Lesson 16 | Transformations and Angle Relationships | 8th Grade Mathematics

Objective

Use properties of similar triangles to model and solve real-world problems.

Common Core Standards

Core Standards

The core standards covered in this lesson

8.G.A.4 — Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

Geometry

8.G.A.4 — Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

Foundational Standards

The foundational standards covered in this lesson

7.RP.A.2

Ratios and Proportional Relationships

7.RP.A.2 — Recognize and represent proportional relationships between quantities.
7.RP.A.3

Ratios and Proportional Relationships

7.RP.A.3 — Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Model real-world situations using similar triangles (MP.4).
Use ratio reasoning to find missing measurements in real-world applications of similar triangles.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Students have not yet covered the angle-angle criterion for similar triangles, but they do not need that for this lesson. Students can understand the similarity between the triangles using dilations and transformations. For example, in Anchor Problem #1, students can see how the small triangle can be reflected over the vertical line through point $$O$$ and then dilated from point $$O$$ to map to the larger triangle.

Lesson Materials

Optional: Calculators (1 per student)

Fishtank Plus

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

You want to determine the approximate height of one of the buildings in the city. You are told that if you place a mirror some distance from yourself so that you can see the top of the building in the mirror, then you can indirectly measure the height using similar triangles. Let point $$O$$ be the location of the mirror so that the person shown can see the top of the building.

If you know that the distance from eye-level straight down to the ground is 5.3 feet, the distance from the person to the mirror is 7.2 feet, and the distance from the mirror to the base of the building is 1,742.8 feet, then what is the approximate height of the building?

Guiding Questions

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References

EngageNY Mathematics Grade 8 Mathematics > Module 3 > Topic B > Lesson 12 — Teacher Version: Exercise 1

Grade 8 Mathematics > Module 3 > Topic B > Lesson 12 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

A geologist wants to determine the distance across the widest part of a nearby body of water. The geologist marked off specific points around the body of water so that the line containing $${\overline{{DE}}}$$ would be parallel to the line containing $${\overline{{BC}}}$$. The segment $${BC}$$ is selected specifically because it is the widest part of the body of water. The segment $${DE}$$ is selected specifically because it is a short enough distance to measure.

The geologist has made the following measurements: $$\left | {DE} \right |$$=5 feet, $${\left | AE \right |}$$= 7 feet, and $${\left | EC \right |}$$ = 15 feet. Can she determine the length across the widest part of the body of water?

Guiding Questions

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References

EngageNY Mathematics Grade 8 Mathematics > Module 3 > Topic B > Lesson 12 — Teacher Version: Exercise 2

Modified by Fishtank Learning, Inc.

Problem Set

A set of suggested resources or problem types that teachers can turn into a problem set

Fishtank Plus Content

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Lily and Ravi are visiting the Redwood Forest in California. They want to measure the height of one of the trees, but they can’t climb it. However, they are able to measure the shadow of the tree. They determine the shadow of the tree is $$583{1\over3}$$ feet long.

If Ravi is 6 feet tall and they determine his shadow is 10 feet long, find the height of the redwood tree.

Student Response

An example response to the Target Task at the level of detail expected of the students.

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Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Spiral in review problems from earlier in the unit.
Challenge: Write your own real-world problem, ensuring it has all the information needed to solve. Swap with a peer and solve your peer's problem.

EngageNY Mathematics Grade 8 Mathematics > Module 3 > Topic B > Lesson 12 — Teacher Version: Exercise #3 and Problem Set

Lesson 15

Lesson 17

Lesson Map

Topic A: Congruence and Rigid Transformations

Understand the rigid transformations that move figures in the plane (translation, reflection, rotation).