Curriculum / Global Curriculum / Grade 10

Grade 10 Mathematics.

10textbooks

250lessons

242competencies

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The whole year, lesson by lesson.

Textbook 1Rational Exponents, Radicals, and Algebraic Structure

4 chapters · 25 lessons

Chapter 1

Rational Exponents and Roots

Chapter 2

Simplifying and Operating with Radicals

Chapter 3

Radical Equations and Extraneous Solutions

Chapter 4

Rational Exponents in Geometry, Science, and Scale

Textbook 2Polynomial Functions, Zeros, and Polynomial Division

4 chapters · 25 lessons

Chapter 1

Polynomial Operations and Factoring

Chapter 2

Polynomial Division and Factor Relationships

Chapter 3

Zeros, Multiplicity, and Polynomial Graphs

Chapter 4

Polynomial Equations and Models

Textbook 3Exponential Functions, Growth, and Decay

4 chapters · 25 lessons

Chapter 1

Recognizing Exponential Change

Chapter 2

Constructing and Graphing Exponential Functions

Chapter 3

Solving Introductory Exponential Equations

Chapter 4

Exponential Models and Long-Term Behavior

Textbook 4Rational Functions and Transformations of Functions

4 chapters · 25 lessons

Chapter 1

Rational Expressions and Restrictions

Chapter 2

Introductory Rational Functions

Chapter 3

Transformations across Function Families

Chapter 4

Comparing Function Families and Models

Textbook 5Right-Triangle Trigonometry and Indirect Measurement

4 chapters · 25 lessons

Chapter 1

Trigonometric Ratios from Similar Right Triangles

Chapter 2

Finding Missing Sides and Angles

Chapter 3

Elevation, Depression, and Indirect Measurement

Chapter 4

Trigonometric Design and Measurement Models

Textbook 6Circle Theorems and Rigorous Geometric Proof

4 chapters · 25 lessons

Chapter 1

Central Angles, Inscribed Angles, Arcs, and Chords

Chapter 2

Tangents, Secants, and Cyclic Quadrilaterals

Chapter 3

Circle Segment Relationships

Chapter 4

Direct and Indirect Proof in Circle Geometry

Textbook 7Solid Geometry, Similar Solids, Precision, and Error

4 chapters · 25 lessons

Chapter 1

Composite, Hollow, and Truncated Solids

Chapter 2

Similar Solids and Scale

Chapter 3

Bounds, Tolerance, and Propagated Error

Chapter 4

Integrated Solid-Measurement Design

Textbook 8Advanced Data Analysis, Regression, and Study Evaluation

4 chapters · 25 lessons

Chapter 1

Comparing Distributions and Standardized Position

Chapter 2

Comparing Linear, Quadratic, and Exponential Models

Chapter 3

Sampling Uncertainty and Informal Inference

Chapter 4

Study Design, Claims, and Evidence

Textbook 9Conditional Probability, Simulations, and Reasoning under Uncertainty

4 chapters · 25 lessons

Chapter 1

Advanced Event Structure and Probability Rules

Chapter 2

Conditional Probability and Dependence

Chapter 3

Repeated Events, Cumulative Risk, and Rare Outcomes

Chapter 4

Simulation Design and Validation

Textbook 10Financial Functions, Optimization, and Integrated Mathematical Modeling

4 chapters · 25 lessons

Chapter 1

Compound Growth as an Exponential Function

Chapter 2

Loan Functions and Repayment Models

Chapter 3

Multi-Period Financial Planning

Chapter 4

Grade 10 Integrated Modeling and Optimization Capstone

Every coin is a 15-minute lesson.

Explained simply, shown with pictures, then put to the test. Stars are chapter reviews. The trophy is the final review.

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Every competency, covered.

The 242 competencies set for Grade 10 Mathematics, and every one has lessons behind it.

Rational Exponents, Radicals, and Algebraic Structure

48 competencies

Use similarity scale factors to compare lengths, surface areas, and volumes.
Determine a scale factor from volume or area information.
Formulate a mathematical problem from a complex unfamiliar situation.
Define variables, functions, parameters, units, constraints, and assumptions.
Select a linear, quadratic, polynomial, exponential, rational, trigonometric, geometric, statistical, probability, or financial model.
Justify why a selected model family is appropriate.
Estimate or calculate model parameters.
Solve the model using algebraic, graphical, numerical, geometric, or technological methods.
Interpret roots, zeros, vertices, intercepts, asymptotes, scale factors, angles, probabilities, and financial balances.
Optimize a supported objective using function behavior, feasible constraints, or geometric structure.
Validate a model using residuals, bounds, independent data, dimensional analysis, or alternate methods.
Conduct simple sensitivity analysis by varying one parameter.
Identify where the model’s assumptions or functional form fail.
Compare candidate models according to fit, interpretability, simplicity, and intended use.
Revise a model when evidence contradicts it.
Quantify or describe uncertainty appropriately.
Distinguish mathematical optimization from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the model, method, findings, assumptions, limitations, and uncertainty coherently.
Interpret a rational exponent as a root, a power, or a combination of both.
Connect (a^{1/n}) with the principal nth root of (a) in the real-number domain where defined.
Connect (a^{m/n}) with an nth root raised to the mth power.
Convert between radical notation and rational-exponent notation.
Apply exponent laws to expressions containing rational exponents.
State real-domain restrictions for even and odd roots.
Simplify radical expressions by extracting perfect-power factors.
Multiply radical expressions.
Divide radical expressions.
Add and subtract like radicals.
Simplify expressions containing several radical terms.
Rationalize supported monomial and binomial denominators.
Solve equations containing one or more radical expressions in supported cases.
Identify restrictions before solving a radical equation.
Check every possible radical-equation solution in the original equation.
Identify extraneous solutions.
Compare exact radical, rational-exponent, and decimal forms.
Select an exact or approximate form appropriate to a context.
Apply rational exponents to geometric, scientific, and scale relationships.
Critique invalid exponent and radical transformations.
Identify the negation of a mathematical claim in supported cases.
Explain why a contradiction invalidates the assumed negation.
Distinguish direct proof, proof by contradiction, counterexample, and verification.
Identify hidden assumptions and domain restrictions.
Identify circular reasoning, converse errors, and reliance on unproved visual appearance.
Determine whether each proof step follows necessarily from prior information.
Repair an incomplete or invalid proof.
Compare synthetic, coordinate, algebraic, and transformational proof methods.
Communicate a rigorous argument in two-column, paragraph, flow, or symbolic form.
State the exact scope and limitations of a proven conclusion.

Polynomial Functions, Zeros, and Polynomial Division

54 competencies

Formulate a mathematical problem from a complex unfamiliar situation.
Define variables, functions, parameters, units, constraints, and assumptions.
Select a linear, quadratic, polynomial, exponential, rational, trigonometric, geometric, statistical, probability, or financial model.
Justify why a selected model family is appropriate.
Estimate or calculate model parameters.
Solve the model using algebraic, graphical, numerical, geometric, or technological methods.
Interpret roots, zeros, vertices, intercepts, asymptotes, scale factors, angles, probabilities, and financial balances.
Optimize a supported objective using function behavior, feasible constraints, or geometric structure.
Validate a model using residuals, bounds, independent data, dimensional analysis, or alternate methods.
Conduct simple sensitivity analysis by varying one parameter.
Identify where the model’s assumptions or functional form fail.
Compare candidate models according to fit, interpretability, simplicity, and intended use.
Revise a model when evidence contradicts it.
Quantify or describe uncertainty appropriately.
Distinguish mathematical optimization from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the model, method, findings, assumptions, limitations, and uncertainty coherently.
Identify polynomial degree, leading coefficient, and constant term.
Write polynomial expressions in standard form.
Add, subtract, and multiply polynomial expressions.
Factor a greatest common polynomial factor.
Factor by grouping in supported cases.
Factor sums and differences of cubes in supported cases.
Factor higher-degree polynomials using known factors and quadratic structure.
Divide a polynomial by a monomial.
Divide a polynomial by a linear factor using long division.
Use introductory synthetic division.
Interpret a polynomial-division remainder.
Use the remainder relationship to evaluate a polynomial at a stated value in supported cases.
Determine whether a linear expression is a factor using a zero or remainder check.
Identify zeros of a polynomial function.
Connect zeros, factors, roots, and horizontal intercepts.
Interpret multiplicity in supported factored forms.
Explain how multiplicity affects whether a graph crosses or touches an axis.
Determine polynomial end behavior from degree and leading coefficient.
Sketch a polynomial graph from zeros, multiplicities, and end behavior.
Use technology to inspect polynomial graphs and refine conjectures.
Solve supported polynomial equations through factoring.
Interpret polynomial zeros and turning behavior in context.
Compare polynomial and quadratic behavior.
Validate polynomial models within a stated domain.
Use algebraic identities to prove numerical or functional claims.
Prove parity and divisibility claims algebraically in supported cases.
Prove geometric claims through congruence, similarity, coordinate geometry, and circle theorems.
Construct an introductory proof by contradiction.
Identify the negation of a mathematical claim in supported cases.
Explain why a contradiction invalidates the assumed negation.
Distinguish direct proof, proof by contradiction, counterexample, and verification.
Identify hidden assumptions and domain restrictions.
Identify circular reasoning, converse errors, and reliance on unproved visual appearance.
Determine whether each proof step follows necessarily from prior information.
Repair an incomplete or invalid proof.
Compare synthetic, coordinate, algebraic, and transformational proof methods.
Communicate a rigorous argument in two-column, paragraph, flow, or symbolic form.
State the exact scope and limitations of a proven conclusion.

