Lesson Plan | Active Learning | First Degree Function: Graph and Table

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 designed to clearly establish the learning goals that students should achieve by the end of the lesson. By focusing on the graphical and tabular representation of linear functions, this section guides students on what is expected of them in understanding and applying different forms of mathematical data presentation. This lays the groundwork for practical activities in the classroom, where they can consolidate and apply the acquired knowledge.

Main Objectives:

1. Empower students to graphically represent a linear function in the Cartesian plane, identifying and interpreting the intersection points with the x and y axes.

2. Develop the ability to interpret and extract relevant information from a table representing a linear function, allowing students to apply these concepts in different mathematical contexts.

Side Objectives:

1. Stimulate students' critical and analytical thinking by comparing and contrasting different representations of a linear function.
2. Foster collaboration and communication among students during practical activities, promoting a participatory learning environment.

Introduction

Duration: (15 - 20 minutes)

The Introduction stage serves to engage students with the content they studied previously, utilizing problem-based situations that encourage them to apply and think critically about the linear function. Additionally, the contextualization helps to understand the relevance of the subject in daily life, enhancing interest and perception of its applicability. This moment prepares students for practical activities, meaningfully connecting theory and practice.

Problem-Based Situations

1. Considering the function f(x) = 2x - 3, ask students to determine the value of f(0) and f(4) and represent these points in the Cartesian plane. Later, question them about the interpretation of these points in relation to the function.

2. Present a table with x values and their respective f(x) for an unknown linear function and ask students to plot these points on the graph and use this information to determine the slope and intercept of the function.

Contextualization

Explain the importance of linear functions with a practical example: imagine a salesperson who started working in a store, and each month he increases his sales by a fixed amount. This situation can be modeled by a linear function, where the x-axis represents time (in months) and the y-axis represents sales. This modeling allows forecasting future sales based on history, aiding the financial planning of the company.

Development

Duration: (70 - 75 minutes)

The Development stage is designed to allow students to practically and contextually apply the concepts of linear functions that they have previously studied. By working in groups, students are encouraged to discuss, collaborate, and think critically, promoting active and meaningful learning. Each proposed activity is designed to be engaging and challenging, using everyday situations or playful scenarios to enhance student interest and understanding of the topic.

Activity Suggestions

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

Activity 1 - Mystery at the Museum: Recovering the Lost Treasure

> Duration: (60 - 70 minutes)

- Objective: Apply skills of graphical representation, table interpretation, and analysis of linear functions in a playful and practical context.

- Description: Students are mathematical detectives who need to solve a mystery at the museum. A famous painting has been stolen, but the thief left mathematical clues. The clues consist of tables with data that represent the thief's movement over time, where each point in the table corresponds to his location in the museum. Students must use this data to create a map of the thief's movement in the Cartesian plane, determine the type of function that describes the movement, and finally discover where the treasure might be hidden.

- Instructions:

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

• Distribute to each group a copy of the movement data table and graph paper.

• Guide students to plot the points from the table on graph paper, creating a chart.

• Ask them to identify the intersection points with the x and y axes and discuss the meaning of these points in the context of the problem.

• Request that based on the graph, they determine whether the function that best describes the movement is a linear function.

• Finally, students should use the acquired knowledge to predict the location of the treasure by applying the linear function to extend the thief's trajectory.

Activity 2 - Building a Linear City

> Duration: (60 - 70 minutes)

- Objective: Develop interpretation skills for tables, graphical representation, and application of linear functions in a practical and relevant scenario.

- Description: In this activity, students are urban planners who must design the layout of a new residential area. They receive a rectangular plot and must distribute houses so that a bus line, which crosses the diagonal of the plot, serves as many houses as possible. The bus line follows a linear trajectory that can be described by a linear function. Students need to use a distance table and a Cartesian plane to optimize the distribution of houses and ensure access to public transport.

- Instructions:

• Organize students into groups of up to 5.

• Give each group a copy of the plot, the distance table, and the Cartesian plane.

• Instruct students to mark the origin and the trajectory of the bus line on the Cartesian plane, using the distance table to determine the points.

• Guide the distribution of houses so that they can be served by the bus line.

• Ask them to apply the concept of linear function to optimize the distribution of houses.

• Request that they justify their choice of house placement and the trajectory of the bus line based on the graph and the function.

Activity 3 - The Great Math Tournament: Linear Race

> Duration: (60 - 70 minutes)

- Objective: Use concepts of linear function to model and forecast real events, promoting a practical and applied understanding of mathematics.

- Description: Students participate in a math tournament where each team must design the route of a relay race in a local park. They receive data on distances and times that the runners must achieve, and they must use this information to create a graph representing the finishing times of the runners in relation to the distances covered. The challenge is to adjust the trajectory to be a linear function, allowing for accurate predictions of the finishing time for any point on the course.

- Instructions:

• Divide the class into groups of no more than 5 students.

• Give each group the distance and time data for each runner.

• Ask students to construct a graph on the Cartesian plane, relating distance and time.

• Guide students to identify a linear function that best fits the data, considering the intersection points with the axes.

• Challenge groups to predict the finishing time of a runner at an unprovided distance using the linear function.

• Conclude with a symbolic race using the predicted data to validate the mathematical model.

Feedback

Duration: (15 - 20 minutes)

The purpose of this stage of the lesson plan is to consolidate students' learning, allowing them to reflect on their experiences and share insights. The group discussion helps reinforce the knowledge gained, allowing for an exchange of ideas and approaches that enrich the understanding of the topic. Furthermore, by answering key questions, students are encouraged to think critically and verbalize their understanding, which is fundamental for applying mathematical concepts in different contexts.

Group Discussion

Start the group discussion with a brief introduction, highlighting the importance of sharing discoveries and challenges faced during the activities. Encourage each group to present a summary of their experience, focusing on the results obtained and the strategies used. Then, promote an open dialogue where students can ask questions and comment on their classmates' work, exploring different approaches and learnings. This moment is crucial for students to articulate what they have learned and critically evaluate their understanding of the content.

Key Questions

1. What were the main challenges in applying the concepts of linear function in the practical activities?

2. How did the graphical representation in the Cartesian plane help solve the proposed problems?

3. What was the importance of understanding and interpreting the intersection points with the x and y axes?

Conclusion

Duration: (10 - 15 minutes)

The Conclusion stage is designed to synthesize and consolidate the lesson's learning, ensuring that students can link theoretical concepts with the practical activities carried out. By summarizing key points and highlighting the applicability of the content, this section helps reinforce students' understanding and recognize the importance of what has been learned in broader contexts. Additionally, it serves as an opportunity for the teacher to assess students' understanding and clarify any remaining questions.

Summary

In this lesson, students explored the linear function through graphical representations in the Cartesian plane and table interpretation. They reviewed how to identify and interpret the intersection points with the x and y axes, essential for understanding the behavior of linear functions. Moreover, they applied these concepts in practical scenarios, such as city planning and solving mathematical mysteries.

Theory Connection

Today's lesson connected mathematical theory with practice through playful and contextualized activities, such as the museum mystery and the linear city construction. These exercises not only reinforced theoretical understanding but also demonstrated the applicability of linear functions in everyday and professional situations, emphasizing the importance of effectively understanding and manipulating mathematical data.

Closing

In conclusion, it is crucial to highlight the relevance of linear functions in everyday life, whether in data forecasting, project planning, or solving practical problems. The ability to interpret and use these mathematical concepts is a fundamental skill in many fields of study and careers, reinforcing the importance of this topic for students.