Lesson Plan | Traditional Methodology | Triangles: Similarity

Objectives

Duration: (10 - 15 minutes)

The purpose of this stage is to introduce the topic of the similarity of triangles, highlighting the skills that students should acquire throughout the lesson. Clearly defining the main objectives helps guide teaching and learning, ensuring that students understand the importance and practical application of the content to be covered.

Main Objectives

1. Understand the concept of similarity of triangles.

2. Calculate the sizes of the sides of similar triangles using proportions.

Introduction

Duration: (10 - 15 minutes)

The purpose of this stage is to introduce the topic of the similarity of triangles, highlighting the skills that students should acquire throughout the lesson. Clearly defining the main objectives helps guide teaching and learning, ensuring that students understand the importance and practical application of the content to be covered.

Context

To start the lesson on Similarity of Triangles, it is important to contextualize students about the relevance of this concept in Mathematics and in various fields of knowledge. The similarity of triangles is a fundamental topic, not just for solving geometric problems, but also for practical applications in areas such as engineering, architecture, and even art. In any situation where we need to scale figures while maintaining their proportions, the similarity of triangles becomes essential.

Curiosities

Did you know that the similarity of triangles is widely used in map making and photography techniques? For example, when taking a photo of a distant object, the camera creates a triangle similar to that formed by the object and the observation point, allowing accurate distance calculations.

Development

Duration: (40 - 50 minutes)

The purpose of this stage is to deepen students' understanding of the similarity of triangles through detailed explanations of concepts, criteria, and properties, as well as to show practical applications. This ensures that students comprehend not only the theory but also how to apply this knowledge to real-world problems. Problem-solving in the classroom reinforces learning and allows students to practice the concepts covered.

Covered Topics

1. Definition of Similarity of Triangles: Explain that two triangles are similar when they have corresponding congruent angles and corresponding proportional sides. Use appropriate mathematical notation to represent this relationship. 2. Criteria for Similarity of Triangles: Present the three criteria for similarity of triangles: AA (Angle-Angle), LAL (Side-Angle-Side), and LLL (Side-Side-Side). Provide practical examples to illustrate each criterion. 3. Properties of Similar Triangles: Detail the properties that result from the similarity of triangles, such as the proportion between corresponding sides and the relationship between the areas of similar triangles. 4. Practical Applications: Demonstrate how the similarity of triangles can be used to solve real-world problems, such as measuring the height of a distant object using shadows or mirrors. Present practical problems and solve them step by step.

Classroom Questions

1. Given two similar triangles, where the sides of one are 3 cm, 4 cm, and 5 cm, and the hypotenuse of the other is 10 cm, calculate the other sides of the larger triangle. 2. In triangle ABC similar to triangle DEF, the sides AB and DE measure 6 cm and 9 cm, respectively. If side BC measures 8 cm, what is the length of side EF? 3. A pole casts a shadow of 12 meters while a 1.80-meter tall person casts a shadow of 2.4 meters. What is the height of the pole?

Questions Discussion

Duration: (20 - 25 minutes)

The purpose of this stage is to review and consolidate learning, allowing students to discuss and reflect on the problems solved. The detailed discussion of the questions helps identify potential doubts and reinforces the understanding of the concepts of similarity of triangles. Student engagement through questions and reflections promotes active and critical learning, encouraging them to think beyond the examples given in class.

Discussion

• ⚙️ Discussion of Question 1: Given two similar triangles, where the sides of one are 3 cm, 4 cm, and 5 cm, and the hypotenuse of the other is 10 cm, calculate the other sides of the larger triangle.

To solve this question, first identify the similarity ratio between the triangles. The hypotenuse of the smaller triangle is 5 cm and that of the larger is 10 cm, so the similarity ratio is 10/5 = 2. Multiply the other sides of the smaller triangle by this ratio to find the sides of the larger triangle: 3 cm * 2 = 6 cm and 4 cm * 2 = 8 cm. Therefore, the sides of the larger triangle are 6 cm and 8 cm.

• ⚙️ Discussion of Question 2: In triangle ABC similar to triangle DEF, the sides AB and DE measure 6 cm and 9 cm, respectively. If side BC measures 8 cm, what is the length of side EF?

To solve this question, identify the similarity ratio. The ratio is 9/6 = 1.5. Multiply side BC by this ratio to find the length of side EF: 8 cm * 1.5 = 12 cm. Therefore, side EF measures 12 cm.

• ⚙️ Discussion of Question 3: A pole casts a shadow of 12 meters while a 1.80-meter tall person casts a shadow of 2.4 meters. What is the height of the pole?

To solve this question, use the similarity ratio between the person's height and their shadow: 1.80 m / 2.4 m. This ratio should be the same for the height of the pole and its shadow. Let x be the height of the pole, then x / 12 m = 1.80 m / 2.4 m. Solve the proportion: x = (1.80 m / 2.4 m) * 12 m = 9 m. Therefore, the height of the pole is 9 meters.

Student Engagement

1.  Questions and Reflections: 2. Why is it important to ensure that the corresponding angles are congruent in similar triangles? 3. How can the similarity of triangles be applied in everyday situations beyond those discussed in class? 4. Can you think of other ways to verify the similarity of triangles besides the AA, LAL, and LLL criteria? 5. If the triangles were not similar, how would that affect the calculations we made? 6. What was the most challenging part of solving the questions about the similarity of triangles?

Conclusion

Duration: (10 - 15 minutes)

The purpose of this stage is to recap and reinforce the main content covered about the similarity of triangles, ensuring that students have a clear and cohesive view of what has been learned. The summary helps consolidate knowledge, while the connections to practice and the relevance of the topic highlight the importance of the content for students' lives.

Summary

• Definition of Similarity of Triangles: Two triangles are similar when they have corresponding congruent angles and proportional corresponding sides.
• Criteria for Similarity of Triangles: There are three main criteria: AA (Angle-Angle), LAL (Side-Angle-Side), and LLL (Side-Side-Side).
• Properties of Similar Triangles: The proportion between corresponding sides and the relationship between the areas of similar triangles.
• Practical Applications: Examples of use in real-world problems such as measuring the height of distant objects using shadows or mirrors.

The lesson connected the theory of similarity of triangles with practice by demonstrating how these concepts can be applied in solving real-world problems. Practical examples were used to show the importance of understanding the similarity of triangles in various everyday and professional situations, such as in engineering and photography.

Knowledge about the similarity of triangles is essential for various aspects of daily life, from map making to photography techniques. The ability to calculate proportions and understand geometric relationships allows for efficient and accurate problem-solving, as well as contributes to the development of students' critical and analytical thinking.