Exponential Functions, Growth, and Decay

45 competencies

Identify an exponential function from an equation, table, graph, or description.
Distinguish exponential change from linear and quadratic change.
Interpret the initial value and growth or decay factor.
Convert a percentage growth rate into a multiplicative growth factor.
Convert a percentage decay rate into a multiplicative decay factor.
Construct an exponential function from an initial value and repeated percentage change.
Generate a table from an exponential function.
Graph an exponential-growth function.
Graph an exponential-decay function.
Interpret horizontal asymptotic behavior informally.
Compare exponential functions through initial value and multiplier.
Solve an exponential equation by matching equivalent bases in supported cases.
Solve an exponential equation graphically or numerically.
Estimate doubling or halving behavior from a table or graph.
Compare linear, quadratic, and exponential growth over a stated domain.
Explain why exponential models eventually outgrow positive linear and quadratic models under suitable conditions.
Interpret exponential parameters in population, depreciation, decay, technology, and finance contexts.
Identify unrealistic extrapolation in an exponential model.
Use technology appropriately to solve and analyze introductory exponential models.
State the limitation that general logarithmic solution methods belong to Grade 11.
Formulate a mathematical problem from a complex unfamiliar situation.
Define variables, functions, parameters, units, constraints, and assumptions.
Select a linear, quadratic, polynomial, exponential, rational, trigonometric, geometric, statistical, probability, or financial model.
Justify why a selected model family is appropriate.
Estimate or calculate model parameters.
Solve the model using algebraic, graphical, numerical, geometric, or technological methods.
Interpret roots, zeros, vertices, intercepts, asymptotes, scale factors, angles, probabilities, and financial balances.
Optimize a supported objective using function behavior, feasible constraints, or geometric structure.
Validate a model using residuals, bounds, independent data, dimensional analysis, or alternate methods.
Conduct simple sensitivity analysis by varying one parameter.
Identify where the model’s assumptions or functional form fail.
Compare candidate models according to fit, interpretability, simplicity, and intended use.
Revise a model when evidence contradicts it.
Quantify or describe uncertainty appropriately.
Distinguish mathematical optimization from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the model, method, findings, assumptions, limitations, and uncertainty coherently.
Describe vertical translations of a function.
Describe horizontal translations of a function.
Describe reflection across the horizontal or vertical axis.
Describe vertical stretches and compressions.
Describe horizontal stretches and compressions at an introductory level.
Apply several transformations to a parent function.
Determine a transformed function from a graph or description.
Graph transformed linear, quadratic, absolute-value, exponential, and reciprocal functions in supported cases.
Explain how transformations affect domain, range, intercepts, vertices, and asymptotes.

Rational Functions and Transformations of Functions

55 competencies

Compare linear, quadratic, and exponential growth over a stated domain.
Explain why exponential models eventually outgrow positive linear and quadratic models under suitable conditions.
Interpret exponential parameters in population, depreciation, decay, technology, and finance contexts.
Identify unrealistic extrapolation in an exponential model.
Use technology appropriately to solve and analyze introductory exponential models.
State the limitation that general logarithmic solution methods belong to Grade 11.
Formulate a mathematical problem from a complex unfamiliar situation.
Define variables, functions, parameters, units, constraints, and assumptions.
Select a linear, quadratic, polynomial, exponential, rational, trigonometric, geometric, statistical, probability, or financial model.
Justify why a selected model family is appropriate.
Estimate or calculate model parameters.
Solve the model using algebraic, graphical, numerical, geometric, or technological methods.
Interpret roots, zeros, vertices, intercepts, asymptotes, scale factors, angles, probabilities, and financial balances.
Optimize a supported objective using function behavior, feasible constraints, or geometric structure.
Validate a model using residuals, bounds, independent data, dimensional analysis, or alternate methods.
Conduct simple sensitivity analysis by varying one parameter.
Identify where the model’s assumptions or functional form fail.
Compare candidate models according to fit, interpretability, simplicity, and intended use.
Revise a model when evidence contradicts it.
Quantify or describe uncertainty appropriately.
Distinguish mathematical optimization from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the model, method, findings, assumptions, limitations, and uncertainty coherently.
Explain why a contradiction invalidates the assumed negation.
Distinguish direct proof, proof by contradiction, counterexample, and verification.
Identify hidden assumptions and domain restrictions.
Identify circular reasoning, converse errors, and reliance on unproved visual appearance.
Determine whether each proof step follows necessarily from prior information.
Repair an incomplete or invalid proof.
Compare synthetic, coordinate, algebraic, and transformational proof methods.
Communicate a rigorous argument in two-column, paragraph, flow, or symbolic form.
State the exact scope and limitations of a proven conclusion.
Identify a rational expression as a quotient of polynomial expressions.
State restrictions arising from denominator zeros.
Simplify rational expressions by factoring and canceling common factors while retaining excluded values.
Multiply and divide supported rational expressions.
Add and subtract rational expressions with like denominators.
Add and subtract rational expressions with straightforward unlike denominators.
Solve supported rational equations.
Check for excluded or extraneous solutions.
Identify vertical asymptotes from noncanceled denominator factors.
Identify removable discontinuities from canceled factors.
Describe horizontal behavior of simple rational functions.
Interpret intercepts and restrictions of an introductory rational function.
Sketch a simple reciprocal-type rational graph.
Interpret rational functions in inverse-variation, rate, and average-cost contexts.
Describe vertical translations of a function.
Describe horizontal translations of a function.
Describe reflection across the horizontal or vertical axis.
Describe vertical stretches and compressions.
Describe horizontal stretches and compressions at an introductory level.
Apply several transformations to a parent function.
Determine a transformed function from a graph or description.
Graph transformed linear, quadratic, absolute-value, exponential, and reciprocal functions in supported cases.
Explain how transformations affect domain, range, intercepts, vertices, and asymptotes.
Compare transformation order in noncommuting cases.

