This guide provides 101 highly effective prompts for educators and students of AP Calculus BC. Each prompt is designed to be used with Generative AI tools like Gemini, ChatGPT, or Claude to enrich the teaching and learning experience. The prompts are based on the 2024-2025 College Board curriculum and are structured to support various stages of the academic journey, from planning and initial understanding to assessment and final revision.
How to Use This Guide:
Simply copy and paste the desired prompt into your preferred AI tool. You can modify the prompts by adding specific details (e.g., a particular function, a specific type of problem, or a desired difficulty level) to get more tailored results.
Section 1 – Educator Prompts
These 50 prompts are designed to assist educators in planning, delivering, assessing, and enriching their AP Calculus BC course.
Planning (15 Prompts)
Syllabus Creation: “Generate a comprehensive, 180-day syllabus for an AP Calculus BC course based on the latest College Board framework. The syllabus should include a unit-by-unit breakdown, key learning objectives for each unit, suggested activities, and assessment timelines.”
Unit Pacing Guide: “Create a detailed pacing guide for the ‘Parametric Equations, Polar Coordinates, and Vector-Valued Functions’ unit in AP Calculus BC. Allocate time for each subtopic and include formative assessment checkpoints.”
Lesson Plan Template: “Design a versatile lesson plan template for a 90-minute AP Calculus BC class. The template should include sections for learning objectives, materials, warm-up activity, direct instruction, guided practice, independent practice, and a closing summary.”
Big Ideas & Enduring Understandings: “List the ‘Big Ideas’ and ‘Enduring Understandings’ for the entire AP Calculus BC curriculum, as defined by the College Board. For each, provide a brief, student-friendly explanation.”
Essential Questions: “Develop a set of essential questions for the ‘Infinite Sequences and Series’ unit that will promote critical thinking and connect the concepts to real-world applications.”
Resource Curation: “Curate a list of high-quality online resources (videos, simulations, articles) to supplement the ‘Integration and Accumulation of Change’ unit. For each resource, provide a brief description and explain how it aligns with the curriculum.”
AP Exam Review Schedule: “Create a 6-week AP exam review schedule that systematically covers all major topics of the course. Include a mix of content review, free-response question (FRQ) practice, and multiple-choice question (MCQ) drills.”
Vertical Alignment: “Explain the vertical alignment of key calculus concepts from Pre-Calculus through AP Calculus AB and into AP Calculus BC. Focus on the progression of limits, derivatives, and integrals.”
Technology Integration: “Suggest innovative ways to integrate graphing calculators (like the TI-84) and computer algebra systems into the ‘Applications of Differentiation’ unit to enhance student understanding.”
Project-Based Learning Ideas: “Brainstorm three project-based learning (PBL) ideas for the ‘Applications of Integration’ unit. Each idea should have a clear objective, a real-world context, and a rubric for assessment.”
Differentiated Instruction Plan: “Outline a plan for differentiating instruction in a mixed-ability AP Calculus BC classroom for the topic of ‘Taylor and Maclaurin Series.’ Provide strategies for supporting struggling learners and challenging advanced students.”
Summer Assignment: “Create a summer assignment for students entering AP Calculus BC. It should review essential Pre-Calculus and Calculus AB topics, including function analysis, limits, and basic differentiation and integration.”
First Day of School Activity: “Design an engaging first-day-of-school activity that introduces students to the core ideas of calculus (change and infinity) without using formal definitions or formulas.”
Substitute Teacher Plans: “Prepare a set of emergency substitute teacher plans for a 3-day absence. The plans should cover the topic of ‘L’Hôpital’s Rule and Improper Integrals’ and include clear instructions, worksheets, and answer keys.”
Lab Activity Design: “Design a hands-on ‘lab’ activity where students can physically model and explore the concept of volumes of revolution using common classroom materials.”
Delivery (15 Prompts)
Concept Explanation Analogy: “Explain the concept of the chain rule using a simple, memorable analogy that a high school student can easily understand.”
Real-World Application: “Provide three real-world applications of vector-valued functions, complete with the mathematical models and explanations suitable for an AP Calculus BC class.”
Interactive Notebook Entry: “Create a template for an interactive notebook entry on the topic of ‘Convergence Tests for Series.’ It should include foldable sections for each test, with definitions, conditions, and examples.”
