# Transformations and Angle Relationships

Students investigate congruence and similarity by studying transformations of figures in the coordinate plane, and apply these transformations to discover new angle relationships.

Math

Unit 3

## Unit Summary

In Unit 3, 8th grade students bridge their understanding of geometry from middle school concepts to high school concepts. Until now, standards in the geometry domain have been either supporting or additional to the major standards. In 8th grade, geometry standards, especially the standards highlighted in this unit, are part of the major work of the grade and play a critical role in setting students up well for success in high school.

In this unit, students begin their work with transformations using patty paper (transparency paper) to experiment with, manipulate, and verify hypotheses around how shapes move under different transformations (MP.5). They use precision in their descriptions of transformations and in their justifications for why two figures may be similar or congruent to each other (MP.6). Students then apply their understanding of transformations to discover new angle relationships in parallel line diagrams and triangles.

Prior to 8th grade, students developed their understanding of geometric figures and learned how to draw them, calculate measurements, and model real-world situations. In 7th grade, students were introduced to the concept of scaling through scale drawings, and they solved for various measurements using proportional reasoning. Students will draw on these prior skills when they investigate dilations and similar triangles.

In high school geometry, students spend significant time studying congruence and similarity in-depth. They build off of the informal proofs and reasoning developed in 8th grade to hone their definitions of transformations, prove geometric theorems, and derive trigonometric ratios.

Pacing: 26 instructional days (22 lessons, 3 flex days, 1 assessment day)

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## Assessment

The following assessments accompany Unit 3.

### Pre-Unit

Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.

### Mid-Unit

Have students complete the Mid-Unit Assessment after lesson 10 .

### Post-Unit

Use the resources below to assess student understanding of the unit content and action plan for future units.

Expanded Assessment Package

Use student data to drive instruction with an expanded suite of assessments. Unlock Pre-Unit and Mid-Unit Assessments, and detailed Assessment Analysis Guides to help assess foundational skills, progress with unit content, and help inform your planning.

## Unit Prep

### Intellectual Prep

Unit Launch

Before you teach this unit, unpack the standards, big ideas, and connections to prior and future content through our guided intellectual preparation process. Each Unit Launch includes a series of short videos, targeted readings, and opportunities for action planning to ensure you're prepared to support every student.

#### Internalization of Standards via the Post-Unit Assessment

• Take the Post-Unit Assessment. Annotate for:
• Standards that each question aligns to
• Strategies and representations used in daily lessons
• Relationship to Essential Understandings of unit
• Lesson(s) that Assessment points to

#### Internalization of Trajectory of Unit

• Read and annotate the Unit Summary.
• Notice the progression of concepts through the unit using the Lesson Map.
• Essential Understandings
• Connection to Post-Unit Assessment questions
• Identify key opportunities to engage students in academic discourse. Read through our Teacher Tool on Academic Discourse and refer back to it throughout the unit.

### Essential Understandings

• Two figures are congruent to each other if there exists a sequence of rigid transformations that will map one figure onto the other.
• Two figures are similar to each other if there exists a sequence of dilations and rigid transformations that will map one figure onto the other.Â
• Certain properties are preserved under rigid transformations (such as angle measurement, line segment length, and parallel line relationships).
• Angle relationships exist in polygons, intersecting lines, and parallel lines that can be used to determine various angle measurements.

### Vocabulary

alternate interior and exterior angles

congruent/ congruence

corresponding angles

dilation

reflection

rigid transformation

rotation

scale factor

similar

translation

vertical angles

To see all the vocabulary for Unit 3 , view our 8th Grade Vocabulary Glossary.

### Materials

• 180Â° Protractor (1 per student)
• Optional: Calculators (1 per student)
• Optional: Graph Paper (2-3 sheets per student)
• Scissors (1 per small group)
• Optional: Ruler (1 per student)
• Optional: Tape (1 per small group)
• Optional: Patty paper (transparency paper)

To see all the materials needed for this course, view our 8th Grade Course Material Overview.

## Lesson Map

Topic A: Congruence and Rigid Transformations

Topic B: Similarity and Dilations

Topic C: Angle Relationships

## Common Core Standards

Key

Major Cluster

Supporting Cluster

### Core Standards

#### Geometry

• 8.G.A.1 — Verify experimentally the properties of rotations, reflections, and translations:
• 8.G.A.1.A — Lines are taken to lines, and line segments to line segments of the same length.
• 8.G.A.1.B — Angles are taken to angles of the same measure.
• 8.G.A.1.C — Parallel lines are taken to parallel lines.
• 8.G.A.2 — Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
• 8.G.A.3 — Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
• 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.
• 8.G.A.5 — Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

• 8.EE.C.7

• 7.G.A.1
• 7.G.A.2
• 7.G.B.5

• 4.MD.C.6

• 7.RP.A.2
• 7.RP.A.3

• G.CO.A.2
• G.CO.A.3
• G.CO.A.4
• G.CO.A.5
• G.CO.B.6
• G.CO.B.7
• G.CO.B.8
• G.CO.C.10
• G.CO.C.9

• G.SRT.A.1
• G.SRT.A.2
• G.SRT.A.3
• G.SRT.B.4
• G.SRT.B.5

### Standards for Mathematical Practice

• CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

• CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

• CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

• CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

• CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

• CCSS.MATH.PRACTICE.MP6 — Attend to precision.

• CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

• CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.

Unit 2

Solving One-Variable Equations

Unit 4

Functions

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