Higher Math

• NTHS Curriculum Syllabus
• Geometry
• Geometry PK Correlation
• New Trier Geometry Textbook Reviews + Answer Keys
• Algebra 2
• New Trier Algebra 2 4-Level Textbook
• New Trier Algebra 2 3-Level Textbook
• Calculus
• Paul's Calculus Notes
• Pre-Calculus
• Analytic Geometry 4-Level Textbook
• Material for teaching matrices

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

Geometry

Geometry PK Correlation

Geometry Mathnasium Material NT Correlation

Unit 1:

Section 1.1 Symmetry

• Line Symmetry - PK3514

• Point Symmetry - 

• Rotational Symmetry - PK3104

• Translational Symmetry - PK3104

Section 1.2 So, What’s the Point

• Definition of a point - PK824

• Definition of a plane - PK824

• Definition of a line - PK824

• Definition of a segment - PK824

• Definition of a ray - PK824

• Union/Intersection - PK824

• Locus of Points/Endpoints - FO2160

Section 1.3 Nspire Intro

Section 1.4 Transformations 

• Isometries {Translation PK3509, Rotation PK3510, Reflection PK3511, Dilation PK3512}

• Dilations

• Pre-image/image (Steps to transformation) - PK3581

Section 1.5 Transformations by Hand

• Translating by vectors

• Reflecting across lines - PK3584

• Rotating around a central point

• Dilating from point - PK3583

Unit 2: Transformations with Coordinates

Section 2.1 Transformations

• Graphing points/ translating points - PK3509

• Reflecting, rotating, dilating on the graph - PK3511, PK3512

Section 2.2 Things I Learned in Algebra

• Midpoints - PK3434

• Length of segments - PK3435

• Slopes of lines - PK3408

Section 2.3 More Geometry and Algebra!

• Altitudes - PK3552

• Perpendicular Bisectors - PK3552

• Medians - PK3552

Section 2.4 One Magic Moment, One Magical Point

• Solving systems of linear equations/finding the intersection of lines - {Substitution PK3417, Elimination PK3418, Graphing PK3416, Problem-Solving PK3452}

• Reflecting points over oblique lines - PK3584

• Determining medians/perpendicular bisectors on shapes graphically

• Distance between two lines - 

Unit 3: Congruent Triangles

Section 3.1 It’s Only Logical

• Conditional Statements - PK3530

Section 3.2 Three is a Magic Number

• AAS, SAS, SSS, ASA - PK4571, PK4570

• Describing transformations

Section 3.3 Congruent Triangles

• Vertical Angles - PK3580, PK3572, PK3573

• Reflexive Property  - PK3533

• Transitive Property - PK3533

• Two-column proofs - PK3533

Section 3.4 Supps and Comps

• Complementary angles - PK3507

• Supplementary angles - PK3507

• Corresponding supp/comp proof reasons

Section 3.5 Addition and Subtraction

• Addition theorem - PK3566

• Subtraction theorem

Section 3.6 Multiplication and Division

• Multiplication theorem

• Division theorem

Unit 4: Constructions

Section 4.1: The Basics

• Copying a segment - FO0023

• Adding segments - FO0023

• Copying an angle - FO0023

• Bisecting an angle - FO0023

• Constructing SSS triangle

• Triangle Inequality Theorem

Section 4.2: Constructing Perpendiculars

• Constructing a perpendicular bisector - FO0023

• Construct a perpendicular to a segment - FO0023

• Construct a perpendicular to a line through a point not on the line

Section 4.3 Points of Concurrency

• Point of concurrency - PK3559

• Incenter - PK3559

• Circumcenter - PK3559

• Centroid - PK3559

• Orthocenter - PK3559

Unit 5: Proving Triangles Congruent is Not the Goal, It’s the Process

Section 5.1: The End of CPCTC

• “CPCTC” - PK3573 (page 6)

• Circles - Congruent Radii

Section 5.2: Turn It Up To 11: Draw a line & overlapping triangles

Section 5.3: Classify Triangles

• The sum of triangle angles equals 180

• Equilateral, Isosceles, Scalene - PK3099

• Acute, Right, Obtuse Triangles - FO111

• The longest side is opposite the largest angle

Section 5.4: My Favorite Theorems

• Isosceles - base angle/base sides

• Isosceles - angle bisector = median = altitude

Section 5.5: RHL

• RHL Proof

Unit 6: The Next Chapter

Section 6.1: The Questions of Mathematics

• If two angles supp and comp -> rt angles

Section 6.2: No Picture? Then Make a Picture

• Draw it out

Section 6.3: Perpendicular Bisector and Equidistance

• If a pt is on the perp bisector, it is central to the endpoints

• If two points are equidistant to the endpoints of a seg -> they form the perp bisector

