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.
Unit Summary
In Unit 3, eighth-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 eighth 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 eighth grade, students developed their understanding of geometric figures and learned how to draw them, calculate measurements, and model real-world situations. In seventh 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 eighth 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)
For guidance on adjusting the pacing for the 2020-2021 school year due to school closures, see our 8th Grade Scope and Sequence Recommended Adjustments.
Assessment
This assessment accompanies Unit 3 and should be given on the suggested assessment day or after completing the unit.
Unit Prep
Intellectual Prep
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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.
- Do all Target Tasks. Annotate the Target Tasks for:
- 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.
Unit-Specific Intellectual Prep
- Read the UnboundEd Mathematics Guide Geometry: Unbound A Guide to Grade 8 Mathematics Standards.
- Read the Progressions for the Common Core State Standards in Mathematics, Geometry, 7-8, High School for standards relevant to this unit.
Essential Understandings
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- 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.
- 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.
Materials
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- GeoGebra (for teachers to make diagrams)
- Patty paper (transparency paper)
- Protractor
- Ruler
- Calculators
- Graph paper
Vocabulary
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alternate interior and exterior angles
corresponding angles
reflection
similar
translation
rigid transformation
rotation
congruent/ congruence
dilation
scale factor
vertical angles
Related Teacher Tools:
Lesson Map
Topic A: Congruence and Rigid Transformations
1
8.G.A.1.A
8.G.A.1.B
8.G.A.1.C
8.G.A.2
Understand the rigid transformations that move figures in the plane (translation, reflection, rotation).
2
8.G.A.1.A
8.G.A.1.B
8.G.A.1.C
8.G.A.2
Describe and perform translations between congruent figures. Use translations to determine if figures are congruent.
3
8.G.A.1.A
8.G.A.1.B
8.G.A.1.C
8.G.A.2
8.G.A.3
Describe and apply properties of translations. Use coordinate points to represent relationships between translated figures.
4
8.G.A.1.A
8.G.A.1.B
8.G.A.1.C
8.G.A.2
Describe and perform reflections between congruent figures. Use reflections to determine if figures are congruent.
5
8.G.A.1.A
8.G.A.1.B
8.G.A.1.C
8.G.A.2
8.G.A.3
Describe sequences of transformations between figures using reflections and translations. Use coordinate points to represent relationships between reflected figures.
6
8.G.A.1.A
8.G.A.1.B
8.G.A.1.C
8.G.A.2
Describe and perform rotations between congruent figures.
7
8.G.A.1.A
8.G.A.1.B
8.G.A.1.C
8.G.A.2
Describe sequences of transformations between figures using rotations and other transformations.
8
8.G.A.1.A
8.G.A.1.B
8.G.A.1.C
8.G.A.2
Describe a sequence of rigid transformations that will map one figure onto another.
9
8.G.A.2
8.G.A.3
Describe multiple rigid transformations using coordinate points.
10
8.G.A.1.A
8.G.A.1.B
8.G.A.1.C
8.G.A.2
8.G.A.3
Review rigid transformations and congruence between two figures.
Topic B: Similarity and Dilations
11
8.G.A.4
Define a dilation as a non-rigid transformation, and understand the impact of scale factor.
12
8.G.A.4
Describe and perform dilations.
13
8.G.A.3
8.G.A.4
Describe a sequence of dilations and rigid motions between two figures. Use coordinate points to represent relationships between similar figures.
14
8.G.A.2
8.G.A.4
Determine and informally prove or disprove if two figures are similar or congruent using transformations.
15
8.G.A.4
Find missing side lengths in similar figures. Find scale factor between similar figures.
16
8.G.A.4
Use properties of similar triangles to model and solve real-world problems.
Topic C: Angle Relationships
17
8.G.A.2
8.G.A.5
Define and identify corresponding angles in parallel line diagrams. Review vertical, supplementary, and complementary angle relationships.
18
8.G.A.2
8.G.A.5
Define and identify alternate interior and alternate exterior angles in parallel line diagrams. Find missing angles in parallel line diagrams.
19
8.G.A.5
Solve for missing angle measures in parallel line diagrams using equations.
20
8.G.A.5
Define and use the interior angle sum theorem for triangles.
21
8.G.A.5
Define and use the exterior angle theorem for triangles.
22
8.G.A.5
Define and use the angle-angle criterion for similar triangles.
Common Core Standards
Key: Major Cluster Supporting Cluster Additional Cluster
Core Standards
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Geometry
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8.G.A.1 — Verify experimentally the properties of rotations, reflections, and translations:
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8.G.A.1.A — Lines are taken to lines, and line segments to line segments of the same length.
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8.G.A.1.B — Angles are taken to angles of the same measure.
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8.G.A.1.C — Parallel lines are taken to parallel lines.
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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.
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8.G.A.3 — Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
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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.
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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.
Foundational Standards
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Expressions and Equations
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8.EE.C.7
Geometry
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7.G.A.1
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7.G.A.2
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7.G.B.5
Measurement and Data
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4.MD.C.6
Ratios and Proportional Relationships
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7.RP.A.2
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7.RP.A.3
Future Standards
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Congruence
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G.CO.A.2
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G.CO.A.3
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G.CO.A.4
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G.CO.A.5
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G.CO.B.6
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G.CO.B.7
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G.CO.B.8
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G.CO.C.10
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G.CO.C.9
Similarity, Right Triangles, and Trigonometry
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G.SRT.A.1
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G.SRT.A.2
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G.SRT.A.3
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G.SRT.B.4
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G.SRT.B.5
Standards for Mathematical Practice
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CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
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CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
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CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
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CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
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CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
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CCSS.MATH.PRACTICE.MP6 — Attend to precision.
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CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
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CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.