Guided Notes: “Generate guided notes for a lesson on ‘Integration by Parts.’ Include clear steps, several examples ranging from simple to complex, and a ‘common mistakes’ section.”
Discussion Prompts: “Develop a set of thought-provoking discussion prompts to use in a Socratic seminar about the Fundamental Theorem of Calculus.”
Demonstration Script: “Write a script for a classroom demonstration explaining the relationship between a function, its first derivative, and its second derivative using a motion detector and a student walking back and forth.”
Error Analysis Activity: “Create an error analysis activity worksheet. Present five solved problems related to finding the interval of convergence for a power series, each with a common conceptual or algebraic error. Students must identify and correct the errors.”
Graphic Organizer: “Design a graphic organizer (e.g., a flowchart or concept map) that helps students decide which integration technique to use (u-substitution, integration by parts, partial fractions, etc.).”
Jigsaw Activity: “Outline a jigsaw activity for the various convergence tests (e.g., Integral Test, Comparison Test, Ratio Test). Assign each group a test to become ‘experts’ on and then have them teach it to their peers.”
Think-Pair-Share Questions: “Generate a sequence of think-pair-share questions for a lesson on polar area. The questions should build in complexity, from finding the area of a simple cardioid to finding the area between two polar curves.”
Visual Aids: “Suggest five powerful visual aids (diagrams, graphs, animations) to explain the concept of a Taylor polynomial approximation.”
Scaffolding Questions: “Provide a list of scaffolding questions to guide a student through a difficult FRQ involving a particle’s motion along a parametric curve.”
Mathematical Language: “Create a glossary of key vocabulary for the ‘Differential Equations’ unit, with student-friendly definitions and examples for terms like ‘slope field,’ ‘separable differential equation,’ and ‘logistic growth.'”
Code for Visualization: “Provide Python code using Matplotlib to visualize a slope field for the differential equation dy/dx = x – y.”
Connecting Concepts: “Explain the connection between Riemann sums, definite integrals, and the concept of accumulation in a way that is intuitive for students.”
Assessment (10 Prompts)
FRQ Creation: “Create an original, AP-style free-response question (FRQ) on the topic of polar coordinates. The question should have three parts, testing area, arc length, and the rate of change of the distance from the origin. Provide a detailed scoring rubric.”
MCQ Creation: “Generate 10 AP-style multiple-choice questions (5 calculator-active, 5 non-calculator) that assess students’ understanding of series convergence and divergence. Include an answer key with detailed explanations for each choice.”
Quiz Generator: “Create a 20-minute quiz on the topic of ‘Parametric Derivatives and Arc Length.’ Include a mix of computational and conceptual questions. Provide an answer key.”
Rubric Design: “Design a rubric for assessing a student project on modeling a real-world phenomenon with a differential equation. The rubric should evaluate mathematical accuracy, clarity of explanation, and the connection to the real-world context.”
Self-Assessment Tool: “Create a student self-assessment checklist for the ‘Applications of Integration’ unit. The checklist should be based on the learning objectives from the course and exam description.”
Peer Review Form: “Develop a peer review form for students to use when evaluating each other’s solutions to a complex FRQ. The form should guide them to provide constructive feedback on clarity, justification, and mathematical correctness.”
Exit Ticket Prompts: “Generate five different exit ticket prompts for a lesson on L’Hôpital’s Rule to quickly gauge student understanding.”
Test Correction Worksheet: “Create a test correction worksheet that requires students to not only correct their mistakes but also explain the conceptual error that led to the mistake and find a similar problem to solve correctly.”
Portfolio Assignment: “Outline a portfolio assignment where students select their best work from each unit to showcase their growth and understanding throughout the course.”
Performance Task: “Design a performance task where students use calculus to optimize a real-world scenario, such as designing a container with maximum volume for a given surface area. The task should require a written report and presentation.”
Enrichment (10 Prompts)
Interdisciplinary Connection: “Explain how the concepts of optimization in calculus are used in the field of economics (e.g., minimizing cost, maximizing profit).”
Historical Context: “Provide a brief historical background on the development of calculus, focusing on the contributions of Newton and Leibniz and the controversy between them.”
Challenge Problems: “Create three challenging problems that go beyond the standard AP curriculum, exploring topics like multivariable calculus or more advanced differential equations, but are solvable with BC concepts.”
