**The following is a comprehensive list of high school and college courses in math offered **

**at Math Academy**

##### High School Level Courses

Arithmetic

Addition, subtraction, multiplication, and division; fractions, percents, and decimals; math with multi-digit numbers; operations with negative numbers; factoring integers; greatest common factor; least common multiple; inequalities; scientific notation and significant digits; place value; word problems

Pre-Algebra

Graphing in the x-y plane; lines and rays; midpoint and distance formulas; absolute value; prime factorization of integers; greatest common factors; square roots and radicals; irrational numbers; polynomials and rational expressions; equations; order of operations; factoring polynomials

Algebra I

Use of variables; polynomials and quadratics; exponents; radical expressions; rational expressions; logarithms; graphing in the x-y plane; coordinate pairs; solving linear equations; systems of equations; inequalities; factoring polynomials; point-slope form and slope-intercept form; parallel and perpendicular lines

Algebra II

Simplifying expressions; radical expressions; complex and imaginary numbers; exponential growth and decay; operations with logarithms; quadratic equations and quadratic formula; techniques for factoring polynomials; completing the square; rational equations; matrix properties; function notation; function transformations and graphing; sequences and series; basic trigonometric functions; conic sections; binomial theorem; partial fraction decomposition

Trigonometry

Trigonometric functions; inverse trigonometric functions; the unit circle; radians and degrees; trigonometric identities; Pythagorean Theorem; right triangle trigonometry; special right triangles; complementary and supplementary angles; non-right triangle trigonometry; double-angle and half-angle formulas; law of sines and cosines; graphing trigonometric functions; polar and spherical coordinates; Pythagorean identities; Triangle Inequality

Geometry I

Graphing in the Cartesian plane; midpoint and distance formulas; the unit circle; right triangle geometry; trigonometric functions and identities; conic sections; area and perimeter formulas; circles, arc lengths, cords, and radii; finding sides and angles of triangles; relationship between angles of polygons; similar triangles; geometric theorems and proofs; volume and surface area of 3D shapes; Pythagorean Theorem

Geometry II

Geometric proofs; proof techniques and logical symbols; constructions with compass; trigonometric identities; congruence of shapes; similar figures; transformations of shapes and functions; solving right triangles; law of sines and cosines; inscribed and circumscribed shapes; vectors and matrices; polar and spherical coordinates; areas of irregular shapes

3-Dimensional Geometry

Platonic solids; solid geometry; graphing in the x-y-z plane; functions in several variables; graphs and intersections of planes; polyhedra and non-polyhedra; hyperplanes and polytopes; conic sections in 3 dimensions; angles between lines and planes; geodesics; cylindrical and quadratic surfaces; change of coordinates; polar, spherical, and cylindrical coordinates; geometry in higher dimensions

Advanced Euclidean Geometry

Constructions; Hilbert’s axioms; Euclid’s postulates; proof methods and classical logic; conic sections; hyperplanes and lattices; vector geometry; parallel postulate; Triangle Angle Sum Theorem; Pythagorean Theorem; Thales’ Theorem; congruence of segments and angles; congruent and similar triangles

New York State Regents

Test preparation; Algebra I (Common Core), Algebra II (Common Core), Geometry (Common Core), Trigonometry/Algebra II, Physics

Pre-Calculus

Function notation; domain and range; inverse functions; graphing and simplifying rational functions; factoring polynomials and finding zeroes; vertical, horizontal, and slant asymptotes; maxima and minima; limits; tangent lines; curve-sketching; trigonometric and inverse trigonometric functions; matrices and vectors; dot and cross products; sequences and series

AP Calculus

Representation of functions; rate of change; limits at a point; infinite limits; two-sided limits; continuity of functions; criteria for differentiability; first and higher derivatives; derivatives of trigonometric functions; product and quotient rule; chain rule; implicit differentiation; optimization; L’Hospital’s Rule and indeterminate forms; Reimann sums; average value theorem; definite and indefinite integrals; antiderivative techniques; u-substitution; trig substitution; solids of revolution; differential equations; Fundamental Theorem of Calculus; modeling physical and biological systems using calculus; Liebniz notation; parametric functions

AP Statistics

Graphical representations of data; measuring center, spread, and position; bivariate data and scatterplots; categorical data; correlation and least-squares; transforming non-linear data using logarithms and power functions; sampling and experimentation; probability; univariate data; normal distribution; t-distribution; chi-square test; statistical inference; tests of significance; sampling and experimentation; methods of data collection; hypothesis testing; population parameters

Math Competitions/Olympiads

Practice problems for math competitions; math puzzles; customized curriculum to prepare for the competitions

