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Arithmetic | |
Whole Numbers | |
Basic Place Value | |
Recognize Place Values to 10,000,000 | |
Greatest and Least Values of Given Digits to 10,000,000 | |
Higher Order Place Value and Number Placement Problems | |
Round Large Numbers | |
Additon and Subtraction | |
Add Whole Numbers | |
Subtract Whole Numbers | |
Estimate Whole Number Sums and Differences | |
Multiplication | |
Multiply Multiple Digits by One Digit | |
Higher Order One Digit Multiplication Word Problems | |
Relate One Digit Number Patterns to Multiplication | |
Multiply Numbers with the Associative Property | |
Multiply Two Digits by Two or More Digits | |
Multiply and Compare with Greater/Less Than | |
Relate Number Patterns to Multiplication | |
Higher Order Multiplication Word Problems | |
Multiply Whole Numbers | |
Mental Multiplication | |
Division | |
Divide Two Digits by One or Two Digits Without Remainders | |
Higher Order Division to 1000 by One Digit | |
Divide More than Two Digits by One Digit with Remainders | |
Relate Division to Multiplication | |
Estimate Whole Number Products and Quotients | |
Long Division | |
Long Division Without Remainders | |
Divide up to Four Digits by Two Digits with Remainders | |
Groups and Remainders Word Problems | |
Long Division Word Problems | |
Divide Whole Numbers | |
Order of Operations | |
Introduction to Order of Operations | |
Divide and Subtract with Remainders in Word Problems | |
Division and Subtraction Word Problems | |
Higher Order Division and Subtraction Word Problems | |
Multiply and Add Word Problems | |
Multiply and Add or Subtract Word Problems | |
Place Operators to Make True Statements | |
Place Operators to Make True Statements that Include Parentheses | |
Place Parentheses to Make True Statements | |
Exponents and Roots | |
Exponents | |
Evaluate and Compare Powers | |
Perfect Square Roots | |
Evaluate Square Roots | |
Factors | |
Prime and Composite Numbers | |
Prime Factorization | |
Identify Factor Pairs | |
Divisibility Rules to Find Factors | |
GCF Greatest Common Factor | |
Common Multiples | |
LCM Least Common Multiple | |
Integers | |
Integers in the Real World | |
Integers on a Number Line | |
Absolute Value of Integers | |
Add Integers with the Same Signs | |
Add Integers with Different Signs | |
Integer Addition | |
Subtract Integers with the Same Sign | |
Subtract Integers with Different Signs | |
Integer Subtraction | |
Multiply Integers | |
Divide Integers | |
Fractions | |
Introduction to Fractions | |
Equivalent Fractions | |
Simplify Fractions | |
Simplify Fractions Word Problems | |
Compare Fractions | |
Compare Fractions using Pictures | |
Compare Fractions that have Common Numerators or Denominators | |
Compare Fractions without Common Numerators or Denominators | |
Compare Fractions without Common Numerators or Denominators Word Problems | |
Estimate and Round Fractions and Mixed Numbers | |
Convert and Compare Mixed Numbers and Improper Fractions | |
Add and Subtract Fractions | |
Add Fractions with Common Denominators | |
Subtract Fractions with Common Denominators | |
Add and Subtract with Common Denominators | |
Add and Subtract Fractions with Common Denominators Word Problems | |
Add Fractions with Different Denominators | |
Subtract Fractions with Different Denominators | |
Add and Subtract Mixed Numbers | |
Add Mixed Numbers with Common Denominators | |
Subtract Mixed Numbers with Common Denominators | |
Subtract Mixed Numbers with Different Denominators | |
Add and Subtract Three Mixed Numbers | |
Add and Subtract Mixed Numbers Word Problems | |
Multiply and Divide Fractions | |
Multiply Whole Numbers and Fractions | |
Multiply Two Fractions | |
Multiply Three or More Fractions and Whole Numbers | |
Multiply Mixed Numbers | |
Multiply Mixed Numbers Word Problems | |
Reciprocal Fractions | |
Divide Fractions and Whole Numbers | |
Divide Fractions | |
Divide Whole Numbers by Mixed Numbers | |
Divide Mixed Numbers | |
Multiply and Divide Fractions and Mixed Numbers | |
Estimate Products of Whole Numbers and Fractions | |
Ratios | |
Definition of a Ratio | |
Equivalent Ratios | |
Ratios in Simplest Form | |
Compare Ratios in Decimal Form | |
Unit Rates | |
Decimals | |
Basic Decimals | |
Place Value Charts and Decimals to Thousandths | |
Equivalent Decimals Ending in Zero | |
Decimals in Words | |
Decimal Place Value | |
Decimals in Expanded Form | |
Compare Decimals | |
Compare and Order Decimals | |
Compare and Compose Decimals and Fractions | |
Compare, Order and Identify Decimal Inequalities | |
Round Decimals | |
Round Decimal Numbers | |
Round Decimals with Place Value | |
Add and Subtract Decimals | |
Add and Subtract Decimal Numbers | |
Mental Math to Add and Subtract Decimals | |
Add and Subtract Decimals with Front-End Estimation | |
Round Decimals to Estimate Sums and Differences | |
Multiply and Divide Decimals | |
Decimal Multiplication | |
Decimal Division | |
Multiply Decimals and Whole Numbers | |
Estimate Products and Quotients of Decimals | |
Powers of Ten and Scientific Notation | |
Multiply Decimals by Powers of Ten | |
Multiplication and Powers of Ten | |
Division and Powers of Ten | |
Scientific Notation | |
Rational and Irrational numbers | |
Rational Numbers | |
Add Rational Numbers | |
Subtract Rational Numbers | |
Multiply Rational Numbers | |
Divide Rational Numbers | |
Irrational Numbers | |
Irrational Square Roots | |
Percents | |
Overview of Percents | |
Percent of a Number | |
Simple Interest | |
Percent of Change | |
Percent of Increase | |
Percent of Decrease | |
Prices Involving Discounts | |
Total Bill Including Tip and Tax | |
Conversions Between Fractions, Decimals, and Percents | |
Convert Percents to Fractions and Decimals | |
Percents as Decimals | |
Percents as Fractions | |
Convert Fractions to Percents and Decimals | |
Fraction and Decimal Conversion | |
Convert Mixed Numbers to Decimals | |
Compare Mixed Numbers and Decimals | |
Convert Between Fractions or Mixed Numbers and Decimals | |
Add Fractions and Convert to Decimals | |
Fractions as Percents | |
Convert Decimals to Percents and Fractions | |
Convert Decimals to Fractions | |
Convert Decimals into Simplified Mixed Numbers | |
Decimals as Percents | |
Compare and Order Fractions and Decimals | |
Convert between Decimals, Fractions, and Percents | |
Algebra | |
Real Numbers Variables, and Expressions | |
The Real Numbers | |
Order Real Numbers | |
Integers | |
Absolute Value of Integers | |
Integer Addition | |
Integer Subtraction | |
Integer Multiplication | |
Integer Division | |
The Order of Operations | |
Order of Operations | |
PEMDAS in Numerical Expressions | |
Algebra Expressions with Exponents | |
Algebra Expressions with Fraction Bars | |
Order of Operations and Variable Substitution | |
Algebra Expressions and Variables | |
Expressions and Variables | |
Evaluate Single Variable Expressions | |
Evaluate Expressions with One or More Variables | |
Words that Describe Mathematical Operations | |
Translate Between English Phrases and Algebraic Expressions | |
Calculator Use with Algebra Expressions | |
Properties and Axioms of Real Numbers | |
Real Number Properties and Axioms | |
Distributive Property | |
Expressions and the Distributive Property | |
When to Use the Distributive Property | |
Distributive Property to Evaluate Formulas with Decimals | |
Additive inverses and Absolute Values | |
Associative and Commutative Property with Decimals | |
Associative and Commutative Property with Fractions | |
