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Differential Equations

Examines first-order and higher-order linear equations using techniques like separation of variables, Laplace transforms, and power series. Connects mathematical models to physical systems such as population growth, fluid dynamics, and electrical circuits.

Math Numbers & Counting Counting Objects Number Names Comparing Numbers Number Operations Counting Sequence Numbers 0-10 Place Value Understanding and System Add and Subtract Within 20 Addition and Subtraction Concepts Addition and Subtraction Equations Addition and Subtraction Problems Foundations for Multiplication Multiplication and Division Properties Multiply and Divide Within 100 Multiplication and Division Problems Factors and Multiples Properties of Operations Patterns and Relationships Generate and Analyze Patterns Multi-Digit Arithmetic Place Value Operations Multi-Digit and Decimal Operations Numerical Expressions Four Operations and Patterns Four Operations Problem Solving Measurement & Data Measurable Attributes Measuring Lengths Measure and Estimate Lengths in Standard Units Relate Addition and Subtraction to Length Classifying and Counting Objects Time and Money Area Concepts and Measurement Perimeter and Area Measures Angle Concepts and Measurement Time, Volume, and Mass Measurement Unit Conversions Geometry Identifying Shapes Shapes and Attributes Shape Attributes and Reasoning Shapes and Composition Classifying 2D Figures Geometric Figures and Relationships Lines, Angles, and Shapes Angle, Area, and Volume Geometric Measurement Volume of 3D Shapes Coordinate Plane Applications Transformations in the Plane Congruence and Similarity Understand congruence in terms of rigid motions Similarity and Transformations Prove Theorems Involving Similarity Pythagorean Theorem Trigonometry for General Triangles Make Geometric Constructions Prove Geometric Theorems Theorems About Circles Find Arc Lengths And Areas of Sectors of Circles Volume Formulas Relationships Between Two-Dimensional and Three-Dimensional Objects Prove Simple Geometric Theorems Algebraically Translate Between Geometric Description and Equation for Conic Section Apply Geometric Concepts in Modeling Situations Fractions & Decimals Fractions as Numbers Building Fractions Fraction Equivalence and Ordering Adding and Subtracting Fractions Multiplying and Dividing Fractions Dividing Fractions Decimal Fractions Multi-Digit Computation and Factors Add, Subtract, Multiply, and Divide Rational Numbers Rational Number System Irrational Numbers and Approximations Ratios Ratios and Proportions Proportional Relationships Unit Rate Algebra Algebraic Expressions Generate Equivalent Expressions Quantitative Relationships Proportional Relationships and Linear Equations Equations and Inequalities Equation Solving and Reasoning Linear Equations and Systems Graph Equations and Inequalities Systems of Equations Real-World Algebraic Problems Quantitative Reasoning with Units Expression Structure Equivalent Expression Forms Radicals and Integer Exponents Rational Exponents Rational and Irrational Numbers Polynomial Operations Polynomial Identities Polynomial Zeros and Factors Rational Expressions Complex Number Operations Complex Numbers in Polynomials Complex Numbers on Plane Statistics & Probability Represent and Interpret Data Data Distributions Statistical Variability Probability Models Compound Event Probabilities Statistical Sampling Interpret Categorical and Quantitative Data Bivariate Data Patterns Interpret Linear Models Comparing Two Populations Random Processes in Statistics Independence and Conditional Probability Expected Values Probability-Based Decision Making Statistical Inference and Conclusions Functions Function Concepts and Notation Define and Compare Functions Interpret Functions in Context Analyze Function Representations Model Relationships with Functions Identify Linear vs Exponential Growth Distinguish Between Function Types Compare Growth Rates Interpret Function Expressions Build Functions from Relationships Construct and Model Functions Transform and Combine Functions Model Comparison and Selection Solve Exponential Equations Trigonometry Trigonometric Ratios Involving Right Triangles Trigonometric Functions and Unit Circle Model with Trigonometric Functions Trigonometric Identities Vectors & Matrices Introduction to Vectors and Matrices Vector Quantities Vector Operations Matrix Operations Calculus Limits and Continuity Derivative Concepts and Notation Derivative Rules and Techniques Applications of Derivatives Optimization Problems Related Rates Curve Sketching and Analysis Integration Concepts and Notation Antiderivatives and Indefinite Integrals Definite Integrals and Area Fundamental Theorem of Calculus Integration Techniques Applications of Integration Differential Equations Sequences and Series Parametric and Polar Functions Vector-Valued Functions

Sequences

Curvilinear Blueprint Slides

Curvilinear Coordinates and Vector Analysis

This mathematical physics sequence explores the coordinate systems necessary for solving problems involving complex shapes, moving beyond Cartesian coordinates to General Curvilinear systems. Students derive scale factors, volume elements, and differential operators, culminating in solving Laplace's equation and understanding metric tensors.

Flight Path Slides

Vector Velocity

An advanced exploration of vector-valued functions and their applications in modeling 2D motion and force, preparing students for multivariable calculus.

Functional Foundations Slides

Path of Least Resistance

A graduate-level exploration of the Calculus of Variations, focusing on optimizing functionals. Students derive the Euler-Lagrange equation and apply it to physics and geometry problems like the Brachistochrone and Isoperimetric challenges.

Topology Foundations Slides

Theoretical Foundations of Convex Optimization

This sequence establishes the rigorous mathematical underpinnings necessary for advanced optimization work, moving beyond procedural calculus to analysis-based proofs. Students explore the intersection of topology, set theory, and multivariate calculus to determine the existence and uniqueness of optimal solutions.

Ordinary Points Lecture Slides

Series Solutions and Special Functions

A comprehensive graduate-level exploration of series solutions for differential equations with variable coefficients, focusing on power series, the Method of Frobenius, and the properties of Bessel and Legendre functions within the framework of Sturm-Liouville theory.

Linear Systems Slides

Qualitative Analysis and Stability of Nonlinear Systems

A graduate-level exploration of dynamical systems, focusing on the qualitative analysis of stability, phase portraits, and topological changes in nonlinear differential equations. Students move from linear classification to advanced stability proofs using Lyapunov functions and bifurcation theory.

Lessons

Doubling Duel Slides

The Doubling Duel

A 12th-grade financial math lesson comparing the 'Rule of 72' shortcut with exact continuous compounding formulas to determine investment doubling time. Students explore the accuracy of mental estimations versus logarithmic calculations across various interest rates.