Lesson Plan | Active Learning | Function: Representations and Applications

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 guiding the focus of students and the teacher, establishing learning goals that will direct classroom activities. By detailing the objectives, students can better organize their prior knowledge and apply it more effectively during proposed activities. Additionally, this section serves to align expectations and ensure that everyone involved has a clear understanding of what is expected by the end of the lesson.

Main Objectives:

1. Empower students to understand the concept of a function, identifying that each input element has a unique output.

2. Develop understanding of the dependency relationships between variables in a function, exemplified with equations like y=2x+3.

Side Objectives:

1. Encourage the ability to analyze graphs and tables to interpret the behavior of functions.

Introduction

Duration: (15 - 20 minutes)

The introduction serves to engage students and connect previously studied content with practical and theoretical applications. Through problem situations, students are encouraged to revisit and apply their knowledge of functions in diverse contexts, stimulating critical thinking. The contextualization, in turn, broadens students' perspectives on the relevance of functions in real situations, increasing interest and understanding of the topic.

Problem-Based Situations

1. Imagine that you are organizing an event and need to calculate the total cost, considering that each guest pays a fixed registration fee and a fee per meal. How would you use a function to model this scenario?

2. Think of a situation where a delivery company needs to optimize its routes to save fuel. They can use functions to calculate the distance and travel time based on variables like traffic and distance. What would this model look like?

Contextualization

Functions are like recipes that transform a set of ingredients (input) into a final dish (output) in a predictable and consistent manner. This concept is crucial not only in mathematics but also in various fields such as economics, engineering, and computer science. For example, in civil engineering, functions are used to predict how different materials behave under different load conditions; or in economics, to understand how interest rates affect investments over time. These real-life applications illustrate the importance of understanding and being able to apply functions in everyday situations.

Development

Duration: (70 - 75 minutes)

The Development stage is designed to provide students with the opportunity to apply and deepen the knowledge acquired about functions in a practical and interactive way. Through playful and contextualized activities, students can explore how mathematical concepts apply to real-world scenarios, developing problem-solving, mathematical modeling, and collaboration skills. This section is essential for consolidating learning and for students to perceive the usefulness and versatility of functions in different contexts.

Activity Suggestions

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

Activity 1 - The Ice Cream Factory

> Duration: (60 - 70 minutes)

- Objective: Apply the concept of a function to model a real production process, developing calculation, analysis, and graphical representation skills.

- Description: Students will be divided into groups of up to 5 people, and each group will represent a team of engineers in an ice cream factory. They must create a function that models the production of ice cream, considering that each flavor has a different proportion of ingredients and each machine has a specific production capacity.

- Instructions:

• Each group must decide what the formula of the function that will represent the production of a flavor of ice cream will be (example: 2x + 3, where x represents the number of hours, and 2 is the production of ice creams per hour).

• Determine the factory's limitations, such as the maximum production capacity per day.

• Use a table to record daily production for each flavor, varying the number of working hours.

• Graphically represent the functions created, showing how the production of ice cream varies with time.

• Each group will present its function and the graphical analysis to the class, explaining how the function models the production process.

Activity 2 - The Delivery Challenge

> Duration: (60 - 70 minutes)

- Objective: Understand and apply the concept of a function in a practical context of logistics and planning, developing analytical and mathematical modeling skills.

- Description: In this activity, students will simulate creating a function to optimize the routes of a delivery service, considering variables such as distance, time, and traffic.

- Instructions:

• In groups, students should discuss and decide which variables are most relevant for calculating an optimized route.

• Each group must propose a function that models the travel time based on these variables.

• Use fictitious data to calculate travel time in different scenarios and test the proposed function.

• Create a graph that represents the function and discuss how changes in variables affect travel time.

• Present the results and the developed function, justifying the choices made for the function's parameters.

Activity 3 - Building the Future: Modeling Investments

> Duration: (60 - 70 minutes)

- Objective: Utilize functions to model financial growth, applying mathematical concepts in a realistic economic scenario.

- Description: Students, in groups, will model the growth of an investment over time using functions. They must consider variables such as interest rate and initial amount invested.

- Instructions:

• Define the variables involved in the investment, such as interest rate and initial amount.

• Develop a function that allows calculating the value of the investment over time.

• Use the function to calculate the value of the investment after different time periods and with different interest rates.

• Present the results in a graph, showing how the investment grows over time.

• Discuss in groups the impacts of changes in interest rates on the investment's growth.

Feedback

Duration: (10 - 15 minutes)

This stage of the lesson plan is essential for consolidating learning, allowing students to articulate what they have learned and reflect on the process of mathematical modeling. Group discussion helps develop communication and critical thinking skills, in addition to providing an opportunity for students to evaluate and learn from their peers' approaches. This collective feedback is fundamental for verifying the understanding of function concepts and their applications, ensuring deeper and more meaningful learning.

Group Discussion

At the end of the activities, promote a group discussion with all students. Start the discussion with a brief introduction: 'Now that everyone has had the opportunity to explore and apply functions in different contexts, let's share our discoveries and challenges. Each group will have a chance to present what they modeled and discuss with the class. Let's take advantage of this opportunity to learn from each other.'

Key Questions

1. What were the main challenges your group faced in modeling the function for the proposed scenario?

2. How did understanding functions help to solve the problem proposed in your activity?

3. Is there any practical application of functions that you had not considered before carrying out the activity?

Conclusion

Duration: (5 - 10 minutes)

The purpose of the Conclusion is to ensure that all the main concepts discussed during the lesson are clearly understood and consolidated. Additionally, it serves to highlight the importance of the content learned, showing how functions are applicable in the real world. This stage is essential for reinforcing learning and ensuring that students can connect the knowledge acquired with their experiences and practical needs.

Summary

To conclude the lesson, the teacher should summarize the main topics covered, reinforcing that a function is a mathematical relationship where each element of one set has exactly one corresponding element in another set. The graphical and analytical representations of functions should be recapped, such as the notation f(x) and equations like y=mx+b, in addition to discussing the practical applications explored in the activities, such as modeling industrial and logistics processes.

Theory Connection

Today's lesson connected the mathematical theory of functions with practical and everyday applications, showing students how mathematical concepts are fundamental to understanding and solving real-world problems. The proposed activities allowed students to apply what they had learned concretely, reinforcing the link between theory and practice.

Closing

Finally, it is important to highlight the relevance of functions in students' daily lives. Understanding these mathematical concepts not only enriches their academic knowledge but also prepares them to face real challenges in various professional and everyday areas, such as administration, engineering, economics, and even simple tasks like optimizing routes or calculating investments.