Unit 1: 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.
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.
Pacing: 21 instructional days (19 lessons, 1 flex day, 1 assessment day)
This assessment accompanies Unit 1 and should be
given on the suggested assessment day or after completing the
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The central mathematical concepts that students will come to understand in this unit
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Topic A: Constructions of Basic Geometric Figures
Describe the precise definition and notation for foundational geometric figures.
Construct an equilateral triangle with only a straight-edge and a compass. Copy a line segment.
Construct, bisect, and copy an angle.
Construct a perpendicular bisector.
Construct the altitudes and perpendicular bisectors of sides of triangles.
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Topic B: Justification and Proof of Angle Measure
Use principles of proof to justify each step in solving an equation.
Use angle relationships around a point to find missing measures. Prove angle relationships around a point using geometric statements and reasons.
Topic C: Translations of Points, Line Segments, and Angles, and Parallel Line Relationships
Describe and identify rigid motions.
Describe rigid motions. Use algebraic rules to translate points and line segments and describe translations on the coordinate plane.
Translate points and line segments not on the coordinate plane using constructions. Describe properties of translations with respect to line segments and angles.
Construct parallel lines. Prove the relationship between corresponding angles. Use this relationship to find missing measures directly and algebraically.
Prove angle relationships in parallel line diagrams.
Construct auxiliary parallel lines and use these in the development of proofs and identification of missing measures.
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.
Use construction and patty paper to reflect a figure not on the coordinate plane. Describe the properties of a reflection.
Perform rotations on a coordinate plane. Identify characteristics and algebraic rules for the rotation.
Use construction and patty paper to rotate a figure not on the coordinate plane. Describe the properties of a rotation.
Describe a sequence of rigid motions that will map a point, line segment, or angle onto another figure.
Describe rigid motions, or sequences of rigid motions that have the same effect on a figure.
The content standards covered in this unit
— 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.
— 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).
— Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
— 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.
— 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.
— 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.
— 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.
— 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.
— Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
— 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.
Standards covered in previous units or grades that are important background for the current unit
— 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.
— Solve linear equations in one variable.
— 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.
— 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.
— 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.
— Verify experimentally the properties of rotations, reflections, and translations:
— 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.
— Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
— 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.
— 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.
Standards in future grades or units that connect to the content in this unit
— Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
— 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.
— Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
— 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.
— 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).
— Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
— Solve problems involving velocity and other quantities that can be represented by vectors.
— Add and subtract vectors.
— Make sense of problems and persevere in solving them.
— Reason abstractly and quantitatively.
— Construct viable arguments and critique the reasoning of others.
— Model with mathematics.
— Use appropriate tools strategically.
— Attend to precision.
— Look for and make use of structure.
— Look for and express regularity in repeated reasoning.
Congruence in Two Dimensions