Addition and Multiplication Properties with Real Numbers | |
Fraction and Mixed Number Applications | |
Simplifying Expressions | |
Simplify Variable Expressions Involving Addition and Subtraction | |
Simplify Variable Expressions Involving Multiplication and Division | |
Simplify Variable Expressions Involving Multiple Operations | |
Simplify Algebraic Expressions | |
Linear Equations | |
Basic Equations | |
Writing Basic Equations | |
Sentences as Single Variable Equations | |
Addition and Subtraction Phrases as Equations | |
Multiplication and Division Phrases as Equations | |
Input-Output Tables | |
Function Rules for Input-Output Tables | |
Input-Output Tables for Function Rules | |
One-Step Equations and the Properties of Equality | |
One-Step Equations and Properties of Equality | |
Single Variable Equations with Addition and Subtraction | |
Properties of Equality with Fractions | |
Single Variable Equations with Multiplication and Division | |
Properties of Equality with Decimals | |
Checking Solutions to Equations | |
Solve Two-Step Equations | |
Two-Step Equations | |
Two-Step Equations from Verbal Models | |
Two-Step Equations with Addition and Multiplication | |
Two-Step Equations with Addition and Division | |
Two-Step Equations with Subtraction and Multiplication | |
Two-Step Equations with Subtraction and Division | |
Applications of Two-Step Equations | |
Solve Multi-Step Equations | |
Multi-Step Equations | |
Multi-Step Equations with Like Terms and Distribution | |
Multi-Step Equations with Fractions | |
Multi-Step Equations with Decimals | |
Multi-Step Equations with Decimals, Fractions, and Parentheses | |
Applications of Multi-Step Equations | |
Equations with Variables on Both Sides | |
Solving for a Variable | |
Absolute Value Equations | |
Linear Equations in the Real World | |
Applications of Linear Equations | |
Problem-Solving Models | |
Guess and Check, Work Backward | |
Applications Using Linear Models | |
Distance, Rate, and Time | |
D = RT | |
Solving for Elapsed Time Related to Rate | |
Finding the Total Time Given a Distance between Two Objects | |
Applications of Finding Time Using Multiple Steps | |
Finding Total Distance Using Multiple Steps Word Problems | |
Find the Rate, Given Time and Distance | |
Solving Length and Distance Problems Involving Time | |
Linear Inequalities | |
Solve One Step Linear Inequalities | |
Inequalities on a Number Line | |
Inequalities that Describe Patterns | |
Inequality Expressions | |
Inequalities with Addition and Subtraction | |
Inequalities with Multiplication and Division | |
Multi-Step Inequalities | |
Checking Solutions to Inequalities | |
Compound Inequalities | |
Graph and Solve Absolute Value Inequalities | |
Intervals and Interval Notation | |
Applications with Inequalities | |
Graphs and functions | |
Graphing in the Coordinate Plane | |
Graphs in the Coordinate Plane | |
Points in the Coordinate Plane | |
Ordered Pairs in Four Quadrants | |
Coordinate Locations on a Map | |
Graphs on a Coordinate Plane | |
Graphs Based on Rules | |
Rules Based on Graphs | |
Identify Types of Linear and Nonlinear Graphs | |
Function Families | |
Functions and Function Notation | |
Function Notation | |
Domain and Range of a Function | |
Applications of Functions | |
Identify Functions and the Vertical Line Test | |
Even and Odd Functions and Function Symmetry | |
Operations on Functions | |
Composition of Functions | |
Graph Linear Equations in Two Variables | |
Graph of a Linear Equation in Two Variables | |
Graphs of Linear Equations | |
Horizontal and Vertical Line Graphs | |
Graph Using Intercepts | |
Problem Solving with Linear Graphs | |
Graphs of Absolute Value Equations | |
Linear and Absolute Value Function Families | |
Graphing Slope | |
Slope | |
Rates of Change | |
Slope of a Line Using Two Points | |
Graph Using Slope-Intercept Form | |
The Forms of a Linear Equation | |
Standard Form of Linear Equations | |
Slope-Intercept Form of Linear Equations | |
Point-Slope Form of Linear Equations | |
Forms of Linear Equations | |
Parallel and Perpendicular Lines | |
Equations of Parallel and Perpendicular Lines | |
Equations of Parallel Lines | |
Equations of Perpendicular Lines | |
Families of Lines | |
The Equation of a Line | |
Determining the Equation of a Line | |
Write an Equation Given the Slope and a Point | |
Write an Equation Given Two Points | |
Write a Function in Slope-Intercept Form | |
Fitting Lines to Data | |
Linear Interpolation and Extrapolation | |
Applications of Linear Interpolation and Extrapolation | |
Systems of Equations and Inequalities | |
Consistent and Inconsistent Linear Systems | |
Checking a Solution for a Linear System | |
Graphs of Linear Systems | |
Systems of Linear Equations in Two Variables | |
Solving Linear Systems | |
Linear Systems by Elimination | |
Solving Systems by Multiplying One Equation | |
Solving Systems by Multiplying Both Equations | |
Linear Systems by Elimination Using Multiplication | |
Systems Using Substitution | |
Comparing Methods for Solving Linear Systems | |
Applications of Linear Systems | |
Mixture Problems | |
Systems of Linear Equations in Three Variables | |
Linear Programming | |
Solving Systems of Linear Inequalities | |
Graphs of Systems of Linear Inequalities in Two Variables | |
Evaluate Solutions for Linear Inequalities in Two Variables | |
Graphing to Check Solutions to Systems of Linear Inequalities | |
Systems of Inequalities | |
Quadratic Inequalities | |
Polynomial and Rational Inequalities | |
Polynomials and Factoring | |
Monomials, Binomials, and Trinomials | |
Polynomials in Standard Form | |
Addition and Subtraction of Polynomials | |
Polynomial Multiplication | |
Multiply Polynomials by Monomials | |
Multiply Binomials by Binomials | |
Multiply Polynomials by Binomials | |
Multiply Polynomials by Polynomials | |
Polynomial Factoring | |
Factoring Polynomials | |
Monomial Factors of Polynomials | |
Polynomial Special Products | |
Special Products of Polynomials | |
Factor Difference of Squares | |
Factor Perfect Square Trinomials | |
Sum and Difference of Cubes | |
Quadratic Expressions | |
Factor Quadratics with a Leading Coefficient of 1 | |
Factor Quadratic Expressions with Negative Coefficients | |
Factor Quadratics | |
Factor by Grouping | |
Zero Product Principle | |
Applications of Factoring | |
Polynomial Division | |
Dividing Polynomials | |
Long Division of Polynomials | |
Synthetic Division of Polynomials | |
Finding Zeros of Polynomials | |
Zeroes and Intercepts of Polynomials | |
Graphing Polynomials | |
Identify Parts of Polynomial Graphs | |
Graphs of Polynomials Using Transformations | |
Graphs of Polynomials Using Zeros | |
Graphing Calculator to Analyze Polynomial Functions | |
Rational Expressions | |
Working with Rational Expressions | |
Simplifying Rational Expressions | |
Excluded Values for Rational Expressions | |
Restricted Domain and Range | |
Multiply and Divide Rational Expressions | |
Products and Quotients of Rational Expressions | |
Multiplication of Rational Expressions | |
Division of Rational Expressions | |
Complex Fractions | |
Add and Subtract Rational Expressions | |
Addition and Subtraction of Rational Expressions | |
Adding and Subtracting Rational Expressions where One Denominator is the LCD | |
Applications of Adding and Subtracting Rational Expressions | |
Percent | |
Percent Problems | |
Simple Interest | |
The Percent Equation | |
Proportions and Percents | |
Proportions | |
Proportions using LCD | |