Right-Triangle Trigonometry and Indirect Measurement

45 competencies

Use right-triangle trigonometry for indirect measurement.
Determine measurement bounds from stated accuracy.
Use upper and lower bounds in multi-step calculations.
Calculate maximum and minimum possible values for supported sums, differences, products, quotients, areas, or volumes.
Calculate absolute, relative, and percentage error.
Compare uncertainty across quantities of different scales.
Analyze how uncertainty in dimensions affects calculated area or volume.
Explain why multiplication and powers may amplify measurement error.
Select a reporting precision consistent with input quality.
Distinguish measurement uncertainty, model error, and computational rounding.
Evaluate whether a design satisfies a stated tolerance.
Formulate a mathematical problem from a complex unfamiliar situation.
Define variables, functions, parameters, units, constraints, and assumptions.
Select a linear, quadratic, polynomial, exponential, rational, trigonometric, geometric, statistical, probability, or financial model.
Justify why a selected model family is appropriate.
Estimate or calculate model parameters.
Solve the model using algebraic, graphical, numerical, geometric, or technological methods.
Interpret roots, zeros, vertices, intercepts, asymptotes, scale factors, angles, probabilities, and financial balances.
Optimize a supported objective using function behavior, feasible constraints, or geometric structure.
Validate a model using residuals, bounds, independent data, dimensional analysis, or alternate methods.
Conduct simple sensitivity analysis by varying one parameter.
Identify where the model’s assumptions or functional form fail.
Compare candidate models according to fit, interpretability, simplicity, and intended use.
Revise a model when evidence contradicts it.
Quantify or describe uncertainty appropriately.
Distinguish mathematical optimization from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the model, method, findings, assumptions, limitations, and uncertainty coherently.
Identify opposite, adjacent, and hypotenuse sides relative to an acute angle.
Define sine as opposite divided by hypotenuse.
Define cosine as adjacent divided by hypotenuse.
Define tangent as opposite divided by adjacent.
Select the appropriate trigonometric ratio from known and unknown sides.
Calculate a missing side using sine, cosine, or tangent.
Use an inverse trigonometric function to calculate an acute angle.
Solve a right triangle from sufficient information.
Check side and angle results using triangle structure and estimation.
Interpret trigonometric ratios as functions of angle size.
Use complementary-angle relationships in right triangles.
Solve angles of elevation and depression.
Construct and interpret a trigonometric measurement diagram.
Apply trigonometry to heights, distances, gradients, navigation bearings in supported planar cases, and design.
Distinguish line-of-sight distance, horizontal distance, and vertical height.
State assumptions involving level ground, right angles, observer height, and straight sight lines.
Retain sufficient precision during calculations and round only at the final stage.
Recognize that unit-circle trigonometry and non-right triangles belong to Grade 11.

Circle Theorems and Rigorous Geometric Proof

39 competencies

Relate central-angle measure to intercepted arc measure.
Relate an inscribed angle to its intercepted arc.
Prove that an inscribed angle is half the corresponding central angle.
Use the theorem that angles subtended by the same chord or arc are equal.
Use the theorem that an angle in a semicircle is a right angle.
Use opposite-angle relationships in a cyclic quadrilateral.
Use exterior-angle relationships of a cyclic quadrilateral.
Prove that a radius is perpendicular to a tangent at the point of tangency.
Prove that tangent segments from the same external point are equal.
Apply tangent–chord angle relationships.
Use relationships among equal chords, equal arcs, and equal central angles.
Apply perpendicular-bisector properties involving chords and the center.
Calculate unknown angles involving chords, tangents, secants, and arcs.
Apply intersecting-chord segment relationships in supported cases.
Apply secant–secant and tangent–secant segment relationships in supported cases.
Write formal proofs involving circle theorems.
Identify necessary auxiliary lines in a circle proof.
Distinguish a theorem from its converse and test the converse.
Solve supported polynomial equations through factoring.
Interpret polynomial zeros and turning behavior in context.
Compare polynomial and quadratic behavior.
Validate polynomial models within a stated domain.
Construct a sustained direct proof from definitions, axioms, properties, and established theorems.
State givens, required conclusions, and necessary intermediate claims.
Select a productive representation or auxiliary construction.
Use algebraic identities to prove numerical or functional claims.
Prove parity and divisibility claims algebraically in supported cases.
Prove geometric claims through congruence, similarity, coordinate geometry, and circle theorems.
Construct an introductory proof by contradiction.
Identify the negation of a mathematical claim in supported cases.
Explain why a contradiction invalidates the assumed negation.
Distinguish direct proof, proof by contradiction, counterexample, and verification.
Identify hidden assumptions and domain restrictions.
Identify circular reasoning, converse errors, and reliance on unproved visual appearance.
Determine whether each proof step follows necessarily from prior information.
Repair an incomplete or invalid proof.
Compare synthetic, coordinate, algebraic, and transformational proof methods.
Communicate a rigorous argument in two-column, paragraph, flow, or symbolic form.
State the exact scope and limitations of a proven conclusion.

Solid Geometry, Similar Solids, Precision, and Error

56 competencies

Calculate surface area and volume of composite solids.
Decompose a composite solid into prisms, cylinders, cones, pyramids, or spheres.
Calculate volumes of hollow or partially filled solids in supported cases.
Calculate volumes of frustums or truncated solids through subtraction of similar solids in supported cases.
Use similarity scale factors to compare lengths, surface areas, and volumes.
Determine a scale factor from volume or area information.
Solve density, capacity, mass, and volume problems involving composite solids.
Model material thickness and internal versus external dimensions in supported cases.
Select appropriate linear, square, or cubic units.
Convert compound and cubic measurement units.
Use right-triangle trigonometry for indirect measurement.
Determine measurement bounds from stated accuracy.
Use upper and lower bounds in multi-step calculations.
Calculate maximum and minimum possible values for supported sums, differences, products, quotients, areas, or volumes.
Calculate absolute, relative, and percentage error.
Compare uncertainty across quantities of different scales.
Analyze how uncertainty in dimensions affects calculated area or volume.
Explain why multiplication and powers may amplify measurement error.
Select a reporting precision consistent with input quality.
Distinguish measurement uncertainty, model error, and computational rounding.
Evaluate whether a design satisfies a stated tolerance.
Formulate a mathematical problem from a complex unfamiliar situation.
Define variables, functions, parameters, units, constraints, and assumptions.
Select a linear, quadratic, polynomial, exponential, rational, trigonometric, geometric, statistical, probability, or financial model.
Justify why a selected model family is appropriate.
Estimate or calculate model parameters.
Solve the model using algebraic, graphical, numerical, geometric, or technological methods.
Interpret roots, zeros, vertices, intercepts, asymptotes, scale factors, angles, probabilities, and financial balances.
Optimize a supported objective using function behavior, feasible constraints, or geometric structure.
Validate a model using residuals, bounds, independent data, dimensional analysis, or alternate methods.
Conduct simple sensitivity analysis by varying one parameter.
Identify where the model’s assumptions or functional form fail.
Compare candidate models according to fit, interpretability, simplicity, and intended use.
Revise a model when evidence contradicts it.
Quantify or describe uncertainty appropriately.
Distinguish mathematical optimization from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the model, method, findings, assumptions, limitations, and uncertainty coherently.
Interpret a rational exponent as a root, a power, or a combination of both.
Connect (a^{1/n}) with the principal nth root of (a) in the real-number domain where defined.
Connect (a^{m/n}) with an nth root raised to the mth power.
Convert between radical notation and rational-exponent notation.
Apply exponent laws to expressions containing rational exponents.
State real-domain restrictions for even and odd roots.
Simplify radical expressions by extracting perfect-power factors.
Multiply radical expressions.
Divide radical expressions.
Add and subtract like radicals.
Simplify expressions containing several radical terms.
Rationalize supported monomial and binomial denominators.
Solve equations containing one or more radical expressions in supported cases.
Identify restrictions before solving a radical equation.
Check every possible radical-equation solution in the original equation.
Identify extraneous solutions.
Compare exact radical, rational-exponent, and decimal forms.
Select an exact or approximate form appropriate to a context.
Apply rational exponents to geometric, scientific, and scale relationships.

