Lesson 15 | Constructions, Proof, and Rigid Motion | 10th Grade Mathematics

Objective

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

Verify that a reflection carries segments onto segments of equal length.
Verify that a reflection carries angles onto angles of equal measure.
Use congruence notation for angles and line segments to note congruent parts.
Discover that each corresponding point on the image and pre-image is equidistant to each point on the line of reflection.
Identify the line of reflection is the perpendicular bisector of the line segment formed by corresponding points of the pre-image and image.
Reflect a figure over a line of reflection by using the line of reflection as the perpendicular bisector between corresponding points.
Locate the line of reflection by determining the perpendicular bisector between corresponding points on the pre-image and the image.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

The last criteria for success, “locate the line of reflection…” is a little beyond the scope of the standards. This is a good extension for the students to find the line using constructions, but should not be taught to mastery.

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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

Watch this video (without sound). What properties of reflections are being used to reflect each point?

Guiding Questions

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Problem 2

Refelect $${\angle ABC }$$ over $${\overleftrightarrow {DE}}$$ using constructions.

Guiding Questions

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Problem 3

Find the line of reflection between the two figures.

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

Reflect angle $${\angle FGH}$$ over $${\overleftrightarrow{JK}}$$.

Describe the relationships between the line segments and angle of the image and the pre-image.
Describe the relationship between the distance of corresponding points to the line of reflection.
Describe the relationship between the line of reflection and line segments formed by corresponding points.

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 the line of reflection goes through the figure, is on the figure, and is outside the figure. Ask comparison questions between each of these configurations of reflections.
Include problems where students are asked to sketch a reflection (in effect, estimating the reflection), and then construct it to verify the sketch.

EngageNY Mathematics Geometry > Module 1 > Topic C > Lesson 14 — Problem Set
Illustrative Mathematics Defining Reflections — This problem asks students to think about the specificity of their language when defining rigid motions. The same kind of problem can be applied to translations.

Lesson 14

Lesson 16

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

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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