Right triangle ratios, unit circle definitions, and fundamental identities applied to angles and geometric relationships. Models periodic phenomena using trigonometric functions to analyze real-world cycles and waveforms.
This sequence introduces students to parametric equations through the lens of particle motion and physics simulations. Students progress from basic plotting and parameter elimination to advanced calculus applications involving derivatives, vectors, and arc length.
An inquiry-based exploration of calculus optimization, focusing on real-world efficiency in travel time, infrastructure cost, and business profit. Students progress from geometric shortest-paths to complex rate-based modeling.
A comprehensive exploration of the unit circle, bridging geometry and trigonometry by scaling triangles, defining radians, and utilizing symmetry to evaluate trigonometric functions.
A comprehensive calculus sequence for undergraduate students focused on the rigorous application of derivatives to industrial, geometric, and economic optimization problems. Students progress from basic modeling to multi-constraint capstone analysis.
This sequence explores matrices as geometric transformations of vectors. Students learn to visualize and calculate how matrices stretch, rotate, reflect, and shear space, culminating in a project where they design a computer graphics animation sequence.
A project-based sequence for 12th Grade students exploring linear transformations through the lens of computer graphics. Students learn to use 2x2 matrices to scale, reflect, shear, and rotate vectors, culminating in a retro video game animation project.
A comprehensive unit on trigonometric substitution in calculus, moving from geometric visualization of radicals to complex integration techniques and algebraic back-substitution. Students learn to map radical expressions onto right triangles and use trigonometric identities to simplify and solve integrals.
This advanced sequence explores related rates through the lens of geometric similarity and trigonometry, focusing on shadows and angular motion. Students move from linear proportions to complex angular derivatives, culminating in a mastery-based problem-solving seminar.
A comprehensive exploration of Related Rates using Pythagorean geometry, moving from basic ladder problems to complex multi-object motion. Students master the calculus of moving triangles through inquiry, digital modeling, and skill-building workshops.
This sequence explores trigonometric integration techniques, from power reduction and identity manipulation to the geometric power of trigonometric substitution. Students learn to bridge the gap between algebraic radicals and right-triangle geometry.
A systematic workshop-style approach to mastering related rates in Calculus. Students progress from foundational implicit differentiation to complex geometric modeling involving Pythagorean theorem, volume expansion, conical constraints, and trigonometric rates.
A high-level calculus sequence for 12th-grade students focused on related rates in complex physical and engineering contexts. Students explore trigonometric rates, multi-variable dependencies like the Ideal Gas Law, relative motion, and conclude with an engineering design project focused on safety protocols.
This sequence guides 9th-grade students through the algebraic representation of vectors. Moving from geometric drawings to coordinate components, students use trigonometry and the Pythagorean theorem to decompose, reconstruct, and add vectors with precision.
A comprehensive introduction to vectors through geometric representation, focusing on the distinction between scalars and vectors, visual addition/subtraction, scalar multiplication, and the transition to component form and magnitude calculation.
A comprehensive 11th-grade unit on vector quantities, moving from conceptual geometric representations to complex algebraic modeling in aviation and navigation contexts. Students master component resolution, vector arithmetic, and resultant force calculations.
A 5-lesson geometry sequence where students move from right-triangle trigonometry to general triangles. They derive the Law of Sines and Law of Cosines through inquiry and verify their accuracy via a hands-on measurement lab.
This sequence explores the 'Ambiguous Case' (SSA) of the Law of Sines through visualization, algebraic proof, and real-world application. Students move from physical constructions to systematic classification and problem-solving.
An advanced trigonometry sequence for 12th-grade students focusing on the Law of Sines and Cosines applied to 3D spatial problems, vector magnitudes, static forces, and structural engineering. Students transition from 2D calculations to analyzing complex 3D systems and physical forces.
This 12th-grade trigonometry sequence explores area calculations for non-right triangles using the Sine Area Formula and Heron's Formula. Students apply these techniques to complex polygon decomposition, area optimization, and a real-world land partitioning case study.
This project-based sequence applies general triangle trigonometry to real-world scenarios in navigation, surveying, and aviation. Students move from abstract geometric figures to interpreting word problems involving bearings, headings, and elevation angles, culminating in a simulated search-and-rescue operation.
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.
A comprehensive unit on trigonometric transformations, focusing on how parameters A, B, C, and D modify the parent sine and cosine functions. Students progress from simple vertical shifts to complex multi-parameter modeling.
A comprehensive unit for 12th Grade Calculus students focusing on the integration of polar functions to find area, arc length, and surface area. Students transition from Cartesian thinking to radial accumulation, mastering the geometry of circular sectors and polar coordinate transformations.
A comprehensive unit for 12th Grade Calculus students focusing on the derivation and application of derivatives in polar coordinates. Students transition from Cartesian slope to polar slope, analyze horizontal and vertical tangency, investigate behavior at the pole, and solve optimization problems involving polar curves.
Students transition from Cartesian to polar coordinates, exploring the geometry of circular grids and the equations that define complex curves like roses and lima\u00e7ons. The unit covers plotting, conversion, and advanced graphing analysis with a focus on symmetry and intersection.
An advanced 11th-grade Calculus unit focusing on the integration of parametric and polar coordinate systems. Students analyze motion, calculate complex areas, perform error analysis, and complete a final synthesis project based on particle kinematics.