Advanced Data Analysis, Regression, and Study Evaluation

48 competencies

Compare distributions using center, spread, shape, outliers, and sample size.
Interpret mean and standard deviation together.
Interpret median and interquartile range together.
Select robust or nonrobust summaries according to distribution shape.
Standardize a value relative to a mean and standard deviation in supported cases.
Interpret a standardized value as relative position, without introducing a complete normal-distribution system.
Compare linear, quadratic, and exponential candidate models using technology.
Interpret parameters of linear, quadratic, and exponential regression models.
Calculate and interpret residuals.
Use a residual plot to identify nonlinearity, unequal spread, or outliers.
Compare correlation strength with model suitability.
Explain why a strong correlation does not guarantee a useful or causal model.
Evaluate the effect of outliers and influential points.
Distinguish interpolation from extrapolation.
Evaluate uncertainty in a prediction.
Distinguish population parameters from sample statistics conceptually.
Explain sampling variability informally.
Interpret an informal margin of error or plausible interval supplied in context.
Evaluate whether a sample design supports generalization.
Evaluate whether an experimental design supports a causal conclusion.
Identify confounding, selection bias, nonresponse, measurement bias, and misleading question wording.
Critique data visualizations for scales, omitted values, area distortion, and unsupported comparisons.
Compare claims from multiple studies with different samples and designs.
State conclusions proportionate to the data and study design.
Select a linear, quadratic, polynomial, exponential, rational, trigonometric, geometric, statistical, probability, or financial model.
Justify why a selected model family is appropriate.
Estimate or calculate model parameters.
Solve the model using algebraic, graphical, numerical, geometric, or technological methods.
Interpret roots, zeros, vertices, intercepts, asymptotes, scale factors, angles, probabilities, and financial balances.
Optimize a supported objective using function behavior, feasible constraints, or geometric structure.
Validate a model using residuals, bounds, independent data, dimensional analysis, or alternate methods.
Conduct simple sensitivity analysis by varying one parameter.
Identify where the model’s assumptions or functional form fail.
Compare candidate models according to fit, interpretability, simplicity, and intended use.
Revise a model when evidence contradicts it.
Design a simulation matching a stated probability structure.
Evaluate whether a simulation preserves dependence or changing probabilities.
Compare simulated, theoretical, and observed results.
Identify the negation of a mathematical claim in supported cases.
Explain why a contradiction invalidates the assumed negation.
Distinguish direct proof, proof by contradiction, counterexample, and verification.
Identify hidden assumptions and domain restrictions.
Identify circular reasoning, converse errors, and reliance on unproved visual appearance.
Determine whether each proof step follows necessarily from prior information.
Repair an incomplete or invalid proof.
Compare synthetic, coordinate, algebraic, and transformational proof methods.
Communicate a rigorous argument in two-column, paragraph, flow, or symbolic form.
State the exact scope and limitations of a proven conclusion.

Conditional Probability, Simulations, and Reasoning under Uncertainty

36 competencies

Formulate a mathematical problem from a complex unfamiliar situation.
Define variables, functions, parameters, units, constraints, and assumptions.
Select a linear, quadratic, polynomial, exponential, rational, trigonometric, geometric, statistical, probability, or financial model.
Justify why a selected model family is appropriate.
Estimate or calculate model parameters.
Solve the model using algebraic, graphical, numerical, geometric, or technological methods.
Interpret roots, zeros, vertices, intercepts, asymptotes, scale factors, angles, probabilities, and financial balances.
Optimize a supported objective using function behavior, feasible constraints, or geometric structure.
Validate a model using residuals, bounds, independent data, dimensional analysis, or alternate methods.
Conduct simple sensitivity analysis by varying one parameter.
Identify where the model’s assumptions or functional form fail.
Compare candidate models according to fit, interpretability, simplicity, and intended use.
Revise a model when evidence contradicts it.
Quantify or describe uncertainty appropriately.
Distinguish mathematical optimization from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the model, method, findings, assumptions, limitations, and uncertainty coherently.
Represent events using set notation, Venn diagrams, trees, and tables.
Apply addition and multiplication probability rules accurately.
Calculate conditional probabilities in multistage settings.
Determine independence through probability comparisons.
Distinguish independence from mutual exclusivity.
Calculate probabilities with and without replacement.
Calculate probabilities involving at least one, exactly one, or none in supported repeated-event settings.
Use complements strategically.
Construct and interpret multistage probability trees.
Use two-way tables for conditional and joint probabilities.
Use total-probability reasoning across branches in supported cases.
Reverse conditional reasoning cautiously without introducing a full Bayes-theorem system.
Design a simulation matching a stated probability structure.
Evaluate whether a simulation preserves dependence or changing probabilities.
Compare simulated, theoretical, and observed results.
Interpret rare events without treating improbability as impossibility.
Calculate repeated-event risk in supported cases.
Explain why cumulative risk may differ from one-period risk.
Identify base-rate, denominator, independence, and sample-space errors.
Communicate probability and uncertainty without overstating certainty.

Financial Functions, Optimization, and Integrated Mathematical Modeling

73 competencies

Identify an exponential function from an equation, table, graph, or description.
Distinguish exponential change from linear and quadratic change.
Interpret the initial value and growth or decay factor.
Convert a percentage growth rate into a multiplicative growth factor.
Convert a percentage decay rate into a multiplicative decay factor.
Construct an exponential function from an initial value and repeated percentage change.
Generate a table from an exponential function.
Graph an exponential-growth function.
Graph an exponential-decay function.
Interpret horizontal asymptotic behavior informally.
Compare exponential functions through initial value and multiplier.
Solve an exponential equation by matching equivalent bases in supported cases.
Solve an exponential equation graphically or numerically.
Estimate doubling or halving behavior from a table or graph.
Compare linear, quadratic, and exponential growth over a stated domain.
Explain why exponential models eventually outgrow positive linear and quadratic models under suitable conditions.
Interpret exponential parameters in population, depreciation, decay, technology, and finance contexts.
Identify unrealistic extrapolation in an exponential model.
Use technology appropriately to solve and analyze introductory exponential models.
State the limitation that general logarithmic solution methods belong to Grade 11.
Represent compound growth using an exponential function.
Calculate future balances under stated periodic compounding.
Compare compounding frequencies.
Calculate total contribution, total interest, and final balance separately.
Calculate percentage gain or loss.
Include fixed or percentage fees in a supported financial model.
Compare gross and net returns mathematically.
Model regular deposits recursively or through tables without requiring an annuity formula.
Compare saving plans with different rates, deposits, or periods.
Model loan growth under a stated interest structure.
Calculate total repayment and total interest.
Track a declining balance through a supplied repayment schedule.
Compare loan options using rate, term, payment amount, fees, and total repayment.
Explain why a lower periodic payment may produce a greater total repayment.
Calculate the effect of an additional payment in a simplified supplied model.
Construct a multi-period financial plan involving income, expenses, saving, debt repayment, and investment growth.
Use linear and exponential functions within the same financial plan.
Test a financial plan under a changed rate, expense, contribution, or duration.
Identify assumptions involving fixed rates, regular payments, fees, taxes, and uninterrupted contributions.
Distinguish mathematical return from risk-adjusted or personally desirable outcomes.
Formulate a mathematical problem from a complex unfamiliar situation.
Define variables, functions, parameters, units, constraints, and assumptions.
Select a linear, quadratic, polynomial, exponential, rational, trigonometric, geometric, statistical, probability, or financial model.
Justify why a selected model family is appropriate.
Estimate or calculate model parameters.
Solve the model using algebraic, graphical, numerical, geometric, or technological methods.
Interpret roots, zeros, vertices, intercepts, asymptotes, scale factors, angles, probabilities, and financial balances.
Optimize a supported objective using function behavior, feasible constraints, or geometric structure.
Validate a model using residuals, bounds, independent data, dimensional analysis, or alternate methods.
Conduct simple sensitivity analysis by varying one parameter.
Identify where the model’s assumptions or functional form fail.
Compare candidate models according to fit, interpretability, simplicity, and intended use.
Revise a model when evidence contradicts it.
Quantify or describe uncertainty appropriately.
Distinguish mathematical optimization from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the model, method, findings, assumptions, limitations, and uncertainty coherently.
Construct a sustained direct proof from definitions, axioms, properties, and established theorems.
State givens, required conclusions, and necessary intermediate claims.
Select a productive representation or auxiliary construction.
Use algebraic identities to prove numerical or functional claims.
Prove parity and divisibility claims algebraically in supported cases.
Prove geometric claims through congruence, similarity, coordinate geometry, and circle theorems.
Construct an introductory proof by contradiction.
Identify the negation of a mathematical claim in supported cases.
Explain why a contradiction invalidates the assumed negation.
Distinguish direct proof, proof by contradiction, counterexample, and verification.
Identify hidden assumptions and domain restrictions.
Identify circular reasoning, converse errors, and reliance on unproved visual appearance.
Determine whether each proof step follows necessarily from prior information.
Repair an incomplete or invalid proof.
Compare synthetic, coordinate, algebraic, and transformational proof methods.
Communicate a rigorous argument in two-column, paragraph, flow, or symbolic form.
State the exact scope and limitations of a proven conclusion.