Physics

Kinematics and the physics of motion; Newton’s laws of motion; position, velocity, acceleration, and jolt; force; free-body diagrams; momentum; work; rotational motion; centripetal and centrifugal force; angular and tangential velocity; angular momentum; torque; elastic and inelastic collisions; kinetic and potential energy; inertia; rotational energy; fluid mechanics; electricity and magnetism; current, voltage, and power; circuit diagrams; Coulomb’s Law; Kirchoff’s Current and Voltage Laws; direct and alternating current; resistance and capacitance; induction; thermodynamics and heat transfer; entropy; thermal energy; waves; diffraction; the Doppler Effect; equations of waves; wavelength, amplitude, and frequency; reflection and refraction; vibrations; light and sound waves; longitudinal waves; polarization; particle physics; subatomic particles; elementary particles; matter and radiation; quantum field theory

AP Computer Science

Java programming language; object-oriented programming; program design; data abstraction and encapsulation; class design and implementation; implementation techniques; procedural abstraction; declaration of variables, constants, and classes; iteration; testing and debugging; finding and correcting errors; number representation in different bases; data structures; lists and arrays; algorithms; traversals, insertions, and deletions; searching and sorting

##### High School Level Test Prep

High School Admissions Test

SAT I

Test prep and test taking strategies; algebra and functions; geometry; numbers and operations; data analysis

Mathematics Levels 1 and 2; Physics

ACT

Test prep and test taking strategies; pre-algebra; elementary and intermediate algebra; coordinate and plane geometry; trigonometry

Reading Comprehension; Logical Reasoning; Scrambled Paragraphs; Mathematics; Multiple Choice Questions; Various mathematical topics; Basic math; Algebra; Factoring; Substitution; Geometry; Basic Coordinate Graphing; Logic; Word Problems

##### College-Undergraduate Level Courses

College Algebra

Solving quadratic equations; solving linear equations; factoring polynomials; completing the square; exponents; logarithms; radical expressions; rational expressions; graphing in the x-y plane; midpoint and distance formulas; parallel and perpendicular lines; solving linear equations; systems of equations; inequalities; functions and function notation; function transformation and graphing; matrix properties; basic trigonometric functions; conic sections

Representation of functions; rate of change; limits at a point; infinite limits; two-sided limits; continuity of functions; criteria for differentiability; first and higher derivatives; derivatives of trigonometric functions; product and quotient rule; chain rule; implicit differentiation; optimization; Liebniz notation; Extreme Value Theorem; Mean Value Theorem; inflection points; first and second derivative tests for finding inflection points

L’Hospital’s Rule and indeterminate forms; Reimann sums; average value theorem; definite and indefinite integrals; antiderivative techniques; u-substitution; trig substitution; integration by parts; partial fraction decomposition; solids of revolution; differential equations; Fundamental Theorem of Calculus; modeling physical and biological systems using calculus; differentiation under the integral

Calculus 3

Parametric functions; calculus in polar, spherical, and cylindrical coordinates; multi-variable calculus; partial derivatives; Green’s Theorem; Stokes’ Theorem; vector calculus; dot and cross products; sequences and series; convergence of series; harmonic, geometric series, and power series; Taylor and Maclaurin series; Fourier series; radius of convergence; ratio test; comparison test; alternating series test; root test

Slope fields; separation of variables; Fourier Transforms; Laplace Transforms; Taylor and Maclaurin series; Bessel Functions; general and particular solution to differential equations; damped harmonic motion; oscillation problems; bifurcations; non-homogeneous differential equations; strategies for non-linear differential equations; boundary value problems; dynamical systems; existence and uniqueness; reduction of order method; Cauchy-Euler Equations

Partial Differential Equations

Distribution theory; Fourier transforms; Laplace transforms; Schrodinger equations; Green’s theorem; Euler-Lagrange equations; boundary condition; boundary value problems; Dirichlet problems; heat and wave equations; Maxwell’s equations; Navier-Stokes equations; Poisson’s equation; Green’s function; applications of partial differential equations; numerical methods for finding solutions

Matrix addition and subtraction; matrix multiplication; matrix row operations; matrix inversion; transpose matrices; skew and zsymmetric matrices; determinants; solving systems of equations; Gaussian elimination; reduced row echelon form; rank of a matrix; Cramer’s Rule; minors and cofactors; eigenvalues and eigenvectors; vector spaces; orthogonality; transformation matrices; vector spaces; subspaces; linear span; projection; matrix decompositions

Advanced Statistics: Fundamentals and axioms; combinatorial probability; conditional probability and independence; binomial, poisson and normal distributions; law of large numbers and the central limit theorem and random variables and generating functions.

