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.
Student investigation worksheet for the Circle Convergence lesson, guiding students through area estimation and the formal derivation of the circle area formula. Consolidated into a clean 2-page layout.
Student reference sheet for calculating regular polygon dimensions using scientific calculators, including apothem multipliers and trigonometric formulas.
Instructional slide deck for Circle Convergence, featuring video integration, vocabulary, and the proof derivation.
Teacher facilitation guide for the Circle Convergence lesson, including pacing, answer keys, and discussion prompts.
A comprehensive teacher facilitation guide for the Frankenstein Functions activity, including pacing, prep checklists, and a sample solution key.
A legal-pad styled reflection journal for students to articulate their understanding of limits and discontinuities through written prompts.
A teacher answer key for the 'Chaos at the Origin' worksheet, including sample student responses, graphing guidance, and instructional tips for the Squeeze Theorem introduction.
A set of activity cards for the 'Limit Lawyer' simulation, featuring various graph types (jump, hole, infinite, continuous) where students act as opposing counsels to determine the existence of an overall limit.
A formal grading rubric for the Frankenstein Functions project, assessing limit properties, graphing accuracy, and piecewise notation with a mad scientist theme.
A slide deck for teaching oscillating discontinuities, featuring a warm-up, video analysis, step-by-step tech activity instructions, and a comparison between persistent and dampened oscillation. All text is optimized for high visibility with sizes of 24px or larger.
A police report styled worksheet for Precalculus students to practice calculating limits from graphs, identifying discontinuities, and applying the formal definition of an overall limit.
A multi-page worksheet for the Frankenstein Functions activity, including a lab report section, requirements checklist, coordinate plane for graphing, and a page of cut-out function 'body parts'.
A courtroom-themed slide deck for a Precalculus lesson on limits, featuring instructional content, embedded video segments, and structured activities to guide students through overall limits and discontinuities.
A student worksheet for a calculus lesson on limits, featuring a warm-up comparison, video note-taking space, and a logic-based 'True/False/Fix' sketching activity.
An exit ticket asking students to synthesize their learning by sketching a single function that incorporates both a removable discontinuity (hole) and a jump discontinuity, followed by a brief written explanation of their design.
A set of interactive sorting cards for students to classify functions as Continuous, Removable, Jump, or Infinite. Includes equation/description cards and corresponding graph cards, some of which are intentionally blank to require student sketching.
A visual anchor chart for calculus students comparing function values and limits, including the formal rule for limit existence and visual examples of discontinuities.
A slide deck for the Frankenstein Functions lesson, featuring a mad scientist theme, warm-up activity, video review, and project instructions. Updated for visual hierarchy and font size compliance.
A comprehensive lab worksheet for students to record warm-up challenges, video notes on discontinuity types, and results from the continuity card sort activity. Features a clean blueprint-inspired layout with ample space for sketching and note-taking.
A comprehensive teacher guide for the Precalculus lesson 'Limit Lawsuit', featuring a minute-by-minute timeline, activity instructions, and key instructional strategies to address student misconceptions.
Teacher answer key for the Golden Function worksheet, including specific numerical values for slopes and conceptual guidance for the discussion questions.
A set of four discussion prompt cards designed to spark higher-order thinking during the AP Calculus activity on Euler's Number and the properties of e^x.
A student worksheet for the AP Calculus lesson on e^x. Includes a warm-up sketching area, a video notes section, an activity table for comparing exponential bases, and a closure reflection.
A presentation deck for AP Calculus students that introduces Euler's number and uses visual aids and a YouTube video to demonstrate the slope, height, and area relationships of e^x.
A detailed lesson plan for AP Calculus teachers outlining the flow of the Euler's Number lesson, including timestamps for the video and specific instructions for the Desmos discovery activity.
A comprehensive answer key for the Speed Expansion Worksheet. It provides final expanded forms for all 10 problems, including step-by-step logic for the 'Boss Battle' questions and notes on common student pitfalls.
A professional rubric for assessing logarithmic expansion skills. It evaluates students on rule mastery, sign consistency, radical conversion, and multi-step expansion logic, with a specific focus on Calculus readiness.
