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How the Toiya question bank is organized

Toiya has 12,300+ practice questions for the SAT, ISEE, AMC 8, 10, and 12, and AP Calculus and Precalculus. Here is how we organize them by domain, skill, and difficulty, and where they come from.

12,300+questions in total
5,000+SAT
3,600+ISEE
2,600+AMC 8 / 10 / 12
980+AP Calculus & Precalculus

How every question is organized

Every question sits in a five-level hierarchy, so tutors and students can zoom from a whole exam down to a single concept:

  1. Exam: SAT, ISEE, AMC, or AP
  2. Subject: for example SAT Math, ISEE Upper Level English, AMC 10, or AP Calculus
  3. Domain: a major content area, such as Advanced Math or Geometry
  4. Skill: the specific thing being tested, such as "Nonlinear functions" or "Chain rule"
  5. Subtopic tags: optional fine-grained labels inside a skill

Questions also come with a difficulty rating, and most include a step-by-step explanation.

ExamWhat's coveredQuestionsDifficulty scale
SATReading & Writing and Math5,0341–3 (easy, medium, hard)
ISEEMiddle and Upper Level, all four multiple-choice sections3,6441–3 (easy, medium, hard)
AMCAMC 8, AMC 10, and AMC 122,6531–5 (contest range)
APCalculus AB/BC and Precalculus, plus an AP Calculus FRQ bank9851–3 (easy, medium, hard)

SAT: the College Board's own domains and skills

SAT questions use the official content domains and skills from the College Board's SAT Suite of Assessments, the same categories shown in the official question bank and on score reports. Reading & Writing has four domains (Information and Ideas, Craft and Structure, Expression of Ideas, Standard English Conventions), and Math has four (Algebra, Advanced Math, Problem-Solving and Data Analysis, Geometry and Trigonometry). Subtopic tags add detail such as grammar rules, passage genres, and question formats.

Math · 2,655 questions

Algebra · 885

  • Linear Equations in One Variable (246)
  • Linear Functions (243)
  • System of Linear Equations (160)
  • Linear Equations in Two Variables (143)
  • Linear Inequalities (93)

Advanced Math · 782

  • Nonlinear Functions (326)
  • Nonlinear Equations and Systems (279)
  • Equivalent Expressions (177)

Geometry and Trigonometry · 499

  • Line, Angles and Triangles (146)
  • Area and Volume (142)
  • Circles (106)
  • Right Triangles and Trigonometry (105)

Problem-Solving and Data Analysis · 489

  • Percents (107)
  • Ratios, Rates, Proportional relationships, and units (106)
  • One-Variable Data (98)
  • Probability (62)
  • Two Variable Data (57)
  • Sample Statistics and Margin of Error (30)
  • Evaluating Statistical Claims (29)
Reading & Writing · 2,379 questions

Craft and Structure · 714

  • Words in Context (405)
  • Text Structure and Purpose (211)
  • Cross-Text Connections (75)
  • Command of Evidence (12)
  • Central Ideas and Details (5)
  • Transitions (2)
  • Boundaries (1)

Information and Ideas · 640

  • Command of Evidence (285)
  • Central Ideas and Details (223)
  • Inferences (131)
  • Cross-Text Connections (1)

Expression of Ideas · 523

  • Rhetorical Synthesis (270)
  • Transitions (253)

Standard English Conventions · 502

  • Form, Structure, and Sense (300)
  • Boundaries (202)

ISEE: organized by level and test section

ISEE questions are organized by level and by the four multiple-choice sections, with skills that mirror what each section asks: synonyms and sentence completion in Verbal Reasoning, quantitative comparison and word problems in Quantitative Reasoning, passage skills in Reading Comprehension, and grade-level math in Mathematics Achievement. See the ISEE practice page for the full Middle and Upper Level skill lists.

AMC: contest areas and StudyToday's lesson sequence

AMC 8, AMC 10, and AMC 12 problems are organized by the core contest areas: Algebra, Geometry, Number Theory, and Counting & Probability (AMC 12 lists Combinatorics and Probability separately). Within each area, skills follow the order of StudyToday's AMC lessons, so a worksheet on a skill maps directly to a lesson. A five-level difficulty scale spans everything from warm-up problems to the final questions on each contest.

