Coordinate Geometry & Transformations

Introduction

This unit bundles student expectations that address geometric explorations of distance, midpoint, slope, and parallel and perpendicular lines in a two-dimensional coordinate system, including determining equations of lines. The student expectations also address rigid and non-rigid transformations both on and off the coordinate plane making connections between algebra and geometry. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

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.