Lesson Plan | Active Learning | Triangles: Similarity

Assumptions: This Active Lesson Plan assumes: a 100-minute class, prior student study with both the Book and the start of Project development, and that only one activity (among the three suggested) will be chosen to be conducted during the class, as each activity is designed to take up a significant portion of the available time.

Objectives

Duration: (5 - 10 minutes)

The objectives stage is crucial to direct the focus of students and the teacher towards the learning goals of the lesson. By clearly establishing what is expected to be achieved, students can mentally prepare and review prior content more effectively. Additionally, this section serves to align expectations and ensure that both the proposed activities and the interaction in the classroom are aimed at fulfilling the specific objectives.

Main Objectives:

1. Empower students to understand the concept of similarity of triangles by identifying their properties and applying them in different mathematical contexts.

2. Develop the skill to calculate side proportions of similar triangles, using this property to solve practical and theoretical problems.

Side Objectives:

1. Encourage critical thinking and logical reasoning skills through solving problems involving the similarity of triangles.

Introduction

Duration: (15 - 20 minutes)

The introduction serves to engage students with the content they studied previously, using problem situations that make them think critically and apply the knowledge of similarity of triangles practically. The contextualization seeks to show the relevance of the theme in the real world, thus increasing interest and awareness of the importance of studying mathematical content in everyday and professional situations.

Problem-Based Situations

1. Imagine you have a scale map where the real distance between two cities is 300 km. On the map, this distance is represented by 5 cm. If you need to calculate the distance on the map between two other cities, how would you use the concept of similarity of triangles to find the answer?

2. Consider a pair of similar triangles where one has a side of 10 cm and the corresponding side of the other measures 5 cm. If the hypotenuse of the first triangle measures 15 cm, what would be the length of the hypotenuse of the second triangle? Use the concept of proportion to solve.

Contextualization

The similarity of triangles is a powerful tool not only in mathematics but also in various practical applications in everyday life, such as engineering, architecture, and even art. For example, architects use principles of similarity to design structures that are visually pleasing and proportional. Furthermore, the history of mathematics is filled with examples where the study of similarity of geometric figures was fundamental to the development of theories and the resolution of complex problems.

Development

Duration: (70 - 80 minutes)

The Development stage is designed to allow students to practically and contextually apply the concepts of similarity of triangles that they studied previously. By working in groups, they not only reinforce collaborative learning and communication but also develop problem-solving skills and critical thinking. This practical and playful approach aims to consolidate theoretical knowledge in a way that is meaningful and engaging for students.

Activity Suggestions

It is recommended to carry out only one of the suggested activities

Activity 1 - Mathematical Expedition: The Treasure of Similar Triangles

> Duration: (60 - 70 minutes)

- Objective: Apply the concept of similarity of triangles in solving practical problems and promote teamwork.

- Description: In this activity, students will be divided into groups of up to 5 people and embark on a 'expedition' around the classroom, where they will find 'clues' that will lead them to discover the hidden 'treasure'. Each clue will be associated with a similar triangle and students will need to use the concept of proportion to solve problems and advance in the expedition.

- Instructions:

• Divide the class into groups of up to 5 students.

• Each group receives an initial envelope containing the first clue and a triangle drawn on a piece of paper.

• Subsequent clues are hidden in different locations around the room, and each of them is associated with a mathematical problem involving the similarity of triangles.

• To advance to the next clue, students must correctly solve the proposed problem and bring the answer to the teacher.

• At the end of the activity, the group that solves all the clues and finds the 'treasure' first wins a small symbolic prize.

Activity 2 - City Builders: Planning with Triangles

> Duration: (60 - 70 minutes)

- Objective: Understand and apply the similarity of triangles in an urban design context, developing presentation skills and creativity.

- Description: Students, organized in groups, will design a city on a large poster board using triangles of different sizes. Each group must ensure that streets, buildings, and public spaces maintain similar proportions, thus applying the concept of similarity of triangles.

- Instructions:

• Organize students into groups of no more than 5 people.

• Distribute kits with paper triangles of varying sizes and a large poster board to each group.

• Explain that each group must design a city on the poster board, positioning the triangles so that the proportions are similar across different parts of the city.

• Each group must present their project at the end, explaining their choices based on the concept of similarity of triangles.

• Conduct a vote to choose the project that best applied the mathematical concepts.

Activity 3 - Detectives of Similarity: Unraveling Mathematical Mysteries

> Duration: (60 - 70 minutes)

- Objective: Engage students in solving practical and complex problems, creatively and playfully utilizing the concept of similarity of triangles.

- Description: In this playful activity, students transform into detectives who need to solve 'mysterious mathematical crimes' using the concept of similarity of triangles. Each 'crime' is a mathematical problem that must be solved to 'capture the criminal'.

- Instructions:

• Divide the class into groups of up to 5 students.

• Present them with a 'crime scene' involving similar triangles, such as a theft in a museum where the only clue is a drawing of triangles left behind by the thief.

• Groups must use the concept of similarity of triangles to solve the problems and advance in the 'investigation'.

• Each correctly solved problem leads to a new 'clue' that brings them closer to the 'criminal'.

• The first group to solve all the problems and 'capture the criminal' is the winner.

Feedback

Duration: (10 - 15 minutes)

The purpose of this feedback stage is to consolidate learning, allowing students to articulate what they learned and hear their peers' experiences. This exchange of ideas and solutions helps reinforce understanding of the concepts of similarity of triangles and promotes mathematical communication and argumentation skills. Additionally, by discussing difficulties and successes, students can gain valuable insights into their own approaches and those of others, thus enriching their mathematical repertoire.

Group Discussion

Initiate the group discussion by inviting each group to share their discoveries and the process of solving the activities. Ask each group to choose a representative to present a brief summary of what they did and what they learned. Encourage students to discuss the different approaches and solutions found, and how the theory of similarity of triangles was applied in each activity.

Key Questions

1. What were the biggest challenges in applying the concept of similarity of triangles in the activities?

2. How did you decide on the best strategy to solve the proposed problems?

3. Was there any moment when the similarity of triangles helped you perceive something unexpected or solve a problem more efficiently?

Conclusion

Duration: (5 - 10 minutes)

The purpose of this conclusion stage is to consolidate the knowledge acquired by students throughout the lesson, linking theory and practice and emphasizing the importance of studying the similarity of triangles in real applications. This recap helps students retain the content and recognize the relevance of what they learned, preparing them for future applications of the concept in their academic and personal lives.

Summary

In this final stage, we recapitulate the concepts addressed regarding the similarity of triangles, highlighting the properties and practical application of the proportions of their sides. We review the theory of similarity and how it applies in various contexts, from maps to architectural drawings and even everyday problems.

Theory Connection

Today's lesson was structured to effectively connect theory and practice. Through interactive activities like 'Mathematical Expedition', 'City Builders', and 'Detectives of Similarity', students were able to directly apply the theoretical concepts studied at home, thus demonstrating the relevance and utility of the similarity of triangles in real and imaginary situations.

Closing

The importance of understanding the similarity of triangles extends beyond the academic realm, reaching into various practical situations in everyday life. From planning the arrangement of furniture in a space to understanding maps and scales, the concept of similarity is an essential tool for solving problems and making everyday decisions.