AMC-8 · 672 questions

Algebra & Arithmetic · 293

  • AS2. Ratio; Proportion; Percentage; Direct & Inverse Variation (56)
  • AS13. Mean and Median Problems; Weighted mean (27)
  • AS7. Linear equation; Consecutive integers; Arithmetic sequence (25)
  • AS10. Ages; Distance, rate, and time; Mixture problems (24)
  • AS3. Unit conversion; Rate; Time problems; Money problems (24)
  • AS1. Fraction; Decimal; Estimation (18)
  • AS5. Place value; Digits; Decimal representation (18)
  • AS14. Logical deduction; Counterexample (17)
  • AS12. Intro to AMC8 Stats; Data and graph interpretations (16)
  • AS16. Invariant; Bounding to limit cases; Work backwards (15)
  • AS4. Exponent; Radical; Order of operations; Custom operations (15)
  • AS15. Pattern recognition; Process simulation; Small cases (15)
  • AS11. System of equations (12)
  • AS8. Inequality; Absolute value (6)
  • AS6. Algebraic manipulation; Distributive property; Substitution; Factoring (5)

Geometry · 167

  • G5. Pythagorean theorem and common triples (26)
  • G9. Circles: sectors, arcs, and tangent-radius perpendicularity (23)
  • G1. Perimeter and basic area: rectangles, squares, triangles, missing lengths (22)
  • G2. Composite area and dissection: cut-outs, rearrangement, overlap (20)
  • G10. 3D geometry: cubes, nets, surface area, volume, hidden dimensions (19)
  • G7. Power scaling (14)
  • G8. Quadrilaterals and polygons: parallelograms, trapezoids, regular polygons (14)
  • G4. Angles and triangle facts: angle sums, exterior angles, isosceles, equilateral, triangle inequality (11)
  • G3. Grid and lattice area: decomposition, Pick's Theorem, Shoelace formula (10)
  • G11. Transformations and visual geometry: reflection, rotation, symmetry, paper folding, counting shapes (6)
  • G6. Similarity and scale factor (2)

Counting & Probability · 110

  • CP1. Basic counting; Tree diagram (21)
  • CP10. Basic probability (16)
  • CP2. Multiplication principle; Casework (13)
  • CP6. Stars and bars; Partitions and compositions (10)
  • CP8. Inclusion-exclusion; Venn diagram (9)
  • CP4. Combination (8)
  • CP5. Arrangements with restrictions (7)
  • CP11. Sampling with replacement; Sampling without replacement; Independent events (7)
  • CP3. Permutations; Multiset permutations (5)
  • CP13. Complementary probability; Geometric probability (4)
  • CP14. Expected value; Dice problems (3)
  • CP9. Pigeonhole principle; Intro to bijection (3)
  • CP7. Complementary counting (2)
  • CP12. Conditional probability (2)

Number Theory · 102

  • NT5. Modular arithmetic I (28)
  • NT2. Prime factorization (25)
  • NT1. Parity; Divisibility (23)
  • NT6. Modular arithmetic II: additive inverses and negative remainders (7)
  • NT4. GCD/LCM relationship (6)
  • NT9. Basic Chinese Remainder Theorem (5)
  • NT3. Euclidean algorithm (4)
  • NT7. Basic linear Diophantine equations (3)
  • NT8. Simon's Favorite Factoring Trick (1)
AMC-10 · 1,132 questions

Algebra · 437

  • A8. Sequences and Series I (Arithmetic/Geometric) (31)
  • A1. Factoring; Factorial Algebra (27)
  • A9. Sequences and Series II (Telescoping/Recursive) (25)
  • A10. Optimization Problems; Inequalities I (Trivial Inequality, Polynomial Inequality) (25)
  • A6. Sigma Summation (24)
  • A11. Inequalities II (AM-GM Inequality, Bounding and Estimations) (23)
  • CP4. Casework; Double counting (22)
  • A3. Vieta’s Formulas (15)
  • A7. Sum and Difference of Powers (12)
  • A5. Polynomial Manipulation; Intro to Rational Functions; Partial Fractions (10)
  • A12. Floor and Ceiling Functions (9)
  • NT7. Repeating Decimals; Number Base (6)
  • A4. Symmetric Sums; Intro to Power Sums (5)
  • G1. Pythagorean Theorem and Similar Triangles (4)
  • CP2. Combinations; Subsets; Counting pairs (3)
  • G6. Coordinate Geometry; Equation of Circle (2)
  • A2. Factor and Remainder Theorem (2)
  • CP8. Pigeonhole principle; Counting integers in a range (2)
  • NT8. Chicken McNugget Theorem; Linear Diophantine Equation (1)
  • G11. 3D Geometry — Cones and Spheres (1)
  • CP10. Conditional probability; Sampling without replacement; Sampling with replacement (1)

