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.
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.
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.
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.