Lesson plan of Second Degree Equations

Objectives (5 - 10 minutes)

1. Understand the concept of second degree equations and their relevance in solving mathematical problems.
2. Develop skills to solve second degree equations effectively, using different methods such as factorization, completing the square and the general formula.
3. Apply the knowledge acquired to solve real-world problems that can be modeled by second degree equations.

Secondary objectives:

• Identify and differentiate the terms of a second degree equation: coefficient of the quadratic term, linear coefficient and independent term.
• Practice solving second degree equations through various exercises, in order to improve the understanding of the topic.
• Develop confidence and critical thinking skills when facing complex mathematical challenges.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher starts the class by reviewing the concepts of 1st degree equations, the importance of solving problems and the application of algebra in everyday life. This is crucial to prepare students for the new content and to ensure that they have the necessary foundation to understand second degree equations. (3-5 minutes)

2. Problem situations: The teacher presents two problem situations that will be solved throughout the class. The first one can be a physics problem involving uniformly varied movement, which can be modeled by a second degree equation. The second one can be a financial mathematics problem, such as the calculation of compound interest, which can also be modeled by a second degree equation. These problem situations will serve to contextualize the content and demonstrate the importance of second degree equations in different areas of knowledge. (2-3 minutes)

3. Contextualization: The teacher highlights the presence of second degree equations in several everyday situations and in different areas of knowledge, such as engineering, economy, physics, among others. For example, the determination of the maximum or minimum point of a function, the solution of optimization problems, the modeling of natural phenomena, among others, can be mentioned. This helps to arouse students' interest in the subject and to show its practical relevance. (2-3 minutes)

4. Introduction to the topic: The teacher presents the concept of second degree equations, explaining that they are equations in which the highest exponent is 2, and that they can have up to two real solutions. To illustrate, the teacher can present some examples of second degree equations and ask students to try to solve them mentally, without using specific methods. This serves to arouse students' curiosity and to assess their level of prior knowledge on the subject. (2-3 minutes)

5. Grabbing the students' attention: To finish the Introduction and arouse the students' interest, the teacher can share some curiosities or interesting applications of second degree equations. For example, he/she can mention that the formula to find the roots of a second degree equation, known as Bhaskara's formula, was developed in India around the 7th century, long before it was discovered in Europe. In addition, he/she can highlight that second degree equations are widely used in computer programming for creating graphs and animations. (2-3 minutes)

Development (20 - 25 minutes)

1. "Second Degree Equations Treasure Hunt" Activity (10 - 15 minutes)

   • Objective: To provide students with a practical and playful experience to understand and solve second degree equations.

   • Description: The teacher divides the class into groups of 4 to 5 students. Each group receives a sheet of paper with a series of tips and clues that will lead them to different second degree equations. The tips can be contextualized problems that, when solved, reveal part of the equation. For example, a tip can be a physics problem that, when solved, reveals the value of the coefficient of the quadratic term of the equation.

   • Step by Step:

     1. The teacher distributes the sheets of paper with the tips to each group.
     2. The students, in their groups, start solving the problems and finding the corresponding equations.
     3. Once all the equations have been found, the students must solve them to find the solution(s).
     4. The first group that solves all the equations correctly and finds the solutions wins the activity.

2. "Applying Second Degree Equations" Activity (10 - 15 minutes)

   • Objective: To apply the knowledge acquired about second degree equations in solving real-world problems.

   • Description: The teacher presents each group with a real-world problem that can be modeled by a second degree equation. The problems can be from different areas, such as physics, economy, engineering, among others. Each group must identify the equation that models the problem and solve it to find the solution(s).

   • Step by Step:

     1. The teacher presents the problems to each group and provides the necessary time for them to discuss and identify the corresponding equation.
     2. Once the equation has been identified, the students must solve it using the methods they have learned.
     3. The students must record the resolution process and the solution found.
     4. After solving the problems, each group must present its solution to the class, explaining the resolution process and the application of the second degree equations.

3. "Building a Formula" Activity (5 - 10 minutes)

   • Objective: To understand Bhaskara's formula and how it is derived.