Proportions using Cross-Multiplication | |
Rational Equations and Proportions | |
Applications of Ratios and Proportions | |
Proportion and Scale | |
Proportions and Scale | |
Dimensional Analysis | |
Applications of Scale and Indirect Measurement | |
Direct Variations | |
Direct Variation | |
Applications Using Direct Variation | |
Graphs of Linear Models of Direct Variation | |
Inverse Variation and Rational Functions | |
Direct and Inverse Variation | |
Inverse Variation Models | |
Estimate Graphs of Rational Functions | |
Horizontal and Vertical Asymptotes | |
Vertical Asymptotes | |
Horizontal Asymptotes | |
Oblique Asymptotes | |
Determining Asymptotes by Division | |
Inverse Variation Problems | |
Joint and Combined Variation | |
Applications Using Rational Equations | |
Roots and Irrational numbers | |
Exponents and Irrational Numbers | |
Negative and Zero Exponents | |
Fractional Exponents | |
Zero, Negative, and Fractional Exponents | |
Operations with Roots and Irrational Numbers | |
The Pythagorean Theorem | |
Pythagorean Theorem and its Converse | |
Pythagorean Triples | |
Converse of the Pythagorean Theorem | |
Solving Equations Using the Pythagorean Theorem | |
Applications Using the Pythagorean Theorem | |
Distance Formula | |
Midpoint Formula | |
Radical Expressions | |
Simplify Expressions with Radicals | |
Multiplication and Division of Radicals | |
Raising a Product or Quotient to a Power | |
Addition and Subtraction of Radicals | |
nth Roots | |
Radical Equations | |
Equations with Radicals | |
Equations with Square Roots | |
Equations with Radicals on Both Sides | |
Equations with Variables Under and not Under a Radical | |
Graphs of Square Root Functions | |
Shifts of Square Root Functions | |
Graphing Cube Root Functions | |
Square and Cube Root Function Families | |
Applications Using Radicals | |
Solving Quadratic and Exponential Equations and Functions | |
Quadratic and Exponential Equations and Functions | |
Solving Quadratic Equations | |
Use Square Roots to Solve Quadratic Equations | |
Square Root Applications | |
Completing the Square | |
Completing the Square when the Leading Coefficient Equals 1 | |
Vertex Form of a Quadratic Equation by Completing the Square | |
Use Graphs and Technology to Solve Quadratic Equations | |
The Quadratic Formula | |
The Discriminant | |
Quadratic Equation Applications | |
Graphing Quadratic Functions and Equations | |
Graph Quadratic Functions and Equations | |
Graphing with the Vertex Form of Quadratic Functions | |
Graphs of Quadratic Functions in Intercept Form | |
Roots to Determine a Quadratic Function | |
Quadratic Functions and Equations | |
Exponential Functions and Properties of Exponents | |
Properties of Exponents in Variable Expressions | |
Exponential Properties Involving Products | |
Exponential Properties Involving Quotients | |
Exponential Terms Raised to an Exponent | |
Exponential Properties in Variable Expressions | |
Exponential Growth and Decay Functions | |
Exponential Functions | |
Solving Equations with Exponents | |
Exponential Growth and Decay | |
Exponential Growth | |
Exponential Decay | |
Geometric Sequences and Exponential Functions | |
Graphs of Exponential Functions | |
Applications of Exponential Functions | |
Logarithms | |
Common and Natural Logarithms | |
Analysis of Logarithmic Graphs | |
Logarithm Properties | |
Change of Base | |
Product and Quotient Properties of Logarithms | |
Power Property of Logarithms | |
Inverse Properties of Logarithms | |
Logistic Functions | |
Linear, Exponential and Quadratic Models | |
Identifying Linear, Exponential, and Quadratic Models | |
Linear, Quadratic, and Cubic Models | |
Cubic Models | |
Applications of Function Models | |
Matrices | |
Matrix Algebra | |
Introduction to Matrices | |
Adding and Subtracting Matrices | |
Multiplying Matrices by a Scalar | |
Matrix Multiplication | |
Matrix Operations | |
Determinants | |
Cramer's Rule | |
Matrix Equations | |
Solving Matrix Equations | |
Augmented Matrices | |
Row Operations and Row Echelon Forms | |
Writing and Solving a Matrix Equation for a Linear System | |
Solving Linear Systems Using Matrices and Technology | |
Inverse Matrices | |
Applications of Matrices | |
Partial Fraction Expansions | |
Geometry | |
Basics of geometry | |
Geometry Terms | |
Shapes | |
2D Shapes | |
Identify Shapes | |
Identify Less Common Shapes | |
Composite Shapes | |
Line Segments | |
Definition of Line Segment | |
Midpoints and Segment Bisectors | |
Midpoint Formula | |
Points that Partition Line Segments | |
Angles | |
Introduction to Angles | |
Identification of Angles by Vertex and Ray | |
Measuring Angles | |
Classifying Angles | |
Congruent Angles and Angle Bisectors | |
Constructions and Bisectors | |
Angle Pairs | |
Angle Properties and Theorems | |
Complementary Angles | |
Supplementary Angles | |
Missing Measures of Complementary and Supplementary Angles | |
Vertical Angles | |
Basic Polygons | |
Classify Polygons | |
Vertices and Sides | |
Polygon Classification in the Coordinate Plane | |
Reasoning and Proof | |
Patterns and Reasoning | |
Basic Visual Patterns | |
Identify Basic Pattern Type | |
Number Patterns | |
Reasoning Types | |
Inductive Reasoning from Patterns | |
Conjectures and Counterexamples | |
Deductive Reasoning | |
Logic Statements | |
Truth Tables | |
And and Or Statements | |
Negative Statements | |
If Then Statements | |
Converse, Inverse, and Contrapositive Statements | |
Properties and Proofs | |
Introduction to Proofs | |
Properties of Equality and Congruence | |
Proofs: Angle Pairs and Segments | |
Proofs involving Parallel and Perpendicular Lines | |
Parallelogram Proofs | |
Proofs Involving Triangles | |
Indirect Proof in Algebra and Geometry | |
Lines | |
Line Types | |
Parallel and Skew Lines | |
Transversals | |
Parallel Lines and Transversals | |
Corresponding Angles | |
Alternate Interior Angles | |
Alternate Exterior Angles | |
Same Side Interior Angles | |
Angles and Perpendicular Lines | |
Lines in the Coordinate Plane | |
Parallel Lines in the Coordinate Plane | |
Perpendicular Lines in the Coordinate Plane | |
Line Construction | |
Triangles | |
Classifying Triangles | |
Classify Triangles | |
Classify Triangles by Angle Measurement | |
Classify Triangles by Side Measurement | |
Isosceles Triangles | |
Equilateral Triangles | |
Triangle Area and Perimeter | |
Area and Perimeter of Triangles | |
Triangle Area | |
Unknown Dimensions of Triangles | |
Triangle Congruency | |
CPCTC | |
Congruence Statements | |
Third Angle Theorem | |
Congruent Triangles | |
SSS | |
SAS | |
ASA and AAS | |
HL | |
Triangle Relationships | |
Triangle Angle Sum Theorem | |
Exterior Angles and Theorems | |
Midsegment Theorem | |
Perpendicular Bisectors | |
Angle Bisectors in Triangles | |
Concurrence and Constructions | |
Medians | |
Altitudes | |
Comparing Angles and Sides in Triangles | |
Triangle Inequality Theorem | |
Right Triangles | |
Pythagorean Theorem | |
The Pythagorean Theorem | |
Basics of Pythagorean Theorem | |
Pythagorean Theorem to Classify Triangles | |
Pythagorean Triples | |
Converse of the Pythagorean Theorem | |
Pythagorean Theorem Applications | |
Right Triangles in Algebra | |
Pythagorean Theorem and its Converse | |
Solving Equations Using the Pythagorean Theorem | |
Applications Using the Pythagorean Theorem | |
Distance Formula | |
Distance and Triangle Classification Using the Pythagorean Theorem | |
Distance Formula and the Pythagorean Theorem | |
Distance Between Parallel Lines | |
