## Geometry and Trigonometry

The geometry and trigonometry strand of Core-Plus Mathematics has several goals. One important goal is to provide mathematical experiences that convey to students the usefulness of knowledge about shapes, shape properties, and relationships between shapes. A second goal is to provide mathematical experiences that allow students to become familiar with a substantial portion of elementary Euclidean geometry of the plane and, to a lesser extent, of space. A third goal is to provide mathematical experiences that allow students to experience both synthetic-graphic and algebraic-symbolic approaches to studying geometric topics. A fourth goal is to introduce students to axiomatic organizations of small parts of Euclidean geometry and to develop reasoning skills in those contexts.

The initial work in geometry comes in Course 1. It is synthetic, begins with three-dimensional shapes, and includes volumes, areas and perimeters of many common shapes. It then considers polygons and their properties. The Pythagorean Theorem is introduced and applied. Symmetry of plane shapes, both bilateral and rotational, is extended to include translational symmetry of infinite strip patterns and infinite plane patterns. All of the content is developed in the context of real-world situations and problems.

An early unit in Course 2, Patterns of Location, Shape, and Size, focuses on the goal of algebraic representation of geometric ideas by introducing coordinate representations of points. Coordinates are used to quantify distance, slope of lines, and to express the numeric representation of the relation of the slopes of two perpendicular lines. Coordinates are further used to model isometries and size transformations and their compositions. Coordinate models of points allow matrices to be introduced as another way to represent a polygon and a transformation that leaves the origin fixed. Matrix representation of shapes and transformations are used to create animations.

The second geometry and trigonometry unit in Course 2, Geometric Form and Its Function, returns to study of three basic plane figures: triangles, quadrilaterals and circles. The fact that a quadrilateral is not rigid is used to introduce linkages and their properties in a variety of contexts. Similar figures are related to a special linkage - the pantograph. Triangles with one side that can vary in length are studied and used to introduce the trigonometric ratios of sine, cosine, and tangent. The sine and cosine functions are developed further in the study of circles and circular motion. Trigonometric concepts and methods are interwoven and extended in each of the three Course 3 algebra and functions units.

Course 3 includes one unit whose primary focus is geometry. Its goal is to consolidate and organize the geometric knowledge of the students more logically and formally. To accomplish this, students learn to reason logically in geometric contexts. Inductive and deductive reasoning patterns are contrasted and the simple geometry of plane angles is presented in a local axiomatic system. Necessary and sufficient conditions (NASC) for parallelism of lines are introduced and applied. The similarity and congruence of triangles is developed and used. The NASC for a quadrilateral to be a parallelogram are included as well as NASC for a few other more specialized parallelograms. Reasoning synthetically and analytically is supported.

Geometry, trigonometry, and algebra become increasingly intertwined in the Course 4 units. In Modeling Motion, two-dimensional vectors are introduced and used to model linear, circular, and other nonlinear motions. Inverse trigonometric functions are introduced and methods for solving trigonometric equations and proving identities are developed in the Functions and Symbolic Reasoning unit. The Space Geometry unit provides college-bound students further work with visualization and representations of three-dimensional shapes and surfaces.

An overview of the sequence and contents of the geometry and trigonometry units in the CPMP four-year curriculum follows.

Course 1

Unit 5 - Patterns in Space and Visualization develops student visualization skills and an understanding of properties of space-shapes including symmetry, area, and volume.

Topics include: Two-and three-dimensional shapes, spatial visualization, perimeter, area, surface area, volume, the Pythagorean Theorem, angle properties, symmetry, isometric transformations (reflections, rotations, translations, glide reflections), one-dimensional strip patterns, tilings of the plane, and the regular (Platonic) solids.

Course 2

Unit 2 - Patterns of Location, Shape, and Size develops student understanding of coordinate methods for representing, and analyzing relations among, geometric shapes and for describing geometric change.

Topics include: Modeling situations with coordinates, including computer-generated graphics, distance, midpoint of a segment, slope, designing and programming algorithms, matrices, systems of equations, coordinate models of isometric transformations (reflections, rotations, translations, glide reflections) and of size transformations, and similarity.

Unit 6 - Geometric Form and Its Function develops student ability to model and analyze physical phenomena with triangles, quadrilaterals, and circles and to use these shapes to investigate trigonometric functions, angular velocity, and periodic change.

Topics include: Parallelogram linkages, pantographs, similarity, triangular linkages (with one side that can change length), sine, cosine, and tangent ratios, indirect measurement, angular velocity, transmission factor, linear velocity, periodic change, radian measure, period, amplitude, and graphs of functions of the form y = A sin Bx, y = A cos Bx.

Course 3

Unit 4 - Shapes and Geometric Reasoning introduces students to formal reasoning and deduction in geometric settings.

Topics include: Inductive and deductive reasoning, counterexamples, the role of assumptions in proof, conclusions concerning supplementary and vertical angles and the angles formed by parallel lines and transversals, conditions insuring similarity and congruence of triangles and their application to quadrilaterals and other shapes, and necessary and sufficient conditions for parallelograms.

Course 4

Unit 2 - Modeling Motion develops student understanding of two-dimensional vectors and their use in modeling linear, circular, and other nonlinear motion.

Topics include: Concept of vector as a mathematical object used to model situations defined by magnitude and direction; equality of vectors, scalar multiples, opposite vectors, sum and difference vectors, position vectors and coordinates; and parametric equations for motion along a line and for motion of projectiles and rotating objects.

Unit 7 - Functions and Symbolic Reasoning extends student ability to manipulate symbolic representations of exponential, logarithmic, and trigonometric functions; to solve exponential and logarithmic equations; to prove or disprove that two trigonometric expressions are identical and to solve trigonometric equations; to reason with complex numbers and complex number operations using geometric representations and to find roots of complex numbers.

Topics include: Equivalent forms of exponential expressions, definition of e and natural logarithms, solving equations using logarithms and solving logarithmic equations; the tangent, cotangent, secant, and cosecant functions; fundamental trigonometric identities, sum and difference identities, double-angle identities; solving trigonometric equations and expression of periodic solutions; rectangular and polar representations of complex numbers, absolute value, DeMoivre's Theorem, and the roots of a complex number.

Unit 8 - Space Geometry extends student ability to visualize and represent nonregular three-dimensional shapes using contours, cross sections and reliefs; to visualize and represent surfaces defined by algebraic equations; to visualize and represent lines in space; and to sketch three-dimensional shapes.

Topics include: Using contours to represent three-dimensional surfaces and developing contour maps from data; conics as planar sections of right circular cones; sketching surfaces from sets of cross sections; three-dimensional rectangular coordinate systems, sketching surfaces using traces, intercepts and cross sections derived from algebraically defined surfaces; cylinders, surfaces of revolution; and describing planes and lines in space.