A comprehensive unit on polar coordinates and functions, moving from basic plotting to complex intersections and symmetry. Students explore the geometric beauty of curves like roses and lima\u00e7ons while mastering the algebraic conversions between rectangular and polar systems.
A comprehensive exploration of the polar coordinate system, covering point plotting, coordinate conversion, and the analysis of complex polar curves including rose curves, limacons, and spirals. Students move from basic radial positioning to deep geometric analysis of symmetry and periodicity.
This sequence explores calculus in the polar coordinate system, focusing on differentiation and integration. Students will master finding slopes of tangent lines, calculating areas of polar regions and intersection areas, and determining arc lengths of polar curves.
This inquiry-driven sequence connects the geometric definitions of the unit circle to algebraic trigonometric identities. Students derive Pythagorean, reciprocal, and quotient identities through visualization and algebraic proof to foster deep conceptual understanding.
A comprehensive unit on graphing trigonometric functions, transitioning from the unit circle to complex transformations. Students explore amplitude, period, phase shifts, and vertical translations for sine, cosine, and tangent functions.
A comprehensive advanced calculus unit exploring the use of vector-valued functions to model and analyze motion in 2D and 3D space. Students will master differentiation, integration, and arc length calculations within a kinematic context, culminating in complex projectile modeling.
A 12th-grade inquiry into complex numbers through the lens of geometry and vector operations. Students transition from algebraic rules to visual intuition, exploring rotations, dilations, and translations in the complex plane.
A comprehensive exploration of complex numbers through a geometric lens, bridging algebraic arithmetic with vector transformations and polynomial theory for undergraduate students.
A 12th-grade calculus unit focusing on advanced integration techniques, including improper integrals, partial fractions, and trigonometric substitution, applied to real-world modeling scenarios like population growth and physics.
An undergraduate-level exploration of the roots of unity, connecting algebraic solutions of polynomial equations to geometric symmetry in the complex plane and the fractal nature of complex iteration.
A high school trigonometry sequence that uses the physics of sound waves to teach modeling with trigonometric functions. Students explore pitch as frequency, volume as amplitude, and the superposition of waves to create harmonies and digital tones.
A comprehensive unit on modeling real-world periodic phenomena using trigonometric functions across physics, biology, and astronomy. Students master identifying amplitude, midline, and period from contextual data to build and solve predictive equations.
A comprehensive unit where students act as data scientists to model real-world environmental phenomena using trigonometric functions. They progress from visual estimation to precise algebraic modeling and technological regression to predict future environmental conditions.
This sequence introduces students to parametric equations as a tool for modeling dynamic systems. Students explore the relationship between independent components, algebraic conversion to Cartesian form, and real-world applications like projectile motion and cycloids.
A comprehensive 11th-grade calculus sequence that synthesizes domain, intercepts, symmetry, asymptotes, derivatives, and concavity into a systematic curve sketching algorithm. Students progress from procedural mastery to critical analysis of technological limitations and a final synthesis project.
An advanced geometry sequence focusing on industrial applications of volume, including frustums, partial cylindrical volumes, displacement, and flow rates. Students integrate trigonometry and calculus-adjacent concepts to solve real-world engineering challenges.
A condensed 3-part Calculus sequence on Related Rates, moving from linear motion models to complex geometric constraints and angular velocity.
This sequence applies circular geometry to a global scale, introducing students to spherical geometry concepts used in navigation and aviation. Students treat the Earth as a sphere and use arc length formulas to calculate 'Great Circle' distances between cities, concluding with a flight path simulation.
A sophisticated sequence for undergraduate students bridging the gap between static geometry (arc length and sector area) and dynamic circular motion. This unit explores linear and angular velocity, Kepler's Second Law, satellite communication footprints, and visual angles.
A comprehensive sequence for undergraduate students focusing on the semantic interpretation of parameters in mathematical models, spanning linear, quadratic, exponential, logistic, and periodic functions.
This sequence immerses students in vector analysis through the lens of navigation and physics. Learners distinguish between scalars and vectors, represent them geometrically, decompose them using trigonometry, and apply these skills to solve complex real-world problems like flight navigation and static equilibrium.
This workshop-style sequence focuses on the mechanics of simplifying complex trigonometric expressions using algebraic techniques like factoring, combining fractions, and Pythagorean substitution. Students learn to treat trigonometric functions as algebraic variables to achieve fluency and precision in preparation for formal proofs.
An advanced exploration of the general second-degree equation, focusing on identifying, rotating, and graphing conics with cross-product terms using both trigonometric and matrix methods.
This undergraduate-level sequence explores the theoretical foundations and analytical applications of trigonometry for oblique triangles. Students derive the Law of Sines and Law of Cosines, analyze the geometric nuances of the SSA ambiguous case, and master advanced area formulas like Heron's, preparing them for calculus and physics.
An advanced undergraduate geometry sequence focusing on the synthetic and metric proofs of concurrency and collinearity. Students master Ceva's and Menelaus' Theorems to explore the deep architecture of triangle centers and the Euler Line.
An advanced exploration of similarity and proportionality in Euclidean geometry, focusing on Menelaus' and Ceva's Theorems, homothety, and the Euler Line. Students move from directed segments and area ratios to complex proofs of collinearity and concurrence suitable for undergraduate mathematics.
A comprehensive series of lessons for undergraduate Calculus II students, focusing on mastering advanced integration techniques including substitution, integration by parts, trigonometric methods, and partial fraction decomposition, culminating in a strategic synthesis workshop.