The Distance Formula and Algebra | |
Applications of the Distance Formula | |
Special Triangles | |
Special Right Triangles and Ratios | |
45-45-90 Right Triangles | |
30-60-90 Right Triangles | |
Quadrilaterals and Polygons | |
Basic Squares and Rectangles | |
Squares | |
Rectangles | |
Square and Rectangle Area and Perimeter | |
Perimeter of Squares and Rectangles | |
Area of Squares and Rectangles | |
Unknown Dimensions of Squares and Rectangles | |
Quadrilaterals | |
Quadrilateral Classification | |
Parallelogram Classification | |
Parallelograms | |
Area of a Parallelogram | |
Unknown Dimensions of Parallelograms | |
Estimation of Parallelogram Area in Scale Drawings | |
Trapezoids | |
Area and Perimeter of Trapezoids | |
Area of Parallelograms: Squares, Rectangles and Trapezoids | |
Kites | |
Area and Perimeter of Rhombuses and Kites | |
Area and Perimeter of Composite Shapes | |
Quadrilateral Classification in the Coordinate Plane | |
Polygons | |
Regular and Irregular Polygons | |
Area of Regular and Irregular Polygons | |
Area and Perimeter of Similar Polygons | |
Construct Regular Polygons | |
Congruent Polygons | |
Corresponding Parts of Congruent Figures | |
Polygon Angle Measures | |
Determine Missing Angle Measures | |
Interior Angles in Convex Polygons | |
Exterior Angles in Convex Polygons | |
Circles | |
Parts of a Circle | |
Semicircles and Quarter Circles | |
Identify Circle Components | |
Use Diameter, Radius and Pi | |
Circumference | |
Diameter or Radius of a Circle Given Circumference | |
Area of a Circle | |
Circle Area | |
Areas of Combined Figures Involving Semicircles | |
Radius or Diameter of a Circle Given Area | |
Arcs and Chords | |
Arcs in Circles | |
Area of Sectors and Segments | |
Arc Length | |
Chords and Central Angle Arcs | |
Segments from Chords | |
Circles and Angles | |
Inscribed Angles in Circles | |
Inscribed Quadrilaterals in Circles | |
Angles On and Inside a Circle | |
Angles Outside a Circle | |
Tangent Lines and Theorems | |
Tangent Lines | |
Intersecting Secants Theorem | |
Tangent Secant Theorem | |
Circles in the Coordinate Plane | |
Similarity | |
Similar Figures | |
Ratio and Proportion in Similar Figures | |
Similar Polygons and Scale Factors | |
Corresponding Parts of Similar Figures | |
Indirect Measurement | |
Indirect Measurement Applications | |
Triangle Similarity | |
AA Similarity | |
SSS Similarity | |
SAS Similarity | |
Triangle Proportionality | |
Proportional Triangles | |
Inscribed Similar Triangles | |
Parallel Lines, Transversals, and Proportionality | |
Proportions and Angle Bisectors | |
Theorems Involving Similarity | |
Dilation | |
Dilation of a Shape | |
Dilation in the Coordinate Plane | |
Mapping Dilations | |
Self-Similarity and Fractals | |
Rigid Transformations | |
Identify Transformation Types | |
Symmetry | |
Lines of Symmetry | |
Reflection Symmetry | |
Rotation Symmetry | |
Translations | |
Geometric Translation | |
Sliding Figures | |
Translation Notation | |
Translation Applications in Circle Similarity | |
Rotations | |
Defining Rotation | |
Rotation Rules | |
Reflections | |
Defining Reflection | |
Rules for Reflections | |
Reflecting Figures | |
Composition of Transformations | |
Composite Transformations | |
Notation for Composite Transformations | |
Tessellations | |
Solid Figures | |
Polyhedrons | |
Faces, Edges, and Vertices of Solids | |
Cross-Sections and Nets | |
Surface Area | |
Volume | |
Cross Sections and Basic Solids of Revolution | |
Composite Solids | |
Area and Volume of Similar Solids | |
Surface Area and Volume Applications | |
Prisms | |
Surface Area and Volume of Prisms | |
Surface Area of Prisms | |
Volume of Prisms | |
Volume of Prisms Using Unit Cubes | |
Volume of Rectangular Prisms | |
Volume of Triangular Prisms | |
Pyramids | |
Surface Area and Volume of Pyramids | |
Volume of Pyramids | |
Cylinders | |
Surface Area and Volume of Cylinders | |
Surface Area of Cylinders | |
Volume of Cylinders | |
Heights of Cylinders Given Surface Area or Volume | |
Cones | |
Surface Area and Volume of Cones | |
Surface Area of Cones | |
Volume of Cones | |
Spheres | |
Surface Area and Volume of Spheres | |
Surface Area of Spheres | |
Volume of Spheres | |
Trigonometry | |
Right triangles and the Pythagorean theorem | |
Pythagorean Theorem | |
Converse of the Pythagorean Theorem | |
Pythagorean Triples | |
Applications Using the Pythagorean Theorem | |
The Pythagorean Theorem for Area and Perimeter | |
Distance Formula and the Pythagorean Theorem | |
Pythagorean Theorem to Classify Triangles | |
Special Right Triangles and Ratios | |
45-45-90 Right Triangles | |
30-60-90 Right Triangles | |
Trigonometric Ratios | |
Trig Functions | |
Right Triangle Trigonometry | |
Calculator Trig Functions | |
SIN | |
COS | |
Sine and Cosine of Complementary Angles | |
TAN | |
SEC CSC COT | |
Solving Triangles | |
The Pythagorean Theorem and Trigonometry | |
Trig Function Applications | |
Angles of Elevation and Depression | |
Right Triangles and Bearings | |
Solve Right Triangles | |
Applications of Inverse Trigonometric Functions | |
Inverse Trig Functions using Algebra | |
Trig in the Unit Circle | |
Trigonometry and the Unit Circle | |
Measuring Rotation | |
Angles of Rotation in Standard Positions | |
Coterminal Angles | |
Signs of Trigonometric Functions | |
Trigonometric Functions and Angles of Rotation | |
Reference Angles and Angles in the Unit Circle | |
Trigonometric Functions of Negative Angles | |
Trigonometric Functions of Angles Greater than 360 Degrees | |
Exact Values for Inverse Sine, Cosine, and Tangent | |
Inverse Trigonometric Functions | |
Inverse Trig Functions | |
Inverses by Mapping | |
Composition of Trig Functions and Their Inverses | |
Definition of Inverse Reciprocal Trig Functions | |
Composition of Inverse Reciprocal Trig Functions | |
Trig Functions as Algebra Expressions | |
Graphing Trigonometric Functions | |
Radians | |
Radian Measure | |
Conversion between Degrees and Radians | |
Trig Functions and Radians with Technology | |
Rotations in Radians | |
Angular Velocity | |
Length of an Arc | |
Area of a Sector | |
Length of a Chord | |
Sine and Cosine Graphs | |
Sine Graph and Cosine Graph | |
Translating Sine and Cosine Functions | |
Vertical Translations | |
Horizontal Translations or Phase Shifts | |
Amplitude, Period, and Frequency | |
Amplitude | |
Period and Frequency | |
Trigonometric Identities and Equations | |
General Sinusoidal Graphs | |
Six Trig Function Graphs | |
Graphing Tangent, Cotangent, Secant and Cosecant | |
Tangent Graphs | |
Tangent and Cotangent Graphs | |
Sine and Cosecant Graphs | |
Cosine and Secant Graphs | |
Graph Inverse Trigonometric Functions | |
Trigonometric Identities | |
Trig Identities | |
Fundamental Trigonometric Identities | |
Quotient Identities | |
Reciprocal Identities | |
Pythagorean Identities | |
Even and Odd Identities | |
Cofunction Identities | |
Basic Trig Identity Applications | |
Trig Identities to Find Exact Trigonometric Values | |
Simplifying Trigonometric Expressions | |
Proofs of Trigonometric Identities | |
Simpler Form of Trigonometric Equations | |
Solving Trigonometric Equations | |
Solving Trigonometric Equations Using Basic Algebra | |
Trigonometric Equations Using the Quadratic Formula | |
Trigonometric Equations Using Factoring | |
Sum and Difference Identities | |
Sum and Difference Formulas | |
Simplifying Trigonometric Expressions using Sum and Difference Formulas | |
Sine Sum and Difference Formulas | |
Cosine Sum and Difference Formulas | |
