Kindergarten Level

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• Representing, Relating, & Operating on Whole Numbers (Using Sets of Objects)
• Number Names & Counting Sequence
• Counting the Number of Objects
• Compare Numbers
• Operations & Algebraic Thinking
• Understand Addition as ‘Putting Together’ & ‘Adding To’
• Understand Subtraction as ‘Taking Apart’ & ‘Taking From’
• Work with Numbers 11–19 to Gain Foundations for Place Value
• Describe & Compare Measurable Attributes
• Classify Objects & Count the Number of Objects in Categories
• Identify & Describe Shapes and Space
• Analyze, Compare, Create, & Compose Shapes

1st Grade Level

• Represent & Solve Problems Involving Addition & Subtraction
• Understand & Apply Properties of Operations
• Relationship Between Addition & Subtraction
• Add & Subtract Within 20
• Work with Addition & Subtraction Equations
• Extend the Counting Sequence
• Understand Place Value
• Use Place Value Understanding & Properties of Operations to Add & Subtract
• Measure Lengths Indirectly & by Iterating Length Units
• Tell & Write Time
• Reason with Shapes & Their Attributes

2nd Grade Level

• Represent & Solve Problems Involving Addition & Subtraction
• Add & Subtract Within 20
• Work with Equal Groups of Objects to Gain Foundations for Multiplication
• Understand Place Value
• Use Place Value Understanding & Properties of Operations to Add & Subtract
• Measurement & Data
• Measure & Estimate Lengths in Standard Units
• Relate Addition & Subtraction to Length
• Work with Time & Money
• Reason with Shapes & Their Attributes

3rd Grade Level

• Represent & Solve Problems Involving Multiplication & Division
• Understand Properties of Multiplication & the Relationship Between Multiplication & Division
• Multiply & Divide Within 100
• Solve Problems Involving the Four Operations, & Identify & Explain Patterns in Arithmetic
• Use Place Value Understanding & Properties of Operations to Perform Multi-Digit Arithmetic
• Develop Understanding of Fractions as Numbers
• Solve Problems Involving Measurement & Estimation of Intervals of Time, Liquid Volumes, & Masses of Objects
• Understand Concepts of Area
• Relate Area to Multiplication & Addition
• Recognize Perimeter as an Attribute of Plane Figures
• Distinguish Between Linear & Area Measures
• Reason with Shapes & Their Attributes

4th Grade Level

• Use the Four Operations with Whole Numbers to Solve Problems
• Gain Familiarity with Factors & Multiples
• Generate & Analyze Patterns
• Generalize Place Value Understanding for Multi- Digit Whole Numbers
• Use Place Value Understanding & Properties of Operations to Perform Multi-Digit Arithmetic
• Extend Understanding of Fraction Equivalence & Ordering
• Build Fractions from Unit Fractions by Applying & Extending Previous Understandings of Operations on Whole Numbers
• Understand Decimal Notation for Fractions, & Compare Decimal Fractions
• Solve Problems Involving Measurement & Conversion of Measurements from A Larger Unit to A Smaller Unit
• Understand Concepts of Angle & Measure Angles
• Draw & Identify Lines & Angles
• Classify Shapes by Properties of Their Lines & Angles

5th Grade Level

• Operations & Algebraic Thinking
• Write & Interpret Numerical Expressions
• Analyze Patterns & Relationships
• Understand the Place Value System
• Perform Operations with Multi-Digit Whole Numbers & with Decimals to Hundredths
• Add & Subtract Fractions
• Multiply & Divide Fractions
• Measurement & Data
• Convert Like Measurement Units Within a Given Measurement System
• Understand Concepts of Volume
• Relate Volume to Multiplication & to Addition
• Graph Points on the Coordinate Plane to Solve Real-World & Mathematical Problems
• Classify 2-D Figures Into Categories Based on Their Properties

6th Grade Level

• Proportional Relationships
• Understand Ratio Concepts & Use Ratio Reasoning to Solve Problems
• The Number System
• Divide Fractions by Fractions
• Compute Fluently with Multi-Digit Numbers & Common Factors & Multiples
• Apply & Extend Previous Understandings of Numbers to the System of Rational Numbers
• Expressions & Equations
• Apply & Extend Previous Understandings of Arithmetic to Algebraic Expressions
• Reason About & Solve One-Variable Equations & Inequalities
• Represent & Analyze Quantitative Relationships Between Dependent & Independent Variables
• Solve Real-World & Mathematical Problems Involving Area, Surface Area, & Volume
• Develop Understanding of Statistical Variability
• Summarize & Describe Distributions

