Exploring Indirect Measurement Using Similar Triangles

Objective:

Students will be able to apply the concept of similar triangles to measure lengths indirectly and use proportions to find missing measurements.

Assessment:

Students will be given a scenario where they have to measure an object indirectly using similar triangles. They will need to set up proportions and solve for the missing measurement.

Key Points:

• Understanding the concept of similar triangles

• Applying proportions to find missing measurements indirectly

• Identifying and measuring lengths using indirect measurement techniques

• An application of similar triangles is to measure lengths indirectly.

• The idea is that you model a situation with similar triangles and then use proportions to find the missing measurement indirectly.

Definitions:

• Similar Triangles: Triangles are similar if they have the same shape but not necessarily the same size. Corresponding angles are congruent, and corresponding sides are in proportion.

Formulas:

• Proportion Formula: If two sets of corresponding sides in similar triangles are in proportion, then the ratios of the corresponding sides are equal. This can be written as: $\frac{a}{A} = \frac{b}{B} = \frac{c}{C}$, where lowercase letters represent the sides of one triangle and uppercase letters represent the sides of the other triangle.

Opening:

• Engage students by presenting a scenario involving two similar triangles and a missing measurement.

• Ask students how they would go about finding the missing measurement without directly measuring it.

Introduction to New Material:

• Define similar triangles and discuss the properties that make them similar.

• Explain how similar triangles can be used to measure lengths indirectly.

• Common Misconception: Students may incorrectly assume that the triangles need to be the same size to be similar.

Guided Practice:

• Provide examples of situations where indirect measurement is necessary.

• Scaffold questioning from easy to hard to guide students in setting up and solving proportions.

• Monitor student performance by circulating the room and providing support as needed.

Independent Practice:

• Assign a worksheet where students have to use similar triangles and proportions to find missing measurements in different scenarios.

• Set expectations for independent work time and encourage students to show all their work clearly.

Closing:

• Have students share their findings from the independent practice assignments.

• Summarise the key concept of using similar triangles for indirect measurement.

Extension Activity:

• For early finishers, provide a more complex scenario involving multiple sets of similar triangles and multiple missing measurements to solve.

Homework:

• Homework suggestion: Students can create their own scenario where they need to use indirect measurement through similar triangles to find a missing length.

Standards Addressed:

• CCSS.MATH.CONTENT.HSG.SRT.A.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

• CCSS.MATH.CONTENT.HSG.SRT.A.3: Prove theorems about triangles.