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Grade 7 Mathematics.

10textbooks

250lessons

195competencies

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Textbook 1Rational Numbers and Signed Operations

4 chapters · 25 lessons

Chapter 1

The Rational-Number System

Chapter 2

Addition and Subtraction of Rational Numbers

Chapter 3

Multiplication and Division of Rational Numbers

Chapter 4

Multi-Operation Rational Expressions

Textbook 2Proportional Relationships, Tables, Graphs, and Equations

4 chapters · 25 lessons

Chapter 1

Recognizing Proportional Relationships

Chapter 2

Equations of Proportional Relationships

Chapter 3

Graphs of Proportional Relationships

Chapter 4

Comparing and Validating Proportional Models

Textbook 3Expressions with Rational Coefficients and Multi-Step Equations

4 chapters · 25 lessons

Chapter 1

Rational Coefficients and Equivalent Expressions

Chapter 2

Multi-Step Equations

Chapter 3

Writing Equations from Contexts

Chapter 4

Critiquing Algebraic Reasoning

Textbook 4Inequalities and Proportional Mathematical Modeling

4 chapters · 25 lessons

Chapter 1

Meaning and Graphs of Inequalities

Chapter 2

Solving Multi-Step Inequalities

Chapter 3

Constraints in Real Situations

Chapter 4

Proportional Modeling and Model Validation

Textbook 5Circles, Circumference, Area, and Compound Measures

4 chapters · 25 lessons

Chapter 1

Radius, Diameter, Circumference, and π

Chapter 2

Area of Circles

Chapter 3

Composite Circular Figures

Chapter 4

Speed, Density, and Compound Units

Textbook 6Scale Drawings, Proportional Measurement, and Geometric Design

4 chapters · 25 lessons

Chapter 1

Reading Scale Drawings

Chapter 2

Creating and Rescaling Drawings

Chapter 3

Length, Area, and Volume under Scaling

Chapter 4

Geometric Design under Constraints

Textbook 7Angle Relationships, Constructions, Cross-Sections, and Informal Geometry

4 chapters · 25 lessons

Chapter 1

Complementary, Supplementary, and Vertical Angles

Chapter 2

Parallel Lines, Transversals, and Polygon Angle Reasoning

Chapter 3

Compass-and-Straightedge Constructions

Chapter 4

Cross-Sections and Three-Dimensional Visualization

Textbook 8Theoretical Probability, Experimental Probability, and Simulations

4 chapters · 25 lessons

Chapter 1

Probability, Outcomes, Events, and Sample Spaces

Chapter 2

Theoretical and Experimental Probability

Chapter 3

Probability Simulations

Chapter 4

Simple Compound Events

Textbook 9Sampling, Population Comparison, and Statistical Arguments

4 chapters · 25 lessons

Chapter 1

Populations, Samples, and Sampling Methods

Chapter 2

Bias, Survey Design, and Evidence Limits

Chapter 3

Comparing Populations with Samples

Chapter 4

Sampling Variability and Simulation

Textbook 10Percent Applications, Simple Interest, and Integrated Modeling

4 chapters · 25 lessons

Chapter 1

Markup, Discount, Tax, and Final Price

Chapter 2

Percent Increase, Decrease, and Error

Chapter 3

Full Simple-Interest Calculations

Chapter 4

Full Grade 7 Modeling Challenges

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Every competency, covered.

The 195 competencies set for Grade 7 Mathematics, and every one has lessons behind it.

