NTHS Curriculum Syllabus

All listed concepts are from the 4-Level track.

Algebra II

Unit 1

Chapter Numbers: 1.1, 2.4, 1.2, 1.3

• Absolute Value Inequalities

• Radicals

• Complex Numbers

• Simplifying Rational Expressions

Unit 2

Chapter Numbers: 3.1, 3.2, 3.3, 3.4, 2.7, 3.5, 3.7

• Odd/Even Functions

• Composite Functions

• Circles (equation)

Unit 3

• Solving high degree polynomial equations

• Solving exponential and radical equations

• Graphing exponential and radical functions

• Solving equations with two absolute values

• Domain, Range, Increasing/Decreasing, zeros

Unit 4

Chapter Numbers: 4.1, 4.2, 4.3

• End behavior (w/o limit notation)

• Descartes’ Rule of Signs

• Rational Root Theorem

• Synthetic Division (optional)

• Multiplicity

Unit 5

Chapter Numbers: 4.5

• Graphing rational functions

• Oblique asymptotes

• Removable discontinuities

• Simplifying rational expressions

Trigonometry

Unit 6

Chapter Numbers: 5.1, 5.2, 5.3, 5.4, 5.5

• Domain/Range of Inverse Functions

• Logarithms

• Graphing logarithmic functions

• Simplifying logarithmic expressions

• Change of Base Formula

• Exponential growth/decay

• Half-life

Unit 7

Chapter Numbers: 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 

• Basic trigonometry functions: sin, cos, tan

• Unit Circle

• Basic trigonometry identities

• Graphing trigonometric functions

• Period, Amplitude, phase-shift

Unit 8

Chapter Numbers: 7.1, 7.2, 7.3, 7.4, 7.6

• Pythagorean Trigonometric Identities

• Cofunction Identities

• Even/Odd Identities

• Sum and Difference Formulae (Trig funcs)

• Double Angle Identities

• Proving Identities

• Solving trigonometric equations

• Inverse trigonometric functions

Analytic Geometry

Unit 1: Conics

• Circles

• Parabolas

• Foci, Directrix, Latus recta

• Ellipses

• Foci, vertices, covertices, major axis, minor axis, latus recta

• Eccentricity

• Hyperbolas

• Vertices, foci, asymptotes, eccentricity, latus recta

• Directrix

Unit 2: Transformations

• Terminology

• Image, pre-image, translated, dilation, invariant, rotation.

• Rotation matrix

• Reflection matrix

• Inverse dilation, inverse rotation, inverse reflection

• Combining transformation matrices.

• Translating axes

Unit 3: Conic Transformations

• Rotating axes

• “Eliminating the xy term”

Unit 4: Polar

• Graphing

• Conversions to Rectangular

• Auxiliary Graphs

• Circles, Parabolas, Ellipses, Hyperbolas in Polar

Unit 5: Vectors

• Head, tail, magnitude

• Dot product

• Cosine dot product theorem

• Cross product

• Sine cross product theorem

• 3D Vectors, 3D lines

• Projections

• 3D planes

• Normal vector

• Distance between points, planes, and lines.

Unit 6: Cycloids

• Parametric equations

• Graphing cycloids

• Epicycloid, hypocycloid

Pre-Calc and Discrete Math

Unit 1: Counting

• Permutations and Combinations

• Nested sums

• Ball and Urn

Unit 2: Probability

• Basic and Conditional Probability

• Probability trees

• Binomial Probability

• Poisson Probability

• Recursive Probability

Unit 3: Induction

• Proof by Induction

• Basis step, inductive step

• Divisibility induction

• Mod function

• Sequences

• Explicit formula, recursive formula

Unit 4: Limits

• Notation

• Removable discontinuities with limits

• End behavior

Unit 5: Graph Theory

• Traveling Salesperson

• Cycles, total routes, unique routes, vertex, edge, optimal solution

• Nearest neighbor method

• Sorted edges method

• Spanning trees

• Prim’s Algorithm

• Christofides Algorithm

• Eulerization

Unit 6: More Limits

• Secants and Tangents

• Instantaneous rate of change

• Difference quotient

AP Calculus BC

Unit 1:

• Limit Definition of a Derivative

• Power Rule

• Product Rule

• Quotient Rule

• Trig Derivatives

• Chain Rule

• Implicit Differentiation

• Linearization

Unit 2:

• Extreme Value Theorem

• Intermediate Value Theorem

• Mean Value Theorem

• End behavior

• Optimization

• Related Rates

Unit 3:

• Derivatives of inverses

• Derivatives of exponential functions

• Derivatives of logarithmic functions

• Derivatives of inverse trigonometry functions

Unit 4:

• Riemann Sums

• Definite Integrals, Indefinite Integrals

• Fundamental Theorem of Calculus

• U-substitution

Unit 5:

• L’ Hospital’s Rule

• Area between curves

• Volume of 3D solids

• Disks, washers

• Shells

• Integral applications

Unit 6:

• Integration by Parts

• Partial Fractions

• Improper Integrals

• Arc Length

• Trig Substitution

Unit 7:

• Slope Fields

• Euler’s Method

• Separation of Variables

• Population Growth

• Logistic Growth, carrying capacity

Unit 8:

• Derivatives of parametric equations

• Vector calculus

• Velocity, acceleration vectors, speed

• Polar Arc Length

Unit 9:

• N-th term test

• Comparison Test

• Integral Test

• Limit Comparison Test

• Alternating Series

• Absolute Convergence

• Ratio Test

Unit 10:

• Power Series

• Interval of convergence, radius of convergence, center

• Taylor and Maclaurin Series

• LaGrange Error Estimation

AP Computer Science

Textbook: Java Concepts Early Objects, Enhanced e-book, 9th edition

Chapters 1, 2, 3

• Constructors

• Default constructor, parameterized constructors

• Methods

• Notation, parameters, arguments, return values

• Overloading methods and constructors

• Implicit this

• Object references, instances

• Getter and setter methods

Chapter 4

• Primitives, classes

• Integer division, modulus, casting doubles to ints, powers

• Edge cases

• Usage of Epsilon when comparing doubles for equality

Chapter 5

• Operators

• Comparing floats

• Comparing strings

• DeMorgan’s Law

• Short-Circuiting

Chapter 6

• For loops

• While loops

• Sentinel values

• Tracing

Chapter 7

• Nested loops

• Enhanced For loops

• Side Effects

Chapter 8,9

•  Object oriented programming

• Polymorphism

• Abstraction

• Encapsulation

• Superclass

• Subclass

• Inheritance

Chapter 13

• Recursion

• Trace and write recursive methods

• Binary and Hexadecimal

• Tail recursion

• Helper methods

Chapter 14

• Selection Sort

• Bubble Sort

• Insertion Sort

• Merge Sort

Multivariable Calculus and Linear Algebra

Unit 1: Review

• Parametric Equations

• Calculus with Parametric Equations

• Polar

• Areas and Lengths in Polar

• Coordinates in 3D/Vectors

• Dot Product

• Cross Product

• Lines and Planes

• Cylinders and Quadric Surfaces

Unit 2: Partial Derivatives

• Functions of Several Variables

• Limits and Continuity

• Partial Derivatives

• Tangent Planes and Linear Approximations

• Chain Rule

• Directional Derivatives, Gradient Vectors

• Extrema

• LaGrange Multipliers

Unit 3: 

• Double Integrals over Rectangles

• Double Integrals over General Regions

• Double Integrals in Polar

• Applications

• Surface Area

• Triple Integrals

• Cartesian, Cylindrical, Spherical

• Change of Variables

• Center of Mass

Unit 4:

• Vector Functions and Space Curves

• Vector Fields

• Line Integrals

• Fundamental Theorem for Line Integrals

• Green’s Theorem

• Curl and Divergence

• Parametric Surfaces and their Areas

• Surface Integrals

• Stokes’ Theorem

• Divergence Theorem

Unit 5:

• Differential Equations

• Second-order linear equations

• Nonhomogeneous Linear Equations

• Applications

End of MV, start of LA

Unit 1:

• Systems of Linear Equations

• Gaussian Elimination

• Matrix Operations

• Inverse and Properties of Matrices

• Elementary Matrices

• Linear Systems and Invertible Matrices

• Diagonal, Triangular, and Symmetric Matrices

• Formal Definition of Functions, Matrix Transformations

• Applications of Linear Systems

Unit 2

• Cofactor Expansion

• Row Reduction

• Determinant Properties and Cramer’s Rule

• Vectors in n-space

• Norm, Dot Product, Distance in R^n

• Orthogonality

• Geometry of Linear Systems

• Cross Product

Unit 3:

• Real Vector Spaces

• Subspaces

• Linear Independence

• Coordinates and Basis

• Change of Basis

• Row, Column, and Null Space

• Rank, Nullity