Curriculum / Global Curriculum / Grade 12

Grade 12 Mathematics.

10textbooks

250lessons

240competencies

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The whole year, lesson by lesson.

Textbook 1Complex Numbers, Advanced Functions, and Calculus Readiness

4 chapters · 25 lessons

Chapter 1

Advanced Complex-Number Operations

Chapter 2

Polar Representation and Geometric Complex Operations

Chapter 3

Advanced Function Review and Domain Analysis

Chapter 4

Average Change and the Need for Calculus

Textbook 2Limits and Continuity

4 chapters · 25 lessons

Chapter 1

Limits from Tables and Graphs

Chapter 2

Algebraic Limit Techniques

Chapter 3

Infinite Limits and Long-Run Behavior

Chapter 4

Continuity and Discontinuity

Textbook 3Derivatives and Differentiation Techniques

4 chapters · 25 lessons

Chapter 1

Derivative from First Principles

Chapter 2

Power, Sum, Product, and Quotient Rules

Chapter 3

Chain Rule and Composite Functions

Chapter 4

Exponential, Logarithmic, Trigonometric, and Implicit Derivatives

Textbook 4Applications of Derivatives and Optimization

4 chapters · 25 lessons

Chapter 1

Tangents, Increasing Behavior, and Extrema

Chapter 2

Concavity, Inflection, and Curve Sketching

Chapter 3

Optimization

Chapter 4

Motion, Related Rates, and Linear Approximation

Textbook 5Antiderivatives, Definite Integrals, and Accumulation

4 chapters · 25 lessons

Chapter 1

Antiderivatives and Indefinite Integrals

Chapter 2

Riemann Sums and Numerical Accumulation

Chapter 3

Definite Integrals and the Fundamental Theorem

Chapter 4

Area, Accumulation Models, and Basic Substitution

Textbook 6Vectors, Parametric Models, and Calculus-Linked Geometry

4 chapters · 25 lessons

Chapter 1

Vector Representation and Operations

Chapter 2

Dot Product and Vector Geometry

Chapter 3

Parametric Equations and Motion

Chapter 4

Vector and Parametric Calculus Applications

Textbook 7Advanced Probability and Discrete Mathematics

4 chapters · 25 lessons

Chapter 1

Advanced Counting and Recurrence

Chapter 2

Introductory Graph Theory and Networks

Chapter 3

Advanced Probability Distributions

Chapter 4

Conditional Updating, Risk, and Simulation

Textbook 8Statistical Inference and Advanced Data Modeling

4 chapters · 25 lessons

Chapter 1

Sampling Distributions and Confidence Intervals

Chapter 2

Hypothesis Testing and Evidence

Chapter 3

Regression Inference and Prediction

Chapter 4

Study Design and Statistical Communication

Textbook 9Present Value, Future Value, Annuities, and Loan Mathematics

4 chapters · 25 lessons

Chapter 1

Time Value, Present Value, and Future Value

Chapter 2

Annuities and Regular Cash Flows

Chapter 3

Loans, Amortization, and Outstanding Balances

Chapter 4

Net Present Value, Return, Risk, and Scenario Analysis

Textbook 10Independent Proof, Mathematical Investigation, and Modeling Capstone

4 chapters · 25 lessons

Chapter 1

Independent Proof and Generalization

Chapter 2

Calculus and Optimization Investigation

Chapter 3

Probability or Statistical Investigation

Chapter 4

Final Mathematical Modeling Capstone

Every coin is a 15-minute lesson.

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Every competency, covered.

The 240 competencies set for Grade 12 Mathematics, and every one has lessons behind it.

Complex Numbers, Advanced Functions, and Calculus Readiness

37 competencies

Review domains, ranges, inverse functions, composition, and transformations.
Identify function behavior relevant to limits and derivatives.
Compare algebraic, graphical, numerical, and technological representations.
Use appropriate function families in calculus and modeling contexts.
Perform arithmetic operations with complex numbers accurately.
Use complex conjugates to simplify quotients.
Represent complex numbers in the complex plane.
Calculate complex modulus.
Interpret complex argument at an introductory level.
Convert supported complex numbers between rectangular and polar forms.
Multiply complex numbers using polar magnitude and angle relationships in supported cases.
Divide complex numbers using polar magnitude and angle relationships in supported cases.
Apply a supported De Moivre-style relationship for integer powers.
Connect complex roots with polynomial equations.
Interpret multiplication by (i) geometrically as a quarter-turn rotation.
Validate complex-number results using rectangular and geometric representations.
Calculate an average rate of change over an interval.
Interpret a difference quotient.
Explain instantaneous rate of change as a limit of average rates.
Formulate a mathematical conjecture independently.
Identify relevant definitions, assumptions, and prior results.
Construct a direct proof.
Construct a proof by contradiction in supported contexts.
Construct a proof by cases in supported contexts.
Use algebraic, geometric, combinatorial, probabilistic, or calculus-based reasoning.
Distinguish proof from numerical verification.
Use a counterexample to disprove a claim.
Identify hidden domain restrictions.
Identify irreversible transformations.
Identify extraneous, excluded, or missing solutions.
Critique a proof for necessity, sufficiency, and logical completeness.
Compare proof methods.
Generalize a result using parameters.
State the scope of a theorem or model precisely.
Revise an invalid argument while preserving valid components.
Communicate an independent sustained mathematical argument.
Explain uncertainty when the conclusion is model-based rather than proven.

Limits and Continuity

50 competencies

Interpret a limit as the value approached by a function.
Estimate a limit from a numerical table.
Estimate a limit from a graph.
Evaluate a limit by direct substitution when valid.
Evaluate supported limits through algebraic simplification.
Evaluate supported limits through factoring and cancellation.
Evaluate supported radical limits through conjugate multiplication.
Distinguish a function value from a limit.
Evaluate one-sided limits.
Determine whether a two-sided limit exists.
Interpret infinite limits as unbounded behavior.
Interpret limits at infinity as long-run behavior.
Connect limits at infinity with horizontal asymptotes in supported functions.
Define continuity at a point through function value and limiting behavior.
Determine intervals of continuity.
Identify removable discontinuities.
Identify jump discontinuities.
Identify infinite discontinuities.
Interpret continuity in a physical or modeling context.
Use technology to investigate limiting behavior without substituting technology for mathematical explanation.
Solve the model analytically, numerically, graphically, recursively, statistically, or technologically.
Interpret intermediate and final results.
Validate dimensional consistency and units.
Validate the model against observed or independent data.
Analyze residuals or discrepancies.
Conduct scenario analysis.
Conduct sensitivity analysis.
Quantify or describe uncertainty.
Identify where assumptions fail.
Compare multiple candidate models.
Refine a model in response to evidence.
State the model’s limitations.
Formulate a mathematical conjecture independently.
Identify relevant definitions, assumptions, and prior results.
Construct a direct proof.
Construct a proof by contradiction in supported contexts.
Construct a proof by cases in supported contexts.
Use algebraic, geometric, combinatorial, probabilistic, or calculus-based reasoning.
Distinguish proof from numerical verification.
Use a counterexample to disprove a claim.
Identify hidden domain restrictions.
Identify irreversible transformations.
Identify extraneous, excluded, or missing solutions.
Critique a proof for necessity, sufficiency, and logical completeness.
Compare proof methods.
Generalize a result using parameters.
State the scope of a theorem or model precisely.
Revise an invalid argument while preserving valid components.
Communicate an independent sustained mathematical argument.
Explain uncertainty when the conclusion is model-based rather than proven.