Rational Numbers and Signed Operations

51 competencies

Formulate a mathematical problem from a real situation involving rational quantities, proportions, equations, inequalities, probability, geometry, statistics, or finance.
Define variables and units.
Identify constraints, assumptions, and relevant information.
Determine whether a proportional model is appropriate.
Represent a situation using equations, inequalities, proportions, ratio tables, graphs, diagrams, simulations, or statistical displays.
Solve the mathematical model accurately.
Interpret the constant of proportionality, coefficients, inequality bounds, probability values, or statistical summaries in context.
Interpret intermediate and final results with units.
Validate a model using observed data, substitution, alternative representations, estimation, or limiting cases.
Evaluate whether a model’s assumptions remain valid across the stated range.
Revise a model when the relationship is nonproportional, a constraint is missing, a sample is biased, or a simulation is invalid.
Compare two possible models and explain which better fits the situation and evidence.
State the practical limitations of a simplified linear or proportional model.
Explain a rational number as a number expressible as a ratio of two integers with a nonzero denominator.
Represent rational numbers as fractions, terminating decimals, repeating decimals, or integers.
Convert among fractions, decimals, and percentages, including signed values.
Locate positive and negative rational numbers on a number line.
Compare and order rational numbers in different forms.
Interpret absolute value as distance from zero for rational numbers.
Determine the distance between two rational numbers on a number line.
Add integers fluently.
Subtract integers fluently.
Multiply and divide integers fluently.
Add and subtract signed fractions.
Multiply and divide signed fractions.
Add and subtract signed decimals.
Multiply and divide signed decimals.
Apply sign rules correctly across rational-number operations.
Evaluate numerical expressions containing several rational-number operations.
Apply the order of operations to signed rational-number expressions.
Use properties of operations to rewrite and simplify rational-number calculations.
Estimate rational-number sums, differences, products, and quotients.
Interpret signed rational-number results in contexts involving change, position, temperature, elevation, balance, or directional quantity.
Explain why a repeating decimal may represent a rational number.
Check rational-number calculations using inverse operations, estimation, or equivalent representations.
Distinguish rational numbers from numerical forms whose status has not yet been established, without formally introducing irrational numbers.
Construct a logical mathematical argument from definitions, properties, equations, diagrams, and calculations.
Distinguish an example supporting a claim from a general argument establishing the claim.
Use a counterexample to disprove a universal claim.
Explain why one counterexample is sufficient to disprove an “always” statement.
Make a conjecture from patterns or examples.
Test a conjecture using additional cases, boundary cases, and counterexamples.
Use rational-number properties to justify transformations.
Justify each equality-preserving or order-preserving step in an equation or inequality.
Identify assumptions in a proportional, geometric, statistical, probability, or financial argument.
Critique a mathematical solution for validity, clarity, completeness, and efficiency.
Identify circular reasoning, unsupported generalization, or reliance on a misleading diagram at an introductory level.
Compare arithmetic, algebraic, graphical, and proportional methods.
Use estimation, bounds, substitution, units, and inverse operations to validate results.
Explain why a valid procedure applies beyond one example.
Communicate an informal proof or general mathematical argument in a logically ordered form.

Proportional Relationships, Tables, Graphs, and Equations

45 competencies

Formulate a mathematical problem from a real situation involving rational quantities, proportions, equations, inequalities, probability, geometry, statistics, or finance.
Define variables and units.
Identify constraints, assumptions, and relevant information.
Determine whether a proportional model is appropriate.
Represent a situation using equations, inequalities, proportions, ratio tables, graphs, diagrams, simulations, or statistical displays.
Solve the mathematical model accurately.
Interpret the constant of proportionality, coefficients, inequality bounds, probability values, or statistical summaries in context.
Interpret intermediate and final results with units.
Validate a model using observed data, substitution, alternative representations, estimation, or limiting cases.
Evaluate whether a model’s assumptions remain valid across the stated range.
Revise a model when the relationship is nonproportional, a constraint is missing, a sample is biased, or a simulation is invalid.
Compare two possible models and explain which better fits the situation and evidence.
State the practical limitations of a simplified linear or proportional model.
Determine whether two quantities have a proportional relationship.
Explain that a proportional relationship has a constant multiplicative comparison.
Calculate the constant of proportionality from a table.
Calculate the constant of proportionality from a verbal description or pair of corresponding values.
Interpret the constant of proportionality as a unit rate.
Write an equation of the form y = kx for a proportional relationship.
Use a proportional equation to determine an unknown value.
Construct a table from a proportional equation.
Graph a proportional relationship in the coordinate plane.
Explain why the graph of a proportional relationship passes through the origin.
Identify the point (1, k) as representing the unit rate in a proportional graph where the scales permit.
Determine whether a graph represents a proportional relationship.
Compare two proportional relationships represented in tables, graphs, equations, or verbal descriptions.
Solve proportions using equivalent ratios, unit rates, equations, or cross-products with explanation.
Distinguish proportional relationships from additive, fixed-fee, curved, or otherwise nonproportional relationships.
Apply proportional reasoning to scale, rates, percentages, measurement, and financial contexts.
State and evaluate assumptions required by a proportional model.
Construct a logical mathematical argument from definitions, properties, equations, diagrams, and calculations.
Distinguish an example supporting a claim from a general argument establishing the claim.
Use a counterexample to disprove a universal claim.
Explain why one counterexample is sufficient to disprove an “always” statement.
Make a conjecture from patterns or examples.
Test a conjecture using additional cases, boundary cases, and counterexamples.
Use rational-number properties to justify transformations.
Justify each equality-preserving or order-preserving step in an equation or inequality.
Identify assumptions in a proportional, geometric, statistical, probability, or financial argument.
Critique a mathematical solution for validity, clarity, completeness, and efficiency.
Identify circular reasoning, unsupported generalization, or reliance on a misleading diagram at an introductory level.
Compare arithmetic, algebraic, graphical, and proportional methods.
Use estimation, bounds, substitution, units, and inverse operations to validate results.
Explain why a valid procedure applies beyond one example.
Communicate an informal proof or general mathematical argument in a logically ordered form.