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Textbook 1Rational Exponents, Radicals, and Algebraic Structure

4 chapters · 25 lessons

Chapter 1

Rational Exponents and Roots

Chapter 2

Simplifying and Operating with Radicals

Chapter 3

Radical Equations and Extraneous Solutions

Chapter 4

Rational Exponents in Geometry, Science, and Scale

Textbook 2Polynomial Functions, Zeros, and Polynomial Division

4 chapters · 25 lessons

Chapter 1

Polynomial Operations and Factoring

Chapter 2

Polynomial Division and Factor Relationships

Chapter 3

Zeros, Multiplicity, and Polynomial Graphs

Chapter 4

Polynomial Equations and Models

Textbook 3Exponential Functions, Growth, and Decay

4 chapters · 25 lessons

Chapter 1

Recognizing Exponential Change

Chapter 2

Constructing and Graphing Exponential Functions

Chapter 3

Solving Introductory Exponential Equations

Chapter 4

Exponential Models and Long-Term Behavior

Textbook 4Rational Functions and Transformations of Functions

4 chapters · 25 lessons

Chapter 1

Rational Expressions and Restrictions

Chapter 2

Introductory Rational Functions

Chapter 3

Transformations across Function Families

Chapter 4

Comparing Function Families and Models

Textbook 5Right-Triangle Trigonometry and Indirect Measurement

4 chapters · 25 lessons

Chapter 1

Trigonometric Ratios from Similar Right Triangles

Chapter 2

Finding Missing Sides and Angles

Chapter 3

Elevation, Depression, and Indirect Measurement

Chapter 4

Trigonometric Design and Measurement Models

Textbook 6Circle Theorems and Rigorous Geometric Proof

4 chapters · 25 lessons

Chapter 1

Central Angles, Inscribed Angles, Arcs, and Chords

Chapter 2

Tangents, Secants, and Cyclic Quadrilaterals

Chapter 3

Circle Segment Relationships

Chapter 4

Direct and Indirect Proof in Circle Geometry

Textbook 7Solid Geometry, Similar Solids, Precision, and Error

4 chapters · 25 lessons

Chapter 1

Composite, Hollow, and Truncated Solids

Chapter 2

Similar Solids and Scale

Chapter 3

Bounds, Tolerance, and Propagated Error

Chapter 4

Integrated Solid-Measurement Design

Textbook 8Advanced Data Analysis, Regression, and Study Evaluation

4 chapters · 25 lessons

Chapter 1

Comparing Distributions and Standardized Position

Chapter 2

Comparing Linear, Quadratic, and Exponential Models

Chapter 3

Sampling Uncertainty and Informal Inference

Chapter 4

Study Design, Claims, and Evidence

Textbook 9Conditional Probability, Simulations, and Reasoning under Uncertainty

4 chapters · 25 lessons

Chapter 1

Advanced Event Structure and Probability Rules

Chapter 2

Conditional Probability and Dependence

Chapter 3

Repeated Events, Cumulative Risk, and Rare Outcomes

Chapter 4

Simulation Design and Validation

Textbook 10Financial Functions, Optimization, and Integrated Mathematical Modeling

4 chapters · 25 lessons

Chapter 1

Compound Growth as an Exponential Function

Chapter 2

Loan Functions and Repayment Models

Chapter 3

Multi-Period Financial Planning

Chapter 4

Grade 10 Integrated Modeling and Optimization Capstone

Every competency, covered.

The 242 competencies set for Grade 10 Mathematics, and every one has lessons behind it.

Rational Exponents, Radicals, and Algebraic Structure

48 competencies

Use similarity scale factors to compare lengths, surface areas, and volumes.
Determine a scale factor from volume or area information.
Formulate a mathematical problem from a complex unfamiliar situation.
Define variables, functions, parameters, units, constraints, and assumptions.
Select a linear, quadratic, polynomial, exponential, rational, trigonometric, geometric, statistical, probability, or financial model.
Justify why a selected model family is appropriate.
Estimate or calculate model parameters.
Solve the model using algebraic, graphical, numerical, geometric, or technological methods.
Interpret roots, zeros, vertices, intercepts, asymptotes, scale factors, angles, probabilities, and financial balances.
Optimize a supported objective using function behavior, feasible constraints, or geometric structure.
Validate a model using residuals, bounds, independent data, dimensional analysis, or alternate methods.
Conduct simple sensitivity analysis by varying one parameter.
Identify where the model’s assumptions or functional form fail.
Compare candidate models according to fit, interpretability, simplicity, and intended use.
Revise a model when evidence contradicts it.
Quantify or describe uncertainty appropriately.
Distinguish mathematical optimization from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the model, method, findings, assumptions, limitations, and uncertainty coherently.
Interpret a rational exponent as a root, a power, or a combination of both.
Connect (a^{1/n}) with the principal nth root of (a) in the real-number domain where defined.
Connect (a^{m/n}) with an nth root raised to the mth power.
Convert between radical notation and rational-exponent notation.
Apply exponent laws to expressions containing rational exponents.
State real-domain restrictions for even and odd roots.
Simplify radical expressions by extracting perfect-power factors.
Multiply radical expressions.
Divide radical expressions.
Add and subtract like radicals.
Simplify expressions containing several radical terms.
Rationalize supported monomial and binomial denominators.
Solve equations containing one or more radical expressions in supported cases.
Identify restrictions before solving a radical equation.
Check every possible radical-equation solution in the original equation.
Identify extraneous solutions.
Compare exact radical, rational-exponent, and decimal forms.
Select an exact or approximate form appropriate to a context.
Apply rational exponents to geometric, scientific, and scale relationships.
Critique invalid exponent and radical transformations.
Identify the negation of a mathematical claim in supported cases.
Explain why a contradiction invalidates the assumed negation.
Distinguish direct proof, proof by contradiction, counterexample, and verification.
Identify hidden assumptions and domain restrictions.
Identify circular reasoning, converse errors, and reliance on unproved visual appearance.
Determine whether each proof step follows necessarily from prior information.
Repair an incomplete or invalid proof.
Compare synthetic, coordinate, algebraic, and transformational proof methods.
Communicate a rigorous argument in two-column, paragraph, flow, or symbolic form.
State the exact scope and limitations of a proven conclusion.

Polynomial Functions, Zeros, and Polynomial Division

54 competencies

Formulate a mathematical problem from a complex unfamiliar situation.
Define variables, functions, parameters, units, constraints, and assumptions.
Select a linear, quadratic, polynomial, exponential, rational, trigonometric, geometric, statistical, probability, or financial model.
Justify why a selected model family is appropriate.
Estimate or calculate model parameters.
Solve the model using algebraic, graphical, numerical, geometric, or technological methods.
Interpret roots, zeros, vertices, intercepts, asymptotes, scale factors, angles, probabilities, and financial balances.
Optimize a supported objective using function behavior, feasible constraints, or geometric structure.
Validate a model using residuals, bounds, independent data, dimensional analysis, or alternate methods.
Conduct simple sensitivity analysis by varying one parameter.
Identify where the model’s assumptions or functional form fail.
Compare candidate models according to fit, interpretability, simplicity, and intended use.
Revise a model when evidence contradicts it.
Quantify or describe uncertainty appropriately.
Distinguish mathematical optimization from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the model, method, findings, assumptions, limitations, and uncertainty coherently.
Identify polynomial degree, leading coefficient, and constant term.
Write polynomial expressions in standard form.
Add, subtract, and multiply polynomial expressions.
Factor a greatest common polynomial factor.
Factor by grouping in supported cases.
Factor sums and differences of cubes in supported cases.
Factor higher-degree polynomials using known factors and quadratic structure.
Divide a polynomial by a monomial.
Divide a polynomial by a linear factor using long division.
Use introductory synthetic division.
Interpret a polynomial-division remainder.
Use the remainder relationship to evaluate a polynomial at a stated value in supported cases.
Determine whether a linear expression is a factor using a zero or remainder check.
Identify zeros of a polynomial function.
Connect zeros, factors, roots, and horizontal intercepts.
Interpret multiplicity in supported factored forms.
Explain how multiplicity affects whether a graph crosses or touches an axis.
Determine polynomial end behavior from degree and leading coefficient.
Sketch a polynomial graph from zeros, multiplicities, and end behavior.
Use technology to inspect polynomial graphs and refine conjectures.
Solve supported polynomial equations through factoring.
Interpret polynomial zeros and turning behavior in context.
Compare polynomial and quadratic behavior.
Validate polynomial models within a stated domain.
Use algebraic identities to prove numerical or functional claims.
Prove parity and divisibility claims algebraically in supported cases.
Prove geometric claims through congruence, similarity, coordinate geometry, and circle theorems.
Construct an introductory proof by contradiction.
Identify the negation of a mathematical claim in supported cases.
Explain why a contradiction invalidates the assumed negation.
Distinguish direct proof, proof by contradiction, counterexample, and verification.
Identify hidden assumptions and domain restrictions.
Identify circular reasoning, converse errors, and reliance on unproved visual appearance.
Determine whether each proof step follows necessarily from prior information.
Repair an incomplete or invalid proof.
Compare synthetic, coordinate, algebraic, and transformational proof methods.
Communicate a rigorous argument in two-column, paragraph, flow, or symbolic form.
State the exact scope and limitations of a proven conclusion.