7th Grade Level

• Analyze Proportional Relationships & Use Them to Solve Real-World & Mathematical Problems
• The Number System
• Apply & Extend Previous Understandings of Operations with Fractions to Add, Subtract, Multiply, & Divide Rational Numbers
• Use Properties of Operations to Generate Equivalent Expressions
• Solve Real-Life & Mathematical Problems Using Numerical & Algebraic Expressions & Equations
• Draw, Construct & Describe Geometrical Figures & Describe the Relationships Between Them
• Solve Real-Life & Mathematical Problems Involving Angle Measure, Area, Surface Area, & Volume
• Use Random Sampling to Draw Inferences About a Population
• Draw Informal Comparative Inferences About Two Populations
• Investigate Chance Processes & Develop, Use, & Evaluate Probability Models

8th Grade Level

• Irrational Numbers & their Approximation by Rational Numbers
• Work with Radicals & Integer Exponents
• Understand the Connections Between Proportional Relationships, Lines, & Linear Equations
• Analyze & Solve Linear Equations & Pairs of Simultaneous Linear Equations
• Define, Evaluate, & Compare Functions
• Use Functions to Model Relationships Between Quantities
• Understand Congruence & Similarity Using Physical Models, Transparencies, or Geometry Software
• Understand & Apply the Pythagorean Theorem
• Solve Real-World & Mathematical Problems Involving Volume of Cylinders, Cones & Spheres
• Investigate Patterns of Association in Bivariate Data

Basic Math

• Whole Numbers
• Signed Numbers
• Fractions
• Decimals
• Conversions
• Measurement
• Basic Algebra
• Proportion & Percent
• Formula & Application Problems
• Powers & Roots

Pre-Algebra

• Review of Natural Number Arithmetic
• Integers, Fractions, Decimals & Negative Numbers
• Factorization of Natural Numbers
• Properties of Operations (Associative, Distributive, etc.)
• Simple (Integer) Roots & Powers
• Order of Operations: P-E-M-D-A-S
• Basics of Equations, Including Rules for Invariant Manipulation of Equations
• Variables & Exponentiation
• Explore & Understand Integers
• Inequalities & One-Step Equations
• Discover Fractions & Factors
• More About the Four Rules of Arithmetic
• More About Equation & Inequalities
• Graphing & Functions
• Ratios & Percent
• Probability & Statistics
• Introducing Geometry
• Area & Volume
• Right Triangles & Algebra

Algebra 1 / College Algebra

• Discovering Expressions, Equations & Functions
• Exploring Real Numbers
• How to Solve Linear Equations
• Visualizing Linear Functions
• Formulating Linear Equations
• Linear Inequalities
• Systems of Linear Equations & Inequalities
• Exponents & Exponential Functions
• Factoring & Polynomials
• Quadratic Equations
• Radical Expressions
• Rational Expressions
• Seeing Structure in Expressions
• Interpret the Structure of Expressions
• Write Expressions in Equivalent Forms to Solve Problems
• Arithmetic with Polynomials & Rational Expressions
• Perform Arithmetic Operations on Polynomials
• Understand the Relationship Between Zeros & Factors of Polynomials
• Use Polynomial Identities to Solve Problems
• Rewrite Rational Expressions
• Creating Equations
• Create Equations That Describe Numbers or Relationships
• Reasoning with Equations & Inequalities
• Understand Solving Equations as A Process of Reasoning & Explain the Reasoning
• Solve Equations & Inequalities in One Variable
• Solve Systems of Equations
• Represent & Solve Equations & Inequalities Graphically

Finite Math

• Linear functions and Mathematical Models
• Systems of Linear Equations
• Matrix arithmetic
• Solve Linear Programming Problems
• Graphing systems of Linear Inequalities
• Sets
• Counting
• Multiplication principle
• Permutations
• Combinations
• Experiments
• Sample spaces and events
• Introduction to Probability
• Rules of Probability
• Counting techniques in probability
• Conditional probability
• Expected Value
• Variance
• Standard Deviation
• Binomial Distribution
• Normal Distribution
• Finance

