Introduction and Contextualization

Theoretical Concept

Mathematics is a discipline that presents a multitude of concepts that we apply in our daily lives, often without realizing it. The chosen theme for this project is the 'Midpoint' - a fundamental concept in analytic geometry.

The midpoint, in the Cartesian plane, is the point that divides a line segment in half, that is, it is equidistant from the two endpoints of the segment. Given a line segment with endpoints A(x1, y1) and B(x2, y2), the midpoint M(xm, ym) is calculated by the following formulas: xm = (x1 + x2)/2 and ym = (y1 + y2)/2.

Importance and Applications

The midpoint is a fundamental tool used in various areas of knowledge, such as in engineering for bridge construction, in computer graphics for image modeling, in physics for motion calculations, and even in cryptography, for data protection.

In addition to its practical use, understanding the midpoint also promotes a greater familiarity with the Cartesian plane and analytic geometry, which are fundamental concepts for the study of many fields of mathematics.

Engineers, architects, designers, and many other professionals use the concept of the midpoint in their work, highlighting the importance and practical application of this concept.

Practical Activity

Activity Title: Lost City Game

Project Objective:

Our goal with this project is to develop a game based on cards and a board that applies the concept of the midpoint in practice. The game will be the 'Lost City', where players will have to find the lost city on the Cartesian plane using the midpoint. This will help you understand the concept of the midpoint, apply it in real life, and also enhance your teamwork, time management, and creativity skills.

Detailed Project Description:

Each group of 3 to 5 students will be responsible for creating a game board on the Cartesian plane, creating 'points of interest' cards with coordinates, and then playing the game to find the lost city. The location of the lost city will be defined as the midpoint of two points of interest.

Required Materials:

• Cardboard or cardstock to create the board
• Blank cards or sturdy paper to create the points of interest cards
• Colored pens or markers
• Ruler
• Calculator

Detailed Step-by-Step for Activity Execution:

1. Board Creation: Each group must create a game board on cardboard or cardstock. The board should be a Cartesian plane ranging from -10 to 10, both on the x and y axes.

2. Card Creation: Each group must create 20 'points of interest' cards, each with an ordered pair of coordinates that are within the board's limits.

3. Choosing Points of Interest: Each turn, the player must pick two points of interest cards and find the midpoint between them. This midpoint represents the possible location of the lost city.

4. Playing the Game: Players must mark each midpoint found on the board. The game ends when all possible midpoints have been marked.

5. Report: After completing the game, students must produce a report documenting the game, explaining how the concept of the midpoint was applied, and discussing the effectiveness of the game as a learning tool. This report should include an introduction, development, conclusion, and bibliography.

Project Deliverables:

1. Game Board and Cards: Students must deliver the game board and used cards. They should be clear and legible, indicating the creativity and effort of the students in creating the game.

2. Final Report: The final report must contain a minimum of 1500 words and must follow the following structure:

   • Introduction: Contextualize the project, discuss the relevance of the midpoint concept, and define the report's objective.
   • Development: Describe the game's development, explaining how the midpoint concept was applied. Also, explain the methodology used in the creation and execution of the game, and discuss any interesting results or issues encountered.
   • Conclusion: Make an overall assessment of the project, discuss what you learned and how it can be applied in other contexts. Give your personal opinions on the project and activities carried out.
   • Bibliography: Cite all the resources you used to complete the project, including books, websites, videos, etc.

We hope this project will give you a better understanding of the midpoint concept and an appreciation of how mathematics can be applied in a creative and fun way.