Exponential Functions, Growth, and Decay

45 competencies

Identify an exponential function from an equation, table, graph, or description.
Distinguish exponential change from linear and quadratic change.
Interpret the initial value and growth or decay factor.
Convert a percentage growth rate into a multiplicative growth factor.
Convert a percentage decay rate into a multiplicative decay factor.
Construct an exponential function from an initial value and repeated percentage change.
Generate a table from an exponential function.
Graph an exponential-growth function.
Graph an exponential-decay function.
Interpret horizontal asymptotic behavior informally.
Compare exponential functions through initial value and multiplier.
Solve an exponential equation by matching equivalent bases in supported cases.
Solve an exponential equation graphically or numerically.
Estimate doubling or halving behavior from a table or graph.
Compare linear, quadratic, and exponential growth over a stated domain.
Explain why exponential models eventually outgrow positive linear and quadratic models under suitable conditions.
Interpret exponential parameters in population, depreciation, decay, technology, and finance contexts.
Identify unrealistic extrapolation in an exponential model.
Use technology appropriately to solve and analyze introductory exponential models.
State the limitation that general logarithmic solution methods belong to Grade 11.
Formulate a mathematical problem from a complex unfamiliar situation.
Define variables, functions, parameters, units, constraints, and assumptions.
Select a linear, quadratic, polynomial, exponential, rational, trigonometric, geometric, statistical, probability, or financial model.
Justify why a selected model family is appropriate.
Estimate or calculate model parameters.
Solve the model using algebraic, graphical, numerical, geometric, or technological methods.
Interpret roots, zeros, vertices, intercepts, asymptotes, scale factors, angles, probabilities, and financial balances.
Optimize a supported objective using function behavior, feasible constraints, or geometric structure.
Validate a model using residuals, bounds, independent data, dimensional analysis, or alternate methods.
Conduct simple sensitivity analysis by varying one parameter.
Identify where the model’s assumptions or functional form fail.
Compare candidate models according to fit, interpretability, simplicity, and intended use.
Revise a model when evidence contradicts it.
Quantify or describe uncertainty appropriately.
Distinguish mathematical optimization from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the model, method, findings, assumptions, limitations, and uncertainty coherently.
Describe vertical translations of a function.
Describe horizontal translations of a function.
Describe reflection across the horizontal or vertical axis.
Describe vertical stretches and compressions.
Describe horizontal stretches and compressions at an introductory level.
Apply several transformations to a parent function.
Determine a transformed function from a graph or description.
Graph transformed linear, quadratic, absolute-value, exponential, and reciprocal functions in supported cases.
Explain how transformations affect domain, range, intercepts, vertices, and asymptotes.

Rational Functions and Transformations of Functions

55 competencies

Compare linear, quadratic, and exponential growth over a stated domain.
Explain why exponential models eventually outgrow positive linear and quadratic models under suitable conditions.
Interpret exponential parameters in population, depreciation, decay, technology, and finance contexts.
Identify unrealistic extrapolation in an exponential model.
Use technology appropriately to solve and analyze introductory exponential models.
State the limitation that general logarithmic solution methods belong to Grade 11.
Formulate a mathematical problem from a complex unfamiliar situation.
Define variables, functions, parameters, units, constraints, and assumptions.
Select a linear, quadratic, polynomial, exponential, rational, trigonometric, geometric, statistical, probability, or financial model.
Justify why a selected model family is appropriate.
Estimate or calculate model parameters.
Solve the model using algebraic, graphical, numerical, geometric, or technological methods.
Interpret roots, zeros, vertices, intercepts, asymptotes, scale factors, angles, probabilities, and financial balances.
Optimize a supported objective using function behavior, feasible constraints, or geometric structure.
Validate a model using residuals, bounds, independent data, dimensional analysis, or alternate methods.
Conduct simple sensitivity analysis by varying one parameter.
Identify where the model’s assumptions or functional form fail.
Compare candidate models according to fit, interpretability, simplicity, and intended use.
Revise a model when evidence contradicts it.
Quantify or describe uncertainty appropriately.
Distinguish mathematical optimization from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the model, method, findings, assumptions, limitations, and uncertainty coherently.
Explain why a contradiction invalidates the assumed negation.
Distinguish direct proof, proof by contradiction, counterexample, and verification.
Identify hidden assumptions and domain restrictions.
Identify circular reasoning, converse errors, and reliance on unproved visual appearance.
Determine whether each proof step follows necessarily from prior information.
Repair an incomplete or invalid proof.
Compare synthetic, coordinate, algebraic, and transformational proof methods.
Communicate a rigorous argument in two-column, paragraph, flow, or symbolic form.
State the exact scope and limitations of a proven conclusion.
Identify a rational expression as a quotient of polynomial expressions.
State restrictions arising from denominator zeros.
Simplify rational expressions by factoring and canceling common factors while retaining excluded values.
Multiply and divide supported rational expressions.
Add and subtract rational expressions with like denominators.
Add and subtract rational expressions with straightforward unlike denominators.
Solve supported rational equations.
Check for excluded or extraneous solutions.
Identify vertical asymptotes from noncanceled denominator factors.
Identify removable discontinuities from canceled factors.
Describe horizontal behavior of simple rational functions.
Interpret intercepts and restrictions of an introductory rational function.
Sketch a simple reciprocal-type rational graph.
Interpret rational functions in inverse-variation, rate, and average-cost contexts.
Describe vertical translations of a function.
Describe horizontal translations of a function.
Describe reflection across the horizontal or vertical axis.
Describe vertical stretches and compressions.
Describe horizontal stretches and compressions at an introductory level.
Apply several transformations to a parent function.
Determine a transformed function from a graph or description.
Graph transformed linear, quadratic, absolute-value, exponential, and reciprocal functions in supported cases.
Explain how transformations affect domain, range, intercepts, vertices, and asymptotes.
Compare transformation order in noncommuting cases.