Expressions with Rational Coefficients and Multi-Step Equations

47 competencies

Identify terms, constants, variables, and rational coefficients in an expression.
Translate verbal statements into expressions containing rational coefficients.
Evaluate expressions containing positive and negative rational values.
Apply the distributive property with positive and negative rational coefficients.
Combine like terms.
Explain why only like terms may be combined through common variable structure.
Factor a common numerical or rational coefficient from an expression.
Determine whether two expressions are equivalent.
Use properties and counterexamples to justify or reject a claim about equivalent expressions.
Solve multi-step addition and subtraction equations.
Solve multi-step multiplication and division equations.
Solve equations containing parentheses using the distributive property.
Solve equations with positive or negative rational coefficients.
Solve equations whose solutions are rational numbers.
Check an equation solution by substitution.
Write a multi-step equation representing a real situation.
Interpret each term and coefficient in a contextual equation.
Compare algebraic and arithmetic solution methods.
Identify the first equality-breaking step in an incorrect equation solution.
Formulate a mathematical problem from a real situation involving rational quantities, proportions, equations, inequalities, probability, geometry, statistics, or finance.
Define variables and units.
Identify constraints, assumptions, and relevant information.
Determine whether a proportional model is appropriate.
Represent a situation using equations, inequalities, proportions, ratio tables, graphs, diagrams, simulations, or statistical displays.
Solve the mathematical model accurately.
Interpret the constant of proportionality, coefficients, inequality bounds, probability values, or statistical summaries in context.
Interpret intermediate and final results with units.
Validate a model using observed data, substitution, alternative representations, estimation, or limiting cases.
Evaluate whether a model’s assumptions remain valid across the stated range.
Revise a model when the relationship is nonproportional, a constraint is missing, a sample is biased, or a simulation is invalid.
Compare two possible models and explain which better fits the situation and evidence.
State the practical limitations of a simplified linear or proportional model.
Construct a logical mathematical argument from definitions, properties, equations, diagrams, and calculations.
Distinguish an example supporting a claim from a general argument establishing the claim.
Use a counterexample to disprove a universal claim.
Explain why one counterexample is sufficient to disprove an “always” statement.
Make a conjecture from patterns or examples.
Test a conjecture using additional cases, boundary cases, and counterexamples.
Use rational-number properties to justify transformations.
Justify each equality-preserving or order-preserving step in an equation or inequality.
Identify assumptions in a proportional, geometric, statistical, probability, or financial argument.
Critique a mathematical solution for validity, clarity, completeness, and efficiency.
Identify circular reasoning, unsupported generalization, or reliance on a misleading diagram at an introductory level.
Compare arithmetic, algebraic, graphical, and proportional methods.
Use estimation, bounds, substitution, units, and inverse operations to validate results.
Explain why a valid procedure applies beyond one example.
Communicate an informal proof or general mathematical argument in a logically ordered form.