Tangent Sum and Difference Formulas | |
Solving Trigonometric Equations using Sum and Difference Formulas | |
Finding Exact Trigonometric Values Using Sum and Difference Formulas | |
Applications of Sum and Difference Formulas | |
Double and Half Angle Identities | |
Double and Half Angle Formulas | |
Double Angle Identities | |
Simplifying Trigonometric Expressions with Double-Angle Identities | |
Solving Equations with Double-Angle Identities | |
Half Angle Formulas | |
Trigonometric Equations Using Half Angle Formulas | |
Sum to Product and Triple Angle Formulas | |
Sum to Product Formulas for Sine and Cosine | |
Product to Sum Formulas for Sine and Cosine | |
Triple-Angle Formulas and Linear Combinations | |
Non-Right Triangle Trigonometry | |
Law of Sines and Law of Cosines | |
Laws of Sines and Cosines | |
Law of Sines | |
Angle-Angle-Side Triangles | |
Angle-Side-Angle Triangles | |
Side-Side-Angle: The Ambiguous Case | |
Law of Cosines | |
Determination of Unknown Angles Using Law of Cosines | |
Identify Accurate Drawings of Triangles | |
Applications of the Law of Cosines | |
Trigonometry Word Problems | |
Area of Non-Right Triangles | |
Area Formula for Non-Right Triangles | |
Introduction to the Triangle Area Formula | |
Determination of Unknown Triangle Measures Given Area | |
Heron's Formula | |
General Solutions of Triangles | |
Polar System and Complex Numbers | |
The Polar Coordinate System | |
The Polar Coordinate System and Graphing | |
Plots of Polar Coordinates | |
Distance Between Two Polar Coordinates | |
Graph Polar Equations | |
Transformations of Polar Graphs | |
Special Polar Equations and Graphs | |
Polar Curves and Rectangular Conversions | |
Polar and Rectangular Conversions | |
Rectangular to Polar Conversions | |
Rectangular to Polar Form for Equations | |
Intersections of Polar Curves | |
Equivalent Polar Curves | |
Systems of Polar Equations | |
Complex Numbers | |
Imaginary and Complex Numbers | |
Imaginary Numbers | |
The Complex Numbers | |
Quadratic Formula and Complex Sums | |
Products and Quotients of Complex Numbers | |
Product and Quotient Theorems | |
Trigonometric Form of Complex Numbers | |
Polar Form of a Complex Number | |
DeMoivre's Theorem and nth Roots | |
DeMoivre's Theorem | |
Equations Using DeMoivre's Theorem | |
Geometry of Complex Roots | |
PreCalculus | |
Fundamentals | |
Real numbers | |
Exponents and radicals | |
Algebraic Expressions | |
Rational Expressions | |
Equations | |
Modeling with equations | |
Inequalities | |
Coordinate Geometry | |
Lines | |
Modeling variation | |
Functions | |
What is a function | |
Graphs of functions | |
Increase and decreasing functions | |
Average rate of change | |
Transformations of functions | |
Quadratic functions | |
Maxima and minima | |
Modeling with functions | |
Combining functions | |
One-to-one functions and their inverses | |
Polynomial and Rational functions | |
Polynomial functions and graphing them | |
Dividing polynomials | |
Real zeros of polynomials | |
Complex numbers | |
Complex zeros and fundamental theorem of algebra | |
Rational functions | |
Exponential and logarithmic functions | |
Exponential functions | |
Logarithmic functions | |
Laws of Logarithms | |
Exponential and Logarithmic equations | |
Modeling with exponential and logarithmic equations | |
Trigonometric Functions of Real Numbers | |
Unit Circle | |
Trigonometric Functions of Real Numbers | |
Trigonometric Graphs | |
Modeling Harmonic Motion | |
Trigonometric Functions of Angles | |
Angle measure | |
Trigonometry of right triangles | |
Trigonometric functions of angles | |
Law of sines | |
Law of cosines | |
Analytic Trigonometry | |
Trigonometric identities | |
Addition and subtraction formulas | |
Double-angle, half-angle, and sum-product formulas | |
Inverse trigonometric functions | |
Trigonometric equations | |
Polar Coordinates and Vectors | |
Polar coordinates | |
Graphs of polar equations | |
Polar form of complex numbers | |
Demoivre's theorem | |
Vectors | |
Dot Product | |
Systems of equations and inequalities | |
Systems of equations | |
Systems of linear equations in two variables | |
two variables | |
several variables | |
matrices | |
Algebra of matrices | |
Inverses of matrices and matrix equations | |
Determinants and Cramer's rule | |
Partial fractions | |
Systems of inequalities | |
Analytic Geometry | |
Parabolas | |
Ellipses | |
Hyperbolas | |
Shifted Conics | |
Rotation of axes | |
Polar equations of conics | |
Plane curves and parametric equations | |
Sequences and series | |
Sequences and summation notation | |
Arithmetic sequences | |
Geometric sequences | |
Mathematical induction | |
Binomial theorem | |
Single Variable Calculus | |
Functions and Models | |
Ways to represent a functions | |
Essential functions | |
New functions from old functions | |
Exponential functions | |
Inverse functions and logarithms | |
Limits and Derivatives | |
Limit of a functions | |
Calculating limits using the limit laws | |
Precise definition of a limit | |
Continuity | |
Limits at infinity: horizontal asymptotes | |
Derivatives and rates of change | |
Derivative as a function | |
Differentiation | |
Derivatives of polynomials and exponential functions | |
Product and quotient rules | |
Derivatives of trigonometric functions | |
Chain rule | |
Implicit differentiation | |
Derivatives of logarithmic functions | |
Exponential growth and decay | |
Linear approximations and differentials | |
Taylor polynomials | |
Hyperbolic functions | |
Applications of Differentiation | |
Max and min values | |
Mean value theorem | |
How derivatives affect the shape of a graph | |
Indeterminate form and l'Hospital's rule | |
Optimization Problems | |
Newton's method | |
Antiderivatives | |
Integrals | |
Areas and distances | |
Definite integrals | |
Fundamental theorem of calculus | |
Indefinite integrals and the Net change theorem | |
Substitution rule | |
Applications of Integration | |
Areas between curves | |
Volumes | |
Volumes by cylindrical shells | |
Average value of a function | |
Techniques of Integration | |
Integration by parts | |
Trigonometric integrals | |
Trigonometric Substitution | |
Integration of rational functions by partial fractions | |
Strategy for integration | |
Approximate integration | |
Improper integrals | |
Further applications of Integration | |
Arc length | |
Area of a surface of revolution | |
Applications to physics,engineering, economics, or biology | |
Probability | |
Differential Equations | |
Modeling with differential equations | |
Direction fields and Euler's method | |
Separable equations | |
Models for population growth | |
Linear equations | |
Parametric Equations | |
Curves defined by parametric equations | |
Calculus with parametric curves | |
Polar Coordinates | |
Areas and lengths in polar coordinates | |
Conic sections in polar coordinates | |
Infinite Sequences and Series | |
Sequences | |
Series | |
Integral test and estimates of sum | |
Comparison tests | |
alternating seres | |
Absolute convergence | |
Ratio test and root test | |
Power series | |
Representing functions as power series | |
Taylor and Maclaurin series | |
Applications of Taylor polynomials | |
Multivariable Calculus | |
Vectors and the geometry of space | |
3d coordinate system | |
Vectors | |
Dot Product | |
Cross Product | |
Equations of lines and planes | |
Cylinders and quadric surfaces | |
Vector Functions | |
Vectors functions and space curves | |
Derivatives and integrals of vector functions | |
