Units

Unit 1: Coordinate Geometry and Transformations

During this unit, students investigate the undefined terms point, line, and plane in a two-dimensional coordinate system in Euclidean and spherical geometries. Using coordinate points, students derive the distance formula and apply the distance formula to determine lengths and congruence of line segments and fractional distances less than one from one end of a line segment to the other. Coordinate points are also used to derive and apply the midpoint formula and slope formula. Slope is applied to define and investigate parallel and perpendicular lines, including comparison of parallel lines in Euclidean and spherical geometry. Students algebraically determine the equation of a line when given a point on the line and a line parallel or perpendicular to the line. In addition, students build upon their knowledge of coordinate geometry to analyze the critical attributes of transformations, including translations, reflections, rotations with points of rotation other than the origin, and dilations where the center of dilation can be any point on the coordinate plane. Students examine patterns to generalize rigid transformations (translations, reflections, and rotations) in the coordinate plane. Students also explore non-rigid transformations or dilations in the coordinate plane using scale factors. They compare and contrast dilations to other geometric transformations and examine relationships in terms of similarity. Students perform rigid transformations, non-rigid transformations, and composite transformations using coordinate notation. Students identify the sequence of transformations performed for a given pre-image or image on or off a coordinate plane. Reflection symmetry and rotational symmetry in plane figures are identified and differentiated.

Vocabulary

Composition of Transformations – combination of two or more transformations
Congruent segments – line segments whose lengths are equal
Degree of Rotation – number of degrees a figure is rotated about the point (center) of rotation
Dilation – a similarity transformation in which a figure is enlarged or reduced using a scale factor and a center of dilation
Euclidean Geometry – the study of plane and solid geometry based on definitions, undefined terms (point, line, plane) and the assumptions of mathematician Euclid
Horizontal Reflectional (Line) Symmetry – reflectional symmetry about a horizontal line of reflection
Image – figure after a transformation
Line of Symmetry – line dividing an image into two congruent parts that are mirror images of each other
Midpoint of a Line Segment – the point halfway between the endpoints of a line segment
Non-Rigid Transformation (Non-isometric) – a transformation that preserves similarity and reductions and enlargements that do not preserve similarity
Parallel Lines – lines in the same plane that never intersect. Slopes of parallel lines are equal, m_y₂ = m_y₁.
Perpendicular Lines – lines in the same plane that intersect at 90 angles whose slopes are opposite reciprocals,
Point of Rotation – point around which a figure is rotated (center of rotation)
Point Symmetry – 180^o rotation around a point
Pre-Image – original figure prior to a transformation
Reflection – rigid transformation in which each point in a geometric figure is at an equal distance on the opposite side of a given line (line of symmetry)
Reflectional Symmetry – symmetry in which one half of the image is a mirror image of the other over a line of reflection
Rigid Transformation (Isometric) – a transformation that preserves the size and shape of a figure
Rotation – a rigid transformation where each point on the figure is rotated about a given point
Rotational Symmetry – symmetry that occurs if a figure can be rotated less than 360° around a central point and still look the same as the original. The number of times a figure fits into itself in one complete rotation is called the order of rotational symmetry.
Spherical Geometry – the study of figures on the two-dimensional curved surface of a sphere
Symmetric Points – two points in a plane such that the line segment joining the points is bisected by a point or center
Transformation – one to one mapping of points in a plane such that each point in the pre-image has a unique image and each point in the image has a pre-image
Translation – rigid transformation moving all points in a geometric figure the same distance and the same direction
Two-Dimensional Coordinate System – two axes at right angles to each other, forming a xy-plane, consisting of points (x, y)
Vertical Reflectional (Line) Symmetry – reflectional symmetry about a vertical line of reflection

Unit 2: Logic and Euclidean Geometry

During this unit, students lay the foundation for geometry by developing an understanding of the structure of a geometric system through examination of the relationship between undefined terms (point, line, and plane), definitions, conjectures, and postulates. Students examine one-dimensional distance relationships in line segments, including fractional distances and midpoints, and make connections to the number line and segment addition. They also examine relationships in rays and angles making connections to the angle measure and angle addition. Constructions are used to explore and make conjectures about congruent geometric relationships in line segments and angles. They connect their understanding of definitions and postulates of lines, angles, and other geometric vocabulary to the context of the real world. Students also investigate logic statements and the conditions that make them true or false. Students explore conditional statements and their related statements (converse, inverse, and contrapositive) in both a real world and mathematical setting to develop an understanding of logic and the role it plays in geometry and the real world. Students are expected to recognize the connection between a biconditional statement and a true conditional statement with a true converse. Deductive reasoning and inductive reasoning are introduced and applied to make conjectures. Students verify that a conjecture is false using counterexamples.

Vocabulary

Biconditional Statement – statement for which both the conditional statement and its converse are true
Conditional Statement – statement composed of a hypothesis (if) and a conclusion (then)
Congruent Angles – angles whose angle measurement is equal
Congruent Segments – line segments whose lengths are equal
Conjecture – statement believed to be true but not yet proven
Contrapositive of a Conditional Statement – statement in which the hypothesis and conclusion of the original conditional statement are interchanged and negated
Converse of a Conditional Statement – statement in which the hypothesis and conclusion of the original conditional statement are interchanged
Counterexample – example used to disprove a statement
Definition – words or terms used to describe new terms/concepts
Geometric Construction – construction of accurate representations of lengths, angles, and geometric figures using only a straight edge and compass
Inverse of a Conditional Statement – statement in which the hypothesis and conclusion of the original conditional statement are negated
Line Segment – part of a line between two points on the line, called endpoints of the segment
Midpoint of a Line Segment – the point halfway between the endpoints of a line segment
One-Dimensional Coordinate System – numbers used as locations of points on a number line
Postulates (Axioms) – true statements not requiring proof
Undefined Terms – terms not formally defined and used to define other terms/concepts in a mathematical system