A student worksheet for the Log Launchpad lesson. It features a scaffolded set of 10 logarithmic expansion problems, ranging from basic rules to complex 'Boss Battle' expressions designed for Calculus preparation.
A visual presentation for the Log Launchpad lesson. It includes a calculus-focused hook showing the utility of log expansion, an embedded instructional video, a summary of expansion rules, and instructions for the Speed Expansion activity.
Teacher answer key for the algebra warm-up, the radical difference quotient video example, and all three relay activity problems, including the final limit conceptualization.
Instructional slide deck guiding students through the lesson objective, conjugate warm-up, Problem 11 video analysis, and relay activity rules, ending with a conceptual preview of limits and derivatives.
A comprehensive student packet containing an algebra review for conjugates, guided notes for the radical difference quotient video example, and a structured three-problem relay activity (linear, quadratic, radical) with space for group collaboration.
Detailed answer key for the 'Rocket Run' activity. Includes full algebraic derivations using the limit definition for multiple rocket height functions, along with peak altitude and impact velocity solutions.
A visual presentation deck for the 'Rocket Run' lesson. Includes warm-up questions, an embedded YouTube video segment, mathematical definitions of the derivative, and a clear overview of the simulation tasks.
A student-facing activity sheet designed as a 'Flight Data Log'. Students use the limit definition of the derivative to find the velocity function of a assigned rocket, then calculate its peak height and impact velocity.
A comprehensive lesson plan for 11th-grade honors students covering the derivative as instantaneous velocity. Includes a warm-up discussion, video guide, and main simulation activity instructions for the teacher.
Simulation worksheet for the Logistic Map, where students calculate terms for different growth rates to observe the transition from stability to bifurcation and chaos.
Visual presentation on the Logistic Map, exploring the transition from stable fixed points to bifurcation and mathematical chaos, concluding the sequence.
Workshop worksheet for students to evaluate the efficiency of sequences used to approximate mathematical constants like Pi and e, featuring partial sum calculations and error analysis.
Visual presentation on mathematical constants, comparing historical and modern sequences for calculating Pi and e, and exploring the concept of series as sequences of partial sums.
A quick-reference guide (scientific calculator assistant) for students to use during the lesson. It provides step-by-step instructions on setting up ARC fractions and tips for calculating rates over shrinking intervals efficiently.
A comprehensive visual slide deck for the lesson on average vs. instantaneous rate of change. Includes warm-up discussion prompts, embedded YouTube video, pause-and-reflect questions, and visual comparisons between secant and tangent lines.
A student worksheet designed to guide students through the warm-up discussion, video analysis of Micah's hike, and a detailed calculation activity where they observe average rates of change approaching an instantaneous value.
A printable exit ticket for the Euler Synthesis lesson, featuring four slips per page for paper-saving. Prompts students to reflect on why e is considered "natural".
Teacher answer key for the Euler synthesis matrix and the formal proof workshop. Includes step-by-step LaTeX derivations and specific teaching points.
Student workshop handout including a synthesis matrix for video note-taking and a structured workspace for the formal proof of e's definitions.
Visual presentation for Undergraduate Math Students covering Euler's Number. Includes warm-up discussion prompts, embedded video analysis, proof scaffolding for the binomial derivation, and an exit ticket prompt.
A visual slide deck for the 'Limit Launchpad' lesson, featuring the warm-up problem, embedded video, conjugate practice, and a guide through the algebraic 'boss battle' problem.
An answer key for the 'Limit Launchpad' worksheet, providing step-by-step solutions for the rationalization examples and a sample reflection response for teachers.
A student worksheet for the 'Limit Launchpad' lesson, guiding students through a warm-up trap, definitions of conjugates, and the three video examples including the complex 'boss battle' rationalization.
A comprehensive teacher's lesson plan for teaching rationalization of the numerator, featuring an instructional timeline, video engagement cues with timestamps, and key vocabulary for 11th/12th grade Pre-Calculus.
An exit ticket designed as a 'forensics case file' where students provide a one-sentence justification for using open intervals at a vertex. Matches the Turning Point Slides theme.
A comprehensive presentation for 11th Grade Pre-Calculus focusing on the justification for excluding turning points from increasing and decreasing intervals. Features thematic 'Forensics' visual design, video integration, and conceptual deep-dives.
