Math Curriculum K-12 and Beyond

Numbers and Counting

• Counting to 20
• Number Recognition (0-20)
• Cardinality (Counting to know "how many")
• Understanding teen numbers (11-19)

Operations and Algebraic Thinking

• Introduction to Addition and Subtraction
• Making 10 with objects or drawings

Measurement and Data

• Comparing lengths and weights
• Sorting and Classifying objects

Shapes and Geometry

• Identifying 2D and 3D shapes
• Understanding positional words (above, below, next to)

Number Sense and Counting

• Counting forward and backward up to 120
• Using terms like "greater than," "less than," and "equal to" to compare numbers
• Beginning to understand the concept of odd and even numbers

Basic Operations: Addition and Subtraction

• Addition and Subtraction within 20
• Solving simple problems and word problems
• Recognizing that addition and subtraction are inverse operations

Place Value

• Understanding the value of tens and ones within two-digit numbers
• Breaking down numbers into tens and ones

Measurement and Data

• Measuring objects using non-standard units
• Telling time to the hour
• Collecting, organizing, and interpreting data

Geometry

• Identifying and describing basic 2D and 3D shapes
• Understanding the properties of shapes
• Creating and recognizing patterns using shapes

Numbers and Operations

• Addition and Subtraction within 1,000 and two-digit numbers
• Place Value (ones, tens, and hundreds)
• Skip Counting by 5s, 10s, and 100s
• Solving word problems with numbers and money

Measurement and Data

• Telling time and working with money
• Measuring objects using standard units (inches, feet, cm, meters)
• Interpreting data in graphs and charts

Geometry

• Recognizing and describing 2D and 3D shapes
• Understanding shape properties (sides, corners)
• Spatial relationships (endpoints, line segments)

Foundational Concepts

• Introduction to fractions (dividing a whole into equal parts)
• Introduction to Multiplication (repeated addition)

Number & Operations

• Multiplication and Division (within 100, using arrays, equal groups)
• Place Value & Rounding (to the nearest 10 or 100)
• Addition and Subtraction (up to 1,000)

Fractions

• Fractions as Parts of a Whole and on a number line
• Equivalency (comparing and finding equivalent fractions)

Measurement and Data

• Area and Perimeter (calculating area and perimeter of polygons)
• Time (telling time to the nearest minute and elapsed time)
• Graphs and Data (interpreting scaled bar and picture graphs)

Geometry

• 2-D Shapes (identifying and categorizing shapes by their properties)

Problem-Solving

• Solving a variety of real-world word problems

Numbers and Operations

• Multi-Digit Arithmetic (multiplication and long division)
• Place Value (to the millions, writing in expanded form)
• Factors, Multiples, and Prime/Composite Numbers
• Rounding whole numbers to any place value

Fractions and Decimals

• Equivalence and Comparison of fractions
• Operations with Fractions (add/subtract with like denominators)
• Introduction to decimal notation and comparison

Geometry

• Angles and Shapes (classifying, measuring, and solving problems)
• Area and Perimeter (calculating and understanding)

Measurement and Data

• Data Interpretation (charts and graphs, mean, median, mode)
• Conversions and Measurement (converting units)

Algebraic Thinking

• Expressions and Equations
• Number and shape patterns

Problem-Solving

• Multi-Step Word Problems

Numbers and Operations

• Whole Numbers (adding, subtracting, multiplying, and dividing)
• Fractions (operations with unlike denominators)
• Decimals (place value, operations)
• Powers of Ten (multiplying and dividing by 10, 100, 1000)
• Algebraic Thinking (solving expressions with variables and parentheses)

Geometry and Measurement

• Area and Perimeter (calculating for various shapes)
• Volume (rectangular prisms)
• Coordinate Plane (graphing points and ordered relationships)
• Geometric Shapes (classifying quadrilaterals)

Data Analysis

• Line Plots (creating and interpreting)

Key Skills to Develop

• Fluency in calculations
• Critical Thinking for complex problems
• Connecting math to real-world applications

Number System & Arithmetic

• Rational Numbers (fractions, decimals, and percentages)
• Positive and Negative Numbers (integers, absolute value, coordinate plane)
• Factors and Multiples (GCF and LCM)