Geometry

• Points, Lines, Planes & Angles
• Proof
• Perpendicular & Parallel
• Triangles
• Similarity
• Right Triangles & Trigonometry
• Quadrilaterals
• Transformations
• Circles
• Area
• Congruence
• Transformations in the Plane
• Understand Congruence in Terms of Rigid Motions
• Prove Geometric Theorems
• Make Geometric Constructions
• Similarity, Right Triangles, & Trigonometry
• Understand Similarity in Terms of Similarity Transformations
• Prove Theorems Involving Similarity
• Define Trigonometric Ratios & Solve Problems Involving Right Triangles
• Apply Trigonometry to General Triangles
• Circles
• Understand & Apply Theorems About Circles
• Find Arc Lengths & Areas of Sectors of Circles
• Expressing Geometric Properties with Equations
• Translate Between the Geometric Description & the Equation for A Conic Section
• Use Coordinates to Prove Simple Geometric Theorems Algebraically
• Geometric Measurement & Dimension
• Explain Volume Formulas & Use Them to Solve Problems
• Visualize Relationships Between 2-D & 3-D Objects
• Modeling with Geometry
• Apply Geometric Concepts in Modeling Situations

Algebra 2 & Trigonometry

• Linear Equations
• Inequalities
• Graphs
• Matrices
• Polynomials & Radical Expressions
• Quadratic Equations
• Functions
• Exponential & Logarithmic Expressions
• Sequences & Series
• Probability
• Trigonometry Equations & Inequalities
• How to Graph Functions & Linear Equations
• How to Solve System of Linear Equations
• Matrices
• Polynomials & Radical Expressions
• Quadratic Functions & Inequalities
• Conic Sections
• Polynomial Functions
• Rational Expressions
• Exponential & Logarithmic Functions
• Sequences & Series
• Discrete Mathematics & Probability

Precalculus

• Find the Inverse of a Function (Algebraically & Graphically)
• Evaluate & Graph Exponential & Logarithmic Functions
• Solve Exponential & Logarithmic Equations
• Change Expressions Between Logarithmic & Exponential Form
• Use Properties of Logarithms to Convert Expressions Involving Logarithms into Different Forms
• Express Angles in Degree or Radian Measure & Convert Between the Two Forms
• Find the Area of the Sector of a Circle
• Find the Linear Speed of an Object Traveling in Circular Motion
• Find the Values of Trigonometric Functions of Acute Angles
• Find the Remaining Trigonometric Functions Given the Value of One
• Understand & Use the Fundamental Trigonometric Identities
• Know the Exact Values of Trigonometric Functions of Certain Acute Angles
• Use Reference Angles to Determine Trigonometric Functions of General Angles
• Express Trigonometric Functions in Terms of the Wrapping Function
• Graph the Six Trigonometric Functions
• Graph Transformations of Sine & Cosine Functions
• Determine the Amplitude, Period & Phase Shift of Sinusoidal Functions
• Find Exact Values of Inverse Sine, Cosine & Tangent Functions for Particular Values
• Find Approximate Values of Inverse Sine, Cosine & Tangent Functions
• Find Exact Values for Expressions Involving Inverse Sine, Cosine & Tangent Functions
• Prove or Disprove Trigonometric Identities
• Use Sum & Difference Formulas to Find Exact Values of Trigonometric Functions
• Use Double-Angle Formulas & Half-Angle Formulas to Find Exact Values
• Solve Equations Involving Trigonometric Functions
• Solve Right Triangles
• Solve Oblique Triangles Using the Law of Sines & the Law of Cosines
• Solve Applications Involving Right Triangles
• Solve Applications Involving Oblique Triangles
• Find the Area of a Triangle Given Two Sides & the Included Angle
• Find the Area of a Triangle Given Three Sides
• Solve a System of Linear Equations Using: Substitution, Elimination & Row Reduction of Matrices