##### Graduate Level Courses

Abstract Algebra

Group theory; abelian groups; cyclic groups and generators; subgroups; simple groups; dihedral groups; finite groups; homomorphisms and isomorphisms; group actions; kernel of a function; Lagrange’s Theorem; ring theory; integral domains; field properties; algebra over a field; field extensions; norms; Gaussian integers; ideals; quaternions; algebraic numbers; polynomial fields; lie algebra; Galois theory

Composite numbers; prime numbers; divisors; Euclid’s algorithm and divisibility; square-free objects; modular arithmetic and congruence; Fermat’s Little Theorem; Euler’s totient function; Chinese Remainder Theorem; quadratic residue; Diophantine equations; prime numbers; Lagrange’s theorem; combinatorical theorem; primality tests; algebraic number theory

Limits of sequences and functions; arithmetic, geometric, and harmonic series; convergence and divergence of series; Cauchy sequence; power series; telescoping series; alternating series; Fourier series; pointwise, uniform, and absolute convergence; tests for convergence; integral test; ratio test; direct comparison test; limit comparison test; root test; alternating series test; Dirichlet’s test; continuity of functions; uniform and absolute continuity; differentiability of functions; derivatives; partial derivatives; integrals and antiderivatives; Fundamental Theorem of Calculus; multiple integrals; differentiating under the integral; Reimann sums; measure theory; compactness; metric spaces; solution of differential equations; change of variables

Analytic functions; complex differentiation and integration; Cauchy integral formula; Laplace transforms; Power series, Taylor series, and Laurent series; residue theory; complex analysis; isometries in the complex plane; conformal mapping; power series; uniform convergence; radius of convergence; poles; zeroes; singularities; branch points; countour integrals; gamma and beta functions; Reimann zeta functions; Reimann surfaces; infinite products; complex dynamics

Topological spaces; uniform and metric spaces; open and closed sets; continuity; bases and subbases; open cover; limit point; Blaire category theorem; compact spaces; connectedness; separation axioms; Kolmogorov, Frechet, and Haudsorff spaces; Banach spaces; Lp spaces; normal space; tensor products; finite products; order theory; homotopy equivalence; algebraic topology; differential topology; topological and differentiable manifolds; polytopes; covering spaces; vector fields; Frobenius Theorem; Lie groups; Stokes’ Theorem; geometric topology; knot theory

Differential Geometry

Curvature; differential geometry of surfaces; first and second fundamental forms; calculus on manifolds; Reimannian manifolds; Finsler manifolds; connections; Levi-Civita connection; tensor analysis; lie groups; bundles and connections; Symplectic geometry; differential topology; critical values; diffeomorphism; character classes; non-Euclidean geometry; geodesics; symmetric spaces; complex manifolds; index theory; homogeneous spaces

Complex manifolds; Reimannian manifolds; geodesics; symmetric spaces; Lie groups; sympletic geometry; conformal geometry; Hodge theory and Fredholm theory; representation theory; projective spaces; non-Euclidean geometry; finite geometry; computational geometry; discrete geometry; algebraic geometry; fractals

Normed vector spaces; Hilbert spaces; Euclidean space; Banach spaces; Lp spaces; duality and separation theory; function spaces; locally convex spaces; Banach-Steinaus theorems; orthonormal basis; uniform boundedness; spectral theory; open mapping theorem; Hausdorff space; operator theory; real and complex algebras; topological vector spaces; wavelets

Iterative methods; rate of convergence; series acceleration; error analysis; interval arithmetic; significant figures; least squares; numerical algorithms; numerical linear algebra; eigenvalues; orthogonalization; approximation formulas; interpolation methods; spline, polynomial, and trigonometric interpolation; linear systems; root-finding algorithms for nonlinear equations; numeric integration and differentiation; numerics for ordinary and partial differential equations; optimization; finite element methods; grids and meshes; Monte Carlo method

Cryptography

Encryption and decryption; cryptanalysis; ciphers; zero knowledge proof; concurrency and protocol security; symmetric-key algorithms; asymmetric-key algorithms; authentication; transport and exchange theory; weak keys; robustness; symmetric algorithms; hash functions; attack models; database security; cryptographic game theory; multivariate cryptography; quantum cryptography; stenography

##### Graduate Level Test Prep

GRE

Quantitative reasoning; arithmetic; algebra; geometry; data analysis; numerical reasoning

GMAT

Quantitative reasoning; data sufficiency; arithmetic; algebra; geometry; word problems; combinatorics; data analysis

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