Objectives (5 - 7 minutes)

1. To introduce the concept of Quadratic Functions: The teacher will explain the definition of quadratic functions and their general form (y = ax^2 + bx + c).

2. To understand the key components of a Quadratic Function: The teacher will illustrate that 'a' is the coefficient of the quadratic term, 'b' is the coefficient of the linear term, and 'c' is the constant term.

3. To learn how to graph a Quadratic Function: The teacher will demonstrate how to plot points and draw a parabolic curve on a coordinate plane using given quadratic functions.

Secondary Objectives:

1. To encourage active participation: The teacher will engage the students in discussions, ask questions, and encourage them to share their thoughts and ideas about the topic.

2. To promote collaborative learning: The teacher will organize group activities where students can work together to solve problems related to quadratic functions.

3. To enhance problem-solving skills: The teacher will facilitate activities that require students to apply their understanding of quadratic functions to solve real-world problems.

By the end of the objective-setting stage, students should have a clear understanding of what they will be learning and what is expected of them during the lesson.

Introduction (10 - 12 minutes)

1. The teacher begins by reminding the students of the previous lesson where they learned about linear functions and their graphs. This will serve as a foundation for the current lesson on quadratic functions. The teacher can ask questions like, "Can anyone recall the formula for a linear function?" or "What does a linear function graph look like?" to jog the students' memory. (2 - 3 minutes)

2. The teacher then presents two problem situations to the class. The first problem could be about finding the maximum height of a ball thrown in the air, and the second problem could be about predicting the profit of a company based on the number of units sold. The teacher emphasizes that these real-world problems can be modeled and solved using quadratic functions. (3 - 4 minutes)

3. To grab the students' attention, the teacher shares two interesting facts or stories related to quadratic functions. For example, the teacher could mention that the ancient Greeks, particularly mathematician Diophantus, were the first to study quadratic equations extensively. The teacher could also share a fun fact that the shape of a parabolic satellite dish is based on the graph of a quadratic function. (2 - 3 minutes)

4. The teacher formally introduces the topic of the day: Quadratic Functions. The teacher explains that quadratic functions are a type of polynomial function where the highest power of the variable is two. The teacher also shows a few examples of quadratic functions and their graphs on a coordinate plane. (2 - 3 minutes)

5. The teacher concludes the introduction by explaining that understanding quadratic functions will not only help them in their math studies but also in real-world applications such as physics, engineering, and economics. The teacher encourages the students to stay engaged and participate actively in the lesson. (1 - 2 minutes)

By the end of the introduction, students should have a clear understanding of what they will be learning and why it is important. The introduction stage also serves the purpose of setting a positive and engaging tone for the rest of the lesson.

Development (20 - 25 minutes)

Activity 1: "Graphing Quadratic Functions" (10 - 12 minutes)

1. The teacher divides the class into groups of five. Each group is provided with a set of quadratic functions in standard form (y = ax^2 + bx + c), a coordinate plane, and colored markers. (1 minute)

2. The teacher instructs each group to choose one quadratic function from their set. They are then required to graph their chosen function on the coordinate plane using the colored markers. (2 minutes)

3. After graphing, students are instructed to identify the vertex and the y-intercept of their graph. They are also asked to identify whether their graph opens up or down based on the value of 'a'. (4 minutes)

4. Once the groups have identified the key points on their graph, the teacher asks each group to explain their findings to the class. This will help reinforce the understanding of these key components of a quadratic function. (3 minutes)

5. To make the exercise more challenging, the teacher can ask the students to swap their graphs with another group and identify the key points on the new graph. This will help develop their ability to analyze a graph without knowing the equation. (2 minutes)

Activity 2: "Quadratic Function Scavenger Hunt" (10 - 12 minutes)

1. The teacher prepares a set of real-world problems related to quadratic functions on different cards. Each card has a problem on one side and a clue leading to the solution on the other side. (2 minutes)

2. The teacher explains the rules of the game: each group must start with one card, solve the problem, and then use the clue on the back of the card to find the next card. The group that solves all the problems and reaches the final card first wins the game. (2 minutes)