Derivatives and Differentiation Techniques

27 competencies

Review domains, ranges, inverse functions, composition, and transformations.
Identify function behavior relevant to limits and derivatives.
Compare algebraic, graphical, numerical, and technological representations.
Use appropriate function families in calculus and modeling contexts.
Calculate an average rate of change over an interval.
Interpret a difference quotient.
Explain instantaneous rate of change as a limit of average rates.
Define the derivative through a difference-quotient limit.
Interpret derivative notation including (f'(x)), (dy/dx), and supported alternatives.
Interpret the derivative as slope of a tangent line.
Estimate a derivative numerically.
Estimate a derivative graphically.
Differentiate constant functions.
Differentiate power functions with integer and supported rational exponents.
Differentiate polynomial functions.
Apply constant-multiple and sum rules.
Apply the product rule.
Apply the quotient rule.
Apply the chain rule.
Differentiate exponential functions.
Differentiate logarithmic functions.
Differentiate sine and cosine functions.
Differentiate tangent and supported trigonometric functions.
Use implicit differentiation in supported cases.
Calculate higher derivatives.
Interpret first and second derivatives contextually.
Check derivative results through graph behavior, numerical estimates, or technology.

Applications of Derivatives and Optimization

61 competencies

Write an equation of a tangent line.
Write an equation of a normal line in supported cases.
Determine intervals where a function is increasing or decreasing.
Identify critical points.
Determine local maxima and minima.
Apply the first-derivative test.
Determine concavity.
Identify inflection points.
Apply the second-derivative test in supported cases.
Sketch a function using derivative information.
Solve calculus-based optimization problems.
Define an objective function and constraints.
Identify an appropriate domain for an optimization model.
Compare critical and boundary values.
Interpret mathematical optima in context.
Model position, velocity, and acceleration.
Distinguish velocity from speed.
Interpret positive and negative velocity.
Analyze changes in motion using first and second derivatives.
Solve supported related-rate problems.
Use linear approximation for nearby values.
Estimate approximation error informally.
Conduct sensitivity analysis using derivatives.
Formulate a mathematical question independently from an unfamiliar situation.
Define variables, functions, parameters, units, constraints, objectives, and assumptions.
Identify relevant and irrelevant information.
Select a calculus, vector, probability, statistical, discrete, or financial model.
Justify the selected model family.
Estimate or calibrate model parameters.
Solve the model analytically, numerically, graphically, recursively, statistically, or technologically.
Interpret intermediate and final results.
Validate dimensional consistency and units.
Validate the model against observed or independent data.
Analyze residuals or discrepancies.
Conduct scenario analysis.
Conduct sensitivity analysis.
Quantify or describe uncertainty.
Identify where assumptions fail.
Compare multiple candidate models.
Refine a model in response to evidence.
State the model’s limitations.
Distinguish a mathematically optimal outcome from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the question, model, method, evidence, conclusions, uncertainty, and limitations clearly.
Formulate a mathematical conjecture independently.
Identify relevant definitions, assumptions, and prior results.
Construct a direct proof.
Construct a proof by contradiction in supported contexts.
Construct a proof by cases in supported contexts.
Use algebraic, geometric, combinatorial, probabilistic, or calculus-based reasoning.
Distinguish proof from numerical verification.
Use a counterexample to disprove a claim.
Identify hidden domain restrictions.
Identify irreversible transformations.
Identify extraneous, excluded, or missing solutions.
Critique a proof for necessity, sufficiency, and logical completeness.
Compare proof methods.
Generalize a result using parameters.
State the scope of a theorem or model precisely.
Revise an invalid argument while preserving valid components.
Communicate an independent sustained mathematical argument.
Explain uncertainty when the conclusion is model-based rather than proven.

Antiderivatives, Definite Integrals, and Accumulation

45 competencies

Explain an antiderivative as a function whose derivative is known.
Calculate basic indefinite integrals.
Include a constant of integration.
Apply reverse power rules in supported cases.
Integrate basic exponential functions.
Integrate basic sine and cosine functions.
Verify an antiderivative by differentiation.
Interpret a definite integral as net accumulation.
Approximate a definite integral using left-endpoint rectangles.
Approximate a definite integral using right-endpoint rectangles.
Approximate a definite integral using midpoint rectangles.
Approximate a definite integral using the trapezoidal rule.
Explain how increasing subdivisions may improve an approximation for suitable functions.
Calculate definite integrals using antiderivatives.
State and apply the Fundamental Theorem of Calculus at an introductory level.
Interpret signed area.
Distinguish signed area from total geometric area.
Calculate area between a curve and an axis in supported cases.
Calculate area between two curves in supported cases.
Use basic substitution in supported integrals.
Interpret an accumulation function.
Recover displacement from velocity.
Recover change in a quantity from a rate function.
Model cumulative cost, flow, growth, or resource use with an integral.
Validate integral results through units, estimates, and differentiation.
Formulate a mathematical question independently from an unfamiliar situation.
Define variables, functions, parameters, units, constraints, objectives, and assumptions.
Identify relevant and irrelevant information.
Select a calculus, vector, probability, statistical, discrete, or financial model.
Justify the selected model family.
Estimate or calibrate model parameters.
Solve the model analytically, numerically, graphically, recursively, statistically, or technologically.
Interpret intermediate and final results.
Validate dimensional consistency and units.
Validate the model against observed or independent data.
Analyze residuals or discrepancies.
Conduct scenario analysis.
Conduct sensitivity analysis.
Quantify or describe uncertainty.
Identify where assumptions fail.
Compare multiple candidate models.
Refine a model in response to evidence.
State the model’s limitations.
Distinguish a mathematically optimal outcome from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the question, model, method, evidence, conclusions, uncertainty, and limitations clearly.

Vectors, Parametric Models, and Calculus-Linked Geometry

29 competencies

Apply the chain rule.
Differentiate exponential functions.
Differentiate logarithmic functions.
Differentiate sine and cosine functions.
Differentiate tangent and supported trigonometric functions.
Use implicit differentiation in supported cases.
Calculate higher derivatives.
Interpret first and second derivatives contextually.
Check derivative results through graph behavior, numerical estimates, or technology.
Represent a vector geometrically by magnitude and direction.
Represent a vector in component form.
Add and subtract vectors.
Multiply a vector by a scalar.
Calculate vector magnitude.
Determine a unit vector in a stated direction.
Resolve a vector into horizontal and vertical components.
Convert between magnitude–direction and component forms in supported cases.
Calculate the dot product.
Interpret the sign of a dot product.
Determine whether vectors are perpendicular.
Calculate the angle between vectors in supported cases.
Use vectors to prove or verify geometric relationships.
Model displacement, velocity, force, or navigation with vectors.
Combine multiple vector effects.
Interpret a parametric equation as coordinates generated by a common parameter.
Convert supported parametric equations to Cartesian form.
Interpret direction and speed in supported parametric motion.
Differentiate supported parametric equations.
Use vectors and parametric representations in geometric or motion models.

