Lesson Plan | Active Learning | Factorization: Second Degree Expressions

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 fundamental to establish a clear direction for the lesson and ensure that both the teacher and students are aligned with the expected outcomes. By defining specific and measurable objectives, this section guides practical activities and classroom interactions to maintain focus on applying students' prior knowledge of factoring quadratic expressions. This ensures that the lesson is effectively dedicated to applying concepts in varied contexts, preparing students for real problem-solving situations.

Main Objectives:

1. Empower students to factor quadratic expressions in the form a(x-r1)(x-r2), identifying and using the roots (r1 and r2) of the polynomial.

2. Develop skills to translate theory into practice by solving exercises involving the factoring of quadratic polynomials.

Side Objectives:

1. Encourage discussion among students for exchanging problem-solving strategies, fostering a collaborative environment.

Introduction

Duration: (15 - 20 minutes)

The Introduction stage serves to engage students and practically revisit the concepts studied about factoring quadratic expressions. By presenting problem-based scenarios and historical contexts, this section seeks to connect mathematical content with the real world, enhancing interest and perceived relevance by students. This initial engagement is crucial to prepare the ground for the practical application of concepts during the lesson.

Problem-Based Situations

1. Ask students to factor the polynomial 2x² - 5x - 3 in class, using their prior knowledge about quadratic expressions. This exercise not only revisits the concept of factoring but also serves as a starting point for the practical application of the polynomial's roots.

2. Request that students solve the equation x² + 6x + 9 = 0 and then factor the polynomial based on the roots found. This activity allows students to directly apply the concept of factoring through the identification and utilization of roots.

Contextualization

Factoring quadratic expressions is an essential tool in mathematics and its applications range from solving equations to studying functions. Interestingly, the technique of completing the square, used to find the roots of a quadratic polynomial, has ancient origins and was independently developed by mathematicians in different parts of the world, such as the Indian mathematician Bhaskara in the 7th century and the Persian mathematician Al-Khwarizmi in the 9th century. This technique revolutionized the field of algebra and is still fundamental in mathematics education.

Development

Duration: (70 - 75 minutes)

The Development stage is designed to allow students to practically and creatively apply their prior knowledge of factoring quadratic expressions. Through group problem-solving, students not only reinforce theoretical content but also develop skills in collaboration, critical thinking, and problem-solving. Each proposed activity aims to solidify students' understanding and provide active and engaging learning.

Activity Suggestions

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

Activity 1 - Magic Roots Challenge

> Duration: (60 - 70 minutes)

- Objective: Develop factoring skills and application of polynomial root concepts while stimulating creativity and teamwork.

- Description: In this activity, students will be challenged to solve a set of quadratic equations, whose roots are 'magical'. The roots will be simple integers or fractions, facilitating algebraic manipulation and the identification of the roots a and b. The task consists of factoring polynomials from the roots, and the 'magic' lies in how these polynomials are presented, with a small narrative involving each of them, stimulating creativity and logical reasoning.

- Instructions:

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

• Give each group a list of five quadratic equations with predetermined roots, but presented creatively, as if they were challenges from a game.

• Set a maximum time of 60 minutes for the groups to solve the equations and factor the polynomials, presenting the result in a standardized factoring form.

• Each group should also create a small story or situation that involves the 'magical' roots for each equation, making the activity more playful and facilitating memorization and understanding of the concept.

Activity 2 - The Mystery of the Lost Roots

> Duration: (60 - 70 minutes)

- Objective: Enhance the ability to find roots of quadratic polynomials and factor complete expressions, while promoting teamwork and effective communication.

- Description: Students will become mathematical detectives in an investigation scenario where they must recover 'lost roots' to solve an enigma. In this enigma, quadratic polynomials are presented without the roots, and students need not only to find the roots but also to factor the complete expressions a(x-r1)(x-r2). The activity incorporates elements of logical and deductive reasoning, as well as teamwork.

- Instructions:

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

• Distribute envelopes containing quadratic polynomials without specified roots.

• Students must use factoring methods and equation-solving to find the roots, and then factor the complete polynomials.

• Each group should prepare a report on the investigation process, explaining the steps taken and the solutions found, and present it to the class at the end of the activity.

Activity 3 - Bridge Builders

> Duration: (60 - 70 minutes)

- Objective: Visualize and solidify the concept of factoring quadratic expressions and their roots through a practical and creative activity that involves manual and spatial skills.

- Description: In this activity, students must 'build bridges' between quadratic equations and their factorizations using recyclable materials. Each group receives a series of equations and must graphically represent the roots and factorizations of each, constructing small bridge models that illustrate these concepts. The activity aims to solidify practical understanding of roots and factorizations.

- Instructions:

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

• Provide each group with colored cards, rubber bands, toothpicks, and small wooden blocks.

• Students must build models representing the factoring of quadratic polynomials, clearly showing the roots and the factored form a(x-r1)(x-r2).

• Each group presents their model to the class, explaining the corresponding equations and how the model represents the roots and the factoring.

Feedback

Duration: (10 - 15 minutes)

The purpose of this feedback section is to consolidate the learning acquired during the practical activities, allowing students to articulate and reflect on the learning process. Through group discussion, students have the opportunity to verbalize their understanding, hear different perspectives, and correct possible misconceptions, contributing to a deeper understanding of the content. Additionally, this stage also serves to assess students' comprehension level and identify areas that may need further review.

Group Discussion

To initiate the group discussion, the teacher can suggest that each group share the most interesting or challenging discoveries they faced during the activities. They can be asked to discuss the different strategies they used to factor the polynomials and how they applied the concept of the roots to solve the proposed problems. The teacher can also guide the discussion for students to reflect on the importance of the factoring method in real situations or in other disciplines.

Key Questions

1. What were the main challenges in factoring the polynomials and how did the group overcome them?

2. How did the roots found help in the factoring of the polynomials and in understanding the concept?

3. Was there any situation during the activities where factoring the polynomials proved useful for solving other problems?

Conclusion

Duration: (5 - 10 minutes)

The aim of the Conclusion stage is to consolidate the learning acquired, ensuring that students have a clear and integrated view of the concepts covered. This moment also serves to reinforce the importance of integrating theory and practice, preparing students to apply their mathematical knowledge of factoring and roots in future situations, both in and out of the classroom.

Summary

In the final stage, it is crucial to summarize and recapitulate the concepts covered about factoring quadratic expressions. This summary helps consolidate learning, ensuring that students have a clear understanding of the roots of polynomials and how to apply factoring to solve equations and identify mathematical patterns.

Theory Connection

During the lesson, the connection between theory and practice was established through interactive activities and contexts that demonstrated the applicability of factoring and roots concepts in real and historical situations. This approach not only facilitated students' understanding but also demonstrated the relevance of mathematical content across various contexts.

Closing

In conclusion, today's lesson was designed not only to convey knowledge about factoring quadratic expressions but also to show how these concepts are fundamental for solving practical problems. It is expected that students have realized the importance of integrating theory and practice for a deeper mathematical understanding and to apply this knowledge in their daily and academic lives.