Objectives (5 - 7 minutes)

1. To understand the basic concepts of rational numbers, including how they are written and their place on a number line.
2. To learn the rules for multiplying and dividing rational numbers, and to understand why these rules work.
3. To apply these rules in solving problems involving the multiplication and division of rational numbers.

Secondary Objectives:

• To enhance critical thinking skills through the analysis and application of the rules for multiplying and dividing rational numbers.
• To promote collaborative learning through group discussions and problem-solving activities.
• To improve communication skills by explaining mathematical concepts and solutions to peers.

Introduction (10 - 12 minutes)

1. The teacher begins the lesson by reminding students of the previous lessons on rational numbers, ensuring they have a clear understanding of what they are and how to represent them on a number line. This can be done by asking a few quick review questions or by having the students share what they remember about rational numbers.

2. The teacher then presents two problem situations that can serve as a starting point for the lesson. For instance, the teacher might ask, "If each of you had 1/2 of a pizza and you decided to share it with a friend, how much of the whole pizza would each of you get?" or "If you have $5 and you want to buy a candy bar that costs $1/2, how many candy bars can you buy?" These problems serve to introduce the concept of dividing and multiplying rational numbers in a real-world context and can help students understand the importance of these operations.

3. The teacher then contextualizes the importance of the topic by explaining its real-world applications. For instance, they might say, "Understanding how to multiply and divide rational numbers can help you when shopping, cooking, or even when you're older and managing your finances."

4. To grab the students' attention, the teacher can share a couple of interesting facts or stories related to the topic. For instance, they might share that the concept of dividing fractions is often considered one of the most difficult concepts in elementary and middle school mathematics, or they might share a fun fact about a famous mathematician who made significant contributions to the study of rational numbers.

5. The teacher then introduces the topic of the lesson: Multiplying and Dividing Rational Numbers. They explain that during the lesson, students will learn the rules for these operations and will apply them in solving problems. They also inform the students that they will be using a flipped classroom methodology, where they will be expected to learn the rules at home and apply them in class. This methodology is explained briefly, with the teacher providing a clear outline of what the students will be expected to do at home and in class.

Development

Pre-Class Activities (10 - 15 minutes)

1. The teacher assigns a video (e.g., Khan Academy's "Multiplying and Dividing Rational Numbers") that explains the basic concepts of multiplying and dividing rational numbers, the rules for performing these operations, and examples of their applications. The video should be engaging and clear, using visual aids and real-life examples to help students understand the concepts.

2. After watching the video, the students are asked to take notes on the main points and any questions or areas of confusion they might have. They should also attempt the practice problems provided in the video and write down their solutions.

3. The teacher also assigns an interactive online quiz, such as one from Quizizz or Kahoot, on multiplying and dividing rational numbers. The quiz should be designed to test the students' understanding of the concepts covered in the video.

In-Class Activities (25 - 30 minutes)

Activity 1: Rational Number Trail (15 - 20 minutes)

1. The teacher begins the in-class session by dividing the students into groups of 4 to 5. Each group is provided with a set of task cards containing problems involving the multiplication and division of rational numbers. The problems are designed to be solved in a specific order, creating a 'trail' that the students must follow.

2. The first card on each trail presents a question related to a real-world scenario, which the students must discuss and translate into a mathematical problem. They then solve this problem, which leads them to the next card in the trail.

3. The students continue in this manner, reading the problem on each card, solving it, and moving on to the next card until they reach the end of their trail.

4. If a group encounters any difficulties, they are encouraged to discuss the problem, ask questions, and seek help from the teacher or their peers. This fosters collaborative learning and problem-solving skills.

5. At the end of the activity, each group should have solved all the problems on their trail, reinforcing their understanding of the rules for multiplying and dividing rational numbers.

Activity 2: Rational Number Relay (10 - 15 minutes)

1. Following the 'Rational Number Trail' activity, the teacher introduces a fun and competitive activity to reinforce the concepts learned—'Rational Number Relay'.

2. The teacher sets up a relay race course, either on the classroom floor or using a digital platform such as ClassDojo. Each 'leg' of the race contains a problem involving the multiplication or division of rational numbers.