Algebraic Thinking

• Expressions (with exponents and order of operations)
• Equations and Inequalities (one-variable)

Ratios and Proportional Relationships

• Ratios and Rates (unit rates)
• Percents (discounts or tips)

Geometry

• Area, Surface Area, and Volume (polygons, three-dimensional solids)

Statistics and Probability

• Data Analysis (mean, median, mode, and variability)
• Graphical Representations (box plots and histograms)

Number Sense and Operations

• Rational Numbers (fractions, decimals, and percentages)
• Integers (addition, subtraction, multiplication, and division)
• Multi-Step Operations (solving word problems with rational numbers)

Algebraic Thinking

• Expressions and Equations (writing and simplifying)
• Solving Equations (one- and two-step linear equations and inequalities)
• Proportional Relationships (ratios, rates, and proportions)

Geometry

• Area and Perimeter (polygons and circles)
• Surface Area and Volume (3-D shapes)
• Scale Drawings (using scales to find measurements)

Data Analysis and Probability

• Statistics (random sampling, comparing two data sets)
• Probability (simple and compound events)

Expressions and Equations

• Linear Equations and Functions (solving and graphing linear equations, slope)
• Exponents and Scientific Notation (powers, negative exponents, very large/small numbers)
• Polynomials (basic operations with polynomials)

Geometry

• Transformations (translation, rotation, and reflection)
• The Pythagorean Theorem (finding unknown side lengths)
• Measurement (volume and surface area of cylinders, cones, and spheres)

Numbers and Operations

• Real Numbers (rational and irrational numbers)
• Roots (square roots and cube roots)
• Properties of Operations (commutative, associative, and distributive properties)

Data Analysis and Probability

• Data Representation (graphs and tables)
• Probability (basic probability concepts)

Chapter 1: Arithmetic to Algebra

• 1A: The Tables of Arithmetic
• 1B: The Number Line
• 1C: The Algorithms of Arithmetic

Chapter 2: Expressions and Equations

• 2A: Expressions
• 2B: Equations
• 2C: Solving Linear Equations
• 2D: Word Problems
• Chapter 2 Extras

Chapter 3: Graphs

• 3A: Introduction to Coordinates
• 3B: Statistical Data
• 3C: Equations and Their Graphs
• 3D: Basic Graphs and Translations

Chapter 4: Lines

• 4A: All About Slope
• 4B: Linear Equations and Graphs
• 4C: Intersections
• 4D: Applications of Lines

Chapter 5: Functions

• 5A: Functions - The Basics
• 5B: Function and Situations (Linear and Exponential)
• 5C: Functions and Situations (Word Problems)

Chapter 6: Exponents and Radicals

• 6A: Exponents
• 6B: Radicals
• 6C: Exponential Expressions and Functions

Chapter 7: Polynomials

• 7A: The Need for Identities - Equivalent Expressions
• 7B: Polynomials and Their Arithmetic
• 7C: Factoring to Solve Quadratics

Chapter 8: Quadratics

• 8A: The Quadratic Formula
• 8B: Quadratic Graphs and Applications
• 8C: Working With Quadratics

Chapter 1: An informal Introduction to Geometry

• 1A: Picturing and Drawing
• 1B: Constructing
• 1C: Geometry Software
• 1D: Invariants - Properties and Values That Don't Change
• 1.0 Habits of Mind (Geom)

Chapter 2: Congruence and Proof

• 2A: The Congruence Relationship
• 2B: Proof and Parallel Lines
• 2C: Writing Proofs
• 2D: Quadrilaterals and Their Properties

Chapter 3: Dissections and Area

• 3A: Cut and Rearrange
• 3B: Area Formulas
• 3C: Proof by Dissection
• 3D: Measuring Solids

Chapter 4: Similarity

• 4A: Scaled Copies
• 4B: Curved or Straight? Just Dilate!
• 4C: The Side-Splitter Theorems
• 4D: Defining Similarity

Chapter 5: Circles

• 5A: Area and Circumference
• 5B: Circles and 3.141592653589793238462643383...
• 5C: Classical Results About Circles
• 5D: Geometric Probability

