Transformations and Angle Relationships

Lesson 6

Unit 3: Transformations and Angle Relationships

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.

Objective

Describe and perform rotations between congruent figures.

Common Core Standards

Core Standards

8.G.A.1.A — Lines are taken to lines, and line segments to line segments of the same length.

Geometry

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.

Geometry

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.

Geometry

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.

Geometry

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.

Foundational Standards

4.MD.C.6

Measurement and Data

4.MD.C.6 — Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

Criteria for Success

Describe rotations with a direction, an angle of rotation, and a point to rotate around.
Use patty (transparency) paper to rotate images.
Understand that in a rotation, every point on an original figure moves the same number of degrees and in the same direction around the same point to create the new figure.
Understand that rotations preserve shape, size, orientation, area, perimeter, and parallel line relationships, but they do not preserve location.

Tips for Teachers

This lesson focuses on rotations in isolation—how to do them and how to talk about them. Upcoming lessons will begin to combine rotations with the other rigid transformations.
Similar to the video from Lesson 4, this video, Rotation, by PBS Learning Media introduces the concept of rotation and includes animation that can be helpful for students to visualize the movements, especially the importance of the point of rotation
The following materials are useful for this lesson: patty paper (or transparency paper) and graph paper.

Remote Learning Guidance

If you need to adapt or shorten this lesson for remote learning, we suggest prioritizing Anchor Problem 1 (benefits from discussion) and Anchor Problem 2 (benefits from worked example). Find more guidance on adapting our math curriculum for remote learning here.

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Problem Set
Student Handout Editor
Google Classrom Integration
Vocabulary Package

Anchor Problems

Problem 1

Triangle $${{LMN}}$$ underwent a single transformation to become triangle $${{PQR}}$$, shown below.

What single transformation maps triangle $${{LMN}}$$ to triangle $${{PQR}}$$? Describe in detail.
Name two things that are the same about both triangles.
Name two things that are different about the triangles.

Guiding Questions

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Problem 2

Perform the following rotations on figure $${{ABCD}}$$ below.

Rotate figure $${{ABCD}}$$ $${90^{\circ}}$$ counter-clockwise around point $$A$$ to create $$A'B'C'D'$$.
Rotate figure $${{ABCD}}$$ $${180^{\circ}}$$ counter-clockwise around point $$A$$ to create $$A''B''C''D''$$.

Guiding Questions

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Problem Set

With Fishtank Plus, you can download a complete problem set and answer key for this lesson. Download Sample

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

Examples where students identify and describe a rotation between two figures (limit to around the origin and around a point on the figure only), similar to Anchor Problem #1
Examples where students perform rotations around either the origin or a point on the figure, similar to Anchor Problem #2
Examples where students identify new coordinate points of a figure after it is rotated
Error analysis of figures that appear to be rotated but the orientation is shifted

Kuta Software Free Geometry Worksheets Transformations - Rotations — strongly suggest having students use patty paper to experiment and verify rotations

EngageNY Mathematics Grade 8 Mathematics > Module 2 > Topic B > Lesson 10 — Teacher Version: Classwork Exercises #1-5, Problem Set #3, skip #1-2

Target Task

Triangle $${QRS}$$ was transformed to create triangle $${Q'R'S'}$$ in the coordinate plane below.

Describe the transformation that maps $${QRS}$$ to $${Q'R'S'}$$.
Rotate figure $${Q'R'S'}$$ $${90^{\circ}}$$ clockwise around the origin and draw the new figure $${{Q''R''S''}}$$. Describe a single transformation that maps triangle $${QRS}$$ to $${{Q''R''S''}}$$.

Mastery Response

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Transformations and Angle Relationships

Lesson 6

Unit 3: Transformations and Angle Relationships

Topic A: Congruence and Rigid Transformations

1

2

3

4

5

6

7

8

9

10

Topic B: Similarity and Dilations

11

12

13

14

15

16

Topic C: Angle Relationships

17

18

19

20

21

22

Objective

Common Core Standards

Core Standards

Geometry

Geometry

Geometry

Geometry

Foundational Standards

Measurement and Data

Criteria for Success

Tips for Teachers

Remote Learning Guidance

Fishtank Plus

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem Set

Target Task

Mastery Response

lesson 5

lesson 7