what they have learned independently. Students should solve a series of problems related to the calculation of the remainder of a division of polynomials. The teacher should circulate around the room, providing support and feedback as needed. The teacher can also use this time to identify any problematic areas that may require additional review.

Return (10 - 15 minutes)

1. Group Discussion (5 - 7 minutes): The teacher should promote a group discussion so that students can share their solutions and insights on the exercises performed. During this discussion, the teacher should:

   • Encourage Participation from Everyone: Ensure that all students have the opportunity to contribute to the discussion. This can be done by asking direct questions to different students or allowing students to volunteer to share their answers.
   • Promote Evidence-Based Discussion: Encourage students to explain their answers, providing justifications based on Ruffini's rule. This helps reinforce students' understanding of the topic and develop their critical thinking skills.
   • Clarify Doubts and Correct Misunderstandings: Use the group discussion as an opportunity to clarify any doubts or correct misunderstandings that may have arisen during the class. The teacher should provide constructive feedback and support students in correcting their errors.

2. Connection with Theory (2 - 3 minutes): After the discussion, the teacher should make the connection between the practice performed and the theory presented at the beginning of the class. This can be done through targeted questions, such as "How did Ruffini's rule help us calculate the remainder of a division of polynomials?" or "What were the steps we followed to apply Ruffini's rule?". This step helps consolidate students' knowledge and reinforce the importance of the concept learned.

3. Individual Reflection (3 - 5 minutes): To conclude the class, the teacher should propose that students reflect individually on what they have learned. The teacher can ask the following reflection questions:

   1. "What was the most important concept you learned today?"
   2. "What questions have not been answered for you yet?"
   3. "How can you apply what you learned today in real-world situations?"
   • Students should have a minute to think about each question. After the reflection, students who wish can share their answers with the class. The teacher should listen carefully to students' answers and use the information to adjust future instruction, if necessary.

4. Teacher Feedback (2 - 3 minutes): Finally, the teacher should provide feedback to students on their participation in the class and their progress in understanding the topic. The teacher should praise students' efforts, highlight strengths, and offer constructive suggestions for future improvements. This step helps motivate students and keep them engaged in the learning process.

Conclusion (5 - 10 minutes)

1. Summary and Recap (2 - 3 minutes): The teacher should recap the main points covered during the class. This includes defining polynomials, polynomial division, Ruffini's rule, and the importance of calculating the remainder of a division of polynomials. The teacher should reinforce how Ruffini's rule can be an effective tool for solving problems involving polynomials and remainders of division.

2. Theory-Practice Connection (1 - 2 minutes): The teacher should explain how the class connected theory, practice, and applications of calculating the remainder of a division of polynomials. This may include discussing how practical examples and problem situations helped illustrate the application of the theory and the importance of the concept in solving real problems.

3. Additional Materials (1 - 2 minutes): The teacher should suggest additional materials for students to deepen their understanding of polynomials and Ruffini's rule. This may include textbooks, online videos, interactive games, and educational websites. The teacher should encourage students to explore these resources and seek additional help if needed.

4. Importance of the Topic (1 - 2 minutes): Finally, the teacher should reinforce the importance of calculating the remainder of a division of polynomials in everyday life and in other disciplines. The teacher can mention how this skill is used in fields such as engineering, computer science, economics, and statistics. The teacher should emphasize that, although calculating the remainder may seem like an abstract concept, it has practical and real applications that can be useful in various contexts.

5. Closure (1 minute): The teacher should end the class by thanking the students for their participation and encouraging them to continue studying and practicing the calculation of the remainder of a division of polynomials. The teacher may remind students of any homework or additional reading that may have been assigned and inform them about the topic of the next class.