Arc length and curvature | |
Velocity and acceleration | |
Partial derivatives | |
Functions of several variables | |
Limits and continuity | |
Partial derivatives | |
Tangent planes and linear approximations | |
Chain rule | |
Directional derivatives and the gradient vector | |
Maximum and minimum values | |
Lagrange multipliers | |
Multiple Integrals | |
Double integrals over rectangles | |
Iterated integrals | |
Double integrals over general regions | |
Double integrals in polar coordinates | |
Application of double integrals | |
Surface area | |
Triple integrals | |
Triple integrals in cylindrical coordinates | |
Triple integrals in spherical coordinates | |
Change of variables in multiple integrals | |
Vector Calculus | |
Vector Fields | |
Line integrals | |
Fundamental theorem for line integrals | |
Green's theorem | |
Curl and divergence | |
Parametric surfaces and their areas | |
Surface integrals | |
Stoke's Theorem | |
Divergence Theorem | |
Second-Order Differential Equations | |
Second-Order Linear Equations | |
Nonhomogenous linear equations | |
Applications of second-order differential eqatuions | |
Series Solutions | |
Linear Algebra | |
Systems of linear equations | |
What is linear algebra? | |
Solving systems of linear equations | |
Reduced row-echelon formulas | |
Types of solutions | |
Homogeneous Systems of Equations | |
Nonsingular matrices | |
Vectors | |
Vector Operations | |
Linear combinations | |
Spanning sets | |
Linear independence | |
Linear dependence and spans | |
Orthogonality | |
Matrices | |
Matrix operations | |
Matrix multiplication | |
Matrix inverses and systems of linear equations | |
Matrix inverses and nonsingular matrices | |
Column and row spaces | |
The four subsets | |
Vector spaces | |
Vector spaces | |
Subspaces | |
Linear independence and spanning sets | |
Bases | |
Dimension | |
Properties of Dimension | |
Determinants | |
Determinant of a matrix | |
Properties of determinants of matrices | |
Eigenvalues | |
Eigenvalues and eigenvectors | |
Properties of eigenvalues and eigenvectors | |
Similarity and diagonalization | |
Linear Transformations | |
Linear transformations | |
Injective linear transformations | |
Surjective linear transformations | |
Invertible linear transformations | |
Representations | |
Vector representations | |
Matrix representations | |
Change of basis | |
Orthonormal Diagonalization | |
Differential Equations | |
Basic Concepts | |
Definitions | |
Direction Fields | |
First Order Differential Equations | |
Linear Equations | |
Separable Equations | |
Exact Equations | |
Bernoulli differential equations | |
Substitutions | |
Intervals of validity | |
Modeling with first order differential equations | |
Equilibrium solutions | |
Euler's method | |
Second order differential equations | |
Basic concepts | |
Real roots | |
Complex roots | |
Repeated roots | |
Reduction of order | |
Fundamental sets of solutions | |
More details about Wronskian | |
Nonhomogenous differential equations | |
Undetermined coefficients | |
Variation of parameters | |
Mechanical Vibrations | |
Laplace Transforms | |
Definition | |
Laplace transform | |
Inverse Laplace transform(IVP) | |
Step functions | |
Solving IVP's with Laplace transforms | |
Nonconstant coefficient IVP's | |
IVP's with step functions | |
Dirac Delta function | |
Convolution Integral | |
Table of Laplace transforms | |
Systems of Differential Equations | |
Systems of differential equations basics | |
Solutions to systems | |
Phase plane | |
Real eigenvalues | |
complex eigenvalues | |
repeated eigenvalues | |
Nonhomogeneous systems | |
Laplace transform to solve a system | |
Series Solutions | |
constructing a series solutions | |
Euler equations | |
Higher order Differential Equations | |
Basic concepts for nth order linear equations | |
Linear homogenous differential equations | |
Undetermined coefficients | |
variation of parameters | |
laplace transforms | |
systems of differential equations | |
series solutions | |
Boundary Value Problems and Fourier Series | |
Boundary Value Problems | |
Eigenvalues and eigenfunctions | |
Periodic functions and orthogonal functions | |
Fourier sine series | |
Fourier cosine series | |
Fourier series | |
Convergence of fourier series | |
Partial Differential Equations | |
Heat equation | |
Wave equation | |
Separation of variables | |
terminology | |
Solving the heat equation | |
Heat equation with non-zero temperature boundaries | |
Laplace's equation | |
on a rectangle | |
on a disk | |
Vibrating string | |
wave equation solution | |
Proofs | |
Fundamentals | |
Sets | |
Introduction | |
Cartesian Products | |
Subsets | |
Power Sets | |
Union, Intersection, Difference | |
Complement | |
Venn Diagrams | |
Indexed Sets | |
Sets that are Number Systems | |
Russell's Paradox | |
Logic | |
Statements | |
And, Or, Not | |
Conditional Statements | |
Biconditional statements | |
Truth tables for statements | |
Logical equivalence | |
Quantifiers | |
Translating English to symbolic logic | |
Negating statements | |
Logical Inference | |
Counting | |
Counting lists | |
Factorials | |
Counting subsets | |
Pascal's triangle and the binomial theorem | |
Inclusion-exclusion | |
How to prove conditional statements | |
Direct Proof | |
Theorems | |
Definitions | |
Direct Proof | |
Using Cases | |
Treating similar cases | |
Contrapositive Proof | |
Contrapositive proof | |
Congruence of integers | |
Mathematical writing | |
Proof by Contradiction | |
Proving statements with contradiction | |
Proving conditional statements by contradiction | |
Combining techniques | |
More on Proofs | |
Proving non-conditional statements | |
If-and-only-If proof | |
Equivalent statements | |
Existence Proofs; Existence and Uniqueness Proofs | |
Constructive versus non-constructive proofs | |
Proofs Involving Sets | |
How to prove a∈A | |
How to prove A⊆B | |
How to prove A=B | |
Examples: Perfect Numbers | |
Disproof | |
Counterexamples | |
Disproving existence statements | |
Disproof by contradiction | |
Mathematical Induction | |
Proof by strong induction | |
Proof by smallest counterexample | |
Fibonacci Numbers | |
Relations, Functions, and Cardinality | |
Relations | |
Properties of relations | |
Equivalence of relations | |
Equivalence classes and partitions | |
The integers modulo n | |
Relations between sets | |
Functions | |
Functions | |
Injective and surjective functions | |
The pigeonhole principles | |
Composition | |
Inverse functions | |
Image and preimage | |
Cardinality of sets | |
Sets with equal cardinalities | |
Countable and uncountable sets | |
Comparing cardinalities | |
The Cantor-Bernstein-Schroeder Theorem | |
Combinatorics | |
Graph Theory | |
Elements of Graph Theory | |
Graph Models | |
Isomorphism | |
Edge Counting | |
Planar graphcs | |
Covering circuits and graph coloring | |
Euler cycles | |
Hamilton circuits | |
Graph coloring | |
Coloring theorems | |
Trees and searching | |
Properties of trees | |
Search trees and spanning trees | |
Traveling salesperson problem | |
Tree analysis of sorting algorithms | |
Network algorithms | |
Shortest paths | |
Minimum spanning trees | |
Network flows | |
Algorithmic matching | |
The Transportation problem | |
Enumeration | |
General counting methods for arrangements and selections | |
Basic counting principles | |
Simple arrangements and selections | |
Arrangements and selections with repetitions | |
Distributions | |
Binomial identities | |
Generating Functions | |
Generating function models | |
Calculating coefficients of generating functions | |
