Lesson Plan | Traditional Methodology | Modular Function: Graph

Objectives

Duration: (10 - 15 minutes)

The aim of this stage of the lesson plan is to clearly establish what is expected for students to acquire by the end of the lesson. By describing the objectives, the teacher guides the focus of the lesson and prepares students for the concepts and skills that will be addressed, ensuring that everyone understands the relevance of the topic and how it will be presented.

Main Objectives

1. Understand and recognize the graph of a modular function.

2. Extract the input and output values from the graph of modular functions.

3. Construct the graph of a modular function.

Introduction

Duration: (10 - 15 minutes)

The aim of this stage of the lesson plan is to spark students' interest and connect the content with practical applications of everyday life. By establishing an initial context and sharing curiosities, the teacher creates a more engaging environment and prepares students for an understanding of the concepts that will be addressed in the lesson.

Context

Start the lesson by asking students if they have encountered situations where negative and positive numbers are treated differently. Explain that today they will study a function that deals exactly with this: the modular function. The modular function is a mathematical tool that helps us understand how absolute values (or modules) of numbers are graphically represented. The module of a number is the distance of that number to zero on the number line, ignoring whether it is positive or negative. This concept is widely used in various fields, such as physics, engineering, and economics, where one works with values that cannot be negative, such as distances and quantities of products.

Curiosities

Did you know that the modular function is often used in computer programming to calculate distances between points and in optimization algorithms? Additionally, in electronics, the module of a signal can indicate the strength of an electric current, regardless of its direction.

Development

Duration: (50 - 60 minutes)

The aim of this stage of the lesson plan is to provide a detailed and practical understanding of the modular function. By addressing specific topics and solving guided problems, students will be able to recognize, analyze, and construct graphs of modular functions. This will ensure they are able to apply this knowledge in various mathematical situations and real-world problems.

Covered Topics

1. Definition of Modular Function: Explain that the modular function, represented by f(x) = |x|, returns the absolute value of x. Detail how the module of a number is always non-negative. 2. Properties of the Modular Function: Discuss important properties of the modular function, such as symmetry concerning the y-axis and non-negativity. Show examples of positive and negative values. 3. Graph of the Modular Function: Draw the graph of the modular function f(x) = |x| on the board. Show how the function is composed of two parts: a straight line with a positive slope for x >= 0 and a straight line with a negative slope for x < 0. 4. Transformations in the Modular Function: Explain how transformations affect the graph of the modular function. For example, f(x) = |x - a| shifts the graph horizontally, while f(x) = |x| + b shifts it vertically. 5. Practical Examples: Provide examples of how to construct the graph of transformed modular functions, such as f(x) = |x - 2| + 3. Draw the graphs on the board and explain each step.

Classroom Questions

1. Draw the graph of the function f(x) = |x - 3|. 2. What is the value of f(x) = |x + 2| when x = -4? 3. Construct the graph of the function f(x) = |2x - 4| and identify its main points.

Questions Discussion

Duration: (20 - 25 minutes)

The aim of this stage of the lesson plan is to review and consolidate the knowledge acquired by students, ensuring they fully understand the concepts discussed. By discussing the answers to the questions and engaging students with reflective questions, the teacher promotes an active and collaborative learning environment, encouraging participation and critical thinking.

Discussion

• Discussion of the Presented Questions:

• 1. Draw the graph of the function f(x) = |x - 3|:
• • Explain that for x >= 3, the function behaves like f(x) = x - 3, resulting in a line with a positive slope starting from the point (3, 0).
• • For x < 3, the function behaves like f(x) = -(x - 3) = -x + 3, forming a line with a negative slope up to the point (3, 0).
• • Highlight that the graph has a 'V' at the vertex at (3, 0), which is the point where the function changes behavior.
• 2. What is the value of f(x) = |x + 2| when x = -4?:
• • Substitute x with -4 in the function: f(-4) = |-4 + 2|.
• • Calculate the value inside the module: -4 + 2 = -2.
• • The absolute value of -2 is 2, so f(-4) = 2.
• 3. Construct the graph of the function f(x) = |2x - 4| and identify its main points:
• • For x >= 2, the function behaves like f(x) = 2x - 4, resulting in a line with a positive slope starting from the point (2, 0).
• • For x < 2, the function behaves like f(x) = -(2x - 4) = -2x + 4, forming a line with a negative slope up to the point (2, 0).
• • Highlight that the graph has a 'V' at the vertex at (2, 0), which is the point where the function changes behavior.

Student Engagement

1. ❓ Questions to Engage Students: 2. What are the notable differences between the graphs of f(x) = |x - 3| and f(x) = |2x - 4|? 3. How does the slope of the lines before and after the vertex influence the shape of the graph of a modular function? 4. If we have the function f(x) = |x + 1| - 2, how would you describe the shift of the graph in relation to the graph of f(x) = |x|? 5.  Reflections: 6. How can understanding the graph of a modular function help in other areas of mathematics and applied sciences? 7. Why is it important to understand how graphs of modular functions are constructed and interpreted?

Conclusion

Duration: (10 - 15 minutes)

The aim of this stage of the lesson plan is to review and consolidate the main concepts addressed during the lesson, ensuring that students have a clear and cohesive view of the content. By recapping the main points and discussing the practical relevance of the topic, the teacher reinforces the importance of the acquired knowledge and its application in real situations, promoting a deeper and lasting understanding.

Summary

• Definition of modular function and its representation as f(x) = |x|.
• Properties of the modular function, including symmetry concerning the y-axis and non-negativity.
• Graph of the modular function f(x) = |x| and the parts that compose it.
• Transformations in the modular function and their effects on the graph, such as horizontal and vertical shifts.
• Practical examples of construction and analysis of graphs of transformed modular functions.

The lesson connected theory with practice by introducing the definition and properties of the modular function and then illustrating how these properties manifest graphically. By drawing and analyzing graphs of modular functions, students were able to see how transformations affect the shape of the graph, facilitating understanding of abstract concepts through concrete visual examples.

Understanding the graph of a modular function is crucial for various practical applications, such as in physics, where one works with distances that cannot be negative, or in computer programming, where modules are used to calculate distances between points. Additionally, knowledge of modular functions and their graphs can be applied in fields such as electronics, engineering, and economics, making it an essential tool in both academic and professional daily life.