Inequalities and Proportional Mathematical Modeling

51 competencies

Explain an inequality as a comparison describing a range of possible values.
Use <, >, ≤, and ≥ accurately.
Represent an inequality solution on a number line.
Interpret open and closed endpoints.
Solve one-step inequalities.
Solve multi-step inequalities involving addition and subtraction.
Solve supported multi-step inequalities involving multiplication or division.
Reverse the inequality direction when multiplying or dividing by a negative number and explain why through number-line or order reasoning.
Check values that satisfy or do not satisfy an inequality.
Write an inequality representing a constraint or range in a real situation.
Interpret inequality solutions in context, including whole-number restrictions where appropriate.
Distinguish an equation’s exact solution from an inequality’s solution set.
Formulate a mathematical problem from a real situation involving rational quantities, proportions, equations, inequalities, probability, geometry, statistics, or finance.
Define variables and units.
Identify constraints, assumptions, and relevant information.
Determine whether a proportional model is appropriate.
Represent a situation using equations, inequalities, proportions, ratio tables, graphs, diagrams, simulations, or statistical displays.
Solve the mathematical model accurately.
Interpret the constant of proportionality, coefficients, inequality bounds, probability values, or statistical summaries in context.
Interpret intermediate and final results with units.
Validate a model using observed data, substitution, alternative representations, estimation, or limiting cases.
Evaluate whether a model’s assumptions remain valid across the stated range.
Revise a model when the relationship is nonproportional, a constraint is missing, a sample is biased, or a simulation is invalid.
Compare two possible models and explain which better fits the situation and evidence.
State the practical limitations of a simplified linear or proportional model.
Determine whether two quantities have a proportional relationship.
Explain that a proportional relationship has a constant multiplicative comparison.
Calculate the constant of proportionality from a table.
Calculate the constant of proportionality from a verbal description or pair of corresponding values.
Interpret the constant of proportionality as a unit rate.
Write an equation of the form y = kx for a proportional relationship.
Use a proportional equation to determine an unknown value.
Construct a table from a proportional equation.
Graph a proportional relationship in the coordinate plane.
Explain why the graph of a proportional relationship passes through the origin.
Identify the point (1, k) as representing the unit rate in a proportional graph where the scales permit.
Determine whether a graph represents a proportional relationship.
Compare two proportional relationships represented in tables, graphs, equations, or verbal descriptions.
Solve proportions using equivalent ratios, unit rates, equations, or cross-products with explanation.
Distinguish proportional relationships from additive, fixed-fee, curved, or otherwise nonproportional relationships.
Apply proportional reasoning to scale, rates, percentages, measurement, and financial contexts.
State and evaluate assumptions required by a proportional model.
Explain why one counterexample is sufficient to disprove an “always” statement.
Justify each equality-preserving or order-preserving step in an equation or inequality.
Identify assumptions in a proportional, geometric, statistical, probability, or financial argument.
Critique a mathematical solution for validity, clarity, completeness, and efficiency.
Identify circular reasoning, unsupported generalization, or reliance on a misleading diagram at an introductory level.
Compare arithmetic, algebraic, graphical, and proportional methods.
Use estimation, bounds, substitution, units, and inverse operations to validate results.
Explain why a valid procedure applies beyond one example.
Communicate an informal proof or general mathematical argument in a logically ordered form.

Circles, Circumference, Area, and Compound Measures

38 competencies

Identify radius, diameter, chord, circumference, center, arc, and sector in a circle.
Explain the relationship between radius and diameter.
Interpret π as the constant ratio of circumference to diameter.
Calculate circle circumference from diameter.
Calculate circle circumference from radius.
Calculate circle area from radius.
Explain the circle-area formula through rearrangement or decomposition.
Determine a missing radius, diameter, circumference, or area in supported cases.
Calculate areas and perimeters of composite figures containing circles or semicircles.
Calculate circumference and area of circles.
Calculate perimeter and area of composite figures involving circular and polygonal parts.
Calculate average speed as distance divided by time.
Calculate distance from speed and time.
Calculate time from distance and speed.
Convert between compatible speed units in supported cases.
Calculate density as mass divided by volume.
Calculate mass or volume from density and the other known quantity.
Interpret compound-unit notation.
Compare compound measurements using common units.
Solve multi-step measurement problems involving scale, circles, speed, density, area, or volume.
Evaluate whether a compound-measure result has reasonable units and magnitude.
Determine whether two quantities have a proportional relationship.
Explain that a proportional relationship has a constant multiplicative comparison.
Calculate the constant of proportionality from a table.
Calculate the constant of proportionality from a verbal description or pair of corresponding values.
Interpret the constant of proportionality as a unit rate.
Write an equation of the form y = kx for a proportional relationship.
Use a proportional equation to determine an unknown value.
Construct a table from a proportional equation.
Graph a proportional relationship in the coordinate plane.
Explain why the graph of a proportional relationship passes through the origin.
Identify the point (1, k) as representing the unit rate in a proportional graph where the scales permit.
Determine whether a graph represents a proportional relationship.
Compare two proportional relationships represented in tables, graphs, equations, or verbal descriptions.
Solve proportions using equivalent ratios, unit rates, equations, or cross-products with explanation.
Distinguish proportional relationships from additive, fixed-fee, curved, or otherwise nonproportional relationships.
Apply proportional reasoning to scale, rates, percentages, measurement, and financial contexts.
State and evaluate assumptions required by a proportional model.