Chapter 6: Using Similarity

• 6A: Similarity Proofs
• 6B: Exploring Right Triangles
• 6C: Volume Formulas

Chapter 7: Coordinates and Vectors

• 7A: Transformations
• 7B: Geometry in the Coordinate Plane
• 7C: Connections to Algebra

Chapter 8: Optimization

• 8A: Making the Least of a Situation
• 8C: Contour Lines
• 8B: Making the Most of a Situation
• 8D: Advanced Optimization

Chapter 1: Fitting Functions to Tables

• 1A: Tables
• 1B: Fitting the Data
• 1C: More About Recursive Models

Chapter 2: Functions and Polynomials

• 2A: About Functions
• 2B: Making it Fit
• 2C: Factors, Roots, and Zeros
• 2D: Advanced Factoring

Chapter 3: Complex Numbers

• 3A: Introduction to Complex Numbers
• 3C: Complex Plane, Graphing, Complex Numbers
• 3B: The Complex Plane

Chapter 4: Linear Algebra

• 4A: Gaussian Elimination
• 4B: Matrix Algebra
• 4C: Applications of Matrix Multiplication

Chapter 5: Exponential and Logarithmic Functions

• 5A: Working with Exponents
• 5B: Exponential Functions
• 5C: Logarithmic Functions

Chapter 6: Graphs and Transformations

• 6A: Transforming Basic Graphs
• 6B: Affine Tansformations
• 6C: Graphing Using Affine Transformations

Chapter 7: Sequences and Series

• 7A: The Need to Sum
• 7B: Sum Identities
• 7C: Arithmetic and Geometric Sequences and Series
• 7D: Pascal's Triangle and the Binomial Theorem

Chapter 1: Analyzing Trigonometric Functions

• 1A: The Cosine and Sine Functions
• 1B: Other Trigonometric Functions
• 1C: Trigonometric Functions and their Graphs

Chapter 2: Complex Numbers and Trigonometry

• 2A: Graphing Complex Numbers
• 2B: Trigonometric Identities
• 2C: DeMoivre's Theorem

Chapter 3: Analysis of Functions

• 3A: Analysis of Polynomial Functions
• 3B: Analysis of Rational Functions
• 3C: Analysis of Exponential and Logarithmic Functions

Chapter 4: Combinatorics

• 4B: Permutations and Combinations
• 4C: Making Connections
• 4A: Learning to Count

Chapter 5: Functions and Tables

• 5A: Mathematical Induction
• 5B: Newton's Difference Formula
• 5C: Closed-form and Recursive Functions

Chapter 6: Analytic Geometry

• 6A: Coordinate Geometry
• 6B: Conic Sections
• 6C: Vector Algebra and Geometry

Chapter 7: Probability and Statistics

• 7A: Probability and Polynomials
• 7B: Expectation and Variation
• 7C: The Normal Distribution

Chapter 8: Ideas of Calculus

• 8A: Finding Areas of Shapes
• 8B: Finding Areas Under Curves
• 8C: A Function Emerges

Limits and Continuity

• Calculating limits and infinite limits
• Defining continuity and the Intermediate Value Theorem

Derivatives

• Definition of the derivative
• Power rule, product rule, and quotient rule
• Chain rule and implicit differentiation

Applications of Derivatives

• Related rates
• Optimization problems
• Curve sketching

Transcendental Functions

• Inverse Functions
• The Inverse Trigonometric Functions
• First Order Differential Equations

Techniques of Integration

• Substitution
• Integration by Parts
• Partial Fractions
• Trigonometric Methods

Indeterminate Forms and Improper Integrals

• L’Hˆopital’s Rule
• Other Indeterminate Forms
• Improper Integrals: Infinite Intervals
• Improper Integrals: Finite Asymptotes

Sequences and Series

• Sequences
• Series
• Tests for Convergence
• Power Series
• Taylor Series

Numerical Methods

• Taylor Approximation
• Newton’s Method
• Numerical Integration

Conics and Polar Coordinates

• Quadratic Relations
• Eccentricity and Foci
• String and Optical Properties of the Conics
• Polar Coordinates
• Calculus in Polar Coordinates

Second Order Linear Differential Equations

• Homogeneous Equations
• Behavior of the Solutions
• Applications
• The Inhomogeneous equation

