Fundamental concepts of limits, derivatives, and integrals for modeling change and motion. Examines techniques for differentiation and integration alongside applications in optimization, area calculation, and differential equations.
A comprehensive prep sequence for the most challenging questions on the ACT Math and Science sections. It focuses on high-level conceptual blueprints for math topics like complex numbers and matrices, alongside speed-reading and data-interpretation strategies for the Science section.
This sequence explores numerical analysis through the lens of sequences, focusing on iterative methods to approximate solutions to complex equations. Students investigate fixed-point iteration, Newton's method, convergence rates, and the transition into chaotic behavior.
An advanced exploration of vector-valued functions and their applications in modeling 2D motion and force, preparing students for multivariable calculus.
This advanced sequence bridges series to function approximation, introducing Power Series and Taylor Polynomials. Students discover how polynomials can mimic complex curves like sine and cosine, moving from simple tangent lines to higher-order polynomials while investigating convergence and approximation error.
A technical skill-building sequence for 11th-grade students focusing on the algebraic processes of finding antiderivatives, from basic power rules to solving initial value problems.
A comprehensive introduction to the formal definition of the derivative, moving from the algebraic construction of the difference quotient to the rigorous limit definition. Students explore differentiability, continuity, and notation fluency.
A foundational calculus unit bridging average and instantaneous rates of change. Students move from physical motion data to geometric visualization and numerical estimation, culminating in the qualitative sketching of derivative graphs and interpretation of notation in real-world contexts.
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.
This sequence guides 11th-grade students through the formal application of derivative tests to analyze function behavior. Students will master the First and Second Derivative Tests, the Extreme Value Theorem, and the analysis of non-differentiable points to find and justify relative and absolute extrema.
An inquiry-based exploration of the geometric relationships between functions and their derivatives. Students progress from visual observation of slope and concavity to algebraic analysis using sign charts, culminating in the ability to sketch complex curves from derivative data.
This sequence explores irrational numbers through the lens of numerical analysis and computer science. Students learn to approximate roots using Newton's Method, transition from manual calculation to algorithmic thinking, and analyze how computers handle infinite decimals.
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 rigorous introduction to the formal definition of the derivative, bridging the gap between average rates of change and instantaneous rates using limits, various notations, and graphical analysis of differentiability.
This sequence bridges the gap between pre-calculus algebra and calculus by rigorously defining the derivative through geometric intuition, the formal limit definition, and varied notation systems. Students transition from average to instantaneous rates of change, explore differentiability, and master the algebraic manipulation required to find derivatives from first principles.
This advanced calculus sequence guides students through the theory and application of vector-valued functions, covering limits, differentiation, integration, and their real-world applications in kinematics and projectile motion.
A comprehensive sequence on analyzing function behavior using calculus. Students move from identifying boundaries with limits to analyzing direction and curvature with derivatives, culminating in the ability to sketch complex functions by hand.
This sequence explores the calculus of polar functions, focusing on differentiation techniques. Students will learn to calculate slopes of tangent lines, identify horizontal and vertical tangents, analyze behavior at the pole, and apply optimization to find maximum and minimum distances from the origin.
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 workshop series on optimization in calculus. Students master the Extreme Value Theorem, learn to translate complex word problems into mathematical models, and apply differentiation to find optimal outcomes in number theory and geometric contexts.
A project-based calculus sequence where students use optimization to design efficient packaging. They transition from physical modeling to algebraic functions and derivative-based solutions to maximize volume and minimize material costs.
This sequence bridges the gap between theoretical calculus operations and applied problem-solving by focusing on optimization in real-world contexts. Students begin by mastering the 'modeling process'—translating verbal constraints into mathematical objective functions. Over five lessons, they progress from simple geometric maximization to complex economic minimization and physical efficiency problems. By the end, students will demonstrate proficiency in using the First and Second Derivative Tests to justify absolute extrema in manufacturing and design scenarios.
A project-based calculus unit where students apply curve sketching and derivative tests to real-world optimization problems, moving from modeling constraints to defending optimized designs.
A project-based unit where students apply polynomial calculus concepts to real-world scenarios like business profits, projectile motion, and engineering design. Students transition from abstract solving to modeling data and optimizing outcomes using regression, intercepts, and extrema.
A high-school geometry sequence focusing on the mathematical relationship between surface area and volume to solve optimization problems in manufacturing and design. Students progress from 2D isoperimetric problems to 3D packaging efficiency analysis.
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.
This sequence explores the calculus of related rates through the lens of 3D geometry and fluid dynamics. Students progress from simple spherical expansion to complex conical substitution and industrial net-flow applications.
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.
A foundational sequence for 11th-grade students on Related Rates in Calculus. Students move from static derivatives to dynamic, time-dependent rates of change, establishing a rigorous 4-step problem-solving protocol.
This sequence explores related rates in calculus through geometric modeling of 3D systems, including fluid dynamics and shadow propagation. Students progress from 2D similar triangle models to complex 3D variable elimination in conical tanks.
This sequence establishes the foundational skills for related rates in Calculus. It covers implicit differentiation with respect to time, translating word problems into notation, and solving problems involving Pythagorean relationships and geometric shapes.
This sequence guides students through the rigorous process of modeling and solving related rates problems. Learners progress from simple geometric expansions to complex multi-variable systems involving fluid dynamics and angular displacement, emphasizing a structured problem-solving protocol.