Section 6.4: If It’s Not This, Then It’s Got To Be This 404

• Proof by Contradiction

Unit 7: Lines, Angles, Quads

Section 7.1: Parallel Lines

• Transversal

• Alternate Interior Angles, Alternate Exterior Angles, Corresponding Angles, Same Same Interior Angles - FO272

Section 7.2: Parallel Lines Keep Going

• Constructing parallel lines

Section 7.3: Quadrilaterals

• Definition of a quadrilateral

• Properties of a parallelogram

• Properties of a kite

• Properties of a rectangle

• Properties of a rhombus

• Properties of a square

• properties of a trapezoid

• Properties of a isosceles trapezoid - PK3099

Section 7.4: More Quadrilaterals

• Solving problems involving properties of quads

Section 7.5: It looks like a parallelogram

• Properties of a parallelogram

• Both pairs of opposite sides are parallel

• Both pairs of opposite sides are congruent

• Both diagonals bisected

• Same pair of opposite sides and congruent = parallelogram

• The sum of angles in quadrilaterals is 360. 

Section 7.6: Special Quads

• Solving problems using obscure properties of quads.

Unit 8: Similarity

Section 8.1: Dilations

Section 8.2: Dilations pt. Deux

Section 8.3: Similarity

Section 8.4: Proving Similar Triangles

Section 8.5: Proportions Without Similar Triangles

Unit 9: Right Triangles

Section 9.1: Rt Triangle inside a Rt Triangle

Section 9.2: Pythagorean Theorem - PK3363

Section 9.3: Right Triangle Families - PK3363

Section 9.4: Rt Triangles in 3D Figures - 

Section 9.5: Special Right Triangles - PK3561

Section 9.6: Right Triangle Trigonometry - PK3916

Law of Cosines - PK3914, PK3915

Law of Sines - PK3913, PK3915

Unit 10: Polygons

Section 10.1: Polygon Formulas - PK3501, PK3550, PK3551

Section 10.2: What is Regular? - PK3501

Section 10.3: Constructing of Regular Polygons - FO 023 (page 29)

Section 10.4: Tessellations - FO 194

Unit 11: Circles

Section 11.1: Chords of Circles - PK3562

Section 11.2: From the Center to the Arc 

Section 11.3: Tangents - PK3569

Section 11.4: Angle-Arc Relationships - PK3568

Section 11.5: Arcs have Length and Measure - PK3588

Section 11.6: Power of the Poin 

Unit 12: Area

Section 12.1: Area Basics - PK3089

Section 12.2: More Area Formulas - PK3362(Rectangles), PK3364(Triangles), PK3365(Circles), PK3523(Parallelograms), PK3524(Trapezoids)

Section 12.3: Regular Polygons - PK3590

Section 12.4: Sectors of Circles - PK3588

Section 12.5: Ratios of Areas - PK3582

Unit 13: 3D

Section 13.1: Lines and Planes in Space - PK0804 (skew lines)

Section 13.2: If You Look At It That Way - ???

Section 13.3: Cross-Sections of 3D Objects - FO 242, FO 188

Unit 14: Surface Area and Volume

Section 14.1: Surface Area - PK3527, PK3595

Section 14.2: SA of Cylinders and Cones - PK3527, PK3595

Section 14.3: Volume Basics - PK3528, PK3596

Section 14.4: Volumes with an Apex - PK3528, PK3596

Section 14.5: Spheres - PK3555

Unit 15: Modeling - Difficult to find any Mathnasium material with geometric monitoring. PK3534 and PK3535 have some geometric problem-solving

Section 15.1: Intro to Modeling

Section 15.2: Art Gallery Watchmen

Section 15.3: Packaging and Packing

Section 15.4: Airports

 

New Trier Geometry Textbook Reviews + Answer Keys

link

Algebra 2

New Trier Algebra 2 4-Level Textbook

link

New Trier Algebra 2 3-Level Textbook

link

Calculus

Paul's Calculus Notes

link

Pre-Calculus

Analytic Geometry 4-Level Textbook

link

Material for teaching matrices

link