Objectives (5 - 7 minutes)

• To understand the concept of complex numbers and their components: real and imaginary parts.
• To comprehend the concept and importance of the complex conjugate of a complex number, in particular, how it is obtained by changing the sign of its imaginary part.
• To apply the knowledge of complex conjugate in performing various operations involving complex numbers, such as finding the modulus and dividing complex numbers.
  • Secondary objectives include developing analytical thinking through the application of concepts learned and fostering collaborative learning through in-class activities.

Introduction (10 - 12 minutes)

• The teacher begins by reviewing the concepts of real numbers and imaginary numbers, which are the components of complex numbers. It is highlighted that the understanding of these concepts is necessary for the comprehension of complex numbers and their conjugates.
• The teacher presents two problem situations that can serve as starters for the development of the theory to follow:
  1. Considering the complex number 3 + 4i, what would be its complex conjugate? And how would this complex conjugate be used to determine the modulus of this complex number?
  2. Given two complex numbers, 2 + 3i and 1 - 2i, how can the complex conjugate be used to accomplish the division of these numbers?
• The teacher then contextualizes the importance of the subject with real-world applications. For instance, complex conjugates are used in physics to describe waves and in electrical engineering to analyze alternating current circuits.
• To introduce the topic in an engaging manner, the teacher might share a few curiosities:
  1. The concept of complex numbers and their conjugates was developed in the 16th century by an Italian mathematician, Gerolamo Cardano.
  2. The term 'imaginary' was coined as a derogatory term because these numbers were once considered fictitious and useless. However, 'imaginary' numbers turned out to be immensely useful in various fields, defying their initial reputation.
• The teacher concludes the introduction by stating the objectives of the lesson and the skills the students are expected to acquire by the end of the session. To maintain the interest of the students, the teacher can pose a question, "Ever wondered how these 'imaginary' numbers that were once considered 'useless' are now crucial in numerous fields? Let's find out together!"

Development

Pre-Class Activities (10 - 15 minutes)

• Students are tasked with watching a short informational video at home on the introduction to complex numbers and their conjugates. The video should be under 20 minutes and engaging while also explaining the concepts clearly and in a simplistic manner. The video should cover the formation of complex numbers using real and imaginary parts plus the concept and applications of complex conjugates.
• After watching the video, students should jot down any questions or points of confusion on a shared online forum, which would be addressed in the next class. Students should also try to find some examples of real-world applications of complex numbers and their conjugates.

In-Class Activities (20 - 25 minutes)

• The teacher starts by revisiting the points raised on the online forum, answering student questions, and addressing their points of confusion.

Activity 1: "The Conjugate Pair Show"

1. The teacher divides the class into groups of four students and provides each group with a deck of "Complex Number Cards".
2. The cards contain both complex numbers and their conjugates. For example: (3 + 4i, 3 - 4i), (2 + 5i, 2 - 5i), etc.
3. Each group will shuffle their deck and distribute the cards evenly among themselves.
4. Simultaneously, each student will flip over the top card of their deck.
5. If the two cards flipped over are a complex number and its conjugate, the first student to shout "Conjugate Pair!" gets to keep both cards.
6. If no conjugate pair is formed, the cards are placed back in their respective decks.
7. The game continues until no cards are left in the deck. The player who has collected the most 'conjugate pairs' wins the game.
8. In the end, the teacher provides a brief recap, stressing how the conjugate of a complex number simply involves switching the sign of its imaginary part.

Activity 2: "Complex Conjugate Calculator"

1. Keeping the same groups, the teacher now invites students to use their knowledge of complex conjugates to solve a real mathematical problem.
2. The teacher will put two complex numbers on the board and ask the groups to work out the division of these complex numbers using the concept of complex conjugates.
3. Each group will work together, sketching out their calculations on a piece of chart paper.
4. Upon completion, each group will present their workings and the teacher will facilitate corrections and explanations as necessary.
5. Through this activity, students will get hands-on practice with using complex conjugates to perform operations on complex numbers.

These activities aim to reinforce the learning from the pre-class video and provide a playful, engaging and collaborative atmosphere to explore and understand the topic of complex numbers and their conjugates. The teacher's role is to guide, correct, and explain each step to the students to ensure comprehension and learning.

Feedback (7 - 10 minutes)

• To start the feedback process, the teacher will facilitate a group discussion, allowing each group to present their solutions or conclusions from the "Complex Conjugates Calculator" activity. Each group is given up to 3 minutes to share their findings, explain their thinking process, and how they successfully used the complex conjugate to divide the provided complex numbers. (4 - 5 minutes)

• After all groups have presented their solutions, the teacher will summarize the key learning points, bridging the gap between the theory and practical application. The teacher will further explain how the use of complex conjugates simplifies the process of performing operations such as division on complex numbers. (2 - 3 minutes)

• Following this, the teacher will guide the students to reflect on what they learned in this lesson. The teacher will ask important reflective questions such as:

  1. What was the most significant concept you learned today?
  2. Do you have any remaining questions about complex numbers or their conjugates?
  3. If you were to explain the concept of complex conjugates to a friend, how would you do it?
  4. Can you think of other real-world applications of complex numbers and their conjugates?

• The teacher will then give students a few minutes to ponder these questions and share their responses. This reflection time is crucial as it helps students to consolidate their learning and appreciate the relevance of the topic. (1 - 2 minutes)

• To conclude the feedback session, the teacher will encourage students to review the lesson at home and practice more problems involving complex numbers and their conjugates. The teacher will also remind students that they can post any further questions or confusions on the online forum for discussion in the next class.

This feedback stage is critical as it serves to reinforce the knowledge acquired, address any lingering doubts, and encourage students to apply the concepts learned in different contexts. It also promotes self-reflection, which is a vital step in the learning process.

Conclusion (5 - 7 minutes)

• The teacher begins the conclusion by summarizing the lesson, emphasizing the main points learned about complex numbers and their conjugates. The teacher restates that a complex number consists of a real part and an imaginary part, and the complex conjugate is obtained by changing the sign of the imaginary part. The teacher also reiterates the importance of complex conjugates in performing operations such as division and finding the modulus of complex numbers.
• The teacher then recaps the activities completed during the lesson, reminding students how they played "The Conjugate Pair Show" to familiarize themselves with the concept of complex conjugates and then applied this knowledge in "Complex Conjugate Calculator" to perform division of complex numbers.
• The teacher further elaborates on how the lesson connected theory, practice, and real-world applications. The theory was presented through the pre-class video, which was then applied in in-class activities. The real-world applications were discussed at various points throughout the lesson, providing students with a broader context for the importance of the subject.
• To ensure students have a firm grasp on the topic, the teacher suggests additional resources for further study. These may include textbooks, websites, and online tutorials that offer more detailed explanations and practice problems on complex numbers and their conjugates. The teacher encourages students to explore these resources at their own pace and bring any questions or doubts to the next class.
• Lastly, the teacher emphasizes the significance of the concepts learned in everyday life. The teacher explains that complex numbers and their conjugates play a vital role in many scientific and engineering fields, such as physics and electrical engineering. By understanding these concepts, students are building a foundation that will support their future studies and potential careers in these and related fields.
• To conclude the lesson, the teacher thanks the students for their active participation and encourages them to continue exploring and learning about complex numbers and their conjugates. The teacher reminds the students that learning is a continuous journey, and every step, no matter how complex, brings them closer to a deeper understanding of the world around them.