Calculus in Art: “Suggest a mini-project where students can explore the use of calculus in creating art, such as using polar equations to generate intricate designs or parametric equations for animations.”
Reading List: “Compile a reading list of accessible books and articles about calculus and its applications for students who want to explore the subject further.”
Career Connections: “List five careers that heavily rely on the concepts learned in AP Calculus BC and explain the specific applications in each career.”
“What if?” Scenarios: “Pose a ‘what if’ scenario to challenge students’ understanding. For example, ‘What if the Fundamental Theorem of Calculus didn’t exist? How would we calculate definite integrals?'”
Proof Exploration: “Guide me through a proof of the formula for the arc length of a curve in parametric form, explaining each step in an intuitive way.”
Advanced Topic Introduction: “Provide a simple, conceptual introduction to the topic of Fourier Series, explaining how it relates to the Taylor Series concepts learned in class.”
Math Competition Problems: “Generate five math competition-style problems that are based on AP Calculus BC topics but require more creative problem-solving skills.”
Section 2 – Student Prompts
These 50 prompts are designed to help students understand concepts, practice skills, revise material, and prepare for assessments in AP Calculus BC.
Understanding (15 Prompts)
Explain a Concept Simply: “Explain the concept of a sequence’s convergence versus a series’ convergence as if you were talking to a 10th grader.”
Analogy for a Topic: “Give me a simple analogy to help me understand the difference between integration by parts and u-substitution.”
Step-by-Step Walkthrough: “Walk me step-by-step through solving a logistic differential equation, explaining the ‘why’ behind each step.”
Visual Explanation: “Can you create a visual explanation or diagram that shows why L’Hôpital’s Rule works for the indeterminate form 0/0?”
Key Differences: “What is the key difference between a Maclaurin series and a Taylor series? When would I use one over the other?”
Conceptual Foundation: “What is the conceptual foundation behind Euler’s Method? Why is it an approximation and not an exact solution?”
Common Misconceptions: “What are the most common misconceptions students have about the Alternating Series Test and its conditions?”
Connect to Prior Knowledge: “How does the concept of finding the area between two polar curves relate to finding the area between two Cartesian curves?”
“Why Does This Matter?”: “Why is it important to find the interval of convergence for a power series? What does it tell me?”
Summarize a Unit: “Summarize the entire ‘Infinite Sequences and Series’ unit in 300 words, focusing on the main ideas and their connections.”
Translate Notation: “Translate the mathematical notation for the Lagrange error bound into plain English. What is each part of the formula actually telling me to do?”
Function of a Formula: “What is the function of the ‘+ C’ in an indefinite integral? Why is it so important?”
Deep Dive: “Do a deep dive into the concept of improper integrals. Explain the two main types and the process for evaluating them using limits.”
Role of a Theorem: “Explain the role of the Mean Value Theorem for Integrals. What does it guarantee?”
Building Blocks: “How do parametric equations and vector-valued functions build upon my understanding of functions from Algebra and Pre-Calculus?”
Practicing (15 Prompts)
Generate Practice Problems: “Generate 10 practice problems on the topic of finding the arc length of a parametric curve. Include a variety of functions and provide a detailed answer key.”
Create a Worksheet: “Create a worksheet with 5 problems on integration using partial fraction decomposition. The problems should increase in difficulty.”
FRQ Practice: “Give me an AP-style FRQ about a particle moving in the xy-plane described by a vector-valued function. I want to practice finding velocity, acceleration, speed, and total distance traveled.”
MCQ Drill: “Give me 10 multiple-choice questions (non-calculator) that test my ability to determine the convergence or divergence of series using various tests.”
Targeted Skill Practice: “I need to practice finding the Taylor series for a function centered at a value other than zero. Give me 5 problems to work on.”
Work Through a Problem with Me: “Let’s work through a difficult integration by parts problem together. Ask me what to do at each step and provide feedback.”
Check My Work: “I solved this problem [paste your solution here]. Can you check my work, identify any errors, and explain how to fix them?”
Vary the Difficulty: “Give me three problems on finding the volume of a solid with a known cross-section. Make one easy, one medium, and one hard.”
Problem from Scratch: “Create a unique problem that requires me to use both the chain rule and the product rule in the context of parametric equations.”
Justify Your Answer: “Give me a problem where I have to use the Limit Comparison Test, and then ask me to write a full justification for my conclusion, just like I would on the AP exam.”