Introduction to Statistics

• Understand how the process of posing a question, collecting data relevant to that question, analyzing data, and interpreting data can help them find answers to real problems from their world.
• Explore Data with Graphs
• Measures of Center and Spread
• Explore the Relationship Between Two Variables
• Regression Analysis
• Gathering Data
• Probability Distributions
• Sampling Distribution of the Sample Proportion
• Sampling Distribution of the Sample Mean
• Confidence Interval for the Population Proportion & Mean
• Significance Test about the Population Proportion
• Comparing Two Independent Proportions & Means
• Comparing Means from Dependent Samples
• Summarize, Represent, & Interpret Data on A Single Count or Measurement Variable
• Summarize, Represent, & Interpret Data on Two Categorical & Quantitative Variables
• Interpret Linear Models
• Understand & Evaluate Random Processes Underlying Statistical Experiments
• Make Inferences & Justify Conclusions from Sample Surveys, Experiments & Observational Studies
• Conditional Probability & the Rules of Probability
• Understand Independence & Conditional Probability; Use them to Interpret Data
• Compute Probabilities of Compound Events in a Uniform Probability Model
• Calculate Expected Values
• Use Probability to Evaluate Outcomes of Decisions
• Counting
• Random Variables, Distributions, Quantiles, Mean Variance
• Conditional Probability
• Bayes' Theorem
• Base Rate Fallacy
• Joint Distributions, Covariance, Correlation, Independence
• Central Limit Theorem
• Pure Applied Probability (Data in an Uncertain World, * Perfect Knowledge of the Uncertainty)
• Bayesian Inference with Known Priors, Probability Intervals
• Conjugate Priors
• Applied Probability (Data in an Uncertain World, Imperfect Knowledge of the Uncertainty)
• Bayesian Inference with Unknown Priors
• Frequentist Significance Tests & Confidence Intervals
• Resampling Methods: Bootstrapping
• Linear Regression
• Computation, Simulation, & Visualization Using R & Applets
• Use Software & Simulation to Do Statistics (R)
• Become an Informed Consumer of Statistical Information
• Use Basic Counting Techniques (Multiplication Rule, Combinations, Permutations) to Compute Probability & Odds
• Use R to Run Basic Simulations of Probabilistic Scenarios
• Compute Conditional Probabilities Directly & Using Bayes' Theorem, & Check for Independence of Events
• Set Up & Work with Discrete Random Variables in Particular, Understand the Bernoulli, Binomial, Geometric & Poisson Distributions
• Work with Continuous Random Variables in Particular, Know the Properties of Uniform, Normal & Exponential Distributions
• Understand the Law of Large Numbers & the Central Limit Theorem
• Compute the Covariance & Correlation Between Jointly Distributed Variables
• Create & Interpret Scatter Plots & Histograms
• Understand the Difference Between Probability & Likelihood Functions, & Find the Maximum Likelihood Estimate for A Model Parameter
• Bayesian Updating with Discrete Priors to Compute Posterior Distributions & Posterior Odds
• Bayesian Updating with Continuous Priors
• Construct Estimates & Predictions Using the Posterior Distribution
• Find Credible Intervals for Parameter Estimates
• Use Null Hypothesis Significance Testing (NHST) to Test the Significance of Results, & Understand & Compute the P-Value for These Tests
• Use Specific Significance Tests Including, Z-Test T-Test (One & Two Sample), Chi-Squared Test
• Find Confidence Intervals for Parameter Estimates
• Use Bootstrapping to Estimate Confidence Intervals
• Compute & Interpret Simple Linear Regression Between Two Variables
• Set Up a Least Squares Fit of Data to A Model
• Random variables, distributions, quantiles, mean variance
• Conditional probability, Bayes' theorem, base rate fallacy
• Joint distributions, covariance, correlation, independence
• Central limit theorem
• Conjugate priors
• Linear regression

Calculus 1

• Functions & Graphs
• Inverse Functions
• Limit of a Function
• Algebraic Computation of Limits
• Continuity
• Exponential & Logarithmic Functions
• Techniques of Differentiation
• Derivatives of Trignometric, Exponential & Log Functions
• Rates of Change Rectilinear Motion
• Chain Rule
• Implicit Differentiation
• Related Rates
• Linear Approximation & Differentials
• Extreme Values of a Continuous Function
• Mean Value Theorem
• Sketching the Graph of a Function
• Curve Sketching with Asymptotes
• L’Hopital’s Rule
• Optimization in Physical Sciences, Etc
• Antidifferentiation
• Area as the Limit of a Sum
• Riemann Sums & the Definite Integral
• Fundamental Theorem of Calculus
• Integration by Substitution
• Introduction to Differential Equations
• Mean Value Theorem for Integrals
• Numerical Integration

Calculus 2

• Applications of Integration A Area Between Two Curves
• Volume: Disk/Washer Method
• Volume: Shell Method
• Average Value of a Function & the Mean Value Theorem for Integrals * Arc Length
• Surface Area
• Hydrostatic Force
• Moments & Centroids
• Integration Techniques A Integration by Parts
• Trigonometric integrals
• Trigonometric substitution
• Partial Fractions
• Integration by Tables & Computer Algebra Systems F Indeterminate Forms & Hospital’s Rule
• Improper Integrals
• Infinite Series A Sequences
• Series and convergence
• The divergence test C Geometric and telescoping series
• The Integral Test
• Comparison & Limit Comparison Tests F Alternating Series
• Absolute Convergence; the Ratio & Root Tests H Power Series
• Representation of Functions as Power Series
• Maclaurin & Taylor Series
• Approximation Using Maclaurin & Taylor Polynomials
• Parametric Equations, Polar Coordinates, & Conic Sections a Parametric Equations & Conic Sections
• Calculus with Parametric Curves
• Polar Coordinates
• Area in Polar Coordinates
• Differential Equations
• Separable Differential Equations, Exponential Growth, & Applications
• First-Order Linear Differential Equations & Applications
• Direction Fields
• Euler’s Method