Advanced Probability and Discrete Mathematics

59 competencies

Formulate a mathematical question independently from an unfamiliar situation.
Define variables, functions, parameters, units, constraints, objectives, and assumptions.
Identify relevant and irrelevant information.
Select a calculus, vector, probability, statistical, discrete, or financial model.
Justify the selected model family.
Estimate or calibrate model parameters.
Solve the model analytically, numerically, graphically, recursively, statistically, or technologically.
Interpret intermediate and final results.
Validate dimensional consistency and units.
Validate the model against observed or independent data.
Analyze residuals or discrepancies.
Conduct scenario analysis.
Conduct sensitivity analysis.
Quantify or describe uncertainty.
Identify where assumptions fail.
Compare multiple candidate models.
Refine a model in response to evidence.
State the model’s limitations.
Distinguish a mathematically optimal outcome from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the question, model, method, evidence, conclusions, uncertainty, and limitations clearly.
Use permutations and combinations in multistage probability.
Apply inclusion–exclusion reasoning in supported counting problems.
Use recurrence relations in supported discrete models.
Model paths and networks using introductory graph theory.
Identify vertices, edges, paths, cycles, and connected components.
Analyze introductory route, scheduling, or network problems.
Construct discrete probability distributions.
Calculate expected value and variance.
Apply binomial distributions in depth.
Apply geometric distributions in supported waiting-time contexts.
Apply normal distributions in developed contexts.
Use standardization to calculate probabilities.
Use normal approximations where assumptions are supplied and justified.
Calculate conditional probability.
Apply Bayes-style updating in supported table or tree contexts.
Distinguish prior, likelihood, and updated probability conceptually.
Model repeated and cumulative risk.
Use simulation to approximate probabilities not easily calculated exactly.
Compare exact, approximate, and simulated probability results.
Evaluate whether a discrete or continuous distribution is appropriate.
Identify invalid independence, distribution, or sample-space assumptions.
Formulate a mathematical conjecture independently.
Identify relevant definitions, assumptions, and prior results.
Construct a direct proof.
Construct a proof by contradiction in supported contexts.
Construct a proof by cases in supported contexts.
Use algebraic, geometric, combinatorial, probabilistic, or calculus-based reasoning.
Distinguish proof from numerical verification.
Use a counterexample to disprove a claim.
Identify hidden domain restrictions.
Identify irreversible transformations.
Identify extraneous, excluded, or missing solutions.
Critique a proof for necessity, sufficiency, and logical completeness.
Compare proof methods.
Generalize a result using parameters.
State the scope of a theorem or model precisely.
Revise an invalid argument while preserving valid components.
Communicate an independent sustained mathematical argument.
Explain uncertainty when the conclusion is model-based rather than proven.

Statistical Inference and Advanced Data Modeling

56 competencies

Validate dimensional consistency and units.
Validate the model against observed or independent data.
Analyze residuals or discrepancies.
Conduct scenario analysis.
Conduct sensitivity analysis.
Quantify or describe uncertainty.
Identify where assumptions fail.
Compare multiple candidate models.
Refine a model in response to evidence.
State the model’s limitations.
Formulate a mathematical conjecture independently.
Identify relevant definitions, assumptions, and prior results.
Construct a direct proof.
Construct a proof by contradiction in supported contexts.
Construct a proof by cases in supported contexts.
Use algebraic, geometric, combinatorial, probabilistic, or calculus-based reasoning.
Distinguish proof from numerical verification.
Use a counterexample to disprove a claim.
Identify hidden domain restrictions.
Identify irreversible transformations.
Identify extraneous, excluded, or missing solutions.
Critique a proof for necessity, sufficiency, and logical completeness.
Compare proof methods.
Generalize a result using parameters.
State the scope of a theorem or model precisely.
Revise an invalid argument while preserving valid components.
Communicate an independent sustained mathematical argument.
Explain uncertainty when the conclusion is model-based rather than proven.
Distinguish population parameters and sample statistics.
Explain sampling distributions.
Calculate or interpret standard error in supported cases.
Explain how sample size affects standard error.
Construct a confidence interval for a population proportion in supported cases.
Construct a confidence interval for a population mean in supported cases.
Interpret confidence level correctly.
Interpret margin of error.
Distinguish random sampling error from systematic bias.
State null and alternative hypotheses in supported contexts.
Calculate or interpret a test statistic using a supplied procedure.
Interpret a p-value as evidence under a null model.
Compare a p-value with a stated significance level.
State a conclusion in context without claiming absolute proof.
Explain Type I error.
Explain Type II error.
Distinguish statistical significance from practical importance.
Evaluate assumptions underlying an inference.
Analyze linear regression using slope, intercept, correlation, and residuals.
Interpret uncertainty in regression parameters conceptually.
Interpret a confidence interval for a regression slope when supplied.
Distinguish a confidence interval for mean response from a prediction interval for an individual outcome conceptually.
Compare linear and nonlinear statistical models.
Use residual analysis to identify model failure.
Evaluate the effects of outliers, leverage, and restricted ranges.
Evaluate observational and experimental studies.
Determine whether generalization or causation is supported.
Communicate statistical conclusions with uncertainty and limitations.

Present Value, Future Value, Annuities, and Loan Mathematics

51 competencies

Explain the time value of money.
Calculate future value of a single cash flow.
Calculate present value of a future cash flow.
Interpret discounting as the inverse of compounding.
Compare cash flows occurring at different times.
Calculate effective periodic and annual rates in supported cases.
Calculate future value of an ordinary annuity.
Calculate present value of an ordinary annuity.
Calculate future or present value of an annuity due in supported cases.
Determine a regular payment required for a stated target.
Determine a regular withdrawal supported by a stated balance and model.
Construct or interpret an amortization schedule.
Separate interest and principal components of a payment.
Calculate outstanding loan balance.
Calculate the effect of additional payments.
Compare loans through present value, total repayment, payment timing, and fees.
Calculate net present value for a supported cash-flow stream.
Interpret positive, negative, and zero net present value mathematically.
Calculate nominal and inflation-adjusted returns.
Compare financial scenarios under different inflation or growth assumptions.
Calculate expected return from a probability distribution.
Calculate or interpret variance and standard deviation of financial returns.
Compare expected return and variability mathematically.
Conduct scenario and sensitivity analysis for rates, payments, durations, and cash flows.
Identify assumptions involving rates, fees, taxes, inflation, defaults, and timing.
Distinguish mathematical financial value from personal preference, ethics, liquidity needs, and risk tolerance.
Validate the model against observed or independent data.
Analyze residuals or discrepancies.
Conduct scenario analysis.
Conduct sensitivity analysis.
Quantify or describe uncertainty.
Identify where assumptions fail.
Compare multiple candidate models.
Refine a model in response to evidence.
State the model’s limitations.
Distinguish a mathematically optimal outcome from a decision requiring values, ethics, policy, or risk tolerance.
Construct discrete probability distributions.
Calculate expected value and variance.
Apply binomial distributions in depth.
Apply geometric distributions in supported waiting-time contexts.
Apply normal distributions in developed contexts.
Use standardization to calculate probabilities.
Use normal approximations where assumptions are supplied and justified.
Calculate conditional probability.
Apply Bayes-style updating in supported table or tree contexts.
Distinguish prior, likelihood, and updated probability conceptually.
Model repeated and cumulative risk.
Use simulation to approximate probabilities not easily calculated exactly.
Compare exact, approximate, and simulated probability results.
Evaluate whether a discrete or continuous distribution is appropriate.
Identify invalid independence, distribution, or sample-space assumptions.

