Lesson plan of Fundamental Principle of Counting

Objectives (5 - 7 min)

1. Develop an understanding of the concept of the Fundamental Principle of Counting and its use or application in problem-solving situations. This includes an ability to identify features of a problem which make the Fundamental Principle of Counting applicable.

2. Develop the ability to apply the Fundamental Principle of Counting to practical situations. Students should be able to identify the number of possibilities in a set of events and calculate the total number of possible outcomes.

3. Practice solving a variety of counting exercises that involve the Fundamental Principle of Counting, to solidify an understanding of the concept. This includes reading a problem, applying the formula appropriately, and producing an answer.

   Secondary Objectives:

   • Encourage higher-level thinking skills like analysis and logic
   • Promote student engagement using collaborative problem-solving activities.
   • Enhance students' ability to communicate mathematically by explaining their reasoning and the solution to a problem clearly and coherently.

Introduction (10 - 15 min)

1. Review of Prior Knowledge:

   • Begin with a brief review of previously learned concepts of combinations and permutations. Understanding these concepts will be vital to understanding the Fundamental Principle of Counting.
   • The instructor can ask quick questions to the class to check for understanding of these concepts. For example, "What is a combination?" or "What is a permutation?"

2. Introductory Problem Situations:

   • The instructor shares two problem situations to introduce the topic:
     1. "Suppose you have 3 shirts (a red one, a blue one, and a yellow one) and 2 pairs of pants (a black one and a white one). How many different outfits can you make with these clothes?"
     2. "If you have 4 different letters (A, B, C, or D) and want to make a sequence of 2 letters, how many different sequences are possible?"
   • Encourage the students to try to solve these problems mentally or on paper, without using the Fundamental Principle of Counting yet.

3. Provide Context:

   • Give context to the value and importance of the Fundamental Principle of Counting by explaining that it is a widely useful tool in many different fields, such as statistics, probability, computer science, and genetics.
   • To illustrate the value of the principle, the instructor can provide real-world examples where it is used. Some examples include determining possible genotypes in a genetic cross, optimizing delivery routes for a shipping company, or many others.

4. Introduce the Principle

   • The instructor formally introduces the Fundamental Principle of Counting as a tool used to find the total number of possible outcomes in a set of events.
   • To spark interest, the instructor can share a brief history of how the Fundamental Principle of Counting was discovered and the ways it revolutionized how people thought about counting.
   • The instructor can also share a fun fact related to the topic. For example, the Fundamental Principle of Counting is the basis for developing encryption algorithms used to protect sensitive information online.

Development (20 - 25 min)

1. Mathematical Planet Walk Activity (10 - 12 min):

   • Divide the class into groups of 4 or 5 students. Explain that each group will receive a "Mathematical Planet" game board that will be used to complete this activity. The game board is a 6x6 grid of squares.
   • In each square of the grid, there is a word problem involving the Fundamental Principle of Counting. The word problems vary in difficulty and complexity.
   • The objective is for the group members to solve the word problems and advance their game piece around the board. Each correctly solved problem will earn the group one space on the board.
   • The instructor should circulate around the room and assist any groups who are struggling with the problems and answer questions. It is important for the instructor to check that the groups are correctly applying the Fundamental Principle of Counting.
   • The first group to reach the end of the board is the winner. More importantly, it is valuable for every student to have had the opportunity to practice using the Fundamental Principle of Counting in a variety of problem situations.

2. Exploring Sequences Activity (10 - 12 min):

   • Have the groups remain intact. Each group will receive a set of cards with different symbols on them. Each symbol represents a possible event in a problem situation.
   • Read a problem situation to the class involving the creation of sequences made from these events. The student teams will then use the cards to help them solve the problem in their groups.
   • For example, "If we have 3 cards: one with a circle, one with a square, and one with a triangle, and want to make a sequence of 2 cards, how many different sequences could we make?"
   • The students should be able to correctly identify the problem situation, select the correct cards, and count the total possible sequences.
   • The instructor should circulate to observe the group work, answer questions, and provide feedback.

