Objectives (5-7 minutes)

1. Understand what a Geometric Progression is: Students will be able to define what a geometric progression is by understanding that they are a sequence where each term is obtained by multiplying the prior term by a constant.

2. Find the nth term in a Geometric Progression: Students will be able to find the nth term in a geometric progression, by using the formula: a_n = a_1 * r^(n-1) where a_n represents the nth term, a_1 represents the first term and r represents the ratio of the sequence.

3. Find the value of a term in a Geometric Progression: Students will be able to determine the value of a term in a geometric progression, given values for other terms and the ratio of the progression. This includes applying the formula: a_n = a_1 * r^(n-1).

   Additional objectives:

   • Develop problem solving skills: Students will have the chance to develop their problem solving skills through practice with geometric progressions.

   • Encourage critical thinking and reasoning: Working with numeric sequences challenges students to think critically and apply mathematical reasoning to solve the problems presented.

The teacher should highlight the importance of this content in the field of mathematics as well as in other disciplines, providing examples of how it can be used in daily life situations and in other subject areas.

Introduction (10-15 minutes)

1. Review of previous content: The lesson begins with the teacher taking a few minutes to review the concepts of numerical sequences and arithmetic progressions which are fundamental for understanding the new content. Students will have an opportunity to participate by sharing what they remember and what they already know about the subject. The teacher will clarify that although a geometric progression and an arithmetic progression share similarities, the difference is that in a geometric progression each term is obtained by multiplying the prior term by a constant, known as the ratio.

2. Problem situations: In order to engage students, the instructor will pose two problems that involve geometric sequences:

   • Situation 1: The teacher poses a challenge where the students are asked to predict how many grains of rice will end up on a chess board if on the first square there is one grain of rice and on each of the following squares, the number of grains is double the previous one. This situation exemplifies a geometric progression.

   • Situation 2: The teacher proposes an investment problem, where the students must find the total amount of money for an investment over time, considering an established interest rate. This situation also illustrates a geometric progression.

3. Contextualization: The instructor will then contextualize the importance of geometric sequences, explaining that they can be found in different areas of knowledge such as engineering, economics, the sciences and computing. For example, geometric sequences are useful for modeling growth in populations, depreciation of assets, growth of investment and more.

4. Topic introduction: The teacher can share some fun facts and interesting uses for geometric sequences as an enticing way of introducing the topic:

   • Fun fact 1: The teacher could share the story of the Indian mathematician, Sissa who according to the legend, invented the game of chess and asked the king to pay him with grains of wheat, one grain on the first square of the chess board, two grains on the second, and so on until he reached square 64. After initially accepting, the king quickly realized that the total number of wheat grains would be more than the entire world’s wheat production at that time!

   • Fun fact 2: The instructor could share that geometric progressions are applied in music to create chords and scales and in the arts to design patterns and sequences.

Through these strategies the instructor can capture the students' interest and motivation for studying geometric progressions.

Development (20-25 minutes)

1. "M&M Growth” activity (10 - 15 minutes)

   • Description: The instructor will propose a hands-on activity using a geometric progression. It will consist of a bag of M&Ms (or other type of candy with different colors) and a sheet of graph paper. Each M&M will represent a term in a sequence and different colored M&Ms will represent different ratios.

   • Step by step:

     1. The instructor should select three different colors of M&M and establish a rule: for each color, the number of M&Ms of the same color in the sequence must be double the previous term. For example, the blue series: 1, 2, 4, 8, ... ; red series: 1, 3, 9, 27, ... ; yellow series: 1, 4, 16, 64, ...
     2. Subsequently, the teacher will distribute the M&Ms on the graph paper following the rules established. For example, for the blue series the instructor will put 1 M&M in cell one, 2 M&Ms in cell two (making a rectangle), 4 M&Ms in cell three (forming a rectangle), and so on.
     3. Students in groups will try to anticipate how many M&Ms are going to be required to fill out a specific amount of cells, (for example, 10 cells), for each of the series.
     4. Once they have hypothesized the amount, the students should physically complete the task of filling the cells with the M&Ms and count how many they needed.
     5. Finally, the groups will compare their hypothesis with their results and discuss any differences. The instructor will facilitate the conversation, guiding the students to notice that the amount of M&Ms needed grows exponentially with the amount of cells, following the geometric progression that was defined.

2. “Investing with geometric progressions” activity (10 - 15 minutes)

   • Description: In this activity students will apply the concept of a geometric progression to model the growth of an investment over time.

   • Step by step:

     1. The instructor should create a story problem: An investor deposits an initial amount into an investment account which has a constant interest rate. Every year, the investor adds to the same account the same amount that they already had, multiplied by the interest rate.
     2. The teacher will give the students the interest rate and initial invested amount, then ask them to identify the total amount in the account after a specific number of years given.
     3. Students working in teams will complete the calculations using the geometric progression formula: a_n = a_1 * r^(n-1), where a_n is the value at n years, a_1 is the initial amount, and r is the rate of interest.
     4. The groups will finally compare and contrast their results, and discuss the strategies used to complete their work.