Partitions | |
Exponential generating functions | |
Recurrence Relations | |
Recurrence relation models | |
Divide-and-conquer relations | |
Solution of linear recurrence relations | |
Solution of inhomogeneous recurrence relations | |
Solutions with generating functions | |
Inclusion-Exclusion | |
Counting with Venn Diagrams | |
Inclusion-Exclusion Formula | |
Restricted positions and rook polynomials | |
Additional topics | |
Polya's Enumeration Formula | |
Equivalence and symmetry groups | |
Burnside's theorems | |
Cycle index | |
Polya's formula | |
Games with graphs | |
Progressively finite games | |
Nim-type games | |
Abstract Algebra | |
Preliminaries | |
Sets and equivalence relations | |
The integers | |
Mathematical induction | |
Division algorithm | |
Groups | |
Integer equivalence classes and symmetries | |
Definitions and examples | |
Subgroups | |
Cyclic groups | |
Cyclic subgroups | |
Multiplicative group of complex numbers | |
Method of repeated squares | |
Permutation groups | |
Definitions and notation | |
Dihedral groups | |
Cosets and Lagrange's Theorem | |
Cosets | |
Lagrange's theorem | |
Fermat's and Euler's Theorems | |
Intro To Cryptography | |
Private Key Cryptography | |
Public key cryptography | |
Algebraic Coding theory | |
Error-detecting and correcting codes | |
Linear codes | |
Parity-check and generator matrices | |
Efficient decoding | |
Isomorphisms | |
Definition and examples | |
Direct products | |
Normal subgroups and factor groups | |
Factor groups and normal subgroups | |
The simplicity of the alternating group | |
Homomorphisms | |
Group homomorphisms | |
The isomorphism theorem | |
Matrix group and symmetry | |
Matrix groups | |
Symmetry | |
Structure of groups | |
Finite Abelian groups | |
Solvable groups | |
Group actions | |
Groups acting on sets | |
The class equation | |
Burnside's Counting Theorem | |
The Sylow Theorems | |
The Sylow Theorem | |
Examples and applications | |
Rings | |
Rings | |
Integral domains and fields | |
Ring homomorphisms and ideals | |
Maximal and prime ideals | |
Polynomials | |
Polynomial rings | |
Division algorithm | |
Irreducible polynomials | |
Integral domains | |
Fields of fractions | |
Factorization in integral domains | |
Lattices and boolean algebras | |
Lattices | |
Boolean algebras | |
Algebra of electrical circuits | |
Fields | |
Extension Fields | |
Splitting Fields | |
Geometric Constructions | |
Finite Fields | |
Structure of a Finite Field | |
Polynomial Codes | |
Galois Theory | |
Field Automorphisms | |
The Fundamental Theorem | |
Applications | |
Number Theory | |
Basic properties of the integers | |
Divisibility and primality | |
Ideals and greatest common divisors | |
Consequences of unique factorization | |
Congruences | |
Equivalence relations | |
Definitions and basic properties of congruences | |
Solving linear congruences | |
The Chinese Remainder Theorem | |
Residue classes | |
Euler's phi function | |
Euler's theorem and Fermat's little theorem | |
Quadratic Residues | |
Computing with large integers | |
Asymptotic notation | |
Machine models and complexity theory | |
Basic integer arithmetic | |
Computing in Zn | |
Faster integer arithmetic | |
Euclid's algorithm | |
Basic Euclidean algorithm | |
Extended Euclidean algorithm | |
Computing modular inverses and Chinese remaindering | |
Speeding up algorithms via modular computation | |
An effective version of Fermat's two squares theorem | |
Rational reconstruction and applications | |
RSA cryptosystem | |
Distribution of primes | |
Chebyshev's Theorem on the density of primes | |
Bertrand's postulate | |
Merten's theorem | |
The sieve of Eratosthenes | |
The prime number theorem | |
Abelian groups | |
Definitions, basic properties, examples | |
Subgroups | |
Cosets and quotient groups | |
Group homomorphisms and isomorphisms | |
Cyclic groups | |
Structure of finite abelian groups | |
Rings | |
Definitions, basic properties, examples | |
Polynomial rings | |
Ideals and quotient rings | |
Ring homomorphisms and isomorphisms | |
The structure of Zn* | |
Finite and discrete probability distributions | |
Basic definitions | |
Conditional probability and independence | |
Random variables | |
Expectation and variance | |
Some useful bounds | |
Balls and bins | |
Hash functions | |
Statistical distance | |
Measures of randomness and the leftover hash lemma | |
Discrete probability distributions | |
Probabilistic algorithms | |
Basic definitions | |
Generating a random number from a given interval | |
Generate and test paradigm | |
Generating a random prime | |
Generating a random non-increasing sequence | |
Generating a random factored number | |
Complexity theory | |
Probabilistic primary testing | |
Trial division | |
Miller-Rabin test | |
Generating random primes using Miller-Rabin test | |
Factoring and computing Euler's phi function | |
Finding generators and discrete logarithms in Zp* | |
Finding a generator for Zp* | |
Computing discrete logarithms in Zp* | |
The Diffie-Hellman key establishment protocol | |
Quadratic reciprocity and computing modular square roots | |
The Legendre symbol | |
The Jacobi symbol | |
Computing the Jacobi symbol | |
Testing quadratic residuosity | |
Computing modular square roots | |
The quadratic residuosity assumption | |
Modules and vector spaces | |
Definitions, basic properties, examples | |
Submodules and quotient modules | |
Module homomorphisms and isomorphisms | |
Linear independence and bases | |
Vector spaces and dimensions | |
Matrices | |
Basic definitions and properties | |
Matrices and linear maps | |
Inverse of a matrix | |
Gaussian elimination | |
Applications of Gaussian elimination | |
Subexponential-time discrete logarithms and factoring | |
Smooth numbers | |
Algorithm for discrete logarithms | |
Algorithm for factoring integers | |
More rings | |
Algebras | |
The field of fractions of an integral domain | |
Unique factorization of polynomials | |
Polynomial congruences | |
Minimal polynomials | |
General properties of extension fields | |
Formal derivatives | |
Formal power series and Laurent series | |
Unique factorization domains | |
Polynomial arithmetic and applications | |
Basic arithmetic | |
Computing minimal polynomials in F[X]/(f)(I) | |
Euclid's algorithm | |
Computing modular inverses and Chinese remaindering | |
Rational function reconstruction and applications | |
Faster polynomial arithmetic | |
Linearly generated sequences and applications | |
Basic definition and properties | |
Computing minimal polynomials: a special case | |
Computing minimal polynomials: a more general case | |
Solving sparse linear systems | |
Computing minimal polynomials in F[X]/(f)(II) | |
The algebra of linear transformations | |
Finite fields | |
Preliminaries | |
Existence of finite fields | |
Subfield structure and uniqueness of finite fields | |
Conjugates, norms and traces | |
Algorithms for finite fields | |
Tests for and constructing irreducible polynomials | |
Computing minimal polynomials in F[X]/(f)(III) | |
Factoring polynomials: square-free decomposition | |
Factoring polynomials: the Cantor-Zassenhaus algorithm | |
Factoring polynomials: Berlekamp's algorithm | |
Deterministic factorization algorithms | |
Deterministic primality testing | |
Basic idea | |
Algorithm and its analysis | |
Cryptography | |
Classical Cryptography | |
Simple Cryptosystems | |
Shift Cipher | |
Substitution cipher | |
Affine cipher | |
Vigenere cipher | |
Hill cipher | |
Permutation cipher | |
Stream ciphers | |