A teacher guide for 'The Change Lab' lesson, featuring pacing instructions, discussion prompts, and a detailed answer key for all group scenarios.
A student worksheet for recording derivative calculations and interpretations during the 'Rate of Change Research' activity, with ample space for the limit definition process.
A handout containing four role-play scenarios (biology, finance, social media, and climate science) with specific functions and 'critical moments' for students to analyze using derivatives.
A visual presentation for 'The Change Lab' lesson, featuring a warm-up brainstorming session, a YouTube video summary on derivatives, group activity instructions, and a closure presentation format.
Teacher answer key for 'The Shrinking Gap' worksheet. Includes full calculations for the investigation table and sample responses for discussion questions.
Student worksheet for 'The Shrinking Gap' lesson. Includes a section for video notes on COVID-19 data and an investigation table for calculating average rates of change as the interval shrinks for the function y = x^2 + 2x.
Slide deck for 'The Shrinking Gap' lesson. It includes the hook, video analysis discussion points, and instructions for the interval-shrinking investigation.
Student worksheet for Lesson 5, focusing on manual random walk simulations and the mathematical application of the stochastic price equation.
Teacher facilitation guide for Lesson 5, detailing the introduction to stochastic sequences, the drift/volatility model, and the concept of Monte Carlo simulations.
Slide deck for Lesson 5 introducing stochastic sequences, simulations of random walks, and their application to stock price modeling.
Student worksheet for Lesson 4, covering bond price calculations and the derivation of Macaulay and Modified duration to assess interest rate sensitivity.
Teacher facilitation guide for Lesson 4, detailing the bond pricing series derivation and the calculus-based sensitivity analysis (duration).
Slide deck for Lesson 4 exploring bond pricing as a finite series and the derivation of Macaulay duration to measure interest rate risk.
Student worksheet for Lesson 3, covering the pricing of perpetuities and the mathematical derivation/application of the Dividend Discount Model.
A comprehensive teacher guide for the Symmetry Shortcuts lesson, including pacing details, discussion prompts, and strategies for addressing common misconceptions.
A structured reflection journal document for calculus students to synthesize their learning on even/odd function properties and their application to definite integrals.
Customized graph paper for sketching even and odd functions, designed to help students visualize positive and negative area under the curve for calculus prep.
A visual slide deck for 12th-grade students that connects function symmetry (even/odd) to calculus integral shortcuts, including an embedded instructional video and guided discussion prompts.
A visual presentation deck for introducing infinite limits, featuring a "grey box" hook, the lesson video, and instructions for the Desmos "Zoom In" activity.
A classroom anchor chart summarizing the concept of infinite limits, featuring clear notation, a vertical asymptote diagram, and a reminder about the "close enough" rule.
A student investigation worksheet for exploring infinite limits via Desmos, featuring tables for recording numerical data near vertical asymptotes and reflection questions on "close enough" logic.
A comprehensive teacher facilitation guide for a high school math lesson on infinite limits, featuring a structured timeline, target functions for exploration, and key discussion prompts.
A teacher facilitation guide for the Computational Powerhouse lesson, providing pacing, syntax cheat sheets, specific facilitation tips, and a full answer key for the worksheet challenges.
A professional 6-slide deck for AP Calculus and Statistics students, facilitating a lesson on Desmos efficiency. It includes a warm-up, an embedded YouTube tutorial, a dual-track (Stats/Calc) challenge, and discussion prompts.
A set of four printable discussion cards for AP Calculus and Statistics students to explore the balance between computational speed and conceptual understanding. The cards prompt critical thinking about verification, danger of black-box technology, and professional application.
Printable, foldable study cards for the warm-up drill, covering arithmetic, finite geometric, and infinite series formulas.
A structured workspace for students to record their calculations from the four rotation stations, ending with a creative decision tree synthesis activity.
Large-print problem cards for classroom stations, featuring four distinct types of summation problems: Manual Expansion, Arithmetic Large Sums, Finite Geometric Sums, and Infinite Geometric Series.
A vibrant, high-contrast slide deck designed for direct instruction and activity facilitation. Includes formula review, video integration, station instructions, and a bridge to AP Calculus Riemann Sums.
