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.

Unit Prep

Essential Understandings

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  • 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

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"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

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  • compass
  • straightedge

Intellectual Prep

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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.

Lesson Map

Topic A: Constructions of Basic Geometric Figures

Topic B: Justification and Proof of Angle Measure

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

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

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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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).

  • 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.

  • 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.

  • 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.

  • 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.

Foundational Standards

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Building Functions
  • F.BF.B.3

Expressions and Equations
  • 8.EE.C.7

Geometry
  • 6.G.A.1

  • 7.G.B.4

  • 7.G.B.5

  • 8.G.A.1

  • 8.G.A.2

  • 8.G.A.3

  • 8.G.A.4

  • 8.G.A.5

Future Standards

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Congruence
  • G.CO.A.3

  • G.CO.B.7

  • G.CO.B.8

  • G.CO.C.11

High School — Number and Quantity
  • N.VM.A.1

  • N.VM.A.2

  • N.VM.A.3

  • N.VM.B.4

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.