Independent Proof, Mathematical Investigation, and Modeling Capstone

158 competencies

Write an equation of a tangent line.
Write an equation of a normal line in supported cases.
Determine intervals where a function is increasing or decreasing.
Identify critical points.
Determine local maxima and minima.
Apply the first-derivative test.
Determine concavity.
Identify inflection points.
Apply the second-derivative test in supported cases.
Sketch a function using derivative information.
Solve calculus-based optimization problems.
Define an objective function and constraints.
Identify an appropriate domain for an optimization model.
Compare critical and boundary values.
Interpret mathematical optima in context.
Model position, velocity, and acceleration.
Distinguish velocity from speed.
Interpret positive and negative velocity.
Analyze changes in motion using first and second derivatives.
Solve supported related-rate problems.
Use linear approximation for nearby values.
Estimate approximation error informally.
Conduct sensitivity analysis using derivatives.
Calculate an average rate of change over an interval.
Interpret a difference quotient.
Explain instantaneous rate of change as a limit of average rates.
Define the derivative through a difference-quotient limit.
Interpret derivative notation including (f'(x)), (dy/dx), and supported alternatives.
Interpret the derivative as slope of a tangent line.
Estimate a derivative numerically.
Estimate a derivative graphically.
Differentiate constant functions.
Differentiate power functions with integer and supported rational exponents.
Differentiate polynomial functions.
Apply constant-multiple and sum rules.
Apply the product rule.
Apply the quotient rule.
Apply the chain rule.
Differentiate exponential functions.
Differentiate logarithmic functions.
Differentiate sine and cosine functions.
Differentiate tangent and supported trigonometric functions.
Use implicit differentiation in supported cases.
Calculate higher derivatives.
Interpret first and second derivatives contextually.
Check derivative results through graph behavior, numerical estimates, or technology.
Explain an antiderivative as a function whose derivative is known.
Calculate basic indefinite integrals.
Include a constant of integration.
Apply reverse power rules in supported cases.
Integrate basic exponential functions.
Integrate basic sine and cosine functions.
Verify an antiderivative by differentiation.
Interpret a definite integral as net accumulation.
Approximate a definite integral using left-endpoint rectangles.
Approximate a definite integral using right-endpoint rectangles.
Approximate a definite integral using midpoint rectangles.
Approximate a definite integral using the trapezoidal rule.
Explain how increasing subdivisions may improve an approximation for suitable functions.
Calculate definite integrals using antiderivatives.
State and apply the Fundamental Theorem of Calculus at an introductory level.
Interpret signed area.
Distinguish signed area from total geometric area.
Calculate area between a curve and an axis in supported cases.
Calculate area between two curves in supported cases.
Use basic substitution in supported integrals.
Interpret an accumulation function.
Recover displacement from velocity.
Recover change in a quantity from a rate function.
Model cumulative cost, flow, growth, or resource use with an integral.
Validate integral results through units, estimates, and differentiation.
Formulate a mathematical question independently from an unfamiliar situation.
Define variables, functions, parameters, units, constraints, objectives, and assumptions.
Identify relevant and irrelevant information.
Select a calculus, vector, probability, statistical, discrete, or financial model.
Justify the selected model family.
Estimate or calibrate model parameters.
Solve the model analytically, numerically, graphically, recursively, statistically, or technologically.
Interpret intermediate and final results.
Validate dimensional consistency and units.
Validate the model against observed or independent data.
Analyze residuals or discrepancies.
Conduct scenario analysis.
Conduct sensitivity analysis.
Quantify or describe uncertainty.
Identify where assumptions fail.
Compare multiple candidate models.
Refine a model in response to evidence.
State the model’s limitations.
Distinguish a mathematically optimal outcome from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the question, model, method, evidence, conclusions, uncertainty, and limitations clearly.
Use permutations and combinations in multistage probability.
Apply inclusion–exclusion reasoning in supported counting problems.
Use recurrence relations in supported discrete models.
Model paths and networks using introductory graph theory.
Identify vertices, edges, paths, cycles, and connected components.
Analyze introductory route, scheduling, or network problems.
Construct discrete probability distributions.
Calculate expected value and variance.
Apply binomial distributions in depth.
Apply geometric distributions in supported waiting-time contexts.
Apply normal distributions in developed contexts.
Use standardization to calculate probabilities.
Use normal approximations where assumptions are supplied and justified.
Calculate conditional probability.
Apply Bayes-style updating in supported table or tree contexts.
Distinguish prior, likelihood, and updated probability conceptually.
Model repeated and cumulative risk.
Use simulation to approximate probabilities not easily calculated exactly.
Compare exact, approximate, and simulated probability results.
Evaluate whether a discrete or continuous distribution is appropriate.
Identify invalid independence, distribution, or sample-space assumptions.
Formulate a mathematical conjecture independently.
Identify relevant definitions, assumptions, and prior results.
Construct a direct proof.
Construct a proof by contradiction in supported contexts.
Construct a proof by cases in supported contexts.
Use algebraic, geometric, combinatorial, probabilistic, or calculus-based reasoning.
Distinguish proof from numerical verification.
Use a counterexample to disprove a claim.
Identify hidden domain restrictions.
Identify irreversible transformations.
Identify extraneous, excluded, or missing solutions.
Critique a proof for necessity, sufficiency, and logical completeness.
Compare proof methods.
Generalize a result using parameters.
State the scope of a theorem or model precisely.
Revise an invalid argument while preserving valid components.
Communicate an independent sustained mathematical argument.
Explain uncertainty when the conclusion is model-based rather than proven.
Distinguish population parameters and sample statistics.
Explain sampling distributions.
Calculate or interpret standard error in supported cases.
Explain how sample size affects standard error.
Construct a confidence interval for a population proportion in supported cases.
Construct a confidence interval for a population mean in supported cases.
Interpret confidence level correctly.
Interpret margin of error.
Distinguish random sampling error from systematic bias.
State null and alternative hypotheses in supported contexts.
Calculate or interpret a test statistic using a supplied procedure.
Interpret a p-value as evidence under a null model.
Compare a p-value with a stated significance level.
State a conclusion in context without claiming absolute proof.
Explain Type I error.
Explain Type II error.
Distinguish statistical significance from practical importance.
Evaluate assumptions underlying an inference.
Analyze linear regression using slope, intercept, correlation, and residuals.
Interpret uncertainty in regression parameters conceptually.
Interpret a confidence interval for a regression slope when supplied.
Distinguish a confidence interval for mean response from a prediction interval for an individual outcome conceptually.
Compare linear and nonlinear statistical models.
Use residual analysis to identify model failure.
Evaluate the effects of outliers, leverage, and restricted ranges.
Evaluate observational and experimental studies.
Determine whether generalization or causation is supported.
Communicate statistical conclusions with uncertainty and limitations.

