PRISMS aims to provide a comprehensive 9-12 high school education with a particular focus on Science, Technology, Engineering and Mathematics (STEM), and a culminating research or development project in an area of Science, Technology, Engineering or Mathematics.

## Mathematics

### Core Mathematics Courses

#### Foundations of Mathematical Thinking

This is the PRISMS 9th grade foundational mathematics course on which the mathematics course sequence is built. Topics include:

The logic of compound statements: Logic connectives: OR, AND, IF...THEN; Translate from English to symbols and vice-versa. Tautologies and contradictions; Operations with sets; Digital Logic circuits: OR-gate, AND-gate, binary system of numeration, halfadder and full-adder; The logic of quantified statements: Quantifiers (all, none, exist), negation of statements, DE Morgan's Laws; Valid and invalid arguments: Proof by contradiction, proof by mathematical induction, example proofs taken from number theory; Algebra notions: Operation on algebraic fractions, powers, radicals, add like terms, polynomial, binomial to a power, accentuate "rules of valid manipulation", and why we should learn them; Equations and inequalities: Linear equation, systems of linear equations, Gaussian elimination, inequalities, systems of linear inequalities, absolute values; Beginning of Geometry: Euclid’s Postulates, logic (deductive reasoning), angles, lines, polygons, circles; Congruent and similar figures, congruence and similarity theorems; Congruence transformations: Translation, rotation, reflection, symmetry; Areas and volumes of solids; A few notions about Probability; Description of some important mathematical ideas: Number axis, absolute value, coordinatization of the entire space (Descartes), Cantor's method of comparing sets.

Typically a 10th grade course which includes: Functions: Definition (relation, correspondence, domain, range), notation, graph, extrema, monotonicity, end behavior; Variation and graphs: Direct variation, inverse variation, their graphs, applications, combined and joint variation; Linear functions: Linear functions (different type of equations), graphs, step functions, arithmetic sequence, basic idea of regression; Quadratic functions: Properties, graphs, quadratic formulas, completing the square, the graph-translation theorem; Power functions: Negative exponents, rational exponents, power functions, graphs, geometric sequence; Inverse functions: Composition of functions, inverse, notation, properties (domain, range, graph); Exponential and logarithmic functions: The graph and properties of exponential functions, applications (exponential growth, compound interests), the definition and properties of logarithms, common logarithm, natural logarithm, the graph and properties of logarithmic functions; Polynomials: Basic concepts of polynomials, operations on polynomials, factoring polynomials, zeros and graphs of polynomial functions; Series and combinations: Arithmetic series, geometric series, combinations, Pascal’s triangle, the binomial theorem, permutations; Solving equations and inequalities by functions: The logic of equation solving, the relationship among equations, inequalities, and functions, the system of linear inequalities; Rational functions: Graphs of rational functions, the behavior near zeros and end behavior, asymptote; Sequence: Explicit formulas, recursive formulas, some famous sequences.

#### Pre-Calculus

Typically an 11th grade course which includes: Trigonometric functions in right triangle: Definition, relations among trigonometric functions, applications; Trigonometric functions with unit circle: Definition, radian measure, observing the properties of trigonometric functions through the unit circle, applications; The law of sines and the law of cosines: The theorems and their proofs, solving triangles by using these theorems, applications; Trigonometric functions: Graphs, properties (periods, symmetry, monotonicity), periodic functions, sine waves; Trigonometric formulas and identities: Symmetry, shifts and periodicity, sum and difference formulas; Polar system: Inverse trigonometric functions, polar coordinates, polar equations and curves; Conic sections: Definitions of ellipse, hyperbola, and parabola, equations for conic sections, directrix and eccentricity, polar equations, applications; Matrices: Notation, addition and multiplication, matrices for transformations, inverses of matrices; Complex numbers: The history from real numbers to imaginary numbers, operations on complex numbers, powers and roots of complex numbers, the number of zeros of a polynomial; Vectors and parametric equations: Examples of parametric equations, applications, vectors, operations on vectors, dot product and the angle between two vectors; Basic derivative; Basic integral

#### AP Calculus BC

This course is a standard, college level, first course in Calculus, using Stewart, Calculus, Early Transcendentals, in which we will convey the excitement of the new concepts one can learn from this branch of mathematics. We will emphasize a practical approach with relevant theory of concepts including continuity, differentiability, differential approximation, optimization and curve sketching of functions and inverse functions of a single variable, the trigonometric functions, the mean value theorem, L’ Hopital’s rule, and an introduction to integration including the fundamental theorem of integral calculus. Applications will be emphasized with many examples drawn from science. The overall objective is to motivate the students to think, to analyze, and to realize and use the power of calculus in their future studies.

### Elective Mathematics Courses

#### Differential Equations (Fall semester only)

This is a standard second-year college mathematics course. Topics studied include: first-order ordinary differential equations (ODEs), higher-order ODEs, Laplace transforms, linear and nonlinear systems, numerical approximations, plane analysis and the stability of equilibria, introduction to Fourier series, and introduction to partial differential equations. Applications are embedded throughout.

#### Probability and Statistics (Fall semester only)

This course will provide a firm foundation in probability theory and a brief introduction to statistical applications. A rigorous treatment will be employed in order to properly prepare the student for advanced applications in mathematics, science, and engineering. Topics to be covered include discrete and continuous random variables, probability density and distribution functions, moment generating functions, multidimensional random variables, functions of random variables, and special distributions, such as the binomial, negative binomial, hypergeometric, multinomial, uniform, Gamma, Beta, and especially the normal distribution (including the Central Limit Theorem).

#### Abstract Mathematical Thinking (Fall semester only)

This course constitutes an introduction to proof and formal mathematics with the idea of preparing students for higher-level mathematics. The emphasis is on understanding and mastering increased levels of rigor, dealing with mathematical notation, learning how to write, present, and analyze proofs. Course content includes axiomatic systems, the principle of mathematical induction, proof by contradiction, existence principles, mathematical logic, elementary set theory, countable and uncountable sets, bijections between sets, combinatorics, and abstract structures and isomorphism.

#### Linear Algebra (Spring semester only)

This college-level course includes the study of systems of linear equations, vector spaces, linear dependence, linear transformations and matrix representation, orthogonal reduction, determinants, eigenvectors and eigenvalues, and a variety of applications.

#### Partial Differential Equations (Spring semester only)

This is a standard course following differential equations. This course will introduce the basic concepts, basic theory and typical solutions of partial differential equations, and use these knowledge to solve the problems of simple partial differential equations from the research of science and engineering technology. It covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods and tools for solving these PDEs, such as separation of variables, Fourier series and transforms, eigenvalue problems, and Green's functions.