Vector Algebra

• Basic Concepts
• Vectors in the Plane
• Vectors in Space
• Lines and Planes in Space

Particles in Motion; Kepler’s Laws

• Vector Functions
• Planar Particle Motion
• Particle Motion in Space
• Derivation of Kepler’s Laws of Planetary Motion from Newton’s Laws

Coordinates and Surfaces

• Change of Coordinates in Two Dimensions
• Special Coordinate Systems
• Surfaces; Graphs and Level Curves
• Cylinders and Surfaces of Revolution
• Quadric Surfaces

Differentiable Functions of Several Variables

• The Differential and Partial Derivatives
• Gradients and Vector Methods
• Theoretical Considerations
• Optimization
• The Method of Lagrange Multipliers

Multiple Integration

• Integration on Planar Regions
• Applications
• Theoretical Considerations
• Integration in Other Coordinates
• Triple Interals
• Integration in Other Coordinates

Vector Calculus

• Vector Fields
• Line Integrals and Work
• Independence of Path
• Green's Theorem in the Plane
• Stokes’ and Gauss’ Theorems in Three Dimensions

Introduction

• Vectors and Matrices
• Lengths and Dot Products

Solving Linear Equations

• Linear Equations
• The Idea of Elimination
• Elimination Using Matrices
• Rules for Matrix Operations
• Inverse Matrices
• Elimination = Factorization: A = LU
• Transposes and Permutations

Vector Spaces and Subspaces

• Spaces of Vectors
• The Nullspace of A: Solving Ax = 0
• The Rank and the Row Reduced Form
• The Complete Solution to Ax=b
• Independence, Basis, and Dimension
• Dimensions of the Four Subspaces

Orthogonality

• Orthogonality of the Four Subspaces
• Projections
• Least Squares Approximations
• Orthogonal Bases and Gram-Schmidt

Determinants

• The Properties of Determinants
• Permutations and Cofactors
• Cramer's Rule, Inverses, and Volumes

Eigenvalues and Eigenvectors

• Introduction to Eigenvalues
• Diagonalizing a Matrix
• Applications to Differential Equations
• Symmetric Matrices
• Positive Definite Matrices
• Similar Matrices
• The Singular Value Decomposition

Linear Transformations

• The Idea of a Linear Transformation
• The Matrix of a Linear Transformation
• Change of Basis
• Diagonalization and the Pseudoinverse

Applications

• Graphs and Networks
• Markov Matrices and Economic Models
• Linear Programming
• Fourier Series: Linear Algebra for Functions
• Computer Graphics

Numerical Linear Algebra

• Gaussian Elimination in Practice
• Norms and Condition Numbers
• Iterative Methods for Linear Algebra

Complex Vectors and Complex Matrices

• Complex Numbers
• Hermitian and Unitary Matrices
• The Fast Fourier Transform

Introduction to Differential Equations

• Simple Equations and Explicit Solutions
• Graphical Solutions and Slope Fields
• Power Series Expansions

First-Order Equations

• Autonomous Equations (Logistic, Simple Models)
• Solving Separable and Homogeneous Equations
• Linear Differential Equations and Models

Second-Order Equations and Systems

• Homogeneous and Nonhomogeneous Equations
• Reduction of Order and Variation of Parameters
• Linear Autonomous Systems and Stability Analysis
• Nonlinear Autonomous Systems and Nullcline Analysis

Advanced Topics and Applications

• Laplace Transforms
• Models (Objects in Motion, Compartmental)
• Applications (Orthogonal Trajectories, Bungee Jumping)

Parametric Equations and Polar Coordinates

• Parametric Equations and Calculus
• Polar Coordinates
• Areas and Lengths in Polar Coordinates
• Conic Sections

Infinite Sequences and Series

• Sequences and Series
• Integral, Comparison, and Alternating Series Tests
• Absolute Convergence, Ratio and Root Tests
• Power Series, Taylor and Maclaurin Series
• Applications of Taylor Polynomials

Vectors and the Geometry of Space

• Three-Dimensional Coordinate Systems
• Vectors, Dot Product, and Cross Product
• Equations of Lines and Planes
• Cylinders and Quadric Surfaces