Unit 3: Lines and Transversals

During this unit, students explore angle relationships formed by one line and one transversal including vertical angles, linear pairs, and adjacent angles. Students construct congruent angles and a line parallel to a given line through a point not on a line using a compass and a straightedge. Students investigate patterns to make conjectures and define angles formed by parallel lines cut by a transversal. Students explore angle relationships formed by two parallel lines and one or more transversal(s) including corresponding angles, same side interior angles, alternate interior angles, and alternate exterior angles. Students use a variety of tools such as patty paper, folding techniques, etc. to investigate these relationships between angle pairs formed when parallel lines are cut by a transversal(s). Students formulate deductive proofs for conjectures about angles formed by parallel lines and transversals and apply these relationships to solve mathematical and real-world problems. Students explore and apply the converse of theorems and postulates for parallel lines cut by a transversal to solve mathematical and real-world problems.

Vocabulary

Congruent angles – angles whose angle measurement are equal
Conjecture – statement believed to be true but not yet proven
Geometric construction – construction of accurate representations of lengths, angles, and geometric figures using only a straight edge and compass
Parallel line – a line parallel to a given point not on a line
Postulates(axioms) – statements accepted as true without requiring proof
Theorems – conjectures that have been proven to be true
Vertical angles – a pair of non-adjacent, non-overlapping angles formed by two intersecting lines creating angles that are opposite and congruent to each other

Unit 4: Triangles

During this unit, students explore patterns and properties of triangles according to sides and angles (interior and exterior angles) using a variety of tools. Students verify theorems involving the sum of the interior angles of a triangle and theorems involving the base angles of isosceles triangles and apply these geometric relationships to solve mathematical and real-world problems. Students compare geometric relationships between Euclidean and spherical geometries, including the sum of the angles in a triangle. Students use constructions to verify the Triangle Inequality theorem and apply the theorem to solve problems. Students construct angle bisectors, segment bisectors, perpendicular lines, and perpendicular bisectors using a compass and a straightedge in order to investigate patterns and make conjectures about geometric relationships of special segments in triangles (altitudes, angle bisectors, medians, perpendicular bisectors, midsegments).Student verify and formalize properties and theorems of special segments and apply the geometric relationships to solve problems. Students analyze patterns of congruent triangles using a variety of methods to identify congruent figures and their corresponding congruent sides and angles. Students use rigid transformations of triangles and constructions to explore triangle congruency. Students formalize a definition of triangle congruency establishing necessary criterion for congruency, as well as formalize postulates and theorems for triangle congruency (Side-Angle-Side, Angle-Side-Angle, Side-Side-Side, Angle-Angle-Side, and Hypotenuse-Leg). Students apply triangle congruency and corresponding parts of congruent triangles are congruent (CPCTC) to prove two triangles are congruent using a variety of proofs. Students apply triangle congruency theorems and CPCTC to solve problems. Students use dilations of triangles and constructions to investigate and explore similarity. Students formalize a definition of triangle similarity establishing corresponding sides of triangles are proportional and corresponding angles of triangles are congruent. Students formalize postulates and theorems to prove triangles are similar using Apply Angle-Angle similarity and the Triangle Proportionality theorem. Students apply triangle similarity to prove two triangles are similar using a variety of proofs. Students apply triangle similarity theorems and proportional understanding to solve problems.

Vocabulary

Altitude of a triangle – line segment drawn from any vertex of a triangle perpendicular to the opposite side
Angle-Angle criterion for triangle similarity – if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar
Angle bisector – line, ray, or segment that divides an angle into two congruent angles
Base angles of a triangle – angles that have one side in common with the base
Congruent angles – angles whose angle measurement is equal
Congruent figures – figures that are the same size and same shape
Congruent segments – line segments whose lengths are equal
Conjecture – statement believed to be true but not yet proven
Corresponding sides and angles – sides and angles in two figures whose relative position is the same
Euclidean geometry – the study of plane and solid geometry based on definitions, undefined terms (point, line, plane) and the assumptions of mathematician Euclid
Exterior angle of a polygon – angle on the outside of a polygon formed by the side of a polygon and an extension of its adjacent side
Geometric construction – construction of accurate representations of lengths, angles, and geometric figures using only a straight edge and compass
Interior angle of a polygon – angle on the inside of a polygon formed by pairs of adjacent sides
Median of a triangle – line segment drawn from any vertex of a triangle to the midpoint of the opposite side
Midsegment of a triangle – a line segment drawn from the midpoints of two sides of the triangle
Parallel line – line parallel to a given line through a point not on a line
Perpendicular bisector of a line segment – line, ray, or segment that divides a line segment into two congruent segments and forms a 90° angle at the point of intersection
Perpendicular bisector of a side of a triangle – line segment that is perpendicular to a side of the triangle at the midpoint of the side
Perpendicular lines – lines in the same plane that intersect at 90° angles whose slopes are opposite reciprocals,
Proportional sides – corresponding side lengths form equivalent ratios
Rigid transformations (isometric transformations, congruent transformations) – transformations where size and shape are preserved
Segment bisector – point, line, ray, or segment that divides a line segment into two congruent segments
Spherical geometry – the study of figures on the two-dimensional curved surface of a sphere
Similar figures – shapes whose angles are congruent and side lengths are proportional (equal scale factor)
Triangle congruence – triangles whose corresponding side lengths and corresponding angle measures are equal

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