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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.