Students investigate congruence and similarity by studying transformations of figures in the coordinate plane, and apply these transformations to discover new angle relationships.
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)
This assessment accompanies Unit 3 and should be given on the suggested assessment day or after completing the unit.
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similar
translation
reflection
rigid transformation
rotation
congruent/ congruence
dilation
scale factor
corresponding angles
alternate interior and exterior angles
vertical angles
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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).
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.
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.
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.
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.
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.
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.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.
8.G.A.2
8.G.A.3
Describe multiple rigid transformations using coordinate points.
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.
8.G.A.4
Define a dilation as a non-rigid transformation, and understand the impact of scale factor.
8.G.A.4
Describe and perform dilations.
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.
8.G.A.2
8.G.A.4
Determine and informally prove or disprove if two figures are similar or congruent using transformations.
8.G.A.4
Find missing side lengths in similar figures. Find scale factor between similar figures.
8.G.A.4
Use properties of similar triangles to model and solve real-world problems.
8.G.A.2
8.G.A.5
Define and identify corresponding angles in parallel line diagrams. Review vertical, supplementary, and complementary angle relationships.
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.
8.G.A.5
Solve for missing angle measures in parallel line diagrams using equations.
8.G.A.5
Define and use the interior angle sum theorem for triangles.
8.G.A.5
Define and use the exterior angle theorem for triangles.
8.G.A.5
Define and use the angle-angle criterion for similar triangles.
Key: Major Cluster Supporting Cluster Additional Cluster
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