Cryptanalysis | |
Cryptanalysis of affine cipher | |
Cryptanalysis of substitution cipher | |
Cryptanalysis of Vigenere cipher | |
Cryptanalysis of Hill cipher | |
Cryptanalysis of LFSR stream cipher | |
Shannons Theory | |
Introduction | |
Elementary Probability theory | |
Perfect secrecy | |
Entropy | |
Huffman Encodings | |
Properties of entropy | |
Spurious keys and unicity distance | |
Product cryptosystems | |
Block ciphers and advanced encryption standard(AES) | |
Introduction | |
Substitution-Permutation networks | |
Linear cryptanalysis | |
Piling-up lemma | |
Linear approximations of S-boxes | |
Linear attack on an SPN | |
Differential Cryptanalysis | |
Data Encryption standard(DES) | |
Description of DES | |
Analysis of DES | |
Advanced encryption standard | |
Description of AES | |
Analysis of AES | |
Modes of operations | |
Cryptographic Hash functions | |
Hash functions and data integrity | |
Security of hash functions | |
Random Oracle model | |
Algorithms in the random oracle model | |
Comparison of security criteria | |
Iterated hash functions | |
Merkle-Damgard Construction | |
Secure hash algorithm | |
Message authentication codes(MAC) | |
Nested MACs and HMAC | |
CBC-MAC and authenticated encryption | |
Unconditionally secure MACs | |
Strongly universal hash families | |
Optimality of deception probabilities | |
RSA cryptosystem and Factoring integers | |
Introduction to public-key cryptography | |
More number theory | |
Euclidean algorithm | |
Chinese remainder theorem | |
RSA cryptosystem | |
Implementing RSA | |
Primality testing | |
Legendre and Jacobi symbols | |
Solovay-Strassen algorithm | |
Miller-Rabin algorithm | |
Square roots modulo n | |
Factoring algorithms | |
Pollard p – 1 algorithm | |
Pollard Rho algorithm | |
Dixon's Random squares algorithm | |
Factoring algorithms in practice | |
Other attacks on RSA | |
Computing φ(n) | |
The decryption exponent | |
Wiener's low decryption exponent attack | |
Rabin cryptosystem | |
Security of Rabin cryptosystem | |
Semantic security of RSA | |
Partial information concerning plaintext bits | |
Optimal asymmetric encryption padding | |
Public-key Cryptography and Discrete Logarithms | |
ElGamal cryptosystem | |
Algorithms for the discrete logarithm problem | |
Shank's algorithm | |
Pollard Rho discrete logarithm algorithm | |
Pohlig-Hellman algorithm | |
Index Calculus method | |
Lower bounds on the complexity of genetic algorithms | |
Finite fields | |
Elliptic curves | |
Elliptic curves over the reals | |
Elliptic curves modulo a prime | |
Properties of elliptic curves | |
Point compression and ECIES | |
Computing point multiples on elliptic curves | |
Discrete logarithm algorithms in practice | |
Security of ElGamal systems | |
Bit security of discrete logarithms | |
Semantic security of ElGamal systems | |
Diffie-Hellman problems | |
Signature Schemes | |
Introduction | |
Security requirements for signature schemes | |
Signatures and hash functions | |
ElGamal signature scheme | |
Security of ElGamal signature scheme | |
Variants of ElGamal signature scheme | |
Schnorr signature scheme | |
Digital signature algorithm | |
Elliptic Curve DSA | |
Provably secure signature schemes | |
One-time signatures | |
Full domain hash | |
Undeniable signatures | |
Fail-stop signatures | |
Pseudo-random number generation | |
Introduction | |
Examples | |
Indistinguishably of probability distributions | |
Next bit predictors | |
Blum-Blum-Shub generator | |
Security of BBS generator | |
Probabilistic Encryption | |
Identification schemes and entity authentication | |
Introduction | |
Challenge-and-response in the secret-key setting | |
Attack model and adversarial goals | |
Mutual authentication | |
Challenge-and-response in the public-key setting | |
Certificates | |
Public-key identification schemes | |
Schnorr identification scheme | |
Security of the Schnorr identification scheme | |
Okamoto identification scheme | |
Guillou-Quisquater identification scheme | |
Identity-based identification schemes | |
Key distribution | |
Introduction | |
Diffie-Hellman Key Predistribution | |
Unconditionally secure key predistribution | |
The Blom Key Predistribution scheme | |
Key distribution patterns | |
Fiat-Naor key distribution patterns | |
Mitchell-Piper key distribution patterns | |
Session key distribution schemes | |
The Needham-Schroeder(NS) Scheme | |
The Denning-Sacco Attack on the NS scheme | |
Kerberos | |
The Bellare-Rogaway scheme | |
Key agreement schemes | |
Introduction | |
Diffie-Hellman key agreement | |
Station-to-station(STS) key agreement scheme | |
Security of STS | |
Known session key attacks | |
MTI key agreement schemes | |
Known session key attacks on MTI/A0 | |
Key agreement using self-certifying keys | |
Encrypted key exchange | |
Conference key agreement schemes | |
Public-key infrastructure | |
Introduction | |
A practical protocol: Secure socket layer | |
Certificates | |
Certificate life-cycle management | |
Trust models | |
Strict hierarchy model | |
Networked PKIs | |
Web Browser model | |
Pretty good privacy | |
Future of PKI | |
Alternatives | |
Identity-based Cryptography | |
The Cocks identity-based encryption scheme | |
Secret sharing schemes | |
Introduction: Shamir Threshold Scheme | |
A simplified (t,t)-threshold scheme | |
Access structures and general secret sharing | |
Monotone circuit construction | |
Formal definitions | |
Information rate and construction of efficient schemes | |
Vector space construction | |
An upper bound on the information rate | |
The decomposition construction | |
Multicast security and copyright protection | |
Introduction | |
Broadcast encryption | |
Improvement using Ramp Schemes | |
Multicast re-keying | |
Blacklisting scheme | |
The Naor-Pinkas Re-keying scheme | |
Logical key hierarchy | |
Copyright protection | |
Fingerprinting | |
Identifiable parent property | |
2-IPP Codes | |
Tracing illegally redistributed keys | |
Probability | |
Discrete Probability Distributions | |
Simulation of discrete probabilities | |
Discrete probability distributions | |
Continuous probability densities | |
Simulation of continuous probabilities | |
Continuous density functions | |
Combinatorics | |
Permutations | |
Combinations | |
Card Shuffling | |
Conditional Probability | |
Discrete conditional probability | |
Continuous conditional probability | |
Paradoxes | |
Distributions and densities | |
Important distributions | |
Important densities | |
Expected Value and Variance | |
Expected value | |
Variance of discrete random variables | |
Continuous random variables | |
Sums of random variables | |
Sums of discrete random variables | |
Sums of continuous random variables | |
Law of large numbers | |
Discrete random variables | |
Continuous random variables | |
Central limit theorem | |
Bernoulli trials | |
Discrete independent trials | |
Continuous independent trials | |
Generating functions | |
Discrete distributions | |
Branching Process | |
Continuous Densities | |
Markov chains | |
Introduction | |
Absorbing Markov Chains | |
Ergodic Markov Chains | |
Fundamental Limit theorem | |
Mean First Passage Time | |
Random Walks | |
Random walks in Euclidean space | |
Gambler's Ruin | |
Arc Sine laws |
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The course is pretty extensive, but I think you should add Modular Arithmetic at the end of the Arithmetic section. It is a fundamental topic that will be used many times in the other sections such as Lagrange's theorem, Euler's theorem, etc.