A detailed facilitation guide for the Sigma Mastery lesson, featuring pacing suggestions, instructional tips, and a full answer key for the station activities.
A detailed answer key and facilitation guide for teachers, providing correct values for the Convergence Lab worksheet and model responses for analysis questions.
A quick-reference guide for AP Calculus BC students to perform complex series summations and error analysis on a graphing calculator during the Convergence Lab activity.
A visual presentation for AP Calculus BC students that guides them through the factorial definition of e, including video prompts, activity instructions, and the introduction to Taylor Series.
A comprehensive worksheet for AP Calculus BC students to explore the factorial series for e, calculate partial sums, and analyze convergence speed compared to the limit definition.
Facilitation guide for teachers, including pacing, answer keys for both Fibonacci and Lucas sequence ratios, discussion prompts for the concept of limits, and differentiation strategies.
A 5-slide visual presentation for the Golden Mean Convergence lesson, including a video embed, AM vs GM comparison, data for the F31 investigation, and a conceptual wrap-up on limits and convergence.
Instructional slide deck for the Golden Blueprint lesson. Features high-impact visuals, an embedded video section, Lucas Sequence challenge instructions, and the formal definition of the Golden Ratio and mathematical limits.
A teacher-facing answer key and instructional guide for the Golden Mean Convergence lesson, providing full worked solutions for the worksheet and conceptual explanations for the geometric mean property.
A set of four pedagogical discussion cards designed for small group debate and critical thinking, covering concepts like sequence oscillation, limits, applied mathematical precision, and alternative recursive sequences.
A comprehensive investigation worksheet where students track Fibonacci ratios from a video and calculate Lucas Sequence ratios to discover the Golden Ratio (Phi). Includes space for calculation tables and synthesis questions about mathematical limits.
A comprehensive worksheet for 11th-12th grade math students to investigate the geometric mean property of the Fibonacci sequence, featuring warm-up problems, video observation notes, and a large-scale calculation task using terms F30 and F32.
A 2-page student worksheet designed for AP Calculus and Statistics students to practice calculating means, standard deviations, and complex definite integrals using Desmos. It includes a manual warm-up, video note-taking space, and a high-complexity computational challenge.
A comprehensive project planner for graduate students to design an inquiry-based lesson plan on limits, featuring sections for hooks, misconceptions, and rigorous bridging.
A slide deck guiding graduate students through the principles of inquiry-based lesson design, specifically applied to the concept of mathematical limits.
A case study handout for graduate students to analyze and diagnose common student misconceptions in limits using transcripts and work samples.
A comprehensive teacher facilitation guide including learning objectives, a detailed pacing guide, a video analysis key, and the full answer key for the Equation Relay activity.
A concise exit ticket featuring a rational-exponent substitution problem and a reflection prompt on the conceptual utility of variable change in advanced mathematics.
A set of four discussion prompts designed to facilitate deep conceptual understanding of variable substitution, domain restrictions, and the connection to future calculus topics.
A tiered 'relay' worksheet where students solve a series of equations in quadratic form. Each station's answer is used to unlock the coefficients of the next problem, covering polynomial, radical, binomial, and rational substitution.
A comprehensive slide deck for teaching non-standard equation solving via substitution, featuring a warm-up drill, video analysis, strategic thinking slides, and a connection to future calculus topics.
A final exit ticket assessing conceptual understanding of weight functions, orthogonality, and the significance of completeness in Sturm-Liouville theory.
A rigorous proof workshop for graduate students to derive the Sturm-Liouville form of Bessel's equation and prove the general orthogonality theorem.
Final lecture slides on Sturm-Liouville theory, defining self-adjoint operators, weight functions, and the orthogonality of eigenfunctions.
A facilitation guide for instructors, connecting Legendre polynomials to physical potential theory and providing discussion strategies for graduate seminars.
A proof-focused worksheet where students use Rodrigues' formula to derive Legendre polynomials and prove their orthogonality and normalization properties.
Lecture slides on Legendre polynomials, including Legendre's equation, Rodrigues' formula, and the generating function used in potential theory.
