Lesson Plan | Active Learning | Second Degree Equation: Coefficients

Assumptions: This Active Lesson Plan assumes: a 100-minute class, prior student study with both the Book and the start of Project development, and that only one activity (among the three suggested) will be chosen to be conducted during the class, as each activity is designed to take up a significant portion of the available time.

Objectives

Duration: (5 - 10 minutes)

The Objectives stage is essential to clearly establish what is expected for students to learn and apply during the lesson. By defining specific objectives, the teacher guides both their planning and the students' learning process, ensuring that the proposed activities align with established educational goals. In this case, the focus is to ensure that students master not only the theory behind the coefficients of quadratic equations but also their practical application in calculating the roots and related concepts.

Main Objectives:

1. Empower students to understand the concept of coefficients in quadratic equations and their importance in determining the roots.

2. Develop the ability to calculate the sum and product of the roots of a quadratic equation from its coefficients and vice versa.

Side Objectives:

1. Encourage mathematical reasoning and problem-solving skills through the manipulation of quadratic equations.

Introduction

Duration: (20 - 25 minutes)

The Introduction stage serves to engage students in the lesson's theme, using problem situations they can solve to activate prior knowledge and contextualize the practical importance of studying coefficients in quadratic equations. This approach not only prepares students for the practical activities in class but also motivates them by showing the applicability of mathematical content in real and theoretical situations. Furthermore, this stage aids in transitioning from theoretical learning to practical application.

Problem-Based Situations

1. Consider the quadratic equation 2x² - 5x + 3 = 0. Ask students to calculate the sum and product of the roots of this equation, using only the coefficients present. Ask them to justify the reasoning behind the operations performed.

2. Present the following problem situation: A flower shop sells two types of arrangements, one with roses and another with tulips. The owner notices that if he adds 3 tulips to each rose arrangement, the total price increases by R$100. If he adds 1 rose to each tulip arrangement, the total price decreases by R$10. Ask students to model this situation with quadratic equations, finding the coefficients and discussing the relationship with the roots.

3. Challenge students to find a number that, when added to its inverse, results in 5, and whose product with its inverse results in 6. This situation will lead students to form a quadratic equation whose roots are the sought numbers.

Contextualization

To contextualize the importance of the coefficients of quadratic equations, the teacher can cite real or historical situations where these concepts have been applied. For example, discuss how solving quadratic equations has allowed the resolution of various problems throughout history, from determining trajectories in classical mechanics to solving differential equations in number theory. Additionally, the teacher can mention curiosities like the origin of Bhaskara's formula and its relevance in modern mathematics and engineering.

Development

Duration: (70 - 75 minutes)

The Development stage is designed to allow students to practically and contextually apply the concepts studied about coefficients in quadratic equations. By engaging them in playful and challenging activities, this section aims to solidify the theoretical understanding acquired while developing problem-solving, collaboration, and communication skills. Each proposed activity is structured to be conducted in groups, promoting an active and interactive learning environment.

Activity Suggestions

It is recommended to carry out only one of the suggested activities

Activity 1 - Investigating Magical Roots

> Duration: (60 - 70 minutes)

- Objective: Apply knowledge about coefficients and properties of roots to solve problems in a practical and collaborative manner.

- Description: In this playful activity, students will be challenged to discover 'magical roots' that meet certain conditions. Each group will receive a set of cards, each containing a quadratic equation with specific coefficients. The equations have been chosen so that their roots form pairs that fit pre-established conditions, such as the sum and product of the roots being both prime numbers, or the sum and product being equal. Students will need to use their knowledge of coefficients and properties of the roots to find the equations corresponding to each condition.

- Instructions:

• Divide the class into groups of up to 5 students.

• Distribute a set of equation cards to each group.

• Each card should be analyzed, and students must determine if the roots meet any of the magical conditions.

• The groups that can discover the highest number of 'magical roots' correctly win the challenge.

• At the end, each group will present one equation and its roots, justifying why they meet the specified condition.

