Distance Formula

Objectives

1. Understanding the Concept: Students will understand the concept of the distance between two points on a plane, recognizing it as the straight line connecting two points.

2. Applying the Formula: Students will learn to apply the distance formula between two points, which is $d = \sqrt{(x\_2 - x\_1)^2 + (y\_2 - y\_1)^2}$.

3. Solving Problems: Students will be able to solve practical problems involving the determination of the distance between two points on a plane.

Introduction (10-15 minutes)

1. Reviewing Previous Content: Begin the class by briefly reviewing the concepts of Cartesian plane and coordinates of a point. Remind students that a Cartesian plane is formed by two intersecting lines (the axes) and that any point on this plane can be defined by an ordered pair of coordinates (x, y).

2. Problem Situation 1: Present the following situation: "Imagine you are in a park and want to know the distance between two trees. How would you do that without a measuring tape?"

3. Problem Situation 2: "Now, suppose you want to know the distance between your house and your friend's house, who lives on the same street. How would you do that?"

4. Contextualization: Explain to students that in both situations, the distance between the points (the trees or the houses) can be determined using a mathematical formula.

5. Introduction to the Topic: Introduce the topic of the lesson, explaining that they will learn how to determine the distance between two points on a Cartesian plane using a formula.

6. Curiosity: To arouse students' interest, share the curiosity that the formula for the distance between two points is derived from the Pythagorean Theorem, a concept they have already encountered in previous studies.

Development (20-25 minutes)

1. Theory: Distance Formula between Two Points (10-12 minutes)

   • Definition: Present the distance formula between two points (d = √((x₂ - x₁)² + (y₂ - y₁)²)) to the students.
   • Explanation: Break down the formula into its components, explaining that (x₁, y₁) and (x₂, y₂) are the coordinates of the two points and that (x₂ - x₁) and (y₂ - y₁) are the differences in the x and y coordinates, respectively.
   • Derivation of the Formula: Briefly explain that the distance formula is derived from the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

2. Application of the Formula (5-7 minutes)

   • Example 1: Present an example of calculating the distance between two points on a Cartesian plane. For example, consider the points A(2, 3) and B(5, 7). Ask students to identify the coordinates of the points and the differences in the coordinates, then to calculate the distance using the formula.
   • Example 2: Present a second example, but this time with negative coordinates. For example, consider the points A(-2, -3) and B(-5, -7). Ask students to repeat the calculation and observe that the formula works in the same way, regardless of the sign of the coordinates.

3. Practical Activities (5-6 minutes)

   • Activity 1: Divide students into groups and give each group a Cartesian plane drawn on a sheet of paper and two points marked. Ask them to calculate the distance between the points using the formula.
   • Activity 2: Ask each group to create a problem situation involving the determination of the distance between two points and to present it to the class. The other groups must solve the presented problem situation using the distance formula.

Feedback (10-15 minutes)

1. Group Discussion (5-7 minutes)

   • Sharing Solutions: Ask each group to share the solutions they found for the practical activities. Encourage students to explain how they applied the distance formula to solve the problem situations.
   • Connection to Theory: After each presentation, ask questions to verify if students are able to connect the solutions found to the theory presented. For example, "How did you decide which were the coordinates of the points in your problem situation?" or "Why did you use the formula d = √((x₂ - x₁)² + (y₂ - y₁)²) to calculate the distance?"

2. Individual Reflection (3-5 minutes)

   • Reflection Questions: Propose that students reflect individually on what they learned in class. Ask questions like:
     1. "What was the most important concept you learned today?"
     2. "What questions have not been answered yet?"
   • Sharing Reflections: After a minute of reflection, ask students to share their answers. Encourage them to express any doubts they may have and to discuss the concepts that were most difficult to understand.

3. Teacher Feedback (2-3 minutes)

   • Reinforcement of Key Concepts: Conclude the class by reinforcing the key concepts that were covered. For example, emphasize the importance of understanding the distance formula and how it can be applied in various situations.
   • Clarifying Doubts: Take the opportunity to clarify any doubts that may have arisen during the class discussion or the individual reflection. Make sure that all students understand the content before ending the class.

Conclusion (5-10 minutes)

1. Summary of Contents (2-3 minutes)

   • Recap the main points covered during the lesson, reinforcing the definition of the distance between two points on a Cartesian plane, the distance formula and how to apply it.
   • Emphasize the importance of understanding how to determine the distance between two points, highlighting its practical applications in various areas, such as geography, physics and engineering.

2. Connection between Theory, Practice and Applications (1-2 minutes)

   • Explain how the lesson connected theory, practice and applications. For example, how the theoretical understanding of the distance formula allowed students to solve practical problems and how this skill can be applied in everyday situations.

3. Extra Materials (1-2 minutes)

   • Suggest additional materials for students who wish to deepen their understanding of the topic. This could include math books, educational websites, explanatory videos, among others.
   • For example, you could suggest that students watch a video explaining the derivation of the distance formula from the Pythagorean Theorem, or that they explore a website that offers interactive exercises on the topic.

4. Relevance of the Topic (1-2 minutes)

   • Finally, emphasize the importance of the lesson topic for students' lives. Explain that the ability to determine the distance between two points is a fundamental skill that can be applied in various everyday situations.
   • For example, you could mention that calculating distances is essential for navigation, whether by car, bike or on foot, or that it can be useful in situations such as measuring distances on a map or determining the route between two places.