A physics-based problem set applying Bessel functions to model the vibrations of a circular drumhead, exploring natural frequencies and nodal lines.
A concise reference sheet for graduate students summarizing generating functions, recurrence relations, and asymptotic properties of Bessel functions.
Lecture slides on Bessel functions, covering the derivation of Bessel's equation, functions of the first and second kind, and their physical properties.
Detailed answer key for the Frobenius Method problem set, providing full derivations for singularity classification and indicial equation solutions.
A problem set focused on classifying singular points and solving the indicial equation using the Method of Frobenius.
Lecture slides on singular points and the Method of Frobenius, including classification of singularities, the Frobenius ansatz, and the indicial equation cases.
Teaching notes for the first lesson on power series, highlighting pedagogical goals, technical pitfalls, and discussion prompts for graduate-level instruction.
A graduate-level workshop involving the derivation of recurrence relations and series solutions for Airy's and Hermite's differential equations.
Introductory slides for power series solutions at ordinary points, covering definitions, the power series ansatz, and the derivation of recurrence relations.
Teacher answer key for Lesson 5, including step-by-step trigonometric substitution for cardioid surface area and geometric verification for spheres.
Student worksheet for Lesson 5, featuring surface area problems for cardioids and circles, plus a challenge involving rose curve rotation.
Slide presentation for Lesson 5 on surface area of revolution in polar coordinates, synthesizing earlier concepts of arc length and coordinate conversion.
A teacher resource for the Doubling Duel lesson, including a pacing guide, a complete answer key for the student worksheet investigation, and instructional talking points for critical analysis.
A classroom anchor chart summarizing the Rule of 72 shortcut vs. the exact logarithmic formula for continuous compounding. It features a bold, visual layout with key formulas, mental math tips, and an accuracy snapshot.
A student investigation worksheet for 12th-grade financial math. It includes a video analysis section, a comparison table for interest rates 1%-25%, critical thinking questions about financial estimations, and a calculus-based extension.
A visual slide deck for 12th Grade Financial Math covering the Rule of 72 versus exact logarithmic calculations. Includes mental math warm-ups, video analysis prompts, investigation instructions, and a Taylor Series extension for advanced students.
Summative assessment covering the entire sequence, including conceptual questions on scale factors, vector operator calculations, and applications to metrics and symmetry.
Discussion and inquiry guide for teachers to explore metric tensors, map distortions, and geodesics through thought experiments and physical demonstrations.
Introductory presentation on the metric tensor, non-Euclidean geometry, geodesics, and the conceptual foundation of General Relativity.
Problem set focused on solving boundary value problems using Legendre polynomials, including hemispherical potential and a sphere in a uniform electric field.
Case study handout connecting the mathematical solutions of Laplace's equation to the physical shapes of atomic orbitals in quantum mechanics.
Presentation on solving Laplace's equation in spherical coordinates using separation of variables and Legendre polynomials, with physical boundary condition examples.
Practice problems for calculating gradient, divergence, curl, and Laplacian in physics-based contexts using curvilinear coordinates.
Comprehensive reference sheet for vector operators (gradient, divergence, curl, Laplacian) in general and specific curvilinear systems.
Visual explanation of vector operators (gradient, divergence, curl, Laplacian) generalized for any orthogonal curvilinear coordinate system.
Detailed answer key for the Jacobian Jigsaw activity, including worked solutions for toroidal volume, mass integration, and surface area.
Problem set for students to calculate volume and area elements for toroidal, spherical, and cylindrical systems, including mass density integration.
Slide deck explaining Jacobian determinants, derivation of volume elements for spherical/cylindrical systems, and integration methods for scalar fields.
Teacher facilitation guide for the "Generalizing Coordinates" lesson, including pacing, common misconceptions, and discussion prompts.
Worksheet for students to derive scale factors for spherical and parabolic cylindrical coordinate systems, including derivation areas and reflection questions.
Visual presentation introducing orthogonal curvilinear coordinates, defining basis vectors and scale factors, and demonstrating the cylindrical case study.
An answer key for the Pumpkin Projectile Worksheet, providing full solutions for the final modeling challenge.
A student worksheet for the final lesson, covering the vector modeling of projectile motion, initial conditions, and key metrics like range and max height.