3. The teacher distributes the first card to each group. The problems could be about finding the maximum height of a ball thrown up in the air, the number of units to be sold to maximize profit, the time taken to reach a certain height when jumping, etc. (4 - 5 minutes)

4. As the groups are working on their problems, the teacher circulates the room to provide assistance, ask probing questions, and ensure that students are applying the concept of quadratic functions correctly. (2 - 3 minutes)

5. Once a group has solved all the problems, they are instructed to write down the solutions and explain their reasoning to the class. This peer-to-peer learning will reinforce the understanding of the concept and its real-world applications. (2 minutes)

By the end of the development stage, the students should have a solid understanding of what quadratic functions are, how to graph them, and how to apply them to solve real-world problems. The development stage also provides an opportunity for the students to collaborate, communicate, and think critically, which are important skills in mathematics and life in general.

Feedback (8 - 10 minutes)

1. The teacher concludes the lesson by bringing all the groups back together. The teacher asks each group to share their solutions or conclusions from the activities and how they arrived at them. Each group is given up to 3 minutes to present their findings. (5 - 6 minutes)

2. The teacher then facilitates a discussion by asking questions and making comments that help the students connect their activities to the theory. For example, the teacher might ask, "How did you determine the vertex of your graph?" or "Can you explain how you used the quadratic function to solve the real-world problem in your scavenger hunt?" The teacher can also make comments like, "I noticed that most groups had their parabolas opening upwards. Can you explain why?" or "It's interesting to see the different approaches you used to solve the same problem. This shows the flexibility of quadratic functions." (2 - 3 minutes)

3. The teacher then asks the students to take a moment to reflect on the lesson and answer the following questions in their notebooks:

   1. What was the most important concept you learned today?
   2. What questions do you still have about quadratic functions?
   3. Can you think of any other real-world problems that can be solved using quadratic functions? (1 minute)

4. The teacher collects the students' notebooks and uses their responses to assess the students' understanding of the lesson and to plan for future lessons. The teacher may also choose to share some of the students' thoughts and questions with the class to promote further discussion and understanding. (1 - 2 minutes)

By the end of the feedback stage, the students should have a clear understanding of what they have learned in the lesson and what they still need to work on. The teacher should have a good sense of the students' understanding of the topic and can use this information to plan future lessons and activities.

Conclusion (5 - 7 minutes)

1. The teacher begins the conclusion by summarizing the main points of the lesson. The teacher reminds the students that a quadratic function is a type of polynomial function where the highest power of the variable is two. The teacher also reiterates the components of a quadratic function: the quadratic term (ax^2), the linear term (bx), and the constant term (c). The teacher discusses how to graph a quadratic function on a coordinate plane, including finding the vertex, the y-intercept, and determining the direction of the parabola. (2 - 3 minutes)

2. The teacher then explains how the lesson connected theory, practice, and real-world applications. The teacher points out that the theoretical part of the lesson was understanding the definition and components of a quadratic function. The practice part was the hands-on activities where the students graphed quadratic functions and solved real-world problems using them. The teacher emphasizes that the real-world problems used in the lesson, such as finding the maximum height of a ball thrown in the air or predicting the profit of a company, showed the practical applications of quadratic functions. (1 - 2 minutes)

3. To further the students' understanding of quadratic functions, the teacher suggests additional materials for study. These could include online resources that provide interactive activities and games for graphing quadratic functions, solving quadratic equations, and exploring more real-world applications. The teacher could also recommend additional textbooks or workbooks that provide more practice problems and explanations. The teacher encourages the students to explore these materials at their own pace to reinforce what they've learned in class. (1 minute)

4. Lastly, the teacher explains the importance of understanding quadratic functions for everyday life. The teacher emphasizes that quadratic functions are not just abstract mathematical concepts, but they are also used in various fields such as physics, engineering, and economics to model and solve real-world problems. The teacher gives examples such as predicting the trajectory of a ball, designing a bridge, or optimizing a business strategy. The teacher encourages the students to keep an eye out for quadratic functions in their everyday life and to think about how they can be used to solve problems. (1 - 2 minutes)

By the end of the conclusion, the students should have a clear and concise summary of what they learned in the lesson. They should also understand the relevance of the topic to their lives and be motivated to continue exploring the topic further.