Introduction

Relevance of the Topic

The study of Problems and Flowcharts is the basis for algorithmic logic, which is fundamental for computer programming and solving more complex mathematical problems. Through flowcharts, we can clearly and systematically visualize the sequence of actions that lead to the solution of a problem. This tool is crucial for logical reasoning and critical thinking, essential skills not only in Mathematics but also in many other disciplines and daily life.

Contextualization

Within the Mathematics curriculum, the topic of Problems and Flowcharts emerges as a natural progression from the study of basic operations and numerical expressions. Problem-solving is a practical application of these skills, and the flowchart, a visual tool that facilitates understanding of the resolution process. In parallel, this topic also prepares the ground for future studies of logic, algebra, geometry, statistics, and probability, where problem-solving is fundamental. Additionally, the algorithmic thinking developed in creating and interpreting flowcharts is transferable to the fields of computer science and technology, where it becomes the basis for creating algorithms and computer programs.

Theoretical Development

Components

• Mathematical Problems: These are situations that, due to their complexity, require us to use certain techniques and strategies to solve. They may involve several mathematical disciplines at the same time, or be based on a single discipline. The fundamental thing is to understand the structure of the problem and which mathematical operations are applicable at each stage.

• Flowcharts: Graphic representations of algorithms or processes, which use standard symbols to indicate stages, decisions, loops, data input, and output. They are a powerful tool for visualization, analysis, and problem-solving, allowing the logical sequence of operations to be clearly represented.

• Mathematical Operations: Include addition, subtraction, multiplication, division, and various others more advanced, depending on the level of study. Understanding basic mathematical operations and the ability to apply the rules correctly are fundamental in problem-solving and creating flowcharts.

Key Terms

• Algorithm: It is a finite sequence of precise and unambiguous instructions, used to solve a problem or perform a task. In mathematical terms, an algorithm is a sequence of operations with a predetermined result.

• Variable: It is an element that can be any value within a certain set. In mathematical problems and algorithms, variables are used to represent unknown quantities or those that may vary during the process.

• Decision: In terms of flowcharts, a decision is a condition that needs to be evaluated, and depending on the result, the program follows one path or another.

Examples and Cases

• Example of a Mathematical Problem: At a party, there are 50 people. If each person shakes hands once with each of the other people, how many handshakes will be given? In this case, the student needs to understand the logic behind the handshakes (each person shakes hands with all the other people) and apply multiplication and subtraction operations (each person shakes hands with 49, excluding the handshake with themselves, resulting in 49 handshakes per person).

• Example of a Flowchart: Let's consider the flowchart for the following problem: "Given a number, if it is even, divide it by 2. If it is odd, multiply it by 3 and add 1. Repeat the process until the number equals 1". Here, the student must understand how the symbols in the flowchart correspond to each stage of the algorithm, and the sequence of actions that lead to the solution of the problem is visually represented.

• Flowchart Case Study: Let's suppose we have a flowchart that represents the process of calculating the total value of a purchase, with a 10% discount if the purchase amount is greater than R$ 100.00. The flowchart allows visualizing the sequence of actions, including data entry (purchase value), the decision (check if the value is greater than R$ 100.00), and the discount calculation operation (10% of the total value). Through the flowchart, the student clearly understands how the final value calculation is done.

Detailed Summary

Relevant Points

• Importance of Logical Reasoning: Flowcharts are tools that, when applied in problem-solving, assist in the development of logical reasoning. They allow students to visualize, in a sequential and structured manner, the actions necessary to reach a solution. This skill will be useful not only in Mathematics but in various other disciplines and everyday situations.

• Definition and Components of a Flowchart: Besides understanding what a flowchart is, it is crucial that students comprehend the components that integrate it. These include, among others, the representative symbols of stages, decisions, loops, data input, and output. Each of these symbols has a specific function in representing the process.

• Ability to Transform Algorithms into Flowcharts (and vice versa): The study of flowcharts is not limited only to their interpretation but also to their creation. Students should be able to transform algorithms (sequences of operations) into flowcharts and, conversely, flowcharts into algorithms. This requires solid skills in understanding and translating ideas into graphic representations.

• Problem Solving Through Flowcharts: After understanding flowcharts, students should learn to use them as tools in problem-solving. Exposing and solving problems through flowcharts helps demystify mathematics, making it more accessible and stimulating.

Conclusions

• Flowcharts as Versatile Tools: Flowcharts have proven to be versatile tools, useful not only in Mathematics but in various other disciplines and practical situations. They allow a clear and organized visualization of processes, facilitating understanding and problem-solving.

• Interconnection between Mathematical Topics: The creation and interpretation of flowcharts involve mathematical concepts that interconnect, such as mathematical operations, logic, algorithms, and variables. Understanding the relationship between these concepts strengthens learning and the application of Mathematics.

• Promotion of Critical and Analytical Thinking: The use of flowcharts in teaching mathematics promotes the development of critical and analytical thinking skills, fundamental not only in Mathematics but in all areas of life.

Exercises

1. Create a flowchart that represents the solution to the following problem: "In a classroom, there are 30 students. Each student has a box with 100 chips. If each student writes the name of all the other students on a chip and deposits it in the respective box, how many chips will there be in total?". Remember to include the stages, decisions, loops, data input, and output.

2. Given the algorithm "If the number is greater than 10, subtract 5. If not, add 10", create a corresponding flowchart.

3. Solve the problem: "In a soccer match, 300 tickets were sold. Tickets cost R$ 10.00 for children under 12 years old, R$ 20.00 for those over 12 years old and under 18 years old, and R$ 30.00 for those over 18 years old. If half of the tickets sold were for children under 12 years old and one third for those over 12 years old and under 18 years old, what was the total revenue from ticket sales?". Design the flowchart that could be used to solve this problem.