Lesson Plan | Active Learning | Notable Cube Products

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 crucial for directing students' focus and clarifying what will be expected of them during the class. Here, the aim is to establish a clear understanding of the main concepts that will be explored, specifically notable products of cubes, and the skills necessary to apply them in different mathematical contexts. This section will help align expectations and ensure that all students are prepared for the practical activities that will follow.

Main Objectives:

1. Empower students to identify and apply notable products involving cubes of algebraic expressions.

2. Develop calculation and logical reasoning skills to solve mathematical problems involving cubes.

Side Objectives:

1. Promote collaboration among students during practical activities, encouraging knowledge sharing and teamwork.

Introduction

Duration: (15 - 20 minutes)

The Introduction stage serves to engage students with the lesson's theme, bridging theoretical knowledge and practical applications. By using problem situations, this phase challenges students to apply their prior knowledge in new and complex contexts, preparing them for more in-depth exploration during the lesson. The contextualization highlights the relevance of learning notable products of cubes, motivating students through the perception of their applicability in the real world.

Problem-Based Situations

1. Imagine you need to calculate the volume of a cube where each side is expressed by the sum of two terms, such as (a + b). How can you apply the notable product of the cube of the sum to find that volume?

2. Consider that a first-degree polynomial is raised to the cube during a physics experiment to calculate potential energy as a function of time. How can the formula for the cube of the binomial simplify this calculation?

Contextualization

Notable products of cubes are essential not only in pure mathematics but also in practical applications such as engineering, physics, and economics. For example, volume calculations and polynomial expansions in physics use these identities to simplify and solve complex problems. Additionally, understanding these concepts helps develop analytical skills necessary for progress in higher mathematics and other sciences.

Development

Duration: (65 - 75 minutes)

The Development stage is designed to allow students to apply in a practical and detailed way the concepts studied at home regarding notable products of cubes. Through creative and engaging activities, this section aims to consolidate learning, encouraging teamwork, quick reasoning, and the ability to connect theory with practice. The activities are designed to engage all students, ensuring that the theory of notable products is understood in a fun and memorable way.

Activity Suggestions

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

Activity 1 - Mathematical Magic Cube

> Duration: (60 - 70 minutes)

- Objective: Visualize and understand the expansion of (a+b)³ through a practical and interactive activity.

- Description: Students will be challenged to build physical models of cubes, visually representing the expansion of (a+b)³. Using colored blocks, each group will create a large cube for each term of the expansion, visually identifying the components as a³, 3a²b, 3ab², and b³.

- Instructions:

• Form groups of up to 5 students.

• Distribute different colored blocks to represent 'a' and 'b'.

• Ask them to build a large cube representing (a+b)³, placing the blocks corresponding to the powers of a and b so that they are visible and orderly.

• Each group must calculate and verify if the volume of the constructed cube corresponds to the theoretical expansion of (a+b)³.

• Present their findings to the class, explaining how each part of the cube represents a term of the expansion.

Activity 2 - Polynomial Adventure in the Park

> Duration: (60 - 70 minutes)

- Objective: Stimulate creativity and understanding of the concept of binomial expansion cubed in a playful and narrative way.

- Description: In this activity, students will create a comic story where characters (terms of a binomial) walk through an amusement park and face challenges that make them 'cubed'. Each challenge faced by the characters must correspond to a term of the expansion of (a+b)³.

- Instructions:

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

• Provide paper, colored pens, and rulers for each group.

• Instruct the students to draw a comic story in which the main characters are 'a' and 'b' from a binomial.

• Each comic strip must represent a term of the expansion of (a+b)³, with the characters facing challenges that make them 'grow cubed'.

• At the end, each group will present their comic story to the class, explaining how each scene represents a term of the expansion.

Activity 3 - Cube Race

> Duration: (60 - 70 minutes)

- Objective: Apply knowledge of notable products of cubes in a physical and competitive activity, reinforcing learning through repetition and time pressure.

- Description: In this dynamic activity, students will compete in a relay race where each stage involves quickly solving parts of the expansion of (a+b)³. Each correct answer allows them to advance to the next stage, and the team that finishes first wins.

- Instructions:

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

• Prepare separate stations for each term of the expansion of (a+b)³.

• At the start, the first student from each group will run to the first station to solve the term a³ and return to pass the baton to the next colleague.

• The process repeats for 3a²b, 3ab², and b³.

• The first team to complete all stations and return to the starting line wins.

Feedback

Duration: (10 - 15 minutes)

The purpose of this stage is to consolidate learning through the sharing of experiences and reflections among the students. By discussing together, students can improve their understanding of the concepts, learn from each other, and recognize the applicability of notable products of cubes in different contexts. This discussion also serves to evaluate students' understanding of the topic and their ability to apply knowledge practically.

Group Discussion

After completing the activities, organize a group discussion with all students. Start this discussion by explaining that the goal is to share learnings and diverse perspectives on the notable products of cubes. Suggest that each group briefly presents what they discovered and how they applied theoretical knowledge in practice. Encourage students to discuss the different approaches and insights that emerged during the activities and to reflect on how these concepts can be applied in real situations or other areas of knowledge.

Key Questions

1. What were the main challenges in applying the notable products of cubes during the activities and how did you overcome them?

2. How did visualization and practical activities help to better understand the concept of notable products of cubes?

3. Is there any everyday situation or in other subjects where you could apply the knowledge about notable products of cubes?

Conclusion

Duration: (5 - 10 minutes)

The purpose of this stage is to ensure that students consolidate the knowledge acquired during the lesson, recognize the applicability of the concepts learned, and perceive mathematics as a living and useful science. Summarizing and recapping key points facilitates long-term retention and prepares students to use these concepts in future academic and everyday situations.

Summary

In the conclusion stage, a review of the concepts of notable products of cubes will be conducted, highlighting the identities (a+b)³ = a³ + 3a²b + 3ab² + b³ and (a-b)³ = a³ - 3a²b + 3ab² - b³. This moment is crucial for reinforcing learning and ensuring that all students have understood the essential points of the lesson.

Theory Connection

The connection between theory and practice was evidenced during the classroom activities, where students applied theoretical knowledge in practical and playful situations. This method not only facilitates the understanding of mathematical concepts but also demonstrates the relevance of notable products of cubes in real and hypothetical contexts, such as volume calculations and polynomial expansions.

Closing

Finally, discussing the importance of notable products of cubes in everyday life, such as in volume calculations and other practical applications, emphasizes to students the practical utility of the mathematics learned. This discussion helps to view mathematics as a useful tool rather than just an academic discipline.