A teacher facilitation guide for Lesson 5, focusing on projectile modeling, initial conditions, and key metrics like range and max height.
Instructional slides for the final lesson, covering the vector modeling of projectile motion, initial conditions, and key metrics like range and max height.
An answer key for the Search Grid Worksheet, providing full solutions for perpendicularity and distance optimization problems.
A student worksheet for Lesson 4, focusing on perpendicularity conditions and distance minimization using the dot product in a search-and-rescue context.
Final teacher guide and assessment key for Lesson 5, including step-by-step solutions for synthesis problems and error analysis pedagogy.
A teacher facilitation guide for Lesson 4, focusing on advanced motion analysis including perpendicularity conditions and distance minimization using vectors.
Error analysis activity for Lesson 5 where students identify and correct common mistakes in a provided "flawed" solution for polar differentiation.
Instructional slides for Lesson 4, focusing on advanced motion analysis including perpendicularity conditions and distance minimization using vectors.
Synthesis problem pack for Lesson 5, featuring complex problems on implicit polar differentiation, boundary optimization, and origin tangency.
Teacher guide for Lesson 4, providing the complete derivation and solution key for the limaçon width investigation, including pedagogy for inner loops.
An answer key for the Race Day Worksheet, providing full solutions for displacement and distance problems on a circular path.
Student worksheet for Lesson 4 focusing on optimizing the dimensions of a limaçon using polar derivatives.
A student worksheet for Lesson 3, covering displacement vectors, distance calculations, and comparative analysis of motion paths.
A comprehensive workshop activity where students use calculus clues (derivatives) to identify a mystery polar curve and perform a full analysis of its tangents and extrema.
Slide deck for Lesson 4 focusing on finding maximum values of r, x, and y for polar curves using optimization techniques.
A teacher facilitation guide for Lesson 3, focusing on scalar vs vector accumulation and integration strategies for arc length.
A quick reference guide for students summarizing all polar differentiation formulas, techniques for finding special tangents, and rules for slopes at the pole and optimization.
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).
A comprehensive answer key for the Rational Blueprint worksheet, providing correct limit notation for all rational function attributes and the analysis of Example 1.
A reflection journal for students to synthesize their understanding of limit notation and connect point-plotting techniques to the Intermediate Value Theorem.
A student activity worksheet that bridges pre-calculus rational function sketching with AP Calculus limit notation, designed with a technical 'blueprint' aesthetic.
A detailed lesson plan for AP Calculus teachers that bridges the gap between pre-calculus rational function rules and formal limit notation, including a timing guide and discussion prompts based on the Justin's video.
A comprehensive teacher facilitation guide for the Roller Coaster Calculus lesson. It includes pacing instructions, discussion prompts, a sample answer key for the criteria cards, and differentiation strategies.
Instructional slide deck for the Roller Coaster Calculus lesson. Includes warm-up on interval notation, embedded YouTube video for Problem 6, activity instructions, and a closure section on common misconceptions.
Set of four distinct "Coaster Specs" cards that provide specific mathematical criteria for students to follow during the Roller Coaster Design activity. Each card outlines domain, increase/decrease intervals, and constant behavior requirements.
A specialized two-part worksheet for the Roller Coaster Design activity. Side A features a coordinate plane for track design, and Side B provides a peer-review section for interval analysis using interval notation.
The teacher answer key for the Calculus Connection Worksheet, providing correct limit evaluations and sample reflection responses.
A student activity worksheet where students evaluate limits for reciprocal functions from the lesson video and connect them to vertical and horizontal asymptotes.
A visually striking anchor chart connecting the limit definitions of vertical and horizontal asymptotes to their graphical behaviors for 12th-grade Pre-Calculus students.
A presentation slide deck for 12th-grade Pre-Calculus connecting limits to asymptotes, including a warm-up, embedded video instruction, and guided practice.
A comprehensive teacher lesson plan for a 12th-grade Pre-Calculus lesson on connecting limits to asymptotes, including pacing, warm-up prompts, and instructional tips.
A comprehensive answer key for the Asymptote Analysis worksheet, providing rigorous calculus limit proofs for all graphical features discussed.
