Lesson Plan | Active Learning | Factorization: Difference of Squares

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 establish a clear basis for what is expected of students to learn and be able to do by the end of the lesson. By setting specific and clear objectives, students can better direct their learning and efforts in class to achieve concrete goals. This section also serves to align the teacher's expectations with the intended outcomes of the lesson, facilitating the assessment of student progress throughout the activities.

Main Objectives:

1. Empower students to factor algebraic expressions that follow the pattern of the difference of squares, applying the formula (a²-b²)=(a+b)(a-b).

2. Develop the ability to identify and differentiate situations where factoring by difference of squares is applicable, contributing to a broader understanding of the concept.

Side Objectives:

1. Encourage active participation from students in identifying practical situations where factoring by difference of squares may be useful.

Introduction

Duration: (15 - 20 minutes)

The Introduction stage aims to engage students with the lesson's theme, using problem situations that encourage the practical application of the content they previously studied. Moreover, by contextualizing theory with practical and historical applications, students can visualize the relevance of what they are learning, thereby increasing interest and motivation for the lesson.

Problem-Based Situations

1. Considering the algebraic expression x² - 25, ask students to factor it using the difference of squares formula. This exercise can be done in pairs to promote discussion and joint reasoning.

2. Ask students to factor the expression y² - 9z², demonstrating the process of identifying perfect squares and applying the formula to obtain the solution. Encourage them to think of multiple ways to arrive at the same result.

Contextualization

To contextualize the importance of factoring by difference of squares, the teacher can explore how this technique is applied in real-world problems and in other areas of knowledge, such as in the simplification of physical equations or the optimization of algorithms. Additionally, they can briefly narrate the history of the development of algebra, highlighting the relevance of discoveries made by mathematicians like Al-Khwarizmi and Bhaskara, who significantly contributed to the evolution of the study of polynomials and their properties.

Development

Duration: (65 - 75 minutes)

The Development section is designed to allow students to practically and interactively apply prior knowledge about factoring by difference of squares. Through the proposed activities, students have the opportunity to explore the concept in depth, developing not only mathematical skills but also teamwork, creativity, and critical thinking skills. This stage aims to consolidate learning dynamically and engagingly, ensuring students can not only understand but also appropriate the content meaningfully.

Activity Suggestions

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

Activity 1 - The Factoring Marathon

> Duration: (60 - 70 minutes)

- Objective: Factor expressions by difference of squares quickly and accurately, promoting teamwork and healthy competition.

- Description: In this activity, students will be divided into groups of up to 5 people and participate in a competition to factor the largest number of expressions by difference of squares correctly in the shortest time possible. Each group will receive a set of algebraic expressions previously prepared by the teacher and must use colored cards to demonstrate the correct factoring. Each correctly factored and demonstrated expression will earn points.

- Instructions:

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

• Distribute the sets of algebraic expressions to each group.

• Explain that they should use the formula (a²-b²)=(a+b)(a-b) to factor the expressions.

• Each group must demonstrate the correct factoring using colored cards, one for each term of the factoring.

• Establish a time limit and start the competition.

• At the end, each group presents their answers and scores are assigned based on the number of correct factorizations and speed.

Activity 2 - Mathematical Theater: The Difference of Squares on Stage

> Duration: (60 - 70 minutes)

- Objective: Develop creativity and deep understanding of the process of factoring by difference of squares, as well as promote presentation and teamwork skills.

- Description: Students, in groups, must create and present a short play that illustrates the concept of factoring by difference of squares. Each group will receive an algebraic expression that must be factored and creatively incorporated into the performance. The goal is not only to factor correctly but also to present it in a fun and memorable way.

- Instructions:

• Organize students into groups of up to 5.

• Assign each group an expression to factor and incorporate into the play.

• Groups will have time to write the script and prepare the presentation.

• Each play must contain the factoring of the expression clearly and creatively.

• After the presentations, allow students to discuss and evaluate the different approaches.

Activity 3 - Mathematical Riddles Challenge

> Duration: (60 - 70 minutes)

- Objective: Promote logical reasoning and the ability to factor by difference of squares in a game context, encouraging collaboration and critical thinking.

- Description: In this challenge, groups of students receive envelopes containing 'riddles' that are, in fact, expressions by difference of squares to be factored. Each correctly solved riddle will lead to a clue for the next one, forming a path that leads to the final 'treasure', which could be an expression that, when factored, reveals a special concept or a hidden message.

- Instructions:

• Divide students into groups and distribute the envelopes.

• Explain that each correctly factored riddle leads to a clue for the next.

• Monitor the progress of the groups and provide hints or assistance when necessary.

• The first group to uncover the complete 'treasure' wins.

• Discuss the solutions and the path taken by the groups to solve the riddles.

Feedback

Duration: (15 - 20 minutes)

This stage of the lesson plan is essential to consolidate learning, allowing students to articulate what they have learned and hear their peers' perspectives. Group discussion helps improve understanding of concepts, promotes idea exchange, and addresses potential misunderstandings. Furthermore, by explaining their reasoning, students can identify and correct mistakes, which is crucial for effective mathematics learning.

Group Discussion

To start the group discussion, the teacher should gather all students in a large circle and ask each group to share their findings and experiences from the activities conducted. Start with a brief introduction, highlighting the importance of reflecting on what was learned and how the practical application of the difference of squares concept can be useful in real situations or other subjects. Encourage students to explain the reasoning behind the factorizations they performed and any challenges they faced during the activities.

Key Questions

1. What were the main challenges your group faced when factoring the expressions by difference of squares?

2. How can factoring by difference of squares be applied in other areas of knowledge or in practical situations?

3. Was there any expression that you initially found difficult to factor, but became clearer after the group discussion? Why?

Conclusion

Duration: (5 - 10 minutes)

The Conclusion stage is vital to ensure students have a clear and consolidated view of what was learned during the lesson. Summarizing the content helps reinforce knowledge, while explaining how theory connects with practical and everyday applications aims to motivate students and highlight the relevance of studying mathematics. Furthermore, by providing an appropriate closure, this stage prepares students for future applications of the concepts learned.

Summary

To conclude the lesson, the teacher should summarize the main concepts covered about factoring by difference of squares, recalling the formula (a²-b²)=(a+b)(a-b) and exemplifying the factored expressions during the activities. It is important to recap the key points to ensure that students have a clear understanding of the content.

Theory Connection

During the lesson, the theory of factoring by difference of squares was linked to real practices and applications through dynamic activities and varied contexts, such as competitions, theater, and challenges. This approach not only helped solidify mathematical knowledge but also showed how theoretical concepts are applicable and relevant in everyday situations and other subjects.

Closing

Finally, the teacher should emphasize the importance of factoring by difference of squares, explaining how this skill can be used not only in mathematical contexts but also in areas such as science and engineering, where the simplification of expressions is crucial for solving practical problems. This connection to the real world helps motivate students and recognize the relevance of what they have learned.