Objectives (5 - 7 minutes)

1. To understand and apply the distance formula in Cartesian Geometry: The students will learn to find the distance between two points in a Cartesian plane using the formula derived from the Pythagorean theorem.

2. To learn the theoretical basis of the distance formula: The students will understand why the formula for finding distance in a Cartesian plane is derived from the Pythagorean theorem.

3. To foster problem-solving skills: The students will apply the distance formula to solve real-world problems, thereby enhancing their ability to analyze and solve mathematical problems in a practical context.

Secondary Objectives:

• Develop critical thinking: The students will be encouraged to think critically in order to understand and apply the distance formula.
• Enhance collaborative skills: The students will engage in group activities and discussions, which will help them improve their collaborative skills.
• Promote creativity: The lesson will incorporate hands-on activities and games that will encourage the students to think creatively and make learning fun.

Introduction (8 - 10 minutes)

1. Review of Cartesian Geometry (2 - 3 minutes): The teacher will start the lesson by reminding the students about Cartesian Geometry, the coordinate plane, and the concepts of x and y coordinates. The teacher will ask a few students to explain these concepts to the class to ensure a quick recap.

2. Problem Situations (3 - 4 minutes): The teacher will pose two problem situations to the class, which will serve as starters for the development of the distance formula. These problems can be:

   • "If a bee flies from point A to point B in a straight line, how can we calculate the distance it covers?"
   • "If a car travels from one city to another on a map, how can we determine the distance it moves?"

3. Real-World Context (1 - 2 minutes): The teacher will then explain the importance of the distance formula in real-world applications. For instance, in navigation, the distance formula is used to determine how far one location is from another. In video games, the distance formula is used to calculate how far a character has moved or how far an object is from the player.

4. Engaging Introduction (2 - 3 minutes): To capture the students' attention, the teacher will share two interesting facts related to the topic:

   • "Did you know that the distance formula is derived from the Pythagorean theorem, a concept you've been studying since middle school?"
   • "The distance formula is not just a mathematical tool; it's also used in physics, engineering, computer science, and many other fields."

By the end of the introduction, the students should have a clear understanding of what they will be learning and why it is important. They should also be excited about exploring the distance formula and its applications.

Development (20 - 25 minutes)

1. Activity 1: Cartesian Geometry Board Game (7 - 10 minutes):

   • The teacher will distribute a Cartesian Geometry Board Game to each group. The game will be designed to have a path on the Cartesian plane from a start point to an end point. The path will include obstacles, represented by polygons or circles. The objective of the game will be to navigate the path using the distance formula, avoiding the obstacles. The team that reaches the end point in the fewest steps (using the distance formula) wins the game.

   • To play the game, each group will take turns rolling a pair of dice. The numbers on the dice will represent the x and y coordinates of a point. The group will then apply the distance formula to determine the number of steps they need to move on the board. If they encounter an obstacle, they will need to find a way around it, again using the distance formula.

   • The teacher will walk around the room, monitoring the groups, and guiding them as needed. After the game, the teacher will lead a discussion about how the distance formula was applied in the game and how it can be used in real-world situations.

2. Activity 2: Building a Cartesian Plane (5 - 7 minutes):

   • Next, the teacher will introduce the "Building a Cartesian Plane" activity. Before the class, the teacher has prepared large graph paper templates and colored stickers to represent points.

   • The students will be divided into groups and given the materials. They will be instructed to create a Cartesian plane using the large graph paper and place the stickers to represent points at random.

   • The teacher will then call out two points on the plane. The groups will use the distance formula to calculate the distance between the two points and write the answer on a separate sheet. The first group to correctly find the distance wins a point.

   • This activity encourages students to visualize the Cartesian plane and see the practical application of the distance formula. It also fosters a sense of competition and a desire to win, making the learning experience more enjoyable.

3. Activity 3: "What's the Distance?" Game (7 - 8 minutes):

   • For the final activity, the teacher will introduce the "What's the Distance?" game. In this game, the teacher will show a pair of points on a large Cartesian plane and ask the students to find the distance between them before a timer runs out.

