Lesson 17 | Constructions, Proof, and Rigid Motion | 10th Grade Mathematics

Objective

Use construction and patty paper to rotate a figure not on the coordinate plane. Describe the properties of a rotation.

Common Core Standards

Core Standards

The core standards covered in this lesson

G.CO.A.2 — Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

Congruence

G.CO.A.2 — Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
G.CO.A.5 — Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Congruence

G.CO.A.5 — Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
G.CO.B.6 — Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Congruence

G.CO.B.6 — Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Foundational Standards

The foundational standards covered in this lesson

8.G.A.1

Geometry

8.G.A.1 — Verify experimentally the properties of rotations, reflections, and translations:
8.G.A.3

Geometry

8.G.A.3 — Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Criteria for Success

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

Verify that a rotation carries segments onto segments of equal length.
Verify that a rotation carries angles onto angles of equal measure.
Use congruence notation for angles and line segments to note congruent parts.
Describe that all corresponding parts have the same distance to the point of rotation and same angle measure to the point of rotation.
Rotate a figure on a coordinate plane given a direction, degree, and center of rotation. Use centers both at the origin and on the figure.
Locate the center of rotation by determining the perpendicular bisector between corresponding points on the pre-image and the image.
Locate the center of rotation by using corresponding points as points on the rays of an angle, with the vertex of an angle at the rotation point.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Criteria for Success #6 and #7 are beyond the scope of the standards. Students should be exposed to these concepts, but they should not be required for mastery of the topic.

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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

Below is a rotation of $${\overline {AB}}$$ to form $${\overline{A'B'}}$$.

What construction steps would you use to transform $${\overline{AB} }$$ to form $${\overline {A'B'}}$$?

Guiding Questions

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Problem 2

Transform the figure below according to the rule shown.

$${R_{C,-45˚}(\angle{ABD})}$$

Guiding Questions

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Problem 3

Find the center and degree of rotation for the figure below.

Guiding Questions

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Target Task

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

Rotate the following line segment counterclockwise by 45° about center A.

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.

Include problems where students need to rotate both line segments and points.
Include error analysis problems where the degree or the direction of rotation is incorrect, and students need to explain why.
Incorporate an introduction to sequences of transformations using a problem that shows two students with two opinions about the transformation to map one figure onto another. Describe who is correct and why.
Include review of other transformations both on and off the coordinate plane.
Include problems where students need to transform a figure by bisecting angles. I.e.: Given a rotation of 60 degrees, show the rotation of 30 degrees.

EngageNY Mathematics Geometry > Module 1 > Topic C > Lesson 13 — Exit Ticket and Problem Set

Lesson 16

Lesson 18

Lesson Map

Topic A: Constructions of Basic Geometric Figures

Describe the precise definition and notation for foundational geometric figures.

G.CO.A.1

Construct an equilateral triangle with only a straight-edge and a compass. Copy a line segment.

G.CO.A.1 G.CO.D.12 G.CO.D.13

Construct, bisect, and copy an angle.

G.CO.A.1 G.CO.D.12

Construct a perpendicular bisector.

G.CO.A.1 G.CO.D.12

Construct the altitudes and perpendicular bisectors of sides of triangles.

G.CO.C.10 G.CO.D.12

Topic B: Justification and Proof of Angle Measure

Use principles of proof to justify each step in solving an equation.

A.REI.A.1

Use angle relationships around a point to find missing measures. Prove angle relationships around a point using geometric statements and reasons.

G.CO.C.9

Topic C: Translations of Points, Line Segments, and Angles, and Parallel Line Relationships

Describe and identify rigid motions.

G.CO.A.1 G.CO.A.2 G.CO.B.6

Describe rigid motions. Use algebraic rules to translate points and line segments and describe translations on the coordinate plane.

G.CO.A.2 G.CO.B.6

Translate points and line segments not on the coordinate plane using constructions. Describe properties of translations with respect to line segments and angles.

G.CO.A.2 G.CO.A.4 G.CO.A.5

Construct parallel lines. Prove the relationship between corresponding angles. Use this relationship to find missing measures directly and algebraically.

G.CO.A.1 G.CO.A.4 G.CO.C.9 G.CO.D.12

Prove angle relationships in parallel line diagrams.

G.CO.A.1 G.CO.C.9

Construct auxiliary parallel lines and use these in the development of proofs and identification of missing measures.

G.CO.C.9 G.CO.D.12

Topic D: Reflections and Rotations of Points, Line Segments, and Angles

Perform reflections on a coordinate plane across axes and other defined lines. Identify characteristics and an algebraic rule for the reflection.

G.CO.A.2 G.CO.A.4 G.CO.A.5 G.CO.B.6

Use construction and patty paper to reflect a figure not on the coordinate plane. Describe the properties of a reflection.

G.CO.A.2 G.CO.A.4 G.CO.A.5 G.CO.B.6

Perform rotations on a coordinate plane. Identify characteristics and algebraic rules for the rotation.

G.CO.A.2 G.CO.A.4 G.CO.A.5 G.CO.B.6

Use construction and patty paper to rotate a figure not on the coordinate plane. Describe the properties of a rotation.

G.CO.A.2 G.CO.A.5 G.CO.B.6

Describe a sequence of rigid motions that will map a point, line segment, or angle onto another figure.

G.CO.A.5 G.CO.B.6

Describe rigid motions, or sequences of rigid motions that have the same effect on a figure.

G.CO.A.5 G.CO.B.6

Constructions, Proof, and Rigid Motion

Objective

Common Core Standards

Core Standards

Congruence

Congruence

Congruence

Foundational Standards

Geometry

Geometry

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Target Task

Additional Practice

Lesson Map

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