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The whole year, lesson by lesson.

Textbook 1Complex Numbers, Advanced Functions, and Calculus Readiness

4 chapters · 25 lessons

Chapter 1

Advanced Complex-Number Operations

Chapter 2

Polar Representation and Geometric Complex Operations

Chapter 3

Advanced Function Review and Domain Analysis

Chapter 4

Average Change and the Need for Calculus

Textbook 2Limits and Continuity

4 chapters · 25 lessons

Chapter 1

Limits from Tables and Graphs

Chapter 2

Algebraic Limit Techniques

Chapter 3

Infinite Limits and Long-Run Behavior

Chapter 4

Continuity and Discontinuity

Textbook 3Derivatives and Differentiation Techniques

4 chapters · 25 lessons

Chapter 1

Derivative from First Principles

Chapter 2

Power, Sum, Product, and Quotient Rules

Chapter 3

Chain Rule and Composite Functions

Chapter 4

Exponential, Logarithmic, Trigonometric, and Implicit Derivatives

Textbook 4Applications of Derivatives and Optimization

4 chapters · 25 lessons

Chapter 1

Tangents, Increasing Behavior, and Extrema

Chapter 2

Concavity, Inflection, and Curve Sketching

Chapter 3

Optimization

Chapter 4

Motion, Related Rates, and Linear Approximation

Textbook 5Antiderivatives, Definite Integrals, and Accumulation

4 chapters · 25 lessons

Chapter 1

Antiderivatives and Indefinite Integrals

Chapter 2

Riemann Sums and Numerical Accumulation

Chapter 3

Definite Integrals and the Fundamental Theorem

Chapter 4

Area, Accumulation Models, and Basic Substitution

Textbook 6Vectors, Parametric Models, and Calculus-Linked Geometry

4 chapters · 25 lessons

Chapter 1

Vector Representation and Operations

Chapter 2

Dot Product and Vector Geometry

Chapter 3

Parametric Equations and Motion

Chapter 4

Vector and Parametric Calculus Applications

Textbook 7Advanced Probability and Discrete Mathematics

4 chapters · 25 lessons

Chapter 1

Advanced Counting and Recurrence

Chapter 2

Introductory Graph Theory and Networks

Chapter 3

Advanced Probability Distributions

Chapter 4

Conditional Updating, Risk, and Simulation

Textbook 8Statistical Inference and Advanced Data Modeling

4 chapters · 25 lessons

Chapter 1

Sampling Distributions and Confidence Intervals

Chapter 2

Hypothesis Testing and Evidence

Chapter 3

Regression Inference and Prediction

Chapter 4

Study Design and Statistical Communication

Textbook 9Present Value, Future Value, Annuities, and Loan Mathematics

4 chapters · 25 lessons

Chapter 1

Time Value, Present Value, and Future Value

Chapter 2

Annuities and Regular Cash Flows

Chapter 3

Loans, Amortization, and Outstanding Balances

Chapter 4

Net Present Value, Return, Risk, and Scenario Analysis

Textbook 10Independent Proof, Mathematical Investigation, and Modeling Capstone

4 chapters · 25 lessons

Chapter 1

Independent Proof and Generalization

Chapter 2

Calculus and Optimization Investigation

Chapter 3

Probability or Statistical Investigation

Chapter 4

Final Mathematical Modeling Capstone

Every competency, covered.

The 240 competencies set for Grade 12 Mathematics, and every one has lessons behind it.

Complex Numbers, Advanced Functions, and Calculus Readiness

37 competencies

Review domains, ranges, inverse functions, composition, and transformations.
Identify function behavior relevant to limits and derivatives.
Compare algebraic, graphical, numerical, and technological representations.
Use appropriate function families in calculus and modeling contexts.
Perform arithmetic operations with complex numbers accurately.
Use complex conjugates to simplify quotients.
Represent complex numbers in the complex plane.
Calculate complex modulus.
Interpret complex argument at an introductory level.
Convert supported complex numbers between rectangular and polar forms.
Multiply complex numbers using polar magnitude and angle relationships in supported cases.
Divide complex numbers using polar magnitude and angle relationships in supported cases.
Apply a supported De Moivre-style relationship for integer powers.
Connect complex roots with polynomial equations.
Interpret multiplication by (i) geometrically as a quarter-turn rotation.
Validate complex-number results using rectangular and geometric representations.
Calculate an average rate of change over an interval.
Interpret a difference quotient.
Explain instantaneous rate of change as a limit of average rates.
Formulate a mathematical conjecture independently.
Identify relevant definitions, assumptions, and prior results.
Construct a direct proof.
Construct a proof by contradiction in supported contexts.
Construct a proof by cases in supported contexts.
Use algebraic, geometric, combinatorial, probabilistic, or calculus-based reasoning.
Distinguish proof from numerical verification.
Use a counterexample to disprove a claim.
Identify hidden domain restrictions.
Identify irreversible transformations.
Identify extraneous, excluded, or missing solutions.
Critique a proof for necessity, sufficiency, and logical completeness.
Compare proof methods.
Generalize a result using parameters.
State the scope of a theorem or model precisely.
Revise an invalid argument while preserving valid components.
Communicate an independent sustained mathematical argument.
Explain uncertainty when the conclusion is model-based rather than proven.

Limits and Continuity

50 competencies

Interpret a limit as the value approached by a function.
Estimate a limit from a numerical table.
Estimate a limit from a graph.
Evaluate a limit by direct substitution when valid.
Evaluate supported limits through algebraic simplification.
Evaluate supported limits through factoring and cancellation.
Evaluate supported radical limits through conjugate multiplication.
Distinguish a function value from a limit.
Evaluate one-sided limits.
Determine whether a two-sided limit exists.
Interpret infinite limits as unbounded behavior.
Interpret limits at infinity as long-run behavior.
Connect limits at infinity with horizontal asymptotes in supported functions.
Define continuity at a point through function value and limiting behavior.
Determine intervals of continuity.
Identify removable discontinuities.
Identify jump discontinuities.
Identify infinite discontinuities.
Interpret continuity in a physical or modeling context.
Use technology to investigate limiting behavior without substituting technology for mathematical explanation.
Solve the model analytically, numerically, graphically, recursively, statistically, or technologically.
Interpret intermediate and final results.
Validate dimensional consistency and units.
Validate the model against observed or independent data.
Analyze residuals or discrepancies.
Conduct scenario analysis.
Conduct sensitivity analysis.
Quantify or describe uncertainty.
Identify where assumptions fail.
Compare multiple candidate models.
Refine a model in response to evidence.
State the model’s limitations.
Formulate a mathematical conjecture independently.
Identify relevant definitions, assumptions, and prior results.
Construct a direct proof.
Construct a proof by contradiction in supported contexts.
Construct a proof by cases in supported contexts.
Use algebraic, geometric, combinatorial, probabilistic, or calculus-based reasoning.
Distinguish proof from numerical verification.
Use a counterexample to disprove a claim.
Identify hidden domain restrictions.
Identify irreversible transformations.
Identify extraneous, excluded, or missing solutions.
Critique a proof for necessity, sufficiency, and logical completeness.
Compare proof methods.
Generalize a result using parameters.
State the scope of a theorem or model precisely.
Revise an invalid argument while preserving valid components.
Communicate an independent sustained mathematical argument.
Explain uncertainty when the conclusion is model-based rather than proven.

