Math

Philosophy

Mathematics is as old as civilization itself, for only the most primitive form of civilization can exist without it. Counting the days until the next planting, the numbers in the herds and flocks, the numbers of mouths to feed are all essential to the survival of a rooted community.

History and mathematics are wedded in this way, and where mathematics has flourished so has the human condition; where mathematics has remained sterile, so too has any form of progress, be it art, literature, science, or thought. It is no accident that the technological revolution of today is the product of the mathematics of sixty years ago. The great irony in all this is that although mathematics has been essential to man’s ascent and knowledge, the reverse is not true.

The goal of the VES Math Department is to bring as much of this form to our students as possible, to teach them the mechanics of how all this works, why all this works, and to a certain degree to prove that all this works. A close but constant secondary goal is to demonstrate at all levels the applications of mathematics in the world around us. Regardless of the particular class, students will be engrossed in problem solving, investigating, predicting, calculating, analyzing, and verifying, followed by a well-reasoned presentation of results. Our math classes focus on discovery, taking chances, and following where mathematics leads.

The normal progression of classes, although this curriculum is certainly fluid as we accommodate our students’ interests and abilities, is Algebra I, Geometry, Algebra II/Trigonometry, Pre-Calculus or Statistics, AP Calculus AB or AP Statistics, AP Calculus BC or Multi-Variable Calculus.

Course Descriptions

 

Algebra I
Algebra I, the introduction to mathematics at VES, is a vast world of functions, graphs, and the fascinating exploration of numbers and their invaluable uses and qualities. The course develops a facility in working with numbers, graphs, inequalities tables, and various equations. Particular emphasis is placed on solving word problems and reading questions carefully. This process builds algebra skills and strengthens the understanding of the need to solve problems in a context, rather than from drill and practice alone. Students learn to use graphing calculators as a problem-solving tool. Topics include the study of equations and graphs (linear and quadratic), linear data versus nonlinear data, exponents, inequalities, radicals, solving fractional equations, special products, and factoring.

Algebra II – Trigonometry
This course provides a continuation and extension of the basic algebraic concepts from Algebra I and geometry. Students discuss, represent, and solve increasingly sophisticated real-world problems using advanced algebraic techniques, bringing opportunities for doing mathematics into focus. Incorporating appropriate technology, they study the properties and the algebra of quadratic, exponential, logarithmic, and rational functions, systems of equations and inequalities, as well as conic sections and applied trigonometry. This course provides a sound understanding of all elementary functions from linear through trigonometric and circular. Prerequisite: Algebra I.

Honors Algebra II - Trigonometry
The main topics of Honors Algebra II – Trigonometry are basic number theory, algebraic properties and proofs, formal notation, word problems, and the algorithms to solve them. As the course advances, students solve higher order equations, formal functions, logarithms, exponentials and more word problem applications. The spring term introduces trigonometry and vectors, including Laws of Sines and Cosines, radian and degree trigonometry, graphs of trig functions, and trigonometric word problem applications. Prerequisites: Honors Geometry or permission of the Department Chair.

Geometry
Familiarity with shapes and spatial relationships and the ability to manipulate them properly are keys to a strong mathematical foundation. Deductive principles are developed throughout this course, using both direct and indirect proofs. Special emphasis is put on inductive reasoning and the creation of original theorems. Mid-year, the emphasis shifts toward the application of algebraic principles to geometric figures. As soon as the Pythagorean Theorem is mastered, numerical answer problems start to take the place of proofs. A brief introduction to trigonometric functions and their roles allows the use of formulas such as the area of a regular polygon in terms of sine and cosine. Prerequisite: Algebra I or permission of the Department Chair.

Honors Geometry
The study of Honors Geometry encompasses far more than its definitions, postulates, and theorems. Students will consistently be challenged to reason analytically. The process of formal proof is emphasized early in the course, and direct and indirect proofs are investigated extensively. Proofs include parallel and perpendicular lines, congruent triangles, parallelograms, and geometric inequalities. The emphasis then shifts to applications. Topics include circles, right triangle trigonometry, coordinate geometry, areas, and volumes. Late in the year, a computer software assisted project is assigned focused on the ideas of construction and locus. Graphing calculators and Geometer’s Sketchpad software are used to demonstrate and model much of the geometry presented within the course. Prerequisite: Algebra I or permission of the Department Chair.

Math Analysis
Math Analysis helps students understand the fundamental concepts of algebra, trigonometry, and analytic geometry. Topics covered in this course are the study of functions (polynomial, rational, trigonometric, exponential, and logarithmic), systems of equations and inequalities, matrices, solving triangles, and conic sections, along with introductory ideas of calculus (determinants and limits). A balance is maintained among the algebraic, numerical, graphical, and verbal methods of representing problems. Students use the graphing calculator daily to visualize topics from a numerical and graphical representation. Prerequisite: Algebra II - Trigonometry.