3. Probability Challenge (5 - 7 min):

   • To end the Development section, pose a challenge question to the teams. Provide each team with a set of cards, numbered 1 - 6.
   • The challenge is for each member of the team to select a card at random, without replacement, within a 1-minute time period. At the end of 1 minute, the team should be able to tell you how many times any given number was selected.
   • The objective of the challenge is for the teams to see that although the individual selection of any card is random, it is possible to predict the frequency with which each number will be drawn.
   • The instructor should emphasize that in order to solve the challenge, they will need to use the Fundamental Principle of Counting to determine the total number of possible outcomes.

These activities give students the opportunity to work collaboratively, apply the Fundamental Principle of Counting in practical scenarios, and develop problem-solving, logical reasoning, and mathematical communication skills.

Debrief (8 - 10 min)

1. Group Discussion (3 - 4 min):

   • Bring the class together as a whole group. Facilitate a class discussion about the solutions or conclusions that each team found.
   • Provide ample opportunity for each team to share their strategies for solving the problems, the challenges they faced, and how they applied the Fundamental Principle of Counting.
   • Ask probing questions during the discussion such as "Why did you choose that strategy?" or "How did you determine that the Fundamental Principle of Counting would work for that problem?"
   • Encourage students to ask each other questions and to offer constructive feedback for each other.

2. Connection to Theory (2 - 3 min):

   • Following the group discussion, take a few minutes to revisit the theory, formally define the Fundamental Principle of Counting, and review how it was used in the activities.
   • This is a good time to clarify any confusion or questions that may have come up during the activities and reinforce the key points of the lesson.

3. Individual Reflection (2 - 3 min):

   • Ask the students to take a moment to individually reflect on what they have learned.
   • Ask questions like "What was the most important concept you learned today?" or "What questions do you still have?"
   • Allow the students a minute or so to think about their responses and then give them the opportunity to share their thoughts with the class if they would like.
   • Encourage the students to voice their opinions and ask questions, reinforcing that learning is a continuous process and that it is okay to still have questions about or not fully understand a concept at this point in time.

4. Instructor Feedback (1 min):

   • Conclude the debrief with a few minutes of instructor feedback on the class's performance. Highlight areas where they excelled as a class and areas where they could improve.
   • The instructor can also provide suggestions for additional practice to help the students master the Fundamental Principle of Counting and solidify their understanding.

At the end of the Debrief, the students will have had a chance to reflect on what they have learned, share their experiences, and hear feedback on their work. This will help to solidify the learning and identify any areas that they may need to revisit or practice further.

Conclusion (5 - 7 min)

1. Review the Lesson (2 - 3 min):

   • Take a few minutes to summarize the most important points that were covered during the class. Reiterate the definition of the Fundamental Principle of Counting and discuss its variety of uses and applications.
   • Discuss the sample problems that were introduced and how they were solved, emphasizing the importance of correctly applying the Fundamental Principle of Counting to reach a solution.
   • Briefly review the hands-on activities that were completed, discussing any challenges that students may have faced and how they were able to work through them.

2. Connect Theory, Practice, and Application (1 - 2 min):

   • Discuss how the lesson has connected the theoretical, practical, and applicable sides of the Fundamental Principle of Counting.
   • Review how the concepts were originally presented and developed, how the hands-on activities allowed the students to apply the concepts, and how the example problems represented real-world applications of the principle.
   • Emphasize that the purpose of the lesson was not just to teach the formula for the Fundamental Principle of Counting, but to develop the students' abilities to identify and solve problems which can be solved using the principle.

3. Share Supplementary Materials (1 min):

   • Provide the students with supplementary materials to use to further their study of the Fundamental Principle of Counting if they choose to do so. This could include books, websites, videos, games, or other resources that provide further explanation and practice for the topic.
   • For example, you might recommend a specific math textbook that includes thorough explanations and worked examples of the Fundamental Principle of Counting. Or, you may share a website that provides a collection of counting problems for students to solve.

4. Reiterate the Importance (1 min):

   • Finally, take a moment to reiterate the importance of the Fundamental Principle of Counting, both in everyday life and in higher-level mathematics.
   • Provide a few specific examples outside of the classroom where the principle may be applied, such as calculating the number of possible combinations in a card game, planning a party with different food and drink options, or analyzing data for a research study.
   • Conclude by reinforcing that the Fundamental Principle of Counting is a powerful tool that can be used to solve a wide range of problems, and that understanding this principle is key to developing strong mathematical and logical reasoning skills.