3. The groups line up, and the first student from each group is given the first problem. They solve the problem and run to the next station to hand over the answer to the next student, who must multiply or divide this number by the next problem's number.

4. This continues until all the students in a group have completed the race. The first group to complete the relay correctly and reach the finish line wins the race.

5. The 'Rational Number Relay' is not only fun but also reinforces the idea that the solution to one problem often becomes the basis for solving the next problem.

By the end of these activities, students will have had a hands-on, collaborative, and competitive experience in applying the rules for multiplying and dividing rational numbers. They would have also had an opportunity to clear up any misconceptions, enhance their problem-solving skills, and build their confidence in their ability to handle these mathematical operations.

Feedback (5 - 7 minutes)

1. The teacher initiates a group discussion, where each group is given up to 3 minutes to share their solutions and the strategies they used to solve the problems in the 'Rational Number Trail' and 'Rational Number Relay' activities. The teacher guides the discussion, ensuring that the focus is on the concepts rather than just the answers. This allows students to learn from each other and see different approaches to solving problems.

2. Following the group discussions, the teacher highlights the link between the activities and the theory learned at home. For instance, they might point out how the rules for multiplying and dividing rational numbers were applied in the problems, and how the order of operations was followed. The teacher also addresses any misconceptions or errors that were common across the groups, providing the correct explanations and solutions.

3. The teacher then assesses the students' understanding of the lesson through a quick formative assessment. This can be done by asking the students to individually solve a few problems similar to the ones they encountered in the activities. The teacher circulates the classroom, observing the students as they work and offering help where needed. This allows the teacher to gauge whether the students have understood the concepts and can apply the rules for multiplying and dividing rational numbers.

4. To conclude the lesson, the teacher encourages the students to reflect on what they have learned. The students are asked to write down the answers to the following questions in their notebooks:

   • What was the most important concept you learned today?
   • Which questions have not yet been answered?

5. The teacher collects these reflections and reviews them later to identify any common areas of confusion or outstanding questions. These can then be addressed in the next class or in individual discussions with the students. This reflection also helps the students consolidate their learning and become more aware of their own understanding of the topic.

6. Finally, the teacher thanks the students for their active participation and reminds them to review the rules for multiplying and dividing rational numbers at home. They also inform the students of the next topic to be covered in the upcoming class. This ensures that the students are prepared and motivated for the next lesson.

Conclusion (5 - 7 minutes)

1. The teacher begins the conclusion by summarizing the main points of the lesson. They remind the students that they have learned the rules for multiplying and dividing rational numbers, and that these rules were applied in the 'Rational Number Trail' and 'Rational Number Relay' activities. The teacher also highlights any particularly important or challenging concepts that were covered during the lesson, such as the importance of following the order of operations when performing these operations.

2. The teacher then explains how the lesson connected theory, practice, and applications. They remind the students that they first learned the theory at home, through the video and online quiz. They then practiced applying this theory in the in-class activities, which were designed to simulate real-world problem-solving situations. The teacher also emphasizes that the real-world applications of multiplying and dividing rational numbers were discussed at the beginning of the lesson, and that the problems in the activities were based on these applications.

3. To further enhance the students' understanding of the topic, the teacher suggests additional resources for self-study. This could include other online videos or tutorials, interactive online games, or printable worksheets. The teacher encourages the students to use these resources to reinforce their understanding of the rules for multiplying and dividing rational numbers and to practice applying these rules in different contexts.

4. The teacher then briefly discusses the importance of the topic for everyday life. They remind the students that understanding how to multiply and divide rational numbers can be useful in many real-life situations, such as when shopping, cooking, or managing finances. They also stress that these operations are fundamental in many areas of mathematics and are often used in more advanced topics, such as algebra and calculus.

5. The teacher concludes the lesson by reminding the students that learning mathematics is a step-by-step process, and that each new concept builds on the ones they have already learned. They encourage the students to keep practicing and to ask questions when they are unsure. They also remind the students that the teacher is there to help and support them in their learning journey.

6. Finally, the teacher thanks the students for their active participation in the lesson and for their hard work. They remind the students to review the rules for multiplying and dividing rational numbers at home and to come prepared for the next class.