Vector properties, magnitudes, and algebraic operations including addition and scalar multiplication. Introduces matrix representations, arithmetic, and computational techniques for solving linear systems.
Students transition from algebraic addition and subtraction of complex numbers to a visual, vector-based representation on the complex plane. This lesson uses a provided video as a foundation for algebraic mastery before extending into graphical verification.
Students will solve systems of linear equations derived from the geometric properties of mutually tangent circles, connecting spatial relationships to algebraic solutions.
Students explore the geometric representation of complex numbers, connecting the modulus formula to the Pythagorean theorem and distance formula through visualization and practice.
An undergraduate-level introduction to visualizing vectors as multi-dimensional data containers. Students explore real-world applications from catering costs to nutritional data, shifting the perspective of vectors from geometric arrows to abstract data structures.
A high school mathematics lesson focused on applying vector addition and bearing calculations to real-world navigation scenarios, specifically airplanes and watercraft.
An introductory lesson on vectors in the coordinate plane. Students distinguish between scalars and vectors, learn component form notation, and practice plotting vectors through a collaborative 'Battleship' style activity.
A lesson for Advanced Precalculus students focusing on expressing vectors as linear combinations of the unit vectors i and j, featuring physics-based force problems and rapid-fire conversion practice.
A high-school geometry or pre-calculus lesson focusing on converting vectors from magnitude and direction to component form through a hands-on 'robot programming' simulation. Students use trigonometry to translate movement commands into x and y displacements.
Students will explore the geometric representation of complex numbers and discover how complex addition corresponds to vector addition and the parallelogram rule.
A Pre-Calculus lesson on solving 3x3 systems of equations to find quadratic models, featuring a manual elimination method and a matrix inverse method using graphing calculators.
A Pre-Calculus lesson connecting algebraic complex number addition to geometric vector addition on the complex plane using a 'Vector Walk' approach. Students visualize addition as head-to-tail movements on a grid.
This lesson introduces the Rule of Sarrus as a shortcut for finding 3x3 determinants, comparing its efficiency against the standard cofactor expansion method through a competitive 'Method Battle'.
Students learn to calculate 3x3 determinants using the Cofactor Expansion method, focusing on minor matrices and the checkerboard sign pattern through video analysis and collaborative problem-solving.
A high-school geometry and linear algebra lesson where students use 3x3 matrix determinants to calculate the area of complex, irregular polygons by triangulating coordinate data.
This lesson introduces students to the determinant of 2x2 matrices. Students will learn the calculation formula, practice with various examples, explore matrices with a determinant of zero, and understand the geometric interpretation of a determinant as the area of a parallelogram.
A lesson where 11th-grade students use matrix determinants to calculate the area of geometric figures in the coordinate plane, culminating in a 'Polygon Surveyors' creative application.
A high school Pre-Calculus lesson that synthesizes vectors and complex numbers, highlighting their structural similarities in the coordinate and complex planes.
Students will bridge the gap between coordinate geometry and linear algebra by connecting the Shoelace Algorithm to matrix determinants. This lesson uses a step-by-step video demonstration followed by algebraic verification of the 3x3 matrix area formula.
A high school math lesson that connects abstract precalculus concepts to real-world engineering and science applications using video analysis and group research.
An advanced introduction to the metric tensor and non-Euclidean geometry, serving as a primer for General Relativity.
A 10th-grade advanced math lesson focused on the difference between standard trigonometric angles and navigational bearings, featuring vector addition applications.
A 45-minute Pre-Calculus lesson where students apply the dot product to real-world business scenarios, specifically calculating total inventory costs using multidimensional vectors. The lesson features a video-based case study on a catering business and a collaborative activity called 'The School Store'.
A 9th-grade Integrated Science and Math lesson where students use vector subtraction (relative velocity) to model and solve algebraic inequalities involving boat travel against river currents.
A 11th-grade lesson focusing on vector magnitude and direction using trigonometry, culminating in a 'Treasure Map' activity where students calculate components to find hidden coordinates.
A lesson exploring vector equality and translation invariance, helping students understand that a vector's identity is defined by its magnitude and direction rather than its position. Includes a warm-up, video analysis, a matching activity, and a reflection journal.
The sequence culminates with a realistic physics modeling lesson. Students set up and analyze parametric equations for projectiles, accounting for gravity and initial velocity vectors.
Students solve complex motion problems, such as finding the time when a particle is moving perpendicular to its position vector or closest to the origin.
A comparative analysis lesson where students rigorously distinguish between the net change in position (displacement vector) and the total scalar distance traveled (integral of speed).
Students extend calculus operations to vector components. They perform component-wise differentiation and integration to find velocity vectors from position and position vectors from velocity.
Students formally define vector-valued functions and explore limits and continuity. They learn to visualize the domain and output as vectors pointing to a path.
Students solve Laplace's equation for systems with spherical symmetry, introducing Legendre polynomials and Spherical Harmonics.
Students translate the Del operator into general curvilinear coordinates and apply these operators to physical vector fields.
Focusing on integration, students construct volume and area elements (Jacobians) for spherical and cylindrical geometries and practice integrating scalar fields over complex 3D domains.
Students derive basis vectors and scale factors for general orthogonal curvilinear coordinates and learn how to define position vectors in non-Cartesian geometries.
Computational estimation of expected payoffs for path-dependent derivatives using Geometric Brownian Motion and Monte Carlo simulations.
Analysis of tail risk through Value at Risk (VaR) and Expected Shortfall, focusing on the limitations of normal distributions.
Introduction to risk-neutral measures and binomial pricing models, using expected values to price options without arbitrage.
Application of expected value to asset returns using matrix algebra to derive the Efficient Frontier and optimize portfolios.
Students contrast mathematical expected value with expected utility to explain decision-making under uncertainty, analyzing different utility functions to model risk-averse behavior.
Exploring iterative methods like Jacobi and Gauss-Seidel for solving massive, sparse systems where elimination is computationally prohibitive.
Examination of ill-conditioned systems, condition numbers, and the impact of rounding errors on numerical stability.
A Pre-Calculus lesson focused on extending 2D distance concepts into 3D space to calculate the distance between a point and a plane. Students use coordinate geometry and algebraic manipulation to solve spatial problems.