Calculus 3 & Multivariable Calculus

• Rectangular Coordinates in 3-Space
• Vectors
• Dot Product; Projections
• Cross Product
• Parametric Equations of Lines
• Planes in 3-Space
• Quadric Surfaces
• Cylindrical & Spherical Coordinates
• Introduction to Vector-Valued Functions
• Calculus of Vector-Valued Functions
• Change of Parameter; Arc Length
• Unit Tangent, Normal & Binormal Vectors
• Curvature
• Motion Along a Curve
• Functions of Two of More Variables
• Limits & Continuity
• Partial Derivatives
• Differentiability
• Chain Rule
• Directional Derivatives & Gradients
• Tangent Planes & Normal Vectors
• Maxima & Minima of Functions of Two Variables
• Lagrange Multipliers
• Double Integrals
• Double Integrals Over Nonrectangular Regions
• Double Integrals in Polar Coordinates
• Surface Area; Parametric Surfaces
• Triple Integrals
• Triple Integrals in Cylindrical & Spherical Coordinates
• Change of Variables
• Jacobians
• Centers of Gravity
• Vector Fields
• Line Integrals
• Independence of Path
• Green’s Theorem
• Surface Integrals
• Flux
• The Divergence Theorem
• Stoke’s Theorem

Applied Calculus

• Limit Definition of a Function
• Limits of Functions Using Numerical, Graphical & Algebraic Methods
• Average Rate of Change & Instantaneous Rate of Change
• Use the Limit of Difference Quotient Definition of Derivative to Find Simple Derivatives
• Find the Derivative of any Elementary Function (Algebraic, Logarithmic or Exponential) or Combination Thereof
• Find Higher Order Derivatives
• Find the Slope of the Graph of a Function
• Find the Tangent Line to the Graph of a Function
• Use Derivatives to Find Marginal Cost & Marginal Revenue Functions
• Find Relative Extrema & Points of Inflection of a Function
• Use Derivatives to Describe the Graph of a Function
• Determine Relative & Absolute Extrema of a Function
• Solve Problems Involving Rectilinear Motion, Velocity & Acceleration
• Solve Optimization Problems from Various Areas, Including Business, Social & Natural Sciences
• Solve Growth & Decay Application Problems
• Use Hospital’s Rule to Determine Indeterminate Limits
• Write & Apply the Definition of an Indefinite Integral
• Find Antiderivatives Using Basic Integration Formulas & Rules
• Use an Initial Condition to Find a Particular Solution to an Integral Equation
• Write & Apply the Definition of a Definite Integral
• State & Apply the Fundamental Theorem of Calculus
• Use Recognition, Substitution, & Integration by Parts to Evaluate Both Definite & Indefinite Integrals

Set Theory

• Sets
• Relations, functions and orderings
• Natural numbers
• Finite, countable and uncountable sets
• Cardinal numbers
• Ordinal numbers
• Alephs
• The Axiom of Choice
• Arithmetic of cardinal numbers
• Sets of real numbers 15. Axiomatic set theory.

Differential Equations

• Solutions of First- & Second-Order Linear Differential Equations
• Solutions of Non-Linear Exact & Separable Equations
• Integrating Factors
• Homogeneous Equations
• Higher-Order Linear Equations
• Initial Value Problems
• Solutions as Functions of the Equation Parameters
• Laplace Transforms

Linear Algebra

• Systems of Linear Equations
• Row Reduction & Echelon Forms
• Matrix Operations, Including Inverses
• Block Matrices
• Linear Dependence & Independence
• Subspaces & Bases & Dimensions
• Orthogonal Bases & Orthogonal Projections
• Gram-Schmidt Process
• Linear Models & Least-Squares Problems
• Determinants & Their Properties
• Cramer's Rule
• Eigenvalues & Eigenvectors
• Diagonalization of A Matrix
• Symmetric Matrices
• Positive Definite Matrices
• Similar Matrices
• Linear Transformations
• Singular Value Decomposition