Constructions, Proof, and Rigid Motion

Students use the properties of circles to construct and understand different geometric figures, and lay the groundwork for constructing mathematical arguments through proof.

Unit Summary

In Unit 1, Constructions, Proof and Rigid Motion, students are introduced to the concept that figures can be created by just using a compass and straightedge using the properties of circles, and by doing so, properties of these figures are revealed. Transformations that preserve angle measure and distance are verified through constructions and practiced on and off the coordinate plane. These rigid motion transformations are introduced through points and line segments in this unit, and provide the foundation for rigid motion and congruence of two-dimensional figures in Unit 2. This unit lays the groundwork for constructing mathematical arguments through proof and use of precise mathematical vocabulary to express relationships.

Unit 1 begins with students identifying important components to define- emphasizing precision of language and notation as well as appropriate use of tools to represent geometric figures. Students are introduced to the concept of a construction, and use the properties of circles to construct basic geometric figures. In Topic B, students formalize understanding developed in middle school geometry of angles around a point, vertical angles, complementary angles, and supplementary angles through organizing statements and reasons for why relationships to construct a viable argument. Topic C merges the concepts of specificity of definitions, constructions, and proof to formalize rigid motions studied in 8th grade. Students learn that rigid motions can be used as a tool to show congruence. Students focus on rigid motions with points, line segments and angles in this unit through transformation both on and off the coordinate plane.

In the next unit, students use the concepts of constructions, proof, and rigid motions to establish congruence with two dimensional figures. Through the establishment of a solid foundation of precise vocabulary and developing arguments in Unit 1, students are able to use these strategies and theorems to identify and describe geometric relationships throughout the rest of the year.

Assessment

This assessment accompanies Unit 1 and should be given on the suggested assessment day or after completing the unit.

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Unit Prep

Intellectual Prep

Internalization of Standards via the Unit Assessment

Take unit assessment. Annotate for:
- Standards to which each question aligns
- Purpose of each question: spiral, foundational, mastery, developing
- Strategies and representations used in daily lessons
- Relationship to Essential Understandings of unit
- Lesson(s) to which the assessment points

Internalization of Trajectory of Unit

Read and annotate "Unit Summary."
Notice the progression of concepts through the unit using “Unit at a Glance”
Do all target tasks. Annotate the target tasks for:
- Essential understandings
- Connection to assessment questions

Unit-specific Intellectual Prep

Math Open ref
Exploring and Writing Geometry: Constructions
Dan Meyer's If Proof Is Aspirin, Then How Do You Create the Headache?
Proofs can be a real challenge for teachers and students. While this unit references two column proofs, it is more important that students understand the thinking that is necessary to develop a logical explanation of why a relationship exists. This document has quite a few helpful strategies to develop a logical explanation.

Essential Understandings

Every geometric definition, property, theorem, or conjecture exists because there was a question about whether a relationship exists and then a subsequent chain of reasoning based on previously known facts, or through geometric constructions, to convince us that this relationship exists. No theorem exists without proof.
Geometric facts are established through proof by determining a sequence of logical statements and reasons that drive toward a conclusion that is the theorem proved. Through this process, we can establish new facts.
Geometric constructions show properties of a figure through the construction of that figure. Constructions are a precise way to create geometric figures with the simple tools of a compass and a straightedge.
Rigid motions can be performed to map one figure to another in single or a composition of transformations. These transformations can be performed using constructions, patty paper, and on a coordinate plane using properties of perpendicular and parallel lines.

Vocabulary

"undefined" terms (point, line, plane)	ray	line segment	polygon	collinear
coplanar	orthocenter	proof	transitive property	supplementary
rigid motions	translation	reflection	congruent	vector
corresponding angles	alternate interior angles	auxiliary lines	regular	construction
equidistant	bisect	perpendicular bisector	altitude	circumcenter
identity	adjacent angles	complementary	angle/distance preservation	rotation
corresponding parts	transformation	converse theorems	same side interior angles	alternate exterior angles

Unit Materials, Representations and Tools

compass
straightedge

Lesson Map

Topic A: Constructions of Basic Geometric Figures

1

G.CO.A.1

Describe the precise definition and notation for foundational geometric figures.