A visual presentation for teaching AP Calculus AB students how to connect limits to graphical asymptotes and holes, featuring embedded video and formal definitions.
A student-facing worksheet for AP Calculus AB students to practice connecting algebraic rational function properties (asymptotes, holes) to limit definitions and notation.
A professional teacher facilitation guide for an AP Calculus lesson connecting limits to graphical asymptotes. Includes a structured timeline, key questions, and pedagogical strategies.
A visual slide deck to facilitate the Asymptote Alchemy lesson, including the warm-up investigation, video embed, activity instructions, and a summary of horizontal asymptote rules.
An exit ticket assessing student mastery of the Alternating Series Test, error bound calculations, and the conceptual distinction between absolute and conditional convergence.
A cumulative decision-making flowchart activity that requires students to synthesize all convergence tests learned in the sequence and apply them to final complex series examples.
Lesson 5 slides on Alternating Series, covering the Leibniz criterion, error bound estimation, and the distinction between absolute and conditional convergence.
Answer key for the Benchmarking Battle worksheet, featuring clear benchmark identification, inequality setups, and limit evaluations for DCT and LCT problems.
A comprehensive problem set for practicing Direct and Limit Comparison Tests, including workspace for inequalities, limits, and a strategic choice challenge.
Lesson 4 slides on Comparison Tests, introducing the Direct and Limit Comparison Tests using a weightlifting analogy and strategies for selecting dominant-term benchmarks.
A sorting activity where students categorize different P-series as convergent or divergent based on their exponents. It includes a reference table for reinforcement.
Lesson 3 slides on P-series and the Harmonic series, illustrating the "infinite overhang" paradox and establishing the p-value criteria for convergence used as benchmarks in future tests.
Detailed answer key for the Integral Test Investigation, providing step-by-step calculus work for improper integrals and logical justifications for convergence conclusions.
An investigation activity that guides students through the geometric derivation of the Integral Test using Riemann sums, followed by two rigorous practice problems involving U-substitution and logarithmic functions.
Worksheet for practicing error analysis in related rates problems, featuring "Crime Scenes" where students must identify and correct common calculus mistakes.
Full solution guide for the "Calculus Gauntlet" seminar in Lesson 5, providing step-by-step mathematical breakdowns for all four challenge stages.
Comprehensive final exam for the Volume Flow Dynamics sequence. Includes problems on spherical balloons, cylindrical reservoirs, and a net-flow conical sieve problem.
Student workspace for the Lesson 5 "Calculus Gauntlet" seminar, designed for teams to record their solutions to the four challenge stages.
Visual presentation for Lesson 5 focusing on common errors like premature substitution, unit misalignment, and sign errors in related rates problems.
Final case study for Lesson 5. Students solve net flow problems involving a cylindrical reactor breach and a conical containment pit, culminating in an engineering recommendation.
Challenge cards for the "Calculus Gauntlet" seminar in Lesson 5, featuring complex, multi-stage related rates problems combining shadows, trigonometry, and geometry.
A final exit ticket to assess student understanding of Pythagorean related rates. Includes a computational problem involving a coordinate path and a conceptual comparison of rates.
Slide deck for Lesson 5 on Net Flow Dynamics. Covers the concept of dV/dt = In - Out, a cylindrical tank emergency scenario, and the added complexity of net flow in conical containers.
Answer key for Lesson 4's "The Shoreline Sweep" activity sheet, with detailed mathematical derivations for Part A, B, and C.
An advanced challenge set for the final lesson, featuring multi-object motion (baseball diamond), accelerating objects, and the classic lamppost shadow problem. Requires synthesis of geometry, calculus, and physics.
Synthesis worksheet for practicing the 4-step Related Rates protocol: Sketch, GFW List, Relate Equation, and Differentiate/Solve.
Worksheet for Lesson 4 exploring the non-linear relationship between a lighthouse's rotation and the speed of its beam as it sweeps across a shoreline.
Final slide deck for Lesson 5, presenting complex word problems like the baseball diamond runner and introduction to searchlight problems. Synthesizes all previous concepts into advanced geometric applications.
Answer key for the Hourglass Hustle workshop, featuring full mathematical proofs for the funnel and sand pile problems.