3. Discussion and Reflection (5 - 7 minutes)

   • Description: After completing the activities the instructor will guide the class through a discussion where students can share their experiences, doubts and discoveries. This is an opportunity for the teacher to clarify any misconceptions as well as to reinforce the relevance of this knowledge in everyday life and its relationship to other subjects.

   • Step by step:

     1. The instructor will invite students to share their answers to the activities and the strategies used.
     2. The teacher should clarify any doubts students may still have and reinforce important concepts.
     3. The instructor should highlight real world applications for geometric progressions such as their use in finance, science, and technology.
     4. The teacher should encourage learners to think about what they have learned and make connections to other subject areas as well as the world around them.

With these activities students will have the chance to understand and apply the concepts of geometric sequences, their general term, and their term calculation. Additionally they will continue to develop problem solving strategies, critical thinking and reasoning skills.

Feedback (8-10 minutes)

1. Group discussion (2 - 3 minutes)

   • Description: The teacher will bring all students together to engage in group discussion about the team solutions to the activities. This provides an opportunity for the students to teach each other and to share strategies, findings and clarify any remaining uncertainties. The instructor will facilitate this conversation, making sure all students participate and that the discussion is constructive.

   • Step by step:

     1. The instructor should ask each team to share the answers and strategies used with the whole class.
     2. The teacher will encourage other students to ask questions and give feedback.
     3. The instructor should clarify any misconceptions and correct any conceptual errors identified during the discussion.

2. Theory Connection (2 - 3 minutes)

   • Description: After group sharing, the teacher will connect the hands-on activities back to the theory presented during class. The goal here is for the students to see how theoretical concepts apply in practice, and to help them see how these concepts can be used in the real world to problem solve.

   • Step by step:

     1. The instructor will highlight the key concepts that were used in the hands on activities, including the geometric sequence terms, general terms, and the term calculation formula.
     2. The teacher will explain how these concepts were used to solve the problems posed during the activities.
     3. The instructor should reflect upon the importance of understanding and being able to apply geometric sequence concepts, emphasizing how they are useful in multiple disciplines and in everyday situations.

3. Final reflection (2 - 3 minutes)

   • Description: The instructor will ask students to take a few moments to reflect on the learning from class. This provides an opportunity for the students to solidify their new knowledge and to identify any gaps in understanding as well as begin to consider how to apply their new learning to other situations.

   • Step by step:

     1. The instructor will pose questions to guide students’ reflections such as: “What was the most important concept you learned today?”, “What questions remain unanswered?”, “How might you use what you learned today in your daily life, or in your other classes?”
     2. Students should have one minute to silently consider their responses.
     3. After reflection time, the teacher may call on some students to share their answers with the whole group, if students feel comfortable doing so.

At the end of the Feedback portion of the class students should be able to clearly identify the concepts of geometric sequences, their general term, and term calculations as well as reflect on why these concepts are valuable and how they may be used in practice. They will have also continued to develop problem-solving skills as well as their creative thinking and critical reasoning skills.

Conclusion (5 - 7 minutes)

1. Summary of main points (1 - 2 minutes)

   • Description: The instructor will recap the significant points of the content covered in the lesson. This will include the definition for geometric progression terms, how to identify nth terms, and how to calculate terms. The instructor will reinforce that these are essential components when understanding and using geometric sequences.

   • Step by step:

     1. The teacher should summarize the geometric progression term definition.
     2. The instructor should restate the general geometric progression formula.
     3. The instructor should summarize the process used to find the value of any term within the geometric sequence, given the ratio and values of other terms.

2. Theory, Practice, and Applications connections (1 - 2 minutes)

   • Description: The instructor will highlight the lesson’s connections to theory, practice and application. The instructor should reflect on the hands on activities, and how they were designed to help illustrate and reinforce theoretical concepts. Additionally the teacher should restate how these concepts have practical applications, providing examples of how geometric progressions are useful in various situations.

   • Step by step:

     1. The instructor will restate how important hands on activities were to understanding the content.
     2. The instructor should summarize the real world applications discussed in class, highlighting the different areas of daily life and disciplines in which geometric progressions can be found.

3. Extension Materials (1 - 2 minutes)

   • Description: The instructor will suggest supplemental resources for students to use to further their study of geometric progressions. These can include textbooks, websites, videos and additional practice problems. Additionally the teacher will encourage the learners to independently find applications of geometric sequences that interest them.

   • Step by step:

     1. The instructor will give a list of supplemental materials including title, author, and link if applicable.
     2. The teacher should briefly explain what each resource is about, and how they can help the students increase their knowledge of geometric sequences.
     3. The instructor should encourage students to use these additional resources independently, as they have time and as they choose.

4. Relevance (1 minute)

   • Description: In closing the lesson the instructor should highlight the importance of understanding geometric sequences not only in the field of Mathematics, but also for other disciplines. The instructor will reinforce that beyond being fundamental components of Mathematics, geometric sequences also have applications in other fields such as the sciences, engineering, economics as well as in music and the arts. Additionally the instructor should summarize that mastering these concepts will contribute to students’ overall critical thinking skill development.