Lesson 14 | Constructions, Proof, and Rigid Motion | 10th Grade Mathematics

Objective

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

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.4 — Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Congruence

G.CO.A.4 — Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
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

Describe that rigid motions describe ways you can move a figure either on or off a coordinate plane without changing size, shape, angles, or relationship between any of the parts.
Describe reflections as a rigid motion of individual points across a line of reflection. To define a reflection, all that is needed is a line of reflection.
Describe that a line of reflection can be in any orientation (horizontal, vertical, or diagonal) and that it can be on a figure, outside a figure, intersect with a figure, or be inside a geometric figure.
Perform a reflection on a coordinate plane by reflecting points over any given line (not just an axis or y=x).
Use an algebraic rule to show the reflection of a figure over an axis or the line y=x. Describe where the general rule is derived from.
Understand that there are an infinite number of fixed points with a reflection, but all fixed points are on the line of reflection.
Describe the relationship between the distance of each point on the original figure and the reflected figure to the line of reflection.

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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

Reflect $${\overline{AC}}$$ over the $$x$$-axis
Reflect $${\overline{A'C'}}$$ over the $$y$$-axis

Guiding Questions

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Problem 2

Reflect $${\angle DAC}$$ over the line $${y=x}$$.

Guiding Questions

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Problem 3

Given the set of points below, what is the line of reflection for the image?

Image: $${E' (-2,1)}$$, $${F' (-3,3)}$$

Pre-image: $${E (1,-2)}$$, $${F (3,-3)}$$

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

Line segment $${\overline{DF}}$$ is shown below. Reflect this over the x-axis.

What is the line of reflection?
Describe the reflection.
What features are “preserved” in a reflection?
What is the algebraic rule for the reflection?

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:
- Find the line of reflection, citing half the distance between corresponding points on the two line segments.
- Reflect an angle given the line of reflection.
- Identify an incorrect reflection.
- Describe a reflection with algebraic notation (if an axis or y=x /y=-x line).
- Reflect a line segment given the line of reflection.
- Translate angles and segments as review.
Include matching cards with lines of reflection, original figures, and final figures. This Mathematics Assessment Project lesson will be used in the next unit, but can be used as a reference for style to create matching cards for this lesson.

CPALMS Angle Transformations — Only ask questions about translations and reflections, not rotations. This resource does a good job of focusing on the distance preservation of rigid motions.
CPALMS Segment Transformations — Only ask questions about translations and reflections, not rotations. This resource does a good job of focusing on the distance and angle preservation of rigid motions.

Lesson 13

Lesson 15

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

Congruence

Foundational Standards

Geometry

Geometry

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Target Task

Additional Practice

Lesson Map

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