Derivatives and Differentiation Techniques

27 competencies

Review domains, ranges, inverse functions, composition, and transformations.
Identify function behavior relevant to limits and derivatives.
Compare algebraic, graphical, numerical, and technological representations.
Use appropriate function families in calculus and modeling contexts.
Calculate an average rate of change over an interval.
Interpret a difference quotient.
Explain instantaneous rate of change as a limit of average rates.
Define the derivative through a difference-quotient limit.
Interpret derivative notation including (f'(x)), (dy/dx), and supported alternatives.
Interpret the derivative as slope of a tangent line.
Estimate a derivative numerically.
Estimate a derivative graphically.
Differentiate constant functions.
Differentiate power functions with integer and supported rational exponents.
Differentiate polynomial functions.
Apply constant-multiple and sum rules.
Apply the product rule.
Apply the quotient rule.
Apply the chain rule.
Differentiate exponential functions.
Differentiate logarithmic functions.
Differentiate sine and cosine functions.
Differentiate tangent and supported trigonometric functions.
Use implicit differentiation in supported cases.
Calculate higher derivatives.
Interpret first and second derivatives contextually.
Check derivative results through graph behavior, numerical estimates, or technology.

Applications of Derivatives and Optimization

61 competencies

Write an equation of a tangent line.
Write an equation of a normal line in supported cases.
Determine intervals where a function is increasing or decreasing.
Identify critical points.
Determine local maxima and minima.
Apply the first-derivative test.
Determine concavity.
Identify inflection points.
Apply the second-derivative test in supported cases.
Sketch a function using derivative information.
Solve calculus-based optimization problems.
Define an objective function and constraints.
Identify an appropriate domain for an optimization model.
Compare critical and boundary values.
Interpret mathematical optima in context.
Model position, velocity, and acceleration.
Distinguish velocity from speed.
Interpret positive and negative velocity.
Analyze changes in motion using first and second derivatives.
Solve supported related-rate problems.
Use linear approximation for nearby values.
Estimate approximation error informally.
Conduct sensitivity analysis using derivatives.
Formulate a mathematical question independently from an unfamiliar situation.
Define variables, functions, parameters, units, constraints, objectives, and assumptions.
Identify relevant and irrelevant information.
Select a calculus, vector, probability, statistical, discrete, or financial model.
Justify the selected model family.
Estimate or calibrate model parameters.
Solve the model analytically, numerically, graphically, recursively, statistically, or technologically.
Interpret intermediate and final results.
Validate dimensional consistency and units.
Validate the model against observed or independent data.
Analyze residuals or discrepancies.
Conduct scenario analysis.
Conduct sensitivity analysis.
Quantify or describe uncertainty.
Identify where assumptions fail.
Compare multiple candidate models.
Refine a model in response to evidence.
State the model’s limitations.
Distinguish a mathematically optimal outcome from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the question, model, method, evidence, conclusions, uncertainty, and limitations clearly.
Formulate a mathematical conjecture independently.
Identify relevant definitions, assumptions, and prior results.
Construct a direct proof.
Construct a proof by contradiction in supported contexts.
Construct a proof by cases in supported contexts.
Use algebraic, geometric, combinatorial, probabilistic, or calculus-based reasoning.
Distinguish proof from numerical verification.
Use a counterexample to disprove a claim.
Identify hidden domain restrictions.
Identify irreversible transformations.
Identify extraneous, excluded, or missing solutions.
Critique a proof for necessity, sufficiency, and logical completeness.
Compare proof methods.
Generalize a result using parameters.
State the scope of a theorem or model precisely.
Revise an invalid argument while preserving valid components.
Communicate an independent sustained mathematical argument.
Explain uncertainty when the conclusion is model-based rather than proven.

Antiderivatives, Definite Integrals, and Accumulation

45 competencies

Explain an antiderivative as a function whose derivative is known.
Calculate basic indefinite integrals.
Include a constant of integration.
Apply reverse power rules in supported cases.
Integrate basic exponential functions.
Integrate basic sine and cosine functions.
Verify an antiderivative by differentiation.
Interpret a definite integral as net accumulation.
Approximate a definite integral using left-endpoint rectangles.
Approximate a definite integral using right-endpoint rectangles.
Approximate a definite integral using midpoint rectangles.
Approximate a definite integral using the trapezoidal rule.
Explain how increasing subdivisions may improve an approximation for suitable functions.
Calculate definite integrals using antiderivatives.
State and apply the Fundamental Theorem of Calculus at an introductory level.
Interpret signed area.
Distinguish signed area from total geometric area.
Calculate area between a curve and an axis in supported cases.
Calculate area between two curves in supported cases.
Use basic substitution in supported integrals.
Interpret an accumulation function.
Recover displacement from velocity.
Recover change in a quantity from a rate function.
Model cumulative cost, flow, growth, or resource use with an integral.
Validate integral results through units, estimates, and differentiation.
Formulate a mathematical question independently from an unfamiliar situation.
Define variables, functions, parameters, units, constraints, objectives, and assumptions.
Identify relevant and irrelevant information.
Select a calculus, vector, probability, statistical, discrete, or financial model.
Justify the selected model family.
Estimate or calibrate model parameters.
Solve the model analytically, numerically, graphically, recursively, statistically, or technologically.
Interpret intermediate and final results.
Validate dimensional consistency and units.
Validate the model against observed or independent data.
Analyze residuals or discrepancies.
Conduct scenario analysis.
Conduct sensitivity analysis.
Quantify or describe uncertainty.
Identify where assumptions fail.
Compare multiple candidate models.
Refine a model in response to evidence.
State the model’s limitations.
Distinguish a mathematically optimal outcome from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the question, model, method, evidence, conclusions, uncertainty, and limitations clearly.

Vectors, Parametric Models, and Calculus-Linked Geometry

29 competencies

Apply the chain rule.
Differentiate exponential functions.
Differentiate logarithmic functions.
Differentiate sine and cosine functions.
Differentiate tangent and supported trigonometric functions.
Use implicit differentiation in supported cases.
Calculate higher derivatives.
Interpret first and second derivatives contextually.
Check derivative results through graph behavior, numerical estimates, or technology.
Represent a vector geometrically by magnitude and direction.
Represent a vector in component form.
Add and subtract vectors.
Multiply a vector by a scalar.
Calculate vector magnitude.
Determine a unit vector in a stated direction.
Resolve a vector into horizontal and vertical components.
Convert between magnitude–direction and component forms in supported cases.
Calculate the dot product.
Interpret the sign of a dot product.
Determine whether vectors are perpendicular.
Calculate the angle between vectors in supported cases.
Use vectors to prove or verify geometric relationships.
Model displacement, velocity, force, or navigation with vectors.
Combine multiple vector effects.
Interpret a parametric equation as coordinates generated by a common parameter.
Convert supported parametric equations to Cartesian form.
Interpret direction and speed in supported parametric motion.
Differentiate supported parametric equations.
Use vectors and parametric representations in geometric or motion models.