Right-Triangle Trigonometry and Indirect Measurement

45 competencies

Use right-triangle trigonometry for indirect measurement.
Determine measurement bounds from stated accuracy.
Use upper and lower bounds in multi-step calculations.
Calculate maximum and minimum possible values for supported sums, differences, products, quotients, areas, or volumes.
Calculate absolute, relative, and percentage error.
Compare uncertainty across quantities of different scales.
Analyze how uncertainty in dimensions affects calculated area or volume.
Explain why multiplication and powers may amplify measurement error.
Select a reporting precision consistent with input quality.
Distinguish measurement uncertainty, model error, and computational rounding.
Evaluate whether a design satisfies a stated tolerance.
Formulate a mathematical problem from a complex unfamiliar situation.
Define variables, functions, parameters, units, constraints, and assumptions.
Select a linear, quadratic, polynomial, exponential, rational, trigonometric, geometric, statistical, probability, or financial model.
Justify why a selected model family is appropriate.
Estimate or calculate model parameters.
Solve the model using algebraic, graphical, numerical, geometric, or technological methods.
Interpret roots, zeros, vertices, intercepts, asymptotes, scale factors, angles, probabilities, and financial balances.
Optimize a supported objective using function behavior, feasible constraints, or geometric structure.
Validate a model using residuals, bounds, independent data, dimensional analysis, or alternate methods.
Conduct simple sensitivity analysis by varying one parameter.
Identify where the model’s assumptions or functional form fail.
Compare candidate models according to fit, interpretability, simplicity, and intended use.
Revise a model when evidence contradicts it.
Quantify or describe uncertainty appropriately.
Distinguish mathematical optimization from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the model, method, findings, assumptions, limitations, and uncertainty coherently.
Identify opposite, adjacent, and hypotenuse sides relative to an acute angle.
Define sine as opposite divided by hypotenuse.
Define cosine as adjacent divided by hypotenuse.
Define tangent as opposite divided by adjacent.
Select the appropriate trigonometric ratio from known and unknown sides.
Calculate a missing side using sine, cosine, or tangent.
Use an inverse trigonometric function to calculate an acute angle.
Solve a right triangle from sufficient information.
Check side and angle results using triangle structure and estimation.
Interpret trigonometric ratios as functions of angle size.
Use complementary-angle relationships in right triangles.
Solve angles of elevation and depression.
Construct and interpret a trigonometric measurement diagram.
Apply trigonometry to heights, distances, gradients, navigation bearings in supported planar cases, and design.
Distinguish line-of-sight distance, horizontal distance, and vertical height.
State assumptions involving level ground, right angles, observer height, and straight sight lines.
Retain sufficient precision during calculations and round only at the final stage.
Recognize that unit-circle trigonometry and non-right triangles belong to Grade 11.

Circle Theorems and Rigorous Geometric Proof

39 competencies

Relate central-angle measure to intercepted arc measure.
Relate an inscribed angle to its intercepted arc.
Prove that an inscribed angle is half the corresponding central angle.
Use the theorem that angles subtended by the same chord or arc are equal.
Use the theorem that an angle in a semicircle is a right angle.
Use opposite-angle relationships in a cyclic quadrilateral.
Use exterior-angle relationships of a cyclic quadrilateral.
Prove that a radius is perpendicular to a tangent at the point of tangency.
Prove that tangent segments from the same external point are equal.
Apply tangent–chord angle relationships.
Use relationships among equal chords, equal arcs, and equal central angles.
Apply perpendicular-bisector properties involving chords and the center.
Calculate unknown angles involving chords, tangents, secants, and arcs.
Apply intersecting-chord segment relationships in supported cases.
Apply secant–secant and tangent–secant segment relationships in supported cases.
Write formal proofs involving circle theorems.
Identify necessary auxiliary lines in a circle proof.
Distinguish a theorem from its converse and test the converse.
Solve supported polynomial equations through factoring.
Interpret polynomial zeros and turning behavior in context.
Compare polynomial and quadratic behavior.
Validate polynomial models within a stated domain.
Construct a sustained direct proof from definitions, axioms, properties, and established theorems.
State givens, required conclusions, and necessary intermediate claims.
Select a productive representation or auxiliary construction.
Use algebraic identities to prove numerical or functional claims.
Prove parity and divisibility claims algebraically in supported cases.
Prove geometric claims through congruence, similarity, coordinate geometry, and circle theorems.
Construct an introductory proof by contradiction.
Identify the negation of a mathematical claim in supported cases.
Explain why a contradiction invalidates the assumed negation.
Distinguish direct proof, proof by contradiction, counterexample, and verification.
Identify hidden assumptions and domain restrictions.
Identify circular reasoning, converse errors, and reliance on unproved visual appearance.
Determine whether each proof step follows necessarily from prior information.
Repair an incomplete or invalid proof.
Compare synthetic, coordinate, algebraic, and transformational proof methods.
Communicate a rigorous argument in two-column, paragraph, flow, or symbolic form.
State the exact scope and limitations of a proven conclusion.

Solid Geometry, Similar Solids, Precision, and Error

56 competencies

Calculate surface area and volume of composite solids.
Decompose a composite solid into prisms, cylinders, cones, pyramids, or spheres.
Calculate volumes of hollow or partially filled solids in supported cases.
Calculate volumes of frustums or truncated solids through subtraction of similar solids in supported cases.
Use similarity scale factors to compare lengths, surface areas, and volumes.
Determine a scale factor from volume or area information.
Solve density, capacity, mass, and volume problems involving composite solids.
Model material thickness and internal versus external dimensions in supported cases.
Select appropriate linear, square, or cubic units.
Convert compound and cubic measurement units.
Use right-triangle trigonometry for indirect measurement.
Determine measurement bounds from stated accuracy.
Use upper and lower bounds in multi-step calculations.
Calculate maximum and minimum possible values for supported sums, differences, products, quotients, areas, or volumes.
Calculate absolute, relative, and percentage error.
Compare uncertainty across quantities of different scales.
Analyze how uncertainty in dimensions affects calculated area or volume.
Explain why multiplication and powers may amplify measurement error.
Select a reporting precision consistent with input quality.
Distinguish measurement uncertainty, model error, and computational rounding.
Evaluate whether a design satisfies a stated tolerance.
Formulate a mathematical problem from a complex unfamiliar situation.
Define variables, functions, parameters, units, constraints, and assumptions.
Select a linear, quadratic, polynomial, exponential, rational, trigonometric, geometric, statistical, probability, or financial model.
Justify why a selected model family is appropriate.
Estimate or calculate model parameters.
Solve the model using algebraic, graphical, numerical, geometric, or technological methods.
Interpret roots, zeros, vertices, intercepts, asymptotes, scale factors, angles, probabilities, and financial balances.
Optimize a supported objective using function behavior, feasible constraints, or geometric structure.
Validate a model using residuals, bounds, independent data, dimensional analysis, or alternate methods.
Conduct simple sensitivity analysis by varying one parameter.
Identify where the model’s assumptions or functional form fail.
Compare candidate models according to fit, interpretability, simplicity, and intended use.
Revise a model when evidence contradicts it.
Quantify or describe uncertainty appropriately.
Distinguish mathematical optimization from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the model, method, findings, assumptions, limitations, and uncertainty coherently.
Interpret a rational exponent as a root, a power, or a combination of both.
Connect (a^{1/n}) with the principal nth root of (a) in the real-number domain where defined.
Connect (a^{m/n}) with an nth root raised to the mth power.
Convert between radical notation and rational-exponent notation.
Apply exponent laws to expressions containing rational exponents.
State real-domain restrictions for even and odd roots.
Simplify radical expressions by extracting perfect-power factors.
Multiply radical expressions.
Divide radical expressions.
Add and subtract like radicals.
Simplify expressions containing several radical terms.
Rationalize supported monomial and binomial denominators.
Solve equations containing one or more radical expressions in supported cases.
Identify restrictions before solving a radical equation.
Check every possible radical-equation solution in the original equation.
Identify extraneous solutions.
Compare exact radical, rational-exponent, and decimal forms.
Select an exact or approximate form appropriate to a context.
Apply rational exponents to geometric, scientific, and scale relationships.