Scale Drawings, Proportional Measurement, and Geometric Design

49 competencies

Interpret a scale drawing and identify drawing and actual dimensions.
Determine a scale factor between a drawing and an actual object.
Calculate actual dimensions from a scale drawing.
Calculate drawing dimensions from actual measurements.
Reproduce a drawing at a different scale.
Explain how lengths, areas, and volumes respond differently to scale factors at an introductory level.
Calculate circumference and area of circles.
Calculate perimeter and area of composite figures involving circular and polygonal parts.
Use scale factors to convert between drawing and actual measurements.
Explain why an area scale factor is the square of a length scale factor in supported cases.
Explain why a volume scale factor is the cube of a length scale factor in introductory supported cases.
Calculate average speed as distance divided by time.
Calculate distance from speed and time.
Calculate time from distance and speed.
Convert between compatible speed units in supported cases.
Calculate density as mass divided by volume.
Calculate mass or volume from density and the other known quantity.
Interpret compound-unit notation.
Compare compound measurements using common units.
Solve multi-step measurement problems involving scale, circles, speed, density, area, or volume.
Evaluate whether a compound-measure result has reasonable units and magnitude.
Formulate a mathematical problem from a real situation involving rational quantities, proportions, equations, inequalities, probability, geometry, statistics, or finance.
Define variables and units.
Identify constraints, assumptions, and relevant information.
Determine whether a proportional model is appropriate.
Represent a situation using equations, inequalities, proportions, ratio tables, graphs, diagrams, simulations, or statistical displays.
Solve the mathematical model accurately.
Interpret the constant of proportionality, coefficients, inequality bounds, probability values, or statistical summaries in context.
Interpret intermediate and final results with units.
Validate a model using observed data, substitution, alternative representations, estimation, or limiting cases.
Evaluate whether a model’s assumptions remain valid across the stated range.
Revise a model when the relationship is nonproportional, a constraint is missing, a sample is biased, or a simulation is invalid.
Compare two possible models and explain which better fits the situation and evidence.
State the practical limitations of a simplified linear or proportional model.
Construct a logical mathematical argument from definitions, properties, equations, diagrams, and calculations.
Distinguish an example supporting a claim from a general argument establishing the claim.
Use a counterexample to disprove a universal claim.
Explain why one counterexample is sufficient to disprove an “always” statement.
Make a conjecture from patterns or examples.
Test a conjecture using additional cases, boundary cases, and counterexamples.
Use rational-number properties to justify transformations.
Justify each equality-preserving or order-preserving step in an equation or inequality.
Identify assumptions in a proportional, geometric, statistical, probability, or financial argument.
Critique a mathematical solution for validity, clarity, completeness, and efficiency.
Identify circular reasoning, unsupported generalization, or reliance on a misleading diagram at an introductory level.
Compare arithmetic, algebraic, graphical, and proportional methods.
Use estimation, bounds, substitution, units, and inverse operations to validate results.
Explain why a valid procedure applies beyond one example.
Communicate an informal proof or general mathematical argument in a logically ordered form.