Advanced Probability and Discrete Mathematics

59 competencies

Formulate a mathematical question independently from an unfamiliar situation.
Define variables, functions, parameters, units, constraints, objectives, and assumptions.
Identify relevant and irrelevant information.
Select a calculus, vector, probability, statistical, discrete, or financial model.
Justify the selected model family.
Estimate or calibrate model parameters.
Solve the model analytically, numerically, graphically, recursively, statistically, or technologically.
Interpret intermediate and final results.
Validate dimensional consistency and units.
Validate the model against observed or independent data.
Analyze residuals or discrepancies.
Conduct scenario analysis.
Conduct sensitivity analysis.
Quantify or describe uncertainty.
Identify where assumptions fail.
Compare multiple candidate models.
Refine a model in response to evidence.
State the model’s limitations.
Distinguish a mathematically optimal outcome from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the question, model, method, evidence, conclusions, uncertainty, and limitations clearly.
Use permutations and combinations in multistage probability.
Apply inclusion–exclusion reasoning in supported counting problems.
Use recurrence relations in supported discrete models.
Model paths and networks using introductory graph theory.
Identify vertices, edges, paths, cycles, and connected components.
Analyze introductory route, scheduling, or network problems.
Construct discrete probability distributions.
Calculate expected value and variance.
Apply binomial distributions in depth.
Apply geometric distributions in supported waiting-time contexts.
Apply normal distributions in developed contexts.
Use standardization to calculate probabilities.
Use normal approximations where assumptions are supplied and justified.
Calculate conditional probability.
Apply Bayes-style updating in supported table or tree contexts.
Distinguish prior, likelihood, and updated probability conceptually.
Model repeated and cumulative risk.
Use simulation to approximate probabilities not easily calculated exactly.
Compare exact, approximate, and simulated probability results.
Evaluate whether a discrete or continuous distribution is appropriate.
Identify invalid independence, distribution, or sample-space assumptions.
Formulate a mathematical conjecture independently.
Identify relevant definitions, assumptions, and prior results.
Construct a direct proof.
Construct a proof by contradiction in supported contexts.
Construct a proof by cases in supported contexts.
Use algebraic, geometric, combinatorial, probabilistic, or calculus-based reasoning.
Distinguish proof from numerical verification.
Use a counterexample to disprove a claim.
Identify hidden domain restrictions.
Identify irreversible transformations.
Identify extraneous, excluded, or missing solutions.
Critique a proof for necessity, sufficiency, and logical completeness.
Compare proof methods.
Generalize a result using parameters.
State the scope of a theorem or model precisely.
Revise an invalid argument while preserving valid components.
Communicate an independent sustained mathematical argument.
Explain uncertainty when the conclusion is model-based rather than proven.

Statistical Inference and Advanced Data Modeling

56 competencies

Validate dimensional consistency and units.
Validate the model against observed or independent data.
Analyze residuals or discrepancies.
Conduct scenario analysis.
Conduct sensitivity analysis.
Quantify or describe uncertainty.
Identify where assumptions fail.
Compare multiple candidate models.
Refine a model in response to evidence.
State the model’s limitations.
Formulate a mathematical conjecture independently.
Identify relevant definitions, assumptions, and prior results.
Construct a direct proof.
Construct a proof by contradiction in supported contexts.
Construct a proof by cases in supported contexts.
Use algebraic, geometric, combinatorial, probabilistic, or calculus-based reasoning.
Distinguish proof from numerical verification.
Use a counterexample to disprove a claim.
Identify hidden domain restrictions.
Identify irreversible transformations.
Identify extraneous, excluded, or missing solutions.
Critique a proof for necessity, sufficiency, and logical completeness.
Compare proof methods.
Generalize a result using parameters.
State the scope of a theorem or model precisely.
Revise an invalid argument while preserving valid components.
Communicate an independent sustained mathematical argument.
Explain uncertainty when the conclusion is model-based rather than proven.
Distinguish population parameters and sample statistics.
Explain sampling distributions.
Calculate or interpret standard error in supported cases.
Explain how sample size affects standard error.
Construct a confidence interval for a population proportion in supported cases.
Construct a confidence interval for a population mean in supported cases.
Interpret confidence level correctly.
Interpret margin of error.
Distinguish random sampling error from systematic bias.
State null and alternative hypotheses in supported contexts.
Calculate or interpret a test statistic using a supplied procedure.
Interpret a p-value as evidence under a null model.
Compare a p-value with a stated significance level.
State a conclusion in context without claiming absolute proof.
Explain Type I error.
Explain Type II error.
Distinguish statistical significance from practical importance.
Evaluate assumptions underlying an inference.
Analyze linear regression using slope, intercept, correlation, and residuals.
Interpret uncertainty in regression parameters conceptually.
Interpret a confidence interval for a regression slope when supplied.
Distinguish a confidence interval for mean response from a prediction interval for an individual outcome conceptually.
Compare linear and nonlinear statistical models.
Use residual analysis to identify model failure.
Evaluate the effects of outliers, leverage, and restricted ranges.
Evaluate observational and experimental studies.
Determine whether generalization or causation is supported.
Communicate statistical conclusions with uncertainty and limitations.

Present Value, Future Value, Annuities, and Loan Mathematics

51 competencies

Explain the time value of money.
Calculate future value of a single cash flow.
Calculate present value of a future cash flow.
Interpret discounting as the inverse of compounding.
Compare cash flows occurring at different times.
Calculate effective periodic and annual rates in supported cases.
Calculate future value of an ordinary annuity.
Calculate present value of an ordinary annuity.
Calculate future or present value of an annuity due in supported cases.
Determine a regular payment required for a stated target.
Determine a regular withdrawal supported by a stated balance and model.
Construct or interpret an amortization schedule.
Separate interest and principal components of a payment.
Calculate outstanding loan balance.
Calculate the effect of additional payments.
Compare loans through present value, total repayment, payment timing, and fees.
Calculate net present value for a supported cash-flow stream.
Interpret positive, negative, and zero net present value mathematically.
Calculate nominal and inflation-adjusted returns.
Compare financial scenarios under different inflation or growth assumptions.
Calculate expected return from a probability distribution.
Calculate or interpret variance and standard deviation of financial returns.
Compare expected return and variability mathematically.
Conduct scenario and sensitivity analysis for rates, payments, durations, and cash flows.
Identify assumptions involving rates, fees, taxes, inflation, defaults, and timing.
Distinguish mathematical financial value from personal preference, ethics, liquidity needs, and risk tolerance.
Validate the model against observed or independent data.
Analyze residuals or discrepancies.
Conduct scenario analysis.
Conduct sensitivity analysis.
Quantify or describe uncertainty.
Identify where assumptions fail.
Compare multiple candidate models.
Refine a model in response to evidence.
State the model’s limitations.
Distinguish a mathematically optimal outcome from a decision requiring values, ethics, policy, or risk tolerance.
Construct discrete probability distributions.
Calculate expected value and variance.
Apply binomial distributions in depth.
Apply geometric distributions in supported waiting-time contexts.
Apply normal distributions in developed contexts.
Use standardization to calculate probabilities.
Use normal approximations where assumptions are supplied and justified.
Calculate conditional probability.
Apply Bayes-style updating in supported table or tree contexts.
Distinguish prior, likelihood, and updated probability conceptually.
Model repeated and cumulative risk.
Use simulation to approximate probabilities not easily calculated exactly.
Compare exact, approximate, and simulated probability results.
Evaluate whether a discrete or continuous distribution is appropriate.
Identify invalid independence, distribution, or sample-space assumptions.