   • The teacher will use an interactive whiteboard or a large paper with a Cartesian plane drawn on it for this game. The students will use small whiteboards or paper to calculate the distance.

   • The students will write their answers and show them to the teacher within the given time. The teacher will reveal the correct answer and award points to the groups with the correct answer.

   • This game is a fun and competitive way to practice calculating the distance using the distance formula. It also helps students to improve their speed and accuracy in applying the formula.

By the end of the development stage, the students should have a solid understanding of the distance formula and how to apply it to find the distance between two points in a Cartesian plane. They should also have had fun engaging in the hands-on activities and games, which will help to reinforce their learning.

Feedback (10 - 12 minutes)

1. Group Discussion (3 - 4 minutes): The teacher will facilitate a group discussion where each group shares their solutions or conclusions from the activities. During this discussion, the teacher will ask guiding questions to ensure the students are understanding the main concepts of the lesson. The teacher should ensure that each group has had the opportunity to express their thoughts and that all students are participating in the discussion.

2. Connecting Theory to Practice (2 - 3 minutes): The teacher will then guide the students to reflect on how the activities connect with the theoretical aspect of the lesson. They will discuss how the distance formula, derived from the Pythagorean theorem, was applied in the hands-on activities and games. The teacher will emphasize that the distance formula is not just a mathematical concept, but a practical tool that can be used to solve real-world problems.

3. Individual Reflection (3 - 4 minutes): The teacher will ask the students to take a moment to reflect on what they have learned in the lesson. They will be prompted to think about the most important concept they learned, any questions they still have, and how they can apply what they learned in their daily life or in other subjects. The teacher can provide them with reflection questions such as:

   • "What was the most important concept you learned today?"
   • "What questions do you still have about the distance formula?"
   • "Can you think of any other real-world situations where the distance formula could be used?"

4. Wrap-up (2 minutes): The teacher will conclude the lesson by summarizing the main points and answering any remaining questions from the students. The teacher will remind the students that the distance formula is a powerful tool in Cartesian Geometry, and mastering it will help them in solving various mathematical problems. The teacher will also remind the students that the lesson was designed to be fun and interactive, and they should feel proud of their participation and effort.

By the end of the feedback stage, the students should have a clear understanding of the distance formula, its application in Cartesian Geometry, and its relevance in real-world situations. They should also feel confident in their ability to use the distance formula and have a positive attitude towards learning mathematics.

Conclusion (5 - 7 minutes)

1. Summary and Recap (2 - 3 minutes): The teacher will start by summarizing the main contents of the lesson. They will recap the concept of the Cartesian plane, the coordinates of points, and the derivation of the distance formula from the Pythagorean theorem. The teacher will also summarize the hands-on activities the students engaged in, highlighting how they applied the distance formula to find the distance between two points in a Cartesian plane.

2. Connecting Theory, Practice, and Applications (1 - 2 minutes): The teacher will then explain how the lesson connected theory, practice, and real-world applications. They will mention how the theoretical knowledge of the Cartesian plane and the Pythagorean theorem was applied in practice through the various activities and games. The teacher will also reiterate the importance of the distance formula in real-world applications such as navigation, video games, and other fields.

3. Additional Materials (1 minute): The teacher will suggest some additional materials for the students to explore to deepen their understanding of the distance formula and its applications. These materials can include online interactive tutorials, videos, and practice exercises. The teacher can also recommend some books or websites that provide more information on Cartesian Geometry and the Pythagorean theorem.

4. Everyday Life Relevance (1 - 2 minutes): Lastly, the teacher will conclude the lesson by discussing the relevance of the distance formula in everyday life. They will emphasize that understanding the distance formula can help in various situations, such as calculating travel distances, finding the shortest path between two points, or in spatial reasoning tasks. The teacher will also encourage the students to look for more examples of the distance formula in their daily life and share them in the next class.

By the end of the conclusion, the students should have a comprehensive understanding of the distance formula and its applications. They should also be aware of the resources available to them for further study and feel motivated to explore more about the topic.