   • Description: The teacher proposes to the students the task of "building" Bhaskara's formula. To do this, the students must complete the square in a second degree equation, following the steps guided by the teacher.

   • Step by Step:

     1. The teacher presents a second degree equation and explains the concept of completing the square.
     2. The students, in their groups, must follow the steps presented by the teacher to complete the square in the equation.
     3. After the Conclusion of the activity, the teacher presents Bhaskara's formula and explains that it is a simplified way to complete the square.
     4. The students must compare Bhaskara's formula with its solution by completing the square, in order to understand its usefulness and effectiveness.

These playful and contextualized activities allow students to understand the importance and application of second degree equations, as well as to develop problem-solving and critical thinking skills.

Feedback (5 - 10 minutes)

1. Group Discussion (3 - 5 minutes)

   • Objective: To facilitate the exchange of ideas, the clarification of doubts and the reflection on the activities carried out.
   • Description: The teacher promotes a group discussion with all the students, where each group shares its solutions or progress in the activities. The teacher encourages the students to explain their problem-solving strategies, what they have learned and what challenges they have faced. This allows students to understand different approaches to solving the same problem and learn from each other. In addition, the teacher takes the opportunity to clarify any doubts that may have arisen during the activities.

2. Connection to the Theory (1 - 2 minutes)

   • Objective: To reinforce the theory through practical and contextualized application.
   • Description: The teacher goes back to the theoretical concepts discussed at the beginning of the class and makes the connection with the practical activities. For example, the teacher can explain how Bhaskara's formula, which was "built" by the students, is an efficient way to solve second degree equations, and how it applies to real-world problems. This helps students consolidate their understanding of the topic and realize the relevance of the theory in solving problems.

3. Individual Reflection (1 - 2 minutes)

   • Objective: To encourage students to reflect on what they have learned and to identify possible doubts or difficulties.
   • Description: The teacher suggests that the students reflect individually on the following questions:
     1. What was the most important concept you learned today?
     2. What questions have not yet been answered?
   • After a minute of reflection, the teacher invites the students to share their answers. This allows the teacher to assess the effectiveness of the class and identify any gaps in the students' understanding that need to be addressed in future classes.

This Feedback moment is essential to consolidate the students' learning, clarify doubts, reinforce the connection between theory and practice, and identify areas that may need review or further development.

Conclusion (5 - 10 minutes)

1. Content Summary (2 - 3 minutes)

   • The teacher recaps the main concepts covered in the class, recalling what second degree equations are, what their components are (coefficient of the quadratic term, linear coefficient and independent term), and the different methods for solving them (factorization, completing the square and the general formula of Bhaskara).
   • He/she also emphasizes the importance of constant practice in solving second degree equations for the Development of student skills and confidence in this topic.

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

   • The teacher reinforces how the class connected the theory of second degree equations with practice, through the playful activities "Second Degree Equations Treasure Hunt" and "Applying Second Degree Equations".
   • He/she explains that these activities allowed students to apply the theoretical knowledge in solving real-world problems, demonstrating the relevance and applicability of this topic.
   • The teacher can also mention again the areas of application of second degree equations, such as physics, engineering, economics, among others, to reinforce the importance of this content.

3. Complementary Materials (1 - 2 minutes)

   • The teacher suggests some additional study materials for students who want to deepen their knowledge about second degree equations. These materials may include textbooks, explanatory online videos, math websites, and equation-solving apps.
   • He/she can also suggest additional exercises of second degree equations for students to practice at home and reinforce what they learned in class.

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

   • Finally, the teacher emphasizes the importance of second degree equations in everyday life, reinforcing that they are used in several real-life situations, from modeling physical movements to solving financial problems.
   • He/she also encourages students to perceive mathematics as a powerful and versatile tool, capable of helping them understand and solve a wide variety of problems.
   • The teacher concludes the class by reinforcing the idea that, with practice and dedication, all students are capable of mastering the solution of second degree equations and applying them effectively and confidently in their academic and professional life.