Angle Relationships, Constructions, Cross-Sections, and Informal Geometry

38 competencies

Solve multi-step addition and subtraction equations.
Solve multi-step multiplication and division equations.
Solve equations containing parentheses using the distributive property.
Solve equations with positive or negative rational coefficients.
Solve equations whose solutions are rational numbers.
Check an equation solution by substitution.
Write a multi-step equation representing a real situation.
Interpret each term and coefficient in a contextual equation.
Compare algebraic and arithmetic solution methods.
Identify the first equality-breaking step in an incorrect equation solution.
Identify complementary and supplementary angle relationships.
Identify and calculate vertical angles.
Calculate unknown angles formed by intersecting lines.
Analyze angle relationships involving parallel lines and a transversal at an informal level.
Use angle sums in triangles and quadrilaterals.
Construct perpendicular bisectors using compass and straightedge or a validated digital tool.
Construct angle bisectors.
Construct triangles from stated side and angle information in feasible cases.
Explain whether supplied measurements determine no triangle, one triangle, or more than one possible triangle informally.
Identify and draw cross-sections of prisms, pyramids, cylinders, cones, and supported solids.
Explain how cross-section shape depends on the direction and position of the slicing plane.
Use cross-sections to reason about three-dimensional structure.
Construct informal geometric arguments using definitions, diagrams, angle relationships, and counterexamples.
Construct a logical mathematical argument from definitions, properties, equations, diagrams, and calculations.
Distinguish an example supporting a claim from a general argument establishing the claim.
Use a counterexample to disprove a universal claim.
Explain why one counterexample is sufficient to disprove an “always” statement.
Make a conjecture from patterns or examples.
Test a conjecture using additional cases, boundary cases, and counterexamples.
Use rational-number properties to justify transformations.
Justify each equality-preserving or order-preserving step in an equation or inequality.
Identify assumptions in a proportional, geometric, statistical, probability, or financial argument.
Critique a mathematical solution for validity, clarity, completeness, and efficiency.
Identify circular reasoning, unsupported generalization, or reliance on a misleading diagram at an introductory level.
Compare arithmetic, algebraic, graphical, and proportional methods.
Use estimation, bounds, substitution, units, and inverse operations to validate results.
Explain why a valid procedure applies beyond one example.
Communicate an informal proof or general mathematical argument in a logically ordered form.

Theoretical Probability, Experimental Probability, and Simulations

46 competencies

Formulate a mathematical problem from a real situation involving rational quantities, proportions, equations, inequalities, probability, geometry, statistics, or finance.
Define variables and units.
Identify constraints, assumptions, and relevant information.
Determine whether a proportional model is appropriate.
Represent a situation using equations, inequalities, proportions, ratio tables, graphs, diagrams, simulations, or statistical displays.
Solve the mathematical model accurately.
Interpret the constant of proportionality, coefficients, inequality bounds, probability values, or statistical summaries in context.
Interpret intermediate and final results with units.
Validate a model using observed data, substitution, alternative representations, estimation, or limiting cases.
Evaluate whether a model’s assumptions remain valid across the stated range.
Revise a model when the relationship is nonproportional, a constraint is missing, a sample is biased, or a simulation is invalid.
Compare two possible models and explain which better fits the situation and evidence.
State the practical limitations of a simplified linear or proportional model.
Explain probability as a numerical measure between 0 and 1 inclusive.
Interpret probabilities of 0, 1, and values between them.
Express probability as a fraction, decimal, or percentage.
Define an outcome, event, and sample space.
List a sample space for a simple experiment.
Calculate theoretical probability when outcomes are equally likely.
Explain why theoretical probability depends on a justified model of possible outcomes.
Conduct or interpret a probability experiment.
Calculate experimental probability from relative frequency.
Compare theoretical and experimental probability.
Explain why experimental results vary across samples.
Explain why experimental probability often becomes more stable over many trials.
Design or use a simulation representing a simple probability situation.
Determine whether a simulation fairly represents the intended probabilities.
Use organized lists, tables, or tree diagrams for simple compound events.
Calculate probabilities of simple two-stage events with small sample spaces.
Calculate the probability of an event not occurring using the complement in simple cases.
Evaluate probability claims for incorrect assumptions about equally likely outcomes.
Construct a logical mathematical argument from definitions, properties, equations, diagrams, and calculations.
Distinguish an example supporting a claim from a general argument establishing the claim.
Use a counterexample to disprove a universal claim.
Explain why one counterexample is sufficient to disprove an “always” statement.
Make a conjecture from patterns or examples.
Test a conjecture using additional cases, boundary cases, and counterexamples.
Use rational-number properties to justify transformations.
Justify each equality-preserving or order-preserving step in an equation or inequality.
Identify assumptions in a proportional, geometric, statistical, probability, or financial argument.
Critique a mathematical solution for validity, clarity, completeness, and efficiency.
Identify circular reasoning, unsupported generalization, or reliance on a misleading diagram at an introductory level.
Compare arithmetic, algebraic, graphical, and proportional methods.
Use estimation, bounds, substitution, units, and inverse operations to validate results.
Explain why a valid procedure applies beyond one example.
Communicate an informal proof or general mathematical argument in a logically ordered form.