Geometry · 306

  • G1. Pythagorean Theorem and Similar Triangles (84)
  • G5. Circle Basics; Chords and Tangents (69)
  • G6. Coordinate Geometry; Equation of Circle (62)
  • G10. 3D Geometry — Polyhedrons (29)
  • G3. Triangle Centers (Incenter, Circumcenter, Centroid) (18)
  • G2. Cevians and Stewart’s Theorem (13)
  • G11. 3D Geometry — Cones and Spheres (13)
  • G4. Mass Points (6)
  • G8. Cyclic Quadrilaterals (6)
  • G7. Power of a Point (4)
  • G9. Descartes Kissing Circles (2)

Counting and Probability · 221

  • CP10. Conditional probability; Sampling without replacement; Sampling with replacement (51)
  • CP4. Casework; Double counting (40)
  • CP3. Arrangements with restrictions; Complementary counting (30)
  • CP8. Pigeonhole principle; Counting integers in a range (23)
  • CP11. Expected value; Geometric probability (20)
  • CP2. Combinations; Subsets; Counting pairs (16)
  • CP7. Inclusion-exclusion; Venn Diagram (12)
  • CP1. Permutations; Multiset permutations; Circular arrangements (9)
  • CP12. Random walk; Recursive probability (Markov-chain models) (7)
  • CP5. Bijection (7)
  • CP6. Stars and bars; Partitions and compositions (4)
  • CP9. Lattice paths; Pascal’s Triangle (2)

Number Theory · 168

  • NT9. Perfect Squares; Legendre’s Formula; Intro to Quadratic Residue (26)
  • NT7. Repeating Decimals; Number Base (23)
  • NT1. Euclidean Algorithm and Bezout’s Identity (22)
  • NT6. Multiplicative Order (19)
  • NT8. Chicken McNugget Theorem; Linear Diophantine Equation (18)
  • NT4. Fermat’s Little Theorem (16)
  • NT5. Totient Function; Euler Totient Theorem (13)
  • NT10. Nonlinear Diophantine Equations (11)
  • NT3. Multiplicative Inverse (10)
  • NT2. Chinese Remainder Theorem (10)
AMC-12 · 849 questions

Algebra · 200

  • Equation Solving (66)
  • Polynomials (36)
  • Sequence and Series (35)
  • Functions and Graphs (19)
  • Inequalities (AM-GM, Cauchy) (16)
  • Systems of Equations (6)
  • Rational Expressions (5)
  • Functional Equations (2)
  • Triangles (1)
  • Discrete Probability (Independent & Dependent Events) (1)
  • Radicals and Roots (1)
  • Exponential and Logarithmic Functions (1)
  • Area and Perimeter (1)

Geometry · 186

  • Triangles (59)
  • Circles (39)
  • Coordinate Geometry (26)
  • Quadrilaterals (25)
  • Area and Perimeter (8)
  • Volume and Surface Area (8)
  • 3D Geometry (4)
  • Sequence and Series (1)
  • Systems of Equations (1)
  • Equation Solving (1)
  • Polygons (1)

Combinatorics · 58

  • Basic Counting (54)
  • Recurrence and Constructive Counting (2)
  • Inclusion–Exclusion Principle (1)

Number Theory · 58

  • Divisibility and Remainders (38)
  • Prime Factorization (7)
  • GCD & LCM (5)
  • Modular Arithmetic (1)

Probability · 49

  • Discrete Probability (Independent & Dependent Events) (39)
  • Geometric Probability (7)
  • Probability Distributions (1)
  • Expected Value (1)

AP: the official course framework

AP questions follow the College Board's AP course and exam descriptions: domains are the official course units, and each skill groups the closely related topics that are tested together. Our AP multiple-choice bank for Calculus and Precalculus is growing, with new units added regularly.

Calculus · 668 questions

Unit 1: Limits and Continuity · 201

  • 1.10–1.13 Continuity and Discontinuities (57)
  • 1.1–1.4, 1.9 Limits from Graphs and Tables (50)
  • 1.14–1.15 Infinite Limits and Limits at Infinity (Asymptotes) (35)
  • 1.5–1.7 Limits Using Algebra (34)
  • 1.16 Intermediate Value Theorem (17)
  • 1.8 Squeeze Theorem (8)