Activity 2 - The Mystery of the Missing Coefficients

> Duration: (60 - 70 minutes)

- Objective: Develop problem-solving skills and a deep understanding of the role of coefficients in quadratic equations.

- Description: Students, in groups, will face a mathematical mystery. They will receive a set of quadratic equations with some 'missing' coefficients. The challenge will be to use the relationships between the sum and product of the roots with the coefficients to determine the missing values. Each group will have a set of clues and tasks that, when completed correctly, will lead to the discovery of the coefficients.

- Instructions:

• Form groups of up to 5 students and distribute materials (cards with equations, clues, and tasks).

• Students should start by trying to identify the roots based on the initial clues.

• As they solve the tasks, they will receive more information to determine the coefficients.

• The first group to correctly determine all the coefficients wins the challenge.

• Each group, at the end, will present how they reached their results and which strategies were most effective.

Activity 3 - Equation Builders

> Duration: (60 - 70 minutes)

- Objective: Encourage creativity and a deep understanding of the relationships between coefficients and properties of roots.

- Description: Students, in groups, will be challenged to create quadratic equations that meet specific criteria, such as having roots that are whole numbers or whose coefficients are multiples of a given number. They will use cards with partially filled coefficients and will need to complete them so that the equation meets the proposed criteria.

- Instructions:

• Divide the class into groups of up to 5 students and distribute the materials.

• Each group will receive cards with partially filled coefficients and criteria to complete.

• Students should work together to fill in the coefficients and form equations that satisfy the criteria.

• At the end, each group will present one or more created equations and explain how they arrived at the chosen coefficients.

• Class discussion on the different strategies and difficulties encountered.

Feedback

Duration: (15 - 20 minutes)

The purpose of this stage of the lesson plan is to consolidate students' practical learning, allowing them to reflect and verbally articulate the acquired knowledge. The group discussion helps identify comprehension gaps and reinforce key concepts. Additionally, by sharing their experiences, students can learn from one another, developing communication and collaboration skills. This stage also serves to assess students' understanding of the topic, allowing the teacher immediate and targeted feedback on the class's progress.

Group Discussion

Start the group discussion with a brief recap of the activities performed, emphasizing the importance of coefficients and their relationships with the roots of quadratic equations. Ask each group to share their most interesting findings and the challenges encountered during the activities. Encourage students to discuss how they applied theoretical knowledge to solve the proposed practical problems and the strategies they used.

Key Questions

1. Which properties of the coefficients of quadratic equations were most crucial in solving the proposed problems?

2. How did manipulating the coefficients help in better understanding the roots of the equations?

3. Was there any moment during the activities when the theory did not apply as expected? How was this resolved?

Conclusion

Duration: (5 - 10 minutes)

The Conclusion stage is vital to ensure that students have consolidated the knowledge acquired during the lesson. By summarizing and recapping key points, the teacher reinforces learning and helps students retain essential information. Moreover, by discussing the interconnection between theory and practice and the real applicability of the studied concepts, students can perceive the importance and utility of what they learned, motivating them to continue exploring and applying this knowledge.

Summary

In conclusion, the teacher should summarize the main concepts covered, highlighting the importance of coefficients in quadratic equations and how they influence the calculation of the roots. It is crucial to recap the properties of the roots, such as the relationship between the sum and product and the coefficients, and how these properties were applied during the practical activities.

Theory Connection

Throughout the lesson, students had the opportunity to connect the theory of coefficients with practice through activities that simulated real situations and mathematical problems. This allowed for a deeper understanding of the content, demonstrating the applicability of the studied concepts. The lesson was structured to facilitate this transition, preparing students for applying knowledge in various contexts.

Closing

Finally, it is essential to emphasize the relevance of studying quadratic equations and their coefficients, not only for their practical applicability in various areas but also for their presence in everyday situations. Understanding these concepts not only broadens students' mathematical reasoning but also empowers them to solve complex problems more effectively.