Sampling, Population Comparison, and Statistical Arguments

51 competencies

Formulate a mathematical problem from a real situation involving rational quantities, proportions, equations, inequalities, probability, geometry, statistics, or finance.
Define variables and units.
Identify constraints, assumptions, and relevant information.
Determine whether a proportional model is appropriate.
Represent a situation using equations, inequalities, proportions, ratio tables, graphs, diagrams, simulations, or statistical displays.
Solve the mathematical model accurately.
Interpret the constant of proportionality, coefficients, inequality bounds, probability values, or statistical summaries in context.
Interpret intermediate and final results with units.
Validate a model using observed data, substitution, alternative representations, estimation, or limiting cases.
Evaluate whether a model’s assumptions remain valid across the stated range.
Revise a model when the relationship is nonproportional, a constraint is missing, a sample is biased, or a simulation is invalid.
Compare two possible models and explain which better fits the situation and evidence.
State the practical limitations of a simplified linear or proportional model.
Conduct or interpret a probability experiment.
Calculate experimental probability from relative frequency.
Compare theoretical and experimental probability.
Explain why experimental results vary across samples.
Explain why experimental probability often becomes more stable over many trials.
Design or use a simulation representing a simple probability situation.
Determine whether a simulation fairly represents the intended probabilities.
Construct a logical mathematical argument from definitions, properties, equations, diagrams, and calculations.
Distinguish an example supporting a claim from a general argument establishing the claim.
Use a counterexample to disprove a universal claim.
Explain why one counterexample is sufficient to disprove an “always” statement.
Make a conjecture from patterns or examples.
Test a conjecture using additional cases, boundary cases, and counterexamples.
Use rational-number properties to justify transformations.
Justify each equality-preserving or order-preserving step in an equation or inequality.
Identify assumptions in a proportional, geometric, statistical, probability, or financial argument.
Critique a mathematical solution for validity, clarity, completeness, and efficiency.
Identify circular reasoning, unsupported generalization, or reliance on a misleading diagram at an introductory level.
Compare arithmetic, algebraic, graphical, and proportional methods.
Use estimation, bounds, substitution, units, and inverse operations to validate results.
Explain why a valid procedure applies beyond one example.
Communicate an informal proof or general mathematical argument in a logically ordered form.
Distinguish a population from a sample.
Explain why a sample may be used to learn about a population.
Identify simple random, convenience, voluntary-response, and systematic sampling methods at an introductory level.
Explain how selection method may introduce bias.
Identify undercoverage, nonresponse, and leading-question concerns in supported examples.
Determine whether a sample is reasonably representative of a stated population.
Use sample statistics to make a cautious estimate about a population.
Compare two populations informally using samples.
Compare measures of center between two samples.
Compare measures of spread between two samples.
Compare visual distributions using dot plots, histograms, or box plots.
Describe overlap and separation between two distributions.
Explain why different random samples from the same population may produce different statistics.
Use repeated samples or simulation to examine sampling variability.
Distinguish a claim supported by sample evidence from a claim that extends beyond the evidence.
Critique a survey or study for population definition, sample method, wording, size, and possible bias without conducting formal inference.

