Geometry and Trigonometry Strand

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.

Geometry, trigonometry, and algebra become increasingly intertwined in the Courses 3 and 4 units. Circles and Circular Functions from Course 3 introduces the sine, cosine, and tangent functions. Inverse Functions extends these trigonometric functions to their inverse functions. In Course 4, Vectors and Motion, two-dimensional vectors are introduced and used to model linear, circular, and other nonlinear motions. Methods for solving trigonometric equations and proving identities are developed in Trigonometric Functions and Equations. The Surfaces and Cross Sections unit provides college-bound students further work with visualization and representations of three-dimensional shapes and surfaces.

Course 1

Unit 6 - Patterns in Shape develops student ability to visualize and describe two- and three-dimensional shapes, to represent them with drawings, to examine shape properties through both experimentation and careful reasoning, and to use those properties to solve problems.

Topics include: Triangle Inequality, congruence conditions for triangles, special quadrilaterals and quadrilateral linkages, Pythagorean Theorem, properties of polygons, tilings of the plane, properties of polyhedra, and the Platonic solids.

Course 2

Unit 3 - Coordinate Methods develops student understanding of coordinate methods for representing and analyzing properties of geometric shapes, for describing geometric change, and for producing animations.

Topics include: representing two-dimensional figures and modeling situations with coordinates, including computer-generated graphics; distance in the coordinate plane, midpoint of a segment, and slope; coordinate and matrix models of rigid transformations (translations, rotations, and line reflections), of size transformations, and of similarity transformations; animation effects.

Unit 7 - Trigonometric Methods develops student understanding of trigonometric functions and the ability to use trigonometric methods to solve triangulation and indirect measurement problems.

Topics include: sine, cosine, and tangent functions of measures of angles in standard position in a coordinate plane and in a right triangle; indirect measurement; analysis of variable-sided triangle mechanisms; Law of Sines and Law of Cosines.

Course 3

Unit 1 - Reasoning and Proof develops student understanding of formal reasoning in geometric contexts and of basic principles that underlie those reasoning strategies.

Topics include: inductive and deductive reasoning strategies; principles of logical reasoning—Affirming the Hypothesis and Chaining Implications; relation among angles formed by two intersecting lines or by two parallel lines and a transversal.

Unit 3 - Similarity and Congruence extends student understanding of similarity and congruence and their ability to use those relations to solve problems and to prove geometric assertions with and without the use of coordinates.

Topics include: connections between Law of Cosines, Law of Sines, and sufficient conditions for similarity and congruence of triangles, centers of triangles, applications of similarity and congruence in real-world contexts, necessary and sufficient conditions for parallelograms, sufficient conditions for congruence of parallelograms, and midpoint connector theorems.

Unit 6 - Circles and Circular Reasoning develops student understanding of relationships among special lines, segments, and angles in circles and the ability to use properties of circles to solve problems; develops student understanding of circular functions and the ability to use these functions to model periodic change; and extends student ability to reason deductively in geometric settings.

Topics include: properties of chords, tangent lines, and central and inscribed angles of circles; linear and angular velocity; radian measure of angles; and circular functions as models of periodic change.

Course 4

Unit 2 - Vectors and 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, dot product of two vectors, position vectors and coordinates; and parametric equations for motion along a line and for motion of projectiles and objects in circular and elliptical orbits.

Unit 4 - Trigonometric Functions and Equations extends student understanding of, and ability to reason with, trigonometric functions to prove or disprove two trigonometric expressions are identical and to solve trigonometric equations; to geometrically represent complex numbers and complex number operations and to find roots of complex numbers.

Topics include: 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 complex numbers.

Unit 6 - Surfaces and Cross Sections extends student ability to visualize and represent three-dimensional shapes using contours, cross sections, and reliefs, and to visualize and represent surfaces and conic sections defined by algebraic equations.

Topics include: using contours to represent three-dimensional surfaces and developing contour maps from data; sketching surfaces from sets of cross sections; conics as planar sections of right circular cones and as locus of points in a plane; three-dimensional rectangular coordinate system; sketching surfaces using traces, intercepts and cross sections derived from algebraically-defined surfaces; and surfaces of revolution and cylindrical surfaces.