Slides for Lesson 4 exploring rotating beams and lighthouses, connecting angular velocity to the linear velocity of a beam across a surface.
Workshop activity for Lesson 4 where students solve complex related rates problems involving conical containers and sand piles. Includes a conceptual reflection on rate acceleration.
Visual presentation for Lesson 4 focusing on the synthesized 4-step framework: Sketch, List Variables (GFW), Relate Equation, and Solve.
A reflection sheet for students to record their observations from the digital modeling lab, including a sketching area for the velocity-time graph and conceptual questions about rate behavior.
A guide for a digital project using Desmos/GeoGebra to model related rates. Students build a dynamic sliding ladder, graph the resulting vertical velocity function, and analyze the non-linear behavior and vertical asymptotes.
Teacher answer key for Lesson 4, featuring detailed trigonometric identity work for cardioid arc length and the formula application for Archimedean spirals.
Student worksheet for Lesson 4, containing arc length problems for circles, cardioids, and spirals, plus a conceptual reflection on the formula's components.
Comprehensive answer key for all student worksheets in the Polar Integration Expedition sequence, including full evaluations for complex area and arc length problems.
Slide presentation for Lesson 4 on polar arc length, covering the derivation from parametric form and application to cardioids and spirals.
Answer key and instructional notes for Lesson 5, focusing on arc length calculations and the distinction between displacement and distance.
Teacher guide for Lesson 3, providing the complete solution key for the rose curve petal worksheet and conceptual pointers for origin dynamics.
Worksheet for distance optimization. Students analyze a limacon flight path, finding absolute extrema of the radius and connecting radial change to tangent slope.
A comprehensive grading rubric for the Synthesis Assessment. Includes a 4-point proficiency scale for path analysis, vector kinematics, and structural geometry, along with a rapid evaluation key for teachers.
Presentation for Lesson 5 on arc length and total distance traveled in parametric form.
The final synthesis performance task for the sequence. Students perform a full analysis of a 4-petal rose curve, including area integration, vector differentiation for speed, and arc length setup.
Worksheet for Lesson 5 on calculating arc length and distinguishing between net displacement and total distance.
Final slide deck for Lesson 5 introducing the "Particle Report" synthesis assessment. Provides the raw kinematic data for a 4-petal rose curve and outlines the multi-step analysis requirements for the final performance task.
Presentation for Lesson 4 covering kinematics in parametric form, including position, velocity, acceleration, and speed.
Teacher resource for the Logic Lock activity. Contains a deep dive into the specific logical flaws provided in the "compromised" student work and the correct multi-step calculus resolutions.
Worksheet for Lesson 4 applying derivatives to particle motion, including velocity vectors, acceleration vectors, and speed.
An escape-room style error analysis worksheet where students identify flawed calculus logic in area and motion problems to "unlock" a terminal code. Focuses on rigorous critique of mathematical reasoning.
Presentation for Lesson 3 detailing the calculation of the second derivative in parametric form and its application to concavity.
Worksheet for Lesson 3 on finding the second derivative of parametric equations and analyzing concavity.
A slide deck for Lesson 4 on error analysis. Categorizes common calculus bugs (formula amnesia, chain rule neglect, domain errors) and introduces a "Bug Hunt" case study for student critique.
Presentation for Lesson 2 covering parametric differentiation, tangent lines, and horizontal/vertical tangents.
Answer key for the Shaded Region Showdown workshop. Provides intersection point derivations, step-by-step integral setup for shared regions, and final evaluated area values for rose and limacon curves.
Worksheet for Lesson 2 focusing on calculating derivatives dy/dx and finding tangent line equations for parametric curves.
An advanced area workshop focusing on shared regions and inner loops of polar curves. Students must identify intersection points and manage negative r-values to calculate precise areas.
Teacher guide for Lesson 1, including learning objectives, instructional steps, and common misconceptions.
A visualization-heavy slide deck for Lesson 3 focusing on overlapping polar areas. Covers intersection point strategies, multi-curve regions, and handling negative r-values to avoid double-tracing errors.
Introductory presentation for Lesson 1, explaining parametric equations, plane curves, and parameter elimination.