Independent Proof, Mathematical Investigation, and Modeling Capstone

158 competencies

Write an equation of a tangent line.
Write an equation of a normal line in supported cases.
Determine intervals where a function is increasing or decreasing.
Identify critical points.
Determine local maxima and minima.
Apply the first-derivative test.
Determine concavity.
Identify inflection points.
Apply the second-derivative test in supported cases.
Sketch a function using derivative information.
Solve calculus-based optimization problems.
Define an objective function and constraints.
Identify an appropriate domain for an optimization model.
Compare critical and boundary values.
Interpret mathematical optima in context.
Model position, velocity, and acceleration.
Distinguish velocity from speed.
Interpret positive and negative velocity.
Analyze changes in motion using first and second derivatives.
Solve supported related-rate problems.
Use linear approximation for nearby values.
Estimate approximation error informally.
Conduct sensitivity analysis using derivatives.
Calculate an average rate of change over an interval.
Interpret a difference quotient.
Explain instantaneous rate of change as a limit of average rates.
Define the derivative through a difference-quotient limit.
Interpret derivative notation including (f'(x)), (dy/dx), and supported alternatives.
Interpret the derivative as slope of a tangent line.
Estimate a derivative numerically.
Estimate a derivative graphically.
Differentiate constant functions.
Differentiate power functions with integer and supported rational exponents.
Differentiate polynomial functions.
Apply constant-multiple and sum rules.
Apply the product rule.
Apply the quotient rule.
Apply the chain rule.
Differentiate exponential functions.
Differentiate logarithmic functions.
Differentiate sine and cosine functions.
Differentiate tangent and supported trigonometric functions.
Use implicit differentiation in supported cases.
Calculate higher derivatives.
Interpret first and second derivatives contextually.
Check derivative results through graph behavior, numerical estimates, or technology.
Explain an antiderivative as a function whose derivative is known.
Calculate basic indefinite integrals.
Include a constant of integration.
Apply reverse power rules in supported cases.
Integrate basic exponential functions.
Integrate basic sine and cosine functions.
Verify an antiderivative by differentiation.
Interpret a definite integral as net accumulation.
Approximate a definite integral using left-endpoint rectangles.
Approximate a definite integral using right-endpoint rectangles.
Approximate a definite integral using midpoint rectangles.
Approximate a definite integral using the trapezoidal rule.
Explain how increasing subdivisions may improve an approximation for suitable functions.
Calculate definite integrals using antiderivatives.
State and apply the Fundamental Theorem of Calculus at an introductory level.
Interpret signed area.
Distinguish signed area from total geometric area.
Calculate area between a curve and an axis in supported cases.
Calculate area between two curves in supported cases.
Use basic substitution in supported integrals.
Interpret an accumulation function.
Recover displacement from velocity.
Recover change in a quantity from a rate function.
Model cumulative cost, flow, growth, or resource use with an integral.
Validate integral results through units, estimates, and differentiation.
Formulate a mathematical question independently from an unfamiliar situation.
Define variables, functions, parameters, units, constraints, objectives, and assumptions.
Identify relevant and irrelevant information.
Select a calculus, vector, probability, statistical, discrete, or financial model.
Justify the selected model family.
Estimate or calibrate model parameters.
Solve the model analytically, numerically, graphically, recursively, statistically, or technologically.
Interpret intermediate and final results.
Validate dimensional consistency and units.
Validate the model against observed or independent data.
Analyze residuals or discrepancies.
Conduct scenario analysis.
Conduct sensitivity analysis.
Quantify or describe uncertainty.
Identify where assumptions fail.
Compare multiple candidate models.
Refine a model in response to evidence.
State the model’s limitations.
Distinguish a mathematically optimal outcome from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the question, model, method, evidence, conclusions, uncertainty, and limitations clearly.
Use permutations and combinations in multistage probability.
Apply inclusion–exclusion reasoning in supported counting problems.
Use recurrence relations in supported discrete models.
Model paths and networks using introductory graph theory.
Identify vertices, edges, paths, cycles, and connected components.
Analyze introductory route, scheduling, or network problems.
Construct discrete probability distributions.
Calculate expected value and variance.
Apply binomial distributions in depth.
Apply geometric distributions in supported waiting-time contexts.
Apply normal distributions in developed contexts.
Use standardization to calculate probabilities.
Use normal approximations where assumptions are supplied and justified.
Calculate conditional probability.
Apply Bayes-style updating in supported table or tree contexts.
Distinguish prior, likelihood, and updated probability conceptually.
Model repeated and cumulative risk.
Use simulation to approximate probabilities not easily calculated exactly.
Compare exact, approximate, and simulated probability results.
Evaluate whether a discrete or continuous distribution is appropriate.
Identify invalid independence, distribution, or sample-space assumptions.
Formulate a mathematical conjecture independently.
Identify relevant definitions, assumptions, and prior results.
Construct a direct proof.
Construct a proof by contradiction in supported contexts.
Construct a proof by cases in supported contexts.
Use algebraic, geometric, combinatorial, probabilistic, or calculus-based reasoning.
Distinguish proof from numerical verification.
Use a counterexample to disprove a claim.
Identify hidden domain restrictions.
Identify irreversible transformations.
Identify extraneous, excluded, or missing solutions.
Critique a proof for necessity, sufficiency, and logical completeness.
Compare proof methods.
Generalize a result using parameters.
State the scope of a theorem or model precisely.
Revise an invalid argument while preserving valid components.
Communicate an independent sustained mathematical argument.
Explain uncertainty when the conclusion is model-based rather than proven.
Distinguish population parameters and sample statistics.
Explain sampling distributions.
Calculate or interpret standard error in supported cases.
Explain how sample size affects standard error.
Construct a confidence interval for a population proportion in supported cases.
Construct a confidence interval for a population mean in supported cases.
Interpret confidence level correctly.
Interpret margin of error.
Distinguish random sampling error from systematic bias.
State null and alternative hypotheses in supported contexts.
Calculate or interpret a test statistic using a supplied procedure.
Interpret a p-value as evidence under a null model.
Compare a p-value with a stated significance level.
State a conclusion in context without claiming absolute proof.
Explain Type I error.
Explain Type II error.
Distinguish statistical significance from practical importance.
Evaluate assumptions underlying an inference.
Analyze linear regression using slope, intercept, correlation, and residuals.
Interpret uncertainty in regression parameters conceptually.
Interpret a confidence interval for a regression slope when supplied.
Distinguish a confidence interval for mean response from a prediction interval for an individual outcome conceptually.
Compare linear and nonlinear statistical models.
Use residual analysis to identify model failure.
Evaluate the effects of outliers, leverage, and restricted ranges.
Evaluate observational and experimental studies.
Determine whether generalization or causation is supported.
Communicate statistical conclusions with uncertainty and limitations.