Objectives (5 - 7 minutes)

1. Students will understand the concept of combinations in mathematics, particularly in the context of selecting items from a larger set without regard to the order of selection.
2. Students will be able to calculate the number of combinations possible when selecting items from a larger set, using the combination formula.
3. Students will apply their understanding of the combination formula to solve practical problems involving combinations, in both theoretical and real-world contexts.

Secondary Objectives:

• Students will develop critical thinking skills as they analyze problems and devise strategies for calculating combinations.
• Students will enhance their collaborative skills as they work together in groups during the hands-on activities.

Introduction (10 - 15 minutes)

1. The teacher reminds students about the concept of permutations, which they have previously learned. They explain that permutations involve the arrangement of objects where order matters, while combinations involve the selection of objects where order does not matter. (2 minutes)

2. The teacher then presents two problem situations to the class:

   • Problem 1: "If you have 3 different colored balls - red, blue, and yellow, and you need to choose 2 balls to give to your friends, how many different combinations can you make?"
   • Problem 2: "If you have 5 different books on a shelf and you want to select 3 books to take on a trip but the order doesn't matter, how many different combinations can you make?" (3 minutes)

3. The teacher then contextualizes the importance of combinations in real-world applications. They explain that combinations are used in probability, genetics, and in many other fields where selection is involved. For instance, in genetics, combinations are used to determine the possible outcomes of genetic crosses. In probability, combinations are used to calculate the number of possible outcomes in a sample space. (3 minutes)

4. The teacher introduces the topic with two intriguing facts:

   • Fact 1: "Did you know that the concept of combinations was first introduced by a French mathematician, Blaise Pascal, in the 17th century? He used them to solve problems of gambling, which is still one of the most common applications of combinations today!"
   • Fact 2: "Combination locks, commonly used to secure lockers and safes, are based on the mathematical concept of combinations. The number of possible combinations on a lock is determined by the number of numbers or letters on the lock and the length of the combination." (2 minutes)

5. To engage the students further, the teacher asks them to think about how many different combinations of outfits they can make using the clothes in their wardrobe. They also ask the students to guess the number of possible combinations on a typical 4-digit locker combination. (3 minutes)

Development (20 - 25 minutes)

Activity 1: Delicious Combinations

1. The teacher divides the class into small groups and distributes a pack of a popular candy (like M&M's) to each group. (1 minute)

2. The teacher explains that each color of the candy represents a different item in a set, and the objective is to make combinations of the candies where the order does not matter. (1 minute)

3. Each group is tasked with selecting different numbers of candies from the pack, and they are to record the number of different combinations they can make with the selected candies. (5 minutes)

4. Afterward, each group is to put their candies back in the pack and exchange their recorded combinations with another group. The new group will then try to verify the recorded combinations. This activity encourages the students to check one another's work and promotes collaborative learning. (5 minutes)

5. The teacher then brings the class back together and asks each group to share their findings. The teacher ensures that the students understand the process of calculating combinations and how to use the combination formula in this context. (3 minutes)

6. To wrap up the activity, the teacher asks each group to propose a real-world situation where the concept of combinations could be applicable, and they briefly explain how they would go about calculating the possible combinations in that situation. (5 minutes)

Activity 2: Lock Combinations

1. The teacher prepares a set of combination locks with different numbers of dials and digits, and distributes one lock to each group. (1 minute)

2. The teacher then explains that each group's task is to determine the number of possible combinations on their lock. They are to use the number of dials and digits on the lock to calculate this. (5 minutes)

3. The teacher provides each group with a worksheet containing the combination formula and a guide on how to use it. The teacher guides the students in understanding and using the formula. (5 minutes)

4. After calculating the number of possible combinations on their lock, each group is to present their findings to the class. The teacher checks their calculations and provides feedback. (3 minutes)

5. To wrap up the activity, the teacher asks the students to hypothesize how the number of possible combinations on a lock would change if the number of dials or digits were different, and why. This encourages the students to think critically and apply their knowledge of combinations. (5 minutes)

Activity 3: "Guess the Combination" Game

1. This activity is a fun way to conclude the lesson and reinforce the concept of combinations. The teacher prepares a "Guess the Combination" game based on the locks used in the previous activity.

2. The teacher randomly sets a combination on each lock and covers the numbers/digits. The students are then challenged to guess the combination based on the clues given and their knowledge of combinations. (5 minutes)

Feedback (8 - 10 minutes)

1. The teacher begins the feedback session by asking each group to share their solutions or conclusions from the activities. They explain how they arrived at their answers and the strategies they used. This gives the students an opportunity to learn from each other and see different approaches to solving the same problem. (3 minutes)

2. The teacher then connects the group's findings from the activities to the theoretical concept of combinations. They highlight how the process of selecting candies or determining lock combinations aligns with the mathematical formula for combinations. This helps to reinforce the students' understanding of the concept and its application. (2 minutes)

3. The teacher then proposes that the students reflect on the activities and the lesson as a whole. They ask the students to think about the most important concept they learned and any questions they still have. The teacher encourages the students to share their reflections and questions with the class. (3 minutes)

4. After the students have shared their reflections, the teacher provides clarification on any misconceptions and answers any remaining questions. The teacher also provides feedback on the students' performance in the activities, highlighting areas of strength and areas for improvement. This helps the students to understand their progress and what they need to work on. (2 minutes)

5. To conclude the feedback session, the teacher asks the students to consider how they can apply what they have learned about combinations in their daily lives or in other subjects. This encourages the students to see the relevance of the concept and its applicability beyond the classroom. (2 minutes)

Conclusion (5 - 7 minutes)

1. The teacher begins the conclusion by summarizing the main points of the lesson. They remind the students that combinations are used to determine the number of ways to select items from a larger set when the order does not matter. They also recap the formula for calculating combinations and its application in the activities. (2 minutes)

2. The teacher then explains how the lesson connected theory, practice, and applications. They note that the lesson started with a theoretical explanation of combinations, which was then applied in hands-on activities. These activities, in turn, were connected to real-world applications of combinations, such as in probability, genetics, and even in the design of combination locks. (2 minutes)

3. The teacher suggests additional materials for the students to further their understanding and practice of combinations. These could include textbooks, online resources, or math games that involve combinations. They also encourage the students to look for other real-world examples of combinations and to try calculating the possible combinations in these situations. (1 minute)

4. Lastly, the teacher emphasizes the importance of understanding combinations for everyday life. They explain that combinations are not just a concept in mathematics, but they are also used in various fields and in our daily activities. For instance, when we choose what to wear, what to eat, or which books to read, we are making combinations. The teacher encourages the students to be mindful of these applications and to continue exploring the world of combinations beyond the classroom. (2 minutes)