Vector Functions

• Vector Functions and Space Curves
• Derivatives and Integrals of Vector Functions
• Arc Length and Curvature
• Motion in Space: Velocity and Acceleration

Partial Derivatives

• Functions of Several Variables
• Limits and Continuity
• Partial Derivatives and Tangent Planes
• The Chain Rule
• Directional Derivatives and the Gradient Vector
• Maximum and Minimum Values
• Lagrange Multipliers

Multiple Integrals

• Double Integrals over Rectangles and General Regions
• Double Integrals in Polar Coordinates
• Applications of Double Integrals
• Triple Integrals in Cylindrical and Spherical Coordinates
• Change of Variables in Multiple Integrals

Vector Calculus

• Vector Fields
• Line Integrals and Work
• The Fundamental Theorem for Line Integrals
• Green's Theorem, Curl, and Divergence
• Parametric Surfaces and Surface Integrals
• Stokes' and The Divergence Theorem

Second-Order Differential Equations

• Second-Order Linear Equations
• Nonhomogeneous Linear Equations
• Applications of Second-Order Differential Equations
• Series Solutions

Part I: Theory of Integration and Functional Spaces

• Uniform Integrability and Vitali Convergence Theorem
• Completeness of L-P spaces (L p (E), 1 < p < infty)
• Weak Sequential Compactness in L p (E) Spaces

Part II: Structures and Operators in Functional Analysis

• Metric and Topological Spaces
• Banach Spaces and Bounded Linear Operators
• Operators in Hilbert Spaces

Part III: Measure Theory and Applications

• General Measure Theory and Integration
• The Dual of C(X)
• von Neumann's Theorem on Invariant Measures
• Ergodic Mappings and Probability Measures

Preliminaries to Complex Analysis

• Complex Numbers and the Complex Plane
• Continuous and Holomorphic Functions
• Power Series
• Integration along Curves

Cauchy's Theorem and its Applications

• Goursat's Theorem and Local Existence of Primitives
• Cauchy's Integral Formulas and Applications
• Schwarz Reflection Principle and Runge's Theorem

Meromorphic Functions and the Logarithm

• Zeros, Poles, and Residue Formula
• The Argument Principle
• Homotopies and Simply Connected Domains
• The Complex Logarithm

The Fourier Transform

• The Class F and its properties
• Paley-Wiener Theorem

Entire Functions

• Jensen's Formula and Functions of Finite Order
• Infinite Products and Weierstrass Factorization Theorem
• Hadamard's Factorization Theorem

The Gamma and Zeta Functions

• The Gamma Function: Analytic Continuation and Properties
• The Zeta Function: Functional Equation and Analytic Continuation

The Zeta Function and Prime Number Theorem

• Zeros of the Zeta Function
• Reduction to the functions v and v1
• Proof of the Asymptotics for v1

Conformal Mappings

• Conformal Equivalence and Examples
• Schwarz Lemma and Automorphisms
• The Riemann Mapping Theorem
• Conformal Mappings onto Polygons and Schwarz-Christoffel Integral

An Introduction to Elliptic Functions

• Elliptic Functions and Liouville's Theorems
• The Weierstrass p Function
• Modular Character and Eisenstein Series

Applications of Theta Functions

• Product Formula for the Jacobi Theta Function
• Generating Functions
• Theorems about Sums of Squares

Foundations and Tangent Spaces

• Introduction to Manifolds (Topological and Smooth)
• Submanifolds of Euclidean Space
• Tangent Spaces and Derivatives
• Vector Fields and Flows, The Lie Bracket
• Lie Groups and Lie Algebras
• Vector Bundles and Submersions
• The Theorem of Frobenius

The Levi-Civita Connection

• Second Fundamental Form and Covariant Derivative
• Parallel Transport and the Frame Bundle
• Christoffel Symbols and Riemannian Metrics

Geodesics and Curvature

• Length, Energy, and Geodesics
• The Exponential Map and Minimal Geodesics
• The Riemann Curvature Tensor
• Gauß–Codazzi Equations and Theorema Egregium
• Gaussian and Sectional Curvature

Advanced Topics

• Isometries and Conjugate Points
• Flat Spaces and Symmetric Spaces
• The Cartan–Hadamard Theorem
• Positive Ricci and Scalar Curvature