Contextualization

The Cartesian plane is a simple but extremely powerful mathematical tool, invented by René Descartes in the 17th century. It consists of two perpendicular axes, one horizontal (x-axis or X-axis) and one vertical (y-axis or Y-axis), allowing the location of any point in space from the intersection of the values of X and Y. These values are called the coordinates of the point.

The concept of distance between two points in the Cartesian plane is one of the foundations of analytical geometry. It is a method to calculate the distance 'in a straight line' between two points, regardless of their location on the plane. It is calculated through the application of the famous distance formula, derived from the Pythagorean theorem, one of the bases of Euclidean geometry.

Determining the midpoint of a line segment in the Cartesian plane is equally fundamental and simple to calculate. The midpoint of a line segment is the point that divides the segment into two equal parts.

It is essential to understand well the distance between points and the midpoint in the Cartesian plane, as these concepts are essential for various areas of knowledge, such as physics, engineering, architecture, graphic design, and computer science, among others. For example, in architecture, correctly determining the distance between points and the midpoint can be the difference between a well-built structure and a collapsing one.

Furthermore, in everyday life, whenever we need to measure distances, we are, in a way, applying these concepts, whether when using a GPS to locate ourselves, when planning a trip, or even when drawing a map. Understanding these concepts well allows us to better understand the world around us and make better decisions.

To deepen your knowledge, you can consult the following resources:

• The book 'Analytical Geometry and Linear Algebra' by Elon Lages Lima.
• Analytical Geometry classes from the YouTube channel 'Me Salva!'
• Analytical Geometry classes from the Khan Academy portal

Practical Activity

Activity Title: 'The Hidden Map: Discovering with the Cartesian Plane'

Objective:

The objective of this activity is for students, working in groups of 3 to 5, to understand and apply the concepts of the midpoint and distance between two points in the Cartesian plane, as well as to work on socio-emotional skills such as communication, problem-solving, and creative thinking.

Project Description:

Each group will receive a series of coordinates (points on the Cartesian plane) representing the location of various 'treasures.' The challenge is to create a map using these coordinates, find the midpoint between two points (representing the 'hidden treasure' equidistant from the two treasures), and calculate the distance between the points (representing the 'distance to travel' to reach the treasures).

The map will be constructed on graph paper, where each square represents a unit on the Cartesian plane. Students will draw the points on the paper, identifying their coordinates. Subsequently, they will determine the midpoints and distances through appropriate mathematical calculations. The adventure concludes with the creation of a 'map legend,' a brief report describing the steps taken, the treasures found, and the path taken.

Required Materials:

1. Graph paper
2. Colored pencils and/or markers
3. Ruler
4. Pencils and erasers
5. Calculator (optional)

Detailed Activity Steps:

1. Divide the class into groups of 3 to 5 students.
2. Provide each group with a list of coordinates of the points (the 'treasures'). Each coordinate should be unique for each group.
3. Briefly explain the concept of the Cartesian plane, coordinates, midpoint, and distance between two points.
4. Ask students to plot the points on the graph paper based on the provided coordinates and to identify each point with a number or letter.
5. Request that the groups calculate the midpoint between two points and represent it on the map.
6. Ask the groups to calculate the distance between two points and record this information on the paper.
7. Encourage students to be creative in representing the map and the 'treasures.' They can add drawings, colors, symbols, among other elements.
8. Request that they create a 'map legend,' explaining what each symbol or color represents.
9. Finally, each group should create a report of their findings and paths, explaining how they calculated the midpoints and distances, and reflecting on what they learned from the activity.

Project Report:

After completing the practical part, the groups should create a written report detailing the project execution process. This report should be structured as follows:

1. Introduction: In this section, students should provide a summary of the project, explain the relevance of the distance between points and the midpoint in the Cartesian plane, and the project's objective.
2. Development: Here, students should explain in detail how they carried out the practical activity, including how they represented the points on paper, how they calculated the midpoint and the distance between two points, how they created the map and its legend, any difficulties they may have encountered and how they overcame them. They should also present the results obtained, such as the values of the midpoints and calculated distances.
3. Conclusion: Students should conclude the work by summarizing the main points of the work, explaining the learnings obtained and the conclusions drawn from the project.
4. Bibliography: Indicate the sources on which they relied to work on the project: the books, web pages, videos, etc. used to understand and apply the concepts necessary for the project execution.

The written report is a crucial part of the project, as it not only allows students to reflect on the work done but also encourages the ability of written communication, analytical and reflective skills, as well as the ability to document processes and results.