Calculator Skills: “What are the most important calculator skills for the AP Calculus BC exam? Give me three practice problems where using a calculator strategically is key.”
Mixed Review: “Create a mixed review set of 5 problems covering topics from the entire ‘Applications of Integration’ unit.”
Speed Drill: “Give me 5 relatively simple derivative problems involving parametric, polar, and vector functions. I want to practice my speed and accuracy.”
Conceptual Questions: “Give me 5 conceptual questions that don’t require much calculation but test my deep understanding of the relationship between a function and its derivatives.”
From Graph to Conclusion: “Provide a graph of f'(x) and ask me questions about the properties of f(x), such as where it is increasing/decreasing, concave up/down, and where it has local extrema.”
Revising (10 Prompts)
Create Flashcards: “Create a set of 20 digital flashcards for the ‘Infinite Sequences and Series’ unit. The front should have a term or a series, and the back should have the definition or the name of the convergence test that applies.”
Study Guide: “Generate a comprehensive study guide for the ‘Differential Equations’ unit. It should include key concepts, formulas, and a checklist of skills I need to master.”
Cheat Sheet: “Create a one-page ‘cheat sheet’ for all the major integration techniques and series convergence tests. Use concise language and clear formatting.”
Mind Map: “Generate a mind map that visually connects all the major topics in AP Calculus BC, starting from limits and branching out to derivatives, integrals, and series.”
Review Quiz: “Create a 15-question review quiz that covers the most commonly tested topics from the entire AP Calculus BC curriculum.”
Explain My Mistakes: “I keep making mistakes with signs when calculating the area of polar curves. Can you explain the common pitfalls and give me a strategy to avoid them?”
Prioritize Topics: “Based on past AP exams, what are the most important topics to focus on in my final week of review?”
Formula Memorization: “Give me some mnemonics or other memory aids to help me remember the formulas for Taylor series of common functions (e^x, sin(x), cos(x), 1/(1-x)).”
Self-Quiz: “Quiz me on the conditions required for each of the series convergence tests. Ask me one by one and don’t give me the answer until I’ve tried.”
FRQ Strategy: “What is the best strategy for tackling the FRQs on the AP exam? How should I manage my time and what should I focus on to maximize my points?”
Preparing for Assessment (10 Prompts)
Simulate an FRQ: “Act as an AP exam grader. I will provide my solution to an FRQ. Please score it using the official rubric and provide feedback on where I lost points.”
Predict Exam Questions: “Based on the latest curriculum and trends from recent exams, what are some likely topics or combinations of topics for this year’s FRQs?”
Time Management Plan: “Help me create a time management plan for the multiple-choice and free-response sections of the AP exam.”
Full Practice Test: “Generate a full-length practice AP Calculus BC exam (both MCQ and FRQ sections) based on the current format. Provide a separate answer key and scoring guide.”
High-Pressure Drill: “Give me a single, complex, multi-part FRQ. I will set a timer for 15 minutes and solve it. Then, I’ll ask you to review my work.”
Justification Practice: “Give me 5 statements about calculus concepts (e.g., ‘If a series converges, its terms must go to zero’). For each, I have to state if it’s true or false and provide a rigorous justification or a counterexample.”
“AP Classroom” Style Problems: “Create 5 problems similar in style and difficulty to the ‘Progress Check’ questions on the AP Classroom platform for the unit on Parametric and Polar functions.”
Final Review Checklist: “Create a final, one-day-before-the-exam checklist of things to review, including key formulas, theorems, and problem-solving strategies.”
Mental Prep: “What are some strategies for staying calm and focused during the AP exam? How can I avoid making careless mistakes under pressure?”
Post-Exam Analysis: “After I take a practice exam, what’s the best way to analyze my results to identify my weaknesses and guide my final days of studying?”
Section 3 – Bonus Universal Prompt
This prompt can be adapted for both educators and students to foster deeper, interdisciplinary thinking.
The Interdisciplinary Connection: “Act as an expert in [insert field, e.g., Physics, Computer Science, Biology, Economics, Music]. Explain how the core concepts of AP Calculus BC (specifically [insert topic, e.g., differential equations, optimization, series expansions]) are fundamental to your field. Provide a specific, detailed example of a problem in your field that would be impossible to solve without calculus. Then, create a simplified version of that problem that could be explored by an AP Calculus BC class.”
