Lesson Plan | Active Learning | Polynomials: Operations

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 aims to clearly establish what is expected for students to achieve by the end of the lesson. By setting specific goals, students can focus their prior studies and classroom participation on the essential skills needed to manipulate polynomials effectively. This section also serves to align expectations between teacher and students, ensuring a targeted and efficient approach during the lesson.

Main Objectives:

1. Empower students to solve the main operations with polynomials, including addition, multiplication, division, and subtraction.

2. Develop skills to apply theoretical knowledge about polynomials in practical problems and varied contexts.

Side Objectives:

1. Encourage critical thinking and problem-solving through practical examples and challenges.
2. Promote collaboration and debate among students during group activities.

Introduction

Duration: (15 - 20 minutes)

The Introduction serves to engage students and reinforce prior knowledge about polynomials by presenting problem situations that students may need to solve in daily life or in practical applications. Furthermore, it contextualizes the importance of polynomials in various fields, demonstrating the relevance of studying the topic and motivating students to understand the theory behind the operations they will perform in class.

Problem-Based Situations

1. Consider the polynomial function f(x) = 3x² - 5x + 2 and the function g(x) = -2x³ + x² + 4. Ask students to solve f(x) + g(x), f(x) - g(x), f(x) * g(x), and f(x) / g(x), using operations with polynomials.

2. Imagine you are designing a new irrigation system for a farm and need to model the behavior of water in the soil throughout different hours of the day. To do this, you use polynomials to describe the variation of soil moisture. How would you combine polynomials to represent the variation of moisture at different times of day?

Contextualization

Polynomials are extensively used in various areas of mathematics and applied sciences, such as engineering, physics, and economics, to model and solve complex problems. For example, in physics, they are used to describe various quantities such as force, velocity, and acceleration. Moreover, practical applications in modern technologies, such as signal processing in computing, heavily rely on techniques involving operations with polynomials. Therefore, understanding and being able to manipulate polynomials is fundamental for students wishing to pursue careers in STEM (Science, Technology, Engineering, and Mathematics).

Development

Duration: (70 - 75 minutes)

The Development stage is designed to allow students to apply and deepen their understanding of operations with polynomials in practical and challenging contexts. Through the proposed activities, students will have an opportunity to work in teams, enhance their problem-solving skills, and explore real-world applications of the mathematical concepts studied. Each activity is structured to maximize student participation and promote active learning, facilitating knowledge retention and the application of mathematical skills in real-world situations.

Activity Suggestions

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

Activity 1 - Mathematical Engineers Challenge

> Duration: (60 - 70 minutes)

- Objective: Apply knowledge of operations with polynomials to solve a practical engineering problem, developing teamwork and logical reasoning skills.

- Description: In this activity, students are challenged to design the plumbing system of a city using polynomials. They must consider different types of pipes and their flow capacities, which are modeled by polynomials. Each group receives a list of requirements such as distances, necessary flow capacities, and budget constraints.

- Instructions:

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

• Distribute the problem statement and the polynomial functions representing the flow capacities of each type of pipe.

• Ask each group to design a plumbing system that meets the requirements, using operations with polynomials to calculate total capacities and verify if they meet the city's needs.

• Each group must present its solution, explaining the choices made and demonstrating the use of polynomials in the operations performed.

Activity 2 - Polynomial Cinema: Creating the Perfect Soundtrack

> Duration: (60 - 70 minutes)

- Objective: Use operations with polynomials to create a cohesive soundtrack, developing creativity and practical application skills of mathematical concepts.

- Description: Students will use polynomials to model the evolution of a musical theme in a movie. Each group will receive different variations of the theme, represented by polynomials, and must combine them to create the ideal soundtrack. The variations must connect smoothly and adhere to continuity and harmony criteria.

- Instructions:

• Organize students into groups of up to 5 participants.

• Provide each group with the polynomials representing the variations of the musical theme.

• Instruct students to use operations with polynomials to combine the variations so that the resulting soundtrack is continuous and harmonious.

• Each group presents the created soundtrack, explaining the operations performed and the intended effect of each combination.

Activity 3 - Polynomial Investigators: The Garden Mystery

> Duration: (60 - 70 minutes)

- Objective: Develop analysis and problem-solving skills using operations with polynomials in a playful and investigative context.

- Description: In this scenario, students are mathematical detectives who must solve a mystery in a garden, where different types of flowers grow in polynomial shapes. Each group receives data about the growth of some flowers, represented by polynomials, and must use operations with polynomials to predict the growth of other flowers and discover what caused an anomaly in one of their growth patterns.

- Instructions:

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

• Provide each group with the polynomial data modeling the growth of different flowers and the problem situation.

• The groups must apply operations with polynomials to analyze the data and solve the mystery, presenting their conclusions and methods used.

• Each group presents its resolution, and the other groups must analyze and discuss the different approaches and results.

Feedback

Duration: (20 - 25 minutes)

The purpose of this stage is to consolidate the learning gained during the practical activities, allowing students to critically reflect on the use of operations with polynomials in real and theoretical situations. The group discussion helps reinforce the understanding of mathematical concepts and promotes communication and argumentation skills. Additionally, it allows the teacher to assess the students' comprehension level and identify areas that need reinforcement or further exploration.

Group Discussion

Start the group discussion with a brief recap of the activities performed, highlighting the importance of operations with polynomials and how they were applied in different contexts. Ask each group to share their findings and challenges faced during the activities. Encourage students to discuss the different approaches and solutions found, emphasizing mutual learning. Use questions like 'What surprised you most during the activity?' and 'How did group collaboration help solve the problem?' to guide the conversation.

Key Questions

1. What were the main challenges in applying operations with polynomials in the practical activities?

2. How did understanding the basic operations with polynomials help in solving the proposed problems?

3. Was there any situation where a polynomial operation provided an unexpected or non-intuitive solution?

Conclusion

Duration: (5 - 10 minutes)

The purpose of the Conclusion stage is to consolidate learning by providing a clear overview of what was discussed and learned during the lesson. This recap helps students solidify content and understand the relevance of polynomials. Additionally, highlighting practical applications and the connection with theory reinforces the importance of studying mathematics, encouraging students to value learning and see the discipline as something integrated and applicable in their lives.

Summary

In the conclusion, the teacher should summarize and recap the main operations with polynomials covered in the lesson: addition, subtraction, multiplication, and division. It should emphasize how each operation is performed, using practical examples given by students during the activities. This will help reinforce the knowledge and ensure that students have a clear understanding of how to manipulate polynomials in different contexts.

Theory Connection

During the lesson, the connection between theory and practice was established through activities that simulated real problems, such as designing an irrigation system or creating a soundtrack. These practical applications showed students how the theoretical concepts of polynomials are essential for solving daily problems and in fields such as engineering and music. This approach helps students see mathematics as a relevant and powerful tool.

Closing

To close, it is crucial to emphasize the importance of polynomials in everyday life and in various professional areas. Understanding and knowing how to manipulate polynomials not only enriches students' mathematical knowledge but also prepares them for practical applications in their future academic and professional careers. This recognition helps motivate students and perceive mathematics as a living and applicable discipline.