Unit 2: Differentiation: Definition and Fundamental Properties · 116

  • 2.8–2.10 Product and Quotient Rules (incl. tan, cot, sec, csc) (57)
  • 2.5–2.7 Power Rule and Basic Derivative Rules (21)
  • 2.1 Average and Instantaneous Rate of Change (17)
  • 2.2 Definition of the Derivative and Tangent Lines (15)
  • 2.3 Estimating Derivatives from Graphs and Tables (4)
  • 2.4 Differentiability and Continuity (2)

Unit 3: Differentiation: Composite, Implicit, and Inverse Functions · 164

  • 3.1, 3.5 Chain Rule (63)
  • 3.2 Implicit Differentiation (42)
  • 3.3 Derivatives of Inverse Functions (23)
  • 3.6 Higher-Order Derivatives (21)
  • 3.4 Derivatives of Inverse Trigonometric Functions (15)

Unit 4: Contextual Applications of Differentiation · 165

  • 4.4–4.5 Related Rates (50)
  • 4.2 Straight-Line Motion: Position, Velocity, Acceleration (43)
  • 4.7 L'Hôpital's Rule (28)
  • 4.1, 4.3 Derivatives in Context and Applied Rates (22)
  • 4.6 Local Linearity and Linearization (22)

Unit 5: Analytical Applications of Differentiation · 7

  • 5.3–5.4 Increasing/Decreasing and First Derivative Test (2)
  • 5.6–5.7 Concavity, Inflection Points, and Second Derivative Test (2)
  • 5.2 Extreme Value Theorem and Critical Points (1)
  • 5.1 Mean Value Theorem (1)
  • 5.8–5.9 Connecting f, f', and f'' (Graphs and Behavior) (1)

Unit 6: Integration and Accumulation of Change · 6

  • 6.4–6.5 FTC and Accumulation Functions (4)
  • 6.7 Evaluating Definite Integrals (FTC) (1)
  • 6.9 Integration by Substitution (1)

Unit 7: Differential Equations · 3

  • 7.1–7.2 Modeling and Verifying Solutions to Differential Equations (2)
  • 7.3–7.4 Slope Fields (1)

Unit 8: Applications of Integration · 6

  • 8.2 Position, Velocity, and Acceleration Using Integrals (4)
  • 8.7–8.8 Volumes with Cross Sections (1)
  • 8.4–8.6 Area Between Curves (1)
Precalculus · 317 questions

Polynomial Functions · 101

  • Polynomial Characteristics (57)
  • Rates of Change (38)
  • Polynomial Models (4)
  • End Behavior (1)
  • Periodic Functions (1)

Exponential Functions · 63

  • Exponential Growth and Characteristics (40)
  • Modeling (19)
  • Sequences and Change (4)

Trigonometric Functions · 60

  • Sinusoidal Functions (29)
  • Sine, Cosine, Tangent Basics (23)
  • Periodic Functions (8)

Logarithmic Functions · 43

  • Logarithmic Properties and Graphs (33)
  • Inverse and Expressions (10)

Rational Functions · 31

  • Asymptotes and Holes (12)
  • Zeros and Factors (6)
  • Equivalent Representations (6)
  • End Behavior (4)
  • Rates of Change (1)

Polar Functions · 1

  • Polar Graphs and Characteristics (1)

AP Calculus AB/BC free-response questions. Our FRQ bank collects past AP Calculus AB and BC free-response questions, organized by question type, such as rates and accumulation, motion, and differential equations.

Browse AP Calculus FRQs by type

Where the questions come from

Keeping it accurate

Frequently asked questions

How precise are the skill filters?

Questions are tagged by the skill they test, so a request like "linear inequalities, medium" returns questions on that skill at that difficulty. Subtopic tags add a finer layer, such as quadratic functions within SAT Advanced Math or a specific grammar rule within Standard English Conventions.

Can I build a worksheet for one narrow topic?

Yes. Combine exam, skill, subtopic, and difficulty filters to target a single concept, then save the questions as a worksheet, share a link, or print a PDF.

How is difficulty decided?

Each question is rated against real test questions on the same skill. For SAT, ISEE, and AP, 1 is easier than a typical test question, 2 is typical, and 3 matches the hardest questions on the test. AMC uses a 1–5 scale to cover the full range of contest problems, from warm-ups to the final questions.

How often is the question bank updated?

Continuously. SAT questions from recent test dates are added after each administration, and new ISEE, AP, and AMC sets are added throughout the year. The counts on this page update automatically.

I found a mistake in a question. What should I do?

Open the question and choose "Report an Issue", or email cas@studytoday.us. Our team reviews reports, and corrections apply everywhere the question is used.

Last updated November 8, 2026.