Honors Math Analysis
The mathematical spectrum heightens as students enter the world of Honors Analysis. This course is aimed at those who have demonstrated excellent mathematics ability in their previous coursework, with the expectation being toward preparing them for Advanced Placement Calculus in the following year.  The first term begins with an emphasis on mathematical reasoning and proof, with a specific focus on general functions and their properties. After a guided tour of the functions, student begin to explore the concepts of series and sequence, complex numbers, exponential and logarithmic functions, polynomial and trigonometric functions, conic sections, matrices, and vectors. The students finish the year delving into topics essential to calculus such as polar coordinates, complex numbers, analytical geometry, and an introduction to limits and continuity.  Prerequisite: Honors Algebra II - Trigonometry and permission of the department chair.

Calculus
Students learn the mechanics behind solving derivatives and integrals both by hand and on the graphing calculator. Interspersed among the lessons throughout the year are application of the course material in the form of physical motion, product package design, architecture, finance, flowing water, medication, populations, swings, springs, see-saws, police radars, wrecking balls, balloons, ballistics, bacteria, and rocket science – to name a few. This is not a class about theorems or mathematical rigor as is the AP Calculus class, but is an excellent basis for college calculus. Prerequisite: Math Analysis or permission of the Department Chair.

AP AB Calculus
This is a rigorous course aimed at building a strong foundation in differential and integral calculus along with its various applications. The course begins with a study of limits, continuity, and parametric equations. Topics include differentiation and integration of polynomial, exponential, and trigonometric functions. Specific applications studied include velocity, acceleration, position, optimization, slope fields, exponential growth and decay, area, and volume. Various techniques of integration are studied with particular emphasis placed upon the Fundamental Theorem of Calculus and its applications. The course prepares students for the College Board Advanced Placement Examination, with the potential for students to begin their college mathematics at a more advanced level of calculus. Prerequisite: Honors Analysis and / or permission of the Department Chair.

AP BC Calculus
This course is highly rigorous and aimed at building a strong foundation in differential and integral calculus, along with its various applications. The AP BC curriculum includes all of the material covered in the AP AB course, with more emphasis on the underlying proofs. Additional topics include the study of Euler’s method, logistical growth models, integration by parts, partial fractions, volumes by cylindrical shells, arc length, and indeterminate forms. Focus is put upon polynomial approximations and series (Taylor and Maclaurin), as well as polar, parametric, and vector functions and the analysis of planar curves. Students prepare for the College Board Advanced Placement Examination, and have the potential to begin their college mathematics at a significantly more advanced level of calculus. Prerequisite: AP AB Calculus or Honors Analysis and permission of the Department Chair.

Statistics
The course concentrates on application rather than formal theory. Students learn to formulate questions that can be addressed with data, and to collect, organize, and display relevant data to answer them. They learn to select and use appropriate statistical methods. Students develop and evaluate inferences and predictions, and apply basic concepts of probability. Prerequisite: Algebra II - Trigonometry or permission of the Department Chair.

AP Statistics
Statistics is the most widely applicable branch of mathematics, used by more people than any other kind of math both in the workplace and by consumers. Students study lists of raw data, graphical displays and charts, rates, probabilities, percentages, averages, forecasts, and trend lines. Advanced Placement Statistics provides the opportunity for students to acquire statistical literacy. This course is designed to be the equivalent of an introductory college level Statistics course. The syllabus has been constructed under the guidelines of the College Board, and will prepare the student to take the Advanced Placement Examination in the spring. Prerequisites: Algebra Two or Honors Algebra Two along with score of 550 on Math portion of SAT or permission of Department Chair. 

Multivariable & Vector Calculus
The course begins with a thorough review of analytic geometry, polar coordinates, and parametric equations, then proceeds to vectors in both 2-space and 3-space. The topics include tangent and normal vectors, curvature, dot product, cross product, curves and planes in 3-space, and quadric surfaces. Further topics include the analysis of cylindrical and spherical coordinates, partial derivatives, gradients, directional derivatives, and double and triple integrals. Stokes’ and Green's theorems, as well as the related underpinnings of vector theory will be discussed and studied as time permits. Prerequisite: AP BC Calculus and permission of the Department Chair.

Linear Algebra
Linear algebra begins with a review of vectors, lines, and planes in space, then moves to solving systems of linear equations using row reduction, Gaussian elimination, LU decomposition, matrices, and various applications. Vector spaces and linear independence are also studied, followed by other topics including: discrete dynamical systems, eigenvalues, complex eigenvalues, eigenvectors, least square models, linear transformations, the Gram-Schmidt process for finding orthogonal bases, the change of basis, and symmetric matrices, and change of basis. Proofs are emphasized through much of the coursework. This course may be taught in an independent study format under the direction of a senior math teacher. Prerequisite: Multivariable Calculus and / or permission of the Department Chair.

Contact Math Department

Will Greene
Department Chair
Phone: 434-385-3824

Jeff Ross
Phone: 434-385-3629

Chip Jones
Phone: 434-385-3617

Mike Salvia
Phone: 434-385-3676
 

Charles Watson
Phone: 434-385-3630
 

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400 VES Road, Lynchburg, VA 24503 | 434-385-3600