Lesson Plan | Traditional Methodology | Notable Cube Products

Objectives

Duration: (10 - 15 minutes)

The purpose of this stage is to provide students with a clear understanding of the objectives of the lesson, ensuring that they know what is expected of them by the end of the explanation and practice. This initial clarity helps to guide the focus and attention of students during the lesson, facilitating the learning of the notable products of cubes and their applications.

Main Objectives

1. Recognize the notable products involving cubes, such as (a + b)³, (a - b)³ and a³ - b³.

2. Understand the formulas associated with the notable products of cubes and their practical applications.

3. Apply the notable products of cubes in the resolution of mathematical problems.

Introduction

Duration: (10 - 15 minutes)

The purpose of this stage is to prepare students for understanding the notable products of cubes, providing a clear and interesting context that sparks their curiosity and engagement. By understanding the importance and applicability of notable products in various areas, students will be more motivated to learn and apply these concepts in the following practical activities.

Context

To begin the lesson on notable products of cubes, start by contextualizing the concept of powers, particularly the cube of a number. Explain that raising a number to the cube means multiplying it by itself three times. For example, 2³ is equal to 2 * 2 * 2, resulting in 8. Next, introduce the idea of algebraic expressions raised to the cube, such as (a + b)³, and how these expressions can be expanded using specific formulas known as notable products. Emphasize that these formulas simplify the expansion process and are powerful tools in many areas of mathematics.

Curiosities

Did you know that notable products are widely used in various areas of science and engineering? For example, in physics, when calculating the volume of complex three-dimensional figures, the use of notable products can facilitate obtaining accurate results more efficiently. Additionally, in computer graphics, the formulas for notable products are used to optimize algorithms that generate three-dimensional images, allowing for more realistic graphics in games and movies.

Development

Duration: (35 - 45 minutes)

The purpose of this stage is to deepen students' understanding of the notable products of cubes, providing detailed explanations and practical examples. By addressing each topic clearly and solving guided problems, students can see the practical application of the formulas and develop skills to use them effectively in different mathematical contexts.

Covered Topics

1. Notable Product (a + b)³: Explain that the formula for (a + b)³ is expanded as a³ + 3a²b + 3ab² + b³. Detail each term of the expansion, emphasizing how each component is obtained from multiplication and how the coefficients arise from the combination of binomial terms. 2. Notable Product (a - b)³: Describe the formula for (a - b)³, which is a³ - 3a²b + 3ab² - b³. Highlight the differences and similarities with the expansion of (a + b)³ and explain the importance of the alternating signs. 3. Difference of Cubes a³ - b³: Present the formula for a³ - b³, which is (a - b)(a² + ab + b²). Explain the decomposition of the polynomial into a product of a binomial and a trinomial, and how this facilitates the factorization of more complex algebraic expressions. 4. Practical Examples: Provide practical examples of each notable product. For example, expand (2 + 3)³ and show the step-by-step expansion, detailing each term. Do the same for (2 - 3)³ and for a³ - 27. 5. Applications in Problems: Demonstrate how these notable products can be applied to simplify and solve mathematical problems. Provide a contextualized problem and solve it step by step, showing the application of the notable product formulas.

Classroom Questions

1. Expand the expression (x + 4)³ and simplify the result. 2. Factor the expression 27 - a³ using the difference of cubes formula. 3. Given the expression (2y - 5)³, expand and simplify each term.

Questions Discussion

Duration: (25 - 30 minutes)

The purpose of this stage is to ensure that students fully understand the content covered, providing a moment for reflection and discussion on the questions addressed. This feedback allows students to review their own resolution processes, clarify doubts, and reinforce learning through the exchange of ideas and detailed explanations.

Discussion

• Expand the expression (x + 4)³ and simplify the result.

• To expand the expression (x + 4)³, use the notable product formula (a + b)³:

• (x + 4)³ = x³ + 3x²(4) + 3x(4²) + 4³

• = x³ + 12x² + 48x + 64

• Thus, the expanded expression is x³ + 12x² + 48x + 64.

• Factor the expression 27 - a³ using the difference of cubes formula.

• To factor the expression 27 - a³, apply the difference of cubes formula a³ - b³ = (a - b)(a² + ab + b²):

• 27 - a³ = 3³ - a³

• = (3 - a)(3² + 3a + a²)

• = (3 - a)(9 + 3a + a²)

• Therefore, the factored expression is (3 - a)(9 + 3a + a²).

• Given the expression (2y - 5)³, expand and simplify each term.

• To expand the expression (2y - 5)³, use the notable product formula (a - b)³:

• (2y - 5)³ = (2y)³ - 3(2y)²(5) + 3(2y)(5²) - 5³

• = 8y³ - 60y² + 150y - 125

• Thus, the expanded expression is 8y³ - 60y² + 150y - 125.

Student Engagement

1. What is the importance of understanding and applying notable products in mathematical problems? 2. How can notable products simplify the resolution of complex algebraic expressions? 3. Can you think of other practical situations, beyond those discussed, where notable products could be useful? 4. What difficulties did you encounter when applying the notable product formulas? 5. How would you verify if the expansion or factorization of an expression is correct?

Conclusion

Duration: (10 - 15 minutes)

The purpose of this stage is to recap the main points of the lesson, reinforce the connection between theory and practice, and highlight the importance and applicability of notable products. This helps students consolidate their learning and recognize the relevance of the studied content in varied contexts.

Summary

• Concept of power and cube of a number.
• Notable products: (a + b)³, (a - b)³ and a³ - b³.
• Expansion and factorization of algebraic expressions using notable products.
• Practical examples and applications of notable products in mathematical problems.
• Discussion and problem-solving to reinforce understanding of notable products.

The lesson connected the theory of notable products of cubes with practice by presenting detailed examples and solving problems step by step. This demonstrated how the theoretical formulas can be applied to simplify and resolve complex algebraic expressions, providing a deeper and practical understanding of the content addressed.

Understanding notable products is crucial not only for solving mathematical problems, but also in various areas of everyday life and science. For instance, in computer graphics and physics, notable products help simplify complex calculations, optimizing algorithms and facilitating the achievement of accurate results. This shows that mathematical knowledge has practical and significant applications.