This sequence introduces advanced volume techniques in calculus, including the Shell Method and solids with known cross-sections. Students move from theoretical derivation to a project-based application where they model and calculate the volume of real-world objects.
A comprehensive 11th-grade calculus unit focused on strategic method selection for complex integration. Students transition from basic procedural fluency to high-level diagnostic thinking and real-world applications in physics and engineering.
A comprehensive 11th-grade calculus unit focusing on Partial Fraction Decomposition for integration. The sequence moves from pure algebraic skill-building to complex integration techniques and real-world logistic growth modeling.
This sequence introduces Integration by Parts as the inverse of the Product Rule, equipping students to handle products of unrelated functions. Through inquiry, students derive the formula, apply the LIATE heuristic, master the Tabular Method for repeated integration, and solve cyclic integrals.
A comprehensive 5-lesson unit for 11th Grade Calculus students focusing on the u-substitution method for integration, emphasizing pattern recognition, definite integral boundary changes, and advanced algebraic manipulation.
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.
This advanced calculus sequence guides students through the systematic application of complex integration techniques including integration by parts, partial fractions, and trigonometric substitution. Students move from basic antiderivatives to analyzing the algebraic structure of functions to determine the most efficient solution pathway.
A series of higher-level mathematics lessons exploring calculus foundations through engaging, thematic activities and visual demonstrations.
This sequence introduces 11th-grade students to the behavior of sequences and series as they approach infinity. Students explore convergence, divergence, summation notation, and the paradoxes of infinite geometric series and fractals.
This advanced sequence introduces powerful tools for analyzing series with factorials and powers, leading to the concept of power series. Students master the Ratio and Root tests, explore absolute versus conditional convergence, and conclude by connecting series to functions through Taylor polynomials.
A comprehensive unit for 11th Grade Calculus exploring geometric series through the lens of financial literacy and fractal geometry. Students transition from finite sums to infinite convergence, applying these models to population growth, Zeno's Paradox, and complex loan amortization.
This sequence introduces 11th-grade students to the fundamental concepts of mathematical sequences, bridging the gap between algebra and calculus by exploring arithmetic and geometric progressions, recursive and explicit notation, and the behavior of sequences as they approach infinity.
This sequence bridges algebra and calculus by formalizing numerical patterns. Students move from identifying arithmetic and geometric patterns to evaluating limits at infinity and applying the Monotonic Convergence Theorem to real-world models.
This sequence explores the behavior of rational functions, focusing on limits, asymptotes, and discontinuities. Students learn to distinguish between removable and non-removable discontinuities, analyze end behavior at infinity, perform polynomial division for slant asymptotes, and synthesize these skills to sketch complex functions.
This inquiry-based sequence explores transcendental numbers like Pi and Euler's number (e) to connect irrationality with real-world phenomena and geometry. Students investigate historical methods of approximation and modern infinite series.
An exploration of how irrational numbers are approximated in science, engineering, and computer science using algorithms like the Bisection Method and Newton's Method.
A comprehensive unit connecting differentiation and integration through the Fundamental Theorem of Calculus. Students transition from visualizing accumulation to mastery of algebraic evaluation, applying these concepts to real-world net change and total area problems.
This sequence guides students through the conceptual transition from geometry to calculus by investigating the Area Problem. Students begin by estimating areas under curves using geometric shapes, discovering the relationship between rectangle width and approximation accuracy. As the sequence progresses, learners formalize these approximations using Sigma notation and limits, ultimately defining the definite integral.
A comprehensive journey from approximating areas under curves with rectangles to the powerful Fundamental Theorem of Calculus. Students explore Riemann sums, the limit definition of the integral, geometric interpretations of area, and the mechanics of antiderivatives.
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.
This sequence explores the intersection of calculus and geometry through infinite series and fractals. Students investigate convergence and divergence using visual area models, fractal dimensions, and physical simulations like block stacking.
A comprehensive 11th Grade Calculus sequence covering applications of integration including arc length, surface area of revolution, centroids, and the theorems of Pappus. Students explore the geometric properties of curves and regions using analytical methods.
This sequence guides 11th-grade students through the transition from 2D area calculations to 3D volume determinations using integral calculus. Students will master vertical and horizontal slicing techniques for area, and progress to the Disk and Washer methods for rotational volumes.
This sequence guides students through the geometric applications of definite integrals, transitioning from two-dimensional area analysis to complex three-dimensional volume modeling and arc length. Students will master techniques including the area between curves, disk, washer, and shell methods, and the rectification of smooth curves.
This calculus sequence guides 11th-grade students through the integration techniques required to calculate area and arc length within polar coordinate systems. From the geometric derivation of the polar sector formula to complex multi-curve regions and boundary measurements, students apply integral calculus to circular geometries.
This sequence transitions students from listing terms to aggregating them, focusing on the rigorous use of summation notation. Through a workshop approach, learners practice manipulating sigma notation, applying properties of sums, and deriving formulas for arithmetic series.
A project-based sequence exploring infinite geometric series through Zeno's paradox, algebraic proofs of convergence, and fractal geometry. Students investigate how infinite additions can result in finite sums and apply these concepts to real-world paradoxes and self-similar shapes.
This sequence bridges the gap between radical notation and exponential notation, establishing a unified system for algebraic manipulation. Students begin by defining rational exponents through the lens of roots and powers, then systematically apply the laws of exponents to simplify expressions containing fractional powers.
