Objectives (5 - 7 minutes)

1. Understand the concept of proportional relationships: The students will learn to identify and understand the fundamental concept of proportional relationships. They will grasp the idea that a proportional relationship is a special type of linear relationship where the ratio of one quantity to another remains constant.

2. Learn to represent proportional relationships in different ways: The students will explore various methods of representing proportional relationships, including tables, graphs, and equations. They will understand how each representation can provide different insights into the relationship.

3. Apply proportional relationships to real-world situations: The students will apply their understanding of proportional relationships to solve real-world problems. This objective will help them see the relevance and practical applications of the concepts they are learning.

Secondary objectives:

• Enhance mathematical reasoning: The students will develop their ability to reason mathematically, making connections between different representations and applying their knowledge to solve problems.

• Encourage collaborative learning: The students will work in groups during the hands-on activities, fostering collaboration and communication skills.

• Promote active learning: The hands-on nature of the activities will engage the students in active learning, enhancing their understanding and retention of the material.

Introduction (8 - 10 minutes)

1. Review of prerequisite knowledge: The teacher starts by reminding students of the basic concepts of ratios and proportions. This includes a quick recap of what ratios are and how they are used to compare quantities. The teacher also reviews the concept of proportions, emphasizing that they are a special type of ratio where the two terms are equal. This review will help students understand the central concept of proportional relationships.

2. Problem situations: The teacher presents two problem situations to the class. The first situation involves a recipe that uses a certain amount of ingredients to make a certain number of servings. The second situation is about a car that travels at a constant speed and the relationship between the time taken and the distance traveled. These situations are used to introduce the concept of proportional relationships in a practical and engaging way.

3. Real-world applications: The teacher explains that proportional relationships are not just mathematical concepts, but they also have real-world applications. They are used in various fields, including cooking, engineering, and finance. For example, in cooking, the amount of each ingredient in a recipe is usually proportional to the number of servings. In engineering, the time it takes to complete a task is often proportional to the number of workers. In finance, the interest on a loan is usually proportional to the amount borrowed.

4. Topic introduction: The teacher then introduces the topic of the day: Proportional Relationships. The teacher explains that the students will learn how to identify, represent, and apply proportional relationships. The teacher also shares a curiosity or a fun fact related to the topic to grab students' attention. For instance, the teacher might mention that the concept of proportions was first used by ancient Egyptians to build the pyramids, and is now used in many modern technologies, such as computer graphics and animation.

5. Problem statement: The teacher concludes the introduction by stating that by the end of the lesson, the students should be able to identify proportional relationships, represent them in different ways, and use them to solve problems. The teacher also emphasizes that the lesson will be hands-on and interactive, allowing the students to explore and discover the concepts for themselves.

Development (25 - 30 minutes)

Activity 1: Proportional Relationships with Measuring Cups (10 - 15 minutes)

1. The teacher divides the students into groups of four and distributes each group with a set of measuring cups, a container of water, and a large empty bowl.

2. The teacher instructs each group to follow the instructions in a given recipe that requires the use of the measuring cups. The recipe specifies a certain number of servings and the corresponding amount of each ingredient.

3. Each group follows the recipe, measuring the ingredients and adding them to the large empty bowl. They note the amount of each ingredient used and the number of servings the recipe yields.

4. After completing the recipe, the students are asked to double or halve the recipe, maintaining the same proportion of ingredients. They will then compare the amount of ingredients used in the original recipe with the amounts in the modified recipe.

5. The teacher then initiates a discussion in the class, asking groups to share their observations. This will help them to understand that the ratio of ingredients to servings remains constant, and hence, the relationship is proportional.

Activity 2: Graphing Proportional Relationships (10 - 15 minutes)

1. The teacher provides each group with a set of cards, each containing a word problem describing a proportional relationship. These problems involve different scenarios like time and distance, speed and travel time, or cost and number of items.

2. The groups are instructed to read each problem, identify the quantities involved, and decide which quantities are proportional.

3. Once the groups have identified the proportional quantities, they are to plot the pairs of values on a graph, with one quantity on the x-axis and the other on the y-axis.

4. After plotting the points, the students are asked to draw a line connecting the points. The teacher then explains that since the points lie on a straight line that passes through the origin, the relationship between the quantities is proportional.

5. The teacher then instructs the students to find the slope of the line, emphasizing that the slope is the constant of proportionality. They should also understand that the unit rate is the value of the slope, which helps to compare quantities.

6. The students are then asked to write an equation representing the relationship, using the slope and one point on the line.

Activity 3: Proportional Relationships in Everyday Life (5 - 10 minutes)

1. The teacher provides each group with a set of real-world scenarios that involve proportional relationships, such as a cell phone plan with a fixed rate per minute, a sale where items are marked down by a certain percentage, or a job that pays a certain amount per hour.

2. The students are instructed to solve the problems using the concepts they have learned, such as finding the constant of proportionality, calculating the unit rate, or creating a table to represent the relationship.

3. The teacher circulates the classroom, providing guidance, and checking for understanding.

4. After solving the problems, each group is asked to share their solutions and explain their reasoning. This will provide an opportunity for the students to learn from each other and reinforce their understanding of proportional relationships.

Feedback (5 - 7 minutes)

1. Group discussion and reflection: The teacher initiates a group discussion where each group shares their solutions, conclusions, and observations from the activities. The teacher encourages all students to participate, ensuring that every group has a chance to share. This discussion provides an opportunity for the students to learn from each other, reinforce their understanding, and see the concepts from different perspectives.

   • The teacher asks each group to share their findings from Activity 1, where they compared the amounts of ingredients in a recipe for different numbers of servings. The teacher encourages the students to explain why they think the relationship is or isn't proportional.

   • The teacher then asks each group to present one of the word problems from Activity 2 and explain how they identified the proportional quantities and represented the relationship on the graph. The teacher also asks the other groups if they agree with the presentation and why.

   • Finally, the teacher asks each group to share one of the real-world scenarios from Activity 3 and explain how they used proportional relationships to solve the problem. The teacher encourages the other groups to ask questions and provide feedback on the solution.

2. Connecting theory with practice: After the group discussions, the teacher summarizes the key points from the activities, highlighting how they relate to the theoretical concepts of proportional relationships. The teacher emphasizes that a proportional relationship is a special type of linear relationship where the ratio of one quantity to another remains constant.

   • The teacher points out how the students observed this in Activity 1, where they found that doubling or halving the recipe maintained the same proportion of ingredients. The teacher also highlights how the students graphed proportional relationships in Activity 2 and used the slope to find the constant of proportionality.

3. Individual reflection: The teacher then asks the students to take a moment to reflect on what they have learned in the lesson. The teacher provides some guiding questions to help the students with their reflection:

   • What was the most important concept you learned today?
   • Was there anything that you found challenging in today's lesson? If so, what was it and how did you overcome it?
   • Can you think of any other real-world situations where proportional relationships might occur?

4. Sharing reflections: The teacher then invites a few students to share their reflections with the class. This will give the students an opportunity to hear different perspectives and insights, and it will allow the teacher to assess the students' understanding and address any remaining questions or misconceptions.

5. Closure: To conclude the lesson, the teacher summarizes the main points of the lesson, reiterating the definition of proportional relationships and their representation and application. The teacher also reminds the students of the real-world applications of proportional relationships, emphasizing their importance in everyday life. The teacher then previews the next lesson, which will build on the concepts learned in this lesson.