Students use the properties of circles to construct and understand different geometric figures, and lay the groundwork for constructing mathematical arguments through proof.
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.
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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 |
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Internalization of Standards via the Unit Assessment
Internalization of Trajectory of Unit
Unit-specific Intellectual Prep
This assessment accompanies Unit 1 and should be given on the suggested assessment day or after completing the unit.
G.CO.A.1
Describe the precise definition and notation for foundational geometric figures.
G.CO.A.1
G.CO.D.12
G.CO.D.13
Construct an equilateral triangle with only a straight-edge and a compass. Copy a line segment.
G.CO.A.1
G.CO.D.12
Construct, bisect, and copy an angle.
G.CO.A.1
G.CO.D.12
Construct a perpendicular bisector.
G.CO.C.10
G.CO.D.12
Construct the altitudes and perpendicular bisectors of sides of triangles.
A.REI.A.1
Use principles of proof to justify each step in solving an equation.
G.CO.C.9
Use angle relationships around a point to find missing measures. Prove angle relationships around a point using geometric statements and reasons.
G.CO.A.1
G.CO.A.2
G.CO.B.6
Describe and identify rigid motions.
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.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.
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.
G.CO.A.1
G.CO.C.9
Prove angle relationships in parallel line diagrams.
G.CO.C.9
G.CO.D.12
Construct auxiliary parallel lines and use these in the development of proofs and identification of missing measures.
G.CO.A.2
G.CO.A.4
G.CO.A.5
G.CO.B.6
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.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.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.
Key: Major Cluster Supporting Cluster Additional Cluster
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