2

G.CO.A.1

G.CO.D.12

G.CO.A.2

G.CO.B.6

Describe and identify rigid motions.

9

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.

10

G.CO.A.2

G.CO.A.4

G.CO.A.5

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

11

G.CO.A.1

G.CO.A.4

G.CO.C.9

G.CO.D.12

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

12

G.CO.A.1

G.CO.C.9

Prove angle relationships in parallel line diagrams.

13

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

19

G.CO.A.5

G.CO.B.6

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

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

Congruence

G.CO.A.1 — Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Congruence

G.CO.A.1 — Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
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.
G.CO.C.9 — Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

Congruence

G.CO.C.9 — Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
G.CO.C.10 — Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Congruence

G.CO.C.10 — Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
G.CO.D.12 — Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Congruence

G.CO.D.12 — Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
G.CO.D.13 — Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Congruence

G.CO.D.13 — Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Reasoning with Equations and Inequalities

A.REI.A.1 — Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Reasoning with Equations and Inequalities

A.REI.A.1 — Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Foundational Standards

Building Functions

F.BF.B.3

Building Functions

F.BF.B.3 — Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Expressions and Equations

8.EE.C.7

Expressions and Equations

8.EE.C.7 — Solve linear equations in one variable.

Geometry

6.G.A.1

Geometry

6.G.A.1 — Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
7.G.B.4

Geometry

7.G.B.4 — Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
7.G.B.5

Geometry

7.G.B.5 — Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
8.G.A.1

Geometry

8.G.A.1 — Verify experimentally the properties of rotations, reflections, and translations:
8.G.A.2

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.
8.G.A.3

Geometry

8.G.A.3 — Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
8.G.A.4

Geometry

8.G.A.4 — Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
8.G.A.5

Geometry

8.G.A.5 — Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

Future Standards

Congruence

G.CO.A.3

Congruence

G.CO.A.3 — Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
G.CO.B.7

Congruence

G.CO.B.7 — Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G.CO.B.8

Congruence

G.CO.B.8 — Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
G.CO.C.11

Congruence

G.CO.C.11 — Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

High School — Number and Quantity

N.VM.A.1

High School — Number and Quantity

N.VM.A.1 — Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
N.VM.A.2

High School — Number and Quantity

N.VM.A.2 — Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
N.VM.A.3

High School — Number and Quantity

N.VM.A.3 — Solve problems involving velocity and other quantities that can be represented by vectors.
N.VM.B.4

High School — Number and Quantity

N.VM.B.4 — Add and subtract vectors.

Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 — Attend to precision.
CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.

Constructions, Proof, and Rigid Motion

Unit Summary

Assessment

Unit Prep

Intellectual Prep

Essential Understandings

Vocabulary

Unit Materials, Representations and Tools

Lesson Map

Topic A: Constructions of Basic Geometric Figures

1

2

3

4

5

Topic B: Justification and Proof of Angle Measure

6

7

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

8

9

10

11

12

13

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

14

15

16

17

18

19

Common Core Standards

Core Standards

Congruence

Congruence

Congruence

Congruence

Congruence

Congruence

Congruence

Congruence

Congruence

Congruence

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

Foundational Standards

Building Functions

Building Functions

Expressions and Equations

Expressions and Equations

Geometry

Geometry

Geometry

Geometry

Geometry

Geometry

Geometry

Geometry

Geometry

Future Standards

Congruence

Congruence

Congruence

Congruence

Congruence

High School — Number and Quantity

High School — Number and Quantity

High School — Number and Quantity

High School — Number and Quantity

High School — Number and Quantity

Standards for Mathematical Practice

Unit 2

Congruence in Two Dimensions