Advanced Data Analysis, Regression, and Study Evaluation

48 competencies

Compare distributions using center, spread, shape, outliers, and sample size.
Interpret mean and standard deviation together.
Interpret median and interquartile range together.
Select robust or nonrobust summaries according to distribution shape.
Standardize a value relative to a mean and standard deviation in supported cases.
Interpret a standardized value as relative position, without introducing a complete normal-distribution system.
Compare linear, quadratic, and exponential candidate models using technology.
Interpret parameters of linear, quadratic, and exponential regression models.
Calculate and interpret residuals.
Use a residual plot to identify nonlinearity, unequal spread, or outliers.
Compare correlation strength with model suitability.
Explain why a strong correlation does not guarantee a useful or causal model.
Evaluate the effect of outliers and influential points.
Distinguish interpolation from extrapolation.
Evaluate uncertainty in a prediction.
Distinguish population parameters from sample statistics conceptually.
Explain sampling variability informally.
Interpret an informal margin of error or plausible interval supplied in context.
Evaluate whether a sample design supports generalization.
Evaluate whether an experimental design supports a causal conclusion.
Identify confounding, selection bias, nonresponse, measurement bias, and misleading question wording.
Critique data visualizations for scales, omitted values, area distortion, and unsupported comparisons.
Compare claims from multiple studies with different samples and designs.
State conclusions proportionate to the data and study design.
Select a linear, quadratic, polynomial, exponential, rational, trigonometric, geometric, statistical, probability, or financial model.
Justify why a selected model family is appropriate.
Estimate or calculate model parameters.
Solve the model using algebraic, graphical, numerical, geometric, or technological methods.
Interpret roots, zeros, vertices, intercepts, asymptotes, scale factors, angles, probabilities, and financial balances.
Optimize a supported objective using function behavior, feasible constraints, or geometric structure.
Validate a model using residuals, bounds, independent data, dimensional analysis, or alternate methods.
Conduct simple sensitivity analysis by varying one parameter.
Identify where the model’s assumptions or functional form fail.
Compare candidate models according to fit, interpretability, simplicity, and intended use.
Revise a model when evidence contradicts it.
Design a simulation matching a stated probability structure.
Evaluate whether a simulation preserves dependence or changing probabilities.
Compare simulated, theoretical, and observed results.
Identify the negation of a mathematical claim in supported cases.
Explain why a contradiction invalidates the assumed negation.
Distinguish direct proof, proof by contradiction, counterexample, and verification.
Identify hidden assumptions and domain restrictions.
Identify circular reasoning, converse errors, and reliance on unproved visual appearance.
Determine whether each proof step follows necessarily from prior information.
Repair an incomplete or invalid proof.
Compare synthetic, coordinate, algebraic, and transformational proof methods.
Communicate a rigorous argument in two-column, paragraph, flow, or symbolic form.
State the exact scope and limitations of a proven conclusion.

Conditional Probability, Simulations, and Reasoning under Uncertainty

36 competencies

Formulate a mathematical problem from a complex unfamiliar situation.
Define variables, functions, parameters, units, constraints, and assumptions.
Select a linear, quadratic, polynomial, exponential, rational, trigonometric, geometric, statistical, probability, or financial model.
Justify why a selected model family is appropriate.
Estimate or calculate model parameters.
Solve the model using algebraic, graphical, numerical, geometric, or technological methods.
Interpret roots, zeros, vertices, intercepts, asymptotes, scale factors, angles, probabilities, and financial balances.
Optimize a supported objective using function behavior, feasible constraints, or geometric structure.
Validate a model using residuals, bounds, independent data, dimensional analysis, or alternate methods.
Conduct simple sensitivity analysis by varying one parameter.
Identify where the model’s assumptions or functional form fail.
Compare candidate models according to fit, interpretability, simplicity, and intended use.
Revise a model when evidence contradicts it.
Quantify or describe uncertainty appropriately.
Distinguish mathematical optimization from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the model, method, findings, assumptions, limitations, and uncertainty coherently.
Represent events using set notation, Venn diagrams, trees, and tables.
Apply addition and multiplication probability rules accurately.
Calculate conditional probabilities in multistage settings.
Determine independence through probability comparisons.
Distinguish independence from mutual exclusivity.
Calculate probabilities with and without replacement.
Calculate probabilities involving at least one, exactly one, or none in supported repeated-event settings.
Use complements strategically.
Construct and interpret multistage probability trees.
Use two-way tables for conditional and joint probabilities.
Use total-probability reasoning across branches in supported cases.
Reverse conditional reasoning cautiously without introducing a full Bayes-theorem system.
Design a simulation matching a stated probability structure.
Evaluate whether a simulation preserves dependence or changing probabilities.
Compare simulated, theoretical, and observed results.
Interpret rare events without treating improbability as impossibility.
Calculate repeated-event risk in supported cases.
Explain why cumulative risk may differ from one-period risk.
Identify base-rate, denominator, independence, and sample-space errors.
Communicate probability and uncertainty without overstating certainty.

Financial Functions, Optimization, and Integrated Mathematical Modeling

73 competencies

Identify an exponential function from an equation, table, graph, or description.
Distinguish exponential change from linear and quadratic change.
Interpret the initial value and growth or decay factor.
Convert a percentage growth rate into a multiplicative growth factor.
Convert a percentage decay rate into a multiplicative decay factor.
Construct an exponential function from an initial value and repeated percentage change.
Generate a table from an exponential function.
Graph an exponential-growth function.
Graph an exponential-decay function.
Interpret horizontal asymptotic behavior informally.
Compare exponential functions through initial value and multiplier.
Solve an exponential equation by matching equivalent bases in supported cases.
Solve an exponential equation graphically or numerically.
Estimate doubling or halving behavior from a table or graph.
Compare linear, quadratic, and exponential growth over a stated domain.
Explain why exponential models eventually outgrow positive linear and quadratic models under suitable conditions.
Interpret exponential parameters in population, depreciation, decay, technology, and finance contexts.
Identify unrealistic extrapolation in an exponential model.
Use technology appropriately to solve and analyze introductory exponential models.
State the limitation that general logarithmic solution methods belong to Grade 11.
Represent compound growth using an exponential function.
Calculate future balances under stated periodic compounding.
Compare compounding frequencies.
Calculate total contribution, total interest, and final balance separately.
Calculate percentage gain or loss.
Include fixed or percentage fees in a supported financial model.
Compare gross and net returns mathematically.
Model regular deposits recursively or through tables without requiring an annuity formula.
Compare saving plans with different rates, deposits, or periods.
Model loan growth under a stated interest structure.
Calculate total repayment and total interest.
Track a declining balance through a supplied repayment schedule.
Compare loan options using rate, term, payment amount, fees, and total repayment.
Explain why a lower periodic payment may produce a greater total repayment.
Calculate the effect of an additional payment in a simplified supplied model.
Construct a multi-period financial plan involving income, expenses, saving, debt repayment, and investment growth.
Use linear and exponential functions within the same financial plan.
Test a financial plan under a changed rate, expense, contribution, or duration.
Identify assumptions involving fixed rates, regular payments, fees, taxes, and uninterrupted contributions.
Distinguish mathematical return from risk-adjusted or personally desirable outcomes.
Formulate a mathematical problem from a complex unfamiliar situation.
Define variables, functions, parameters, units, constraints, and assumptions.
Select a linear, quadratic, polynomial, exponential, rational, trigonometric, geometric, statistical, probability, or financial model.
Justify why a selected model family is appropriate.
Estimate or calculate model parameters.
Solve the model using algebraic, graphical, numerical, geometric, or technological methods.
Interpret roots, zeros, vertices, intercepts, asymptotes, scale factors, angles, probabilities, and financial balances.
Optimize a supported objective using function behavior, feasible constraints, or geometric structure.
Validate a model using residuals, bounds, independent data, dimensional analysis, or alternate methods.
Conduct simple sensitivity analysis by varying one parameter.
Identify where the model’s assumptions or functional form fail.
Compare candidate models according to fit, interpretability, simplicity, and intended use.
Revise a model when evidence contradicts it.
Quantify or describe uncertainty appropriately.
Distinguish mathematical optimization from a decision requiring values, ethics, policy, or risk tolerance.
Communicate the model, method, findings, assumptions, limitations, and uncertainty coherently.
Construct a sustained direct proof from definitions, axioms, properties, and established theorems.
State givens, required conclusions, and necessary intermediate claims.
Select a productive representation or auxiliary construction.
Use algebraic identities to prove numerical or functional claims.
Prove parity and divisibility claims algebraically in supported cases.
Prove geometric claims through congruence, similarity, coordinate geometry, and circle theorems.
Construct an introductory proof by contradiction.
Identify the negation of a mathematical claim in supported cases.
Explain why a contradiction invalidates the assumed negation.
Distinguish direct proof, proof by contradiction, counterexample, and verification.
Identify hidden assumptions and domain restrictions.
Identify circular reasoning, converse errors, and reliance on unproved visual appearance.
Determine whether each proof step follows necessarily from prior information.
Repair an incomplete or invalid proof.
Compare synthetic, coordinate, algebraic, and transformational proof methods.
Communicate a rigorous argument in two-column, paragraph, flow, or symbolic form.
State the exact scope and limitations of a proven conclusion.