Percent Applications, Simple Interest, and Integrated Modeling

74 competencies

Solve multi-step addition and subtraction equations.
Solve multi-step multiplication and division equations.
Solve equations containing parentheses using the distributive property.
Solve equations with positive or negative rational coefficients.
Solve equations whose solutions are rational numbers.
Check an equation solution by substitution.
Write a multi-step equation representing a real situation.
Interpret each term and coefficient in a contextual equation.
Compare algebraic and arithmetic solution methods.
Identify the first equality-breaking step in an incorrect equation solution.
Calculate markup as a percentage of cost.
Calculate a selling price after markup.
Calculate discount amount and sale price.
Calculate tax, tip, service charge, or another percentage addition.
Calculate a final price involving markup, discount, tax, or a fixed charge in supported multi-step cases.
Calculate percent increase.
Calculate percent decrease.
Determine an original amount from a final amount and a single percentage increase or decrease in supported cases.
Compare percentage changes involving different original amounts.
Calculate simple interest using principal, annual or period rate, and time.
Use I = Prt in supported simple-interest calculations.
Calculate principal, rate, or time when the other simple-interest quantities are known in supported cases.
Calculate the final amount under simple interest.
Represent simple-interest growth using a table, graph, or proportional-style equation.
Compare two simple-interest arrangements mathematically.
Explain why simple-interest growth is additive rather than compound.
Solve multi-step financial problems involving budgets, markup, discount, tax, and simple interest.
Validate financial calculations using estimation, equations, percentages, and constraints.
Distinguish financial calculation from personal judgment about saving, borrowing, spending, or risk.
Formulate a mathematical problem from a real situation involving rational quantities, proportions, equations, inequalities, probability, geometry, statistics, or finance.
Define variables and units.
Identify constraints, assumptions, and relevant information.
Determine whether a proportional model is appropriate.
Represent a situation using equations, inequalities, proportions, ratio tables, graphs, diagrams, simulations, or statistical displays.
Solve the mathematical model accurately.
Interpret the constant of proportionality, coefficients, inequality bounds, probability values, or statistical summaries in context.
Interpret intermediate and final results with units.
Validate a model using observed data, substitution, alternative representations, estimation, or limiting cases.
Evaluate whether a model’s assumptions remain valid across the stated range.
Revise a model when the relationship is nonproportional, a constraint is missing, a sample is biased, or a simulation is invalid.
Compare two possible models and explain which better fits the situation and evidence.
State the practical limitations of a simplified linear or proportional model.
Determine whether two quantities have a proportional relationship.
Explain that a proportional relationship has a constant multiplicative comparison.
Calculate the constant of proportionality from a table.
Calculate the constant of proportionality from a verbal description or pair of corresponding values.
Interpret the constant of proportionality as a unit rate.
Write an equation of the form y = kx for a proportional relationship.
Use a proportional equation to determine an unknown value.
Construct a table from a proportional equation.
Graph a proportional relationship in the coordinate plane.
Explain why the graph of a proportional relationship passes through the origin.
Identify the point (1, k) as representing the unit rate in a proportional graph where the scales permit.
Determine whether a graph represents a proportional relationship.
Compare two proportional relationships represented in tables, graphs, equations, or verbal descriptions.
Solve proportions using equivalent ratios, unit rates, equations, or cross-products with explanation.
Distinguish proportional relationships from additive, fixed-fee, curved, or otherwise nonproportional relationships.
Apply proportional reasoning to scale, rates, percentages, measurement, and financial contexts.
State and evaluate assumptions required by a proportional model.
Construct a logical mathematical argument from definitions, properties, equations, diagrams, and calculations.
Distinguish an example supporting a claim from a general argument establishing the claim.
Use a counterexample to disprove a universal claim.
Explain why one counterexample is sufficient to disprove an “always” statement.
Make a conjecture from patterns or examples.
Test a conjecture using additional cases, boundary cases, and counterexamples.
Use rational-number properties to justify transformations.
Justify each equality-preserving or order-preserving step in an equation or inequality.
Identify assumptions in a proportional, geometric, statistical, probability, or financial argument.
Critique a mathematical solution for validity, clarity, completeness, and efficiency.
Identify circular reasoning, unsupported generalization, or reliance on a misleading diagram at an introductory level.
Compare arithmetic, algebraic, graphical, and proportional methods.
Use estimation, bounds, substitution, units, and inverse operations to validate results.
Explain why a valid procedure applies beyond one example.
Communicate an informal proof or general mathematical argument in a logically ordered form.

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The whole year, lesson by lesson.

Textbook 1Rational Numbers and Signed Operations

4 chapters · 25 lessons

Chapter 1

The Rational-Number System

Chapter 2

Addition and Subtraction of Rational Numbers

Chapter 3

Multiplication and Division of Rational Numbers

Chapter 4

Multi-Operation Rational Expressions

Textbook 2Proportional Relationships, Tables, Graphs, and Equations

4 chapters · 25 lessons

Chapter 1

Recognizing Proportional Relationships

Chapter 2

Equations of Proportional Relationships

Chapter 3

Graphs of Proportional Relationships

Chapter 4

Comparing and Validating Proportional Models

Textbook 3Expressions with Rational Coefficients and Multi-Step Equations

4 chapters · 25 lessons

Chapter 1

Rational Coefficients and Equivalent Expressions

Chapter 2

Multi-Step Equations

Chapter 3

Writing Equations from Contexts

Chapter 4

Critiquing Algebraic Reasoning

Textbook 4Inequalities and Proportional Mathematical Modeling

4 chapters · 25 lessons

Chapter 1

Meaning and Graphs of Inequalities

Chapter 2

Solving Multi-Step Inequalities

Chapter 3

Constraints in Real Situations

Chapter 4

Proportional Modeling and Model Validation

Textbook 5Circles, Circumference, Area, and Compound Measures

4 chapters · 25 lessons

Chapter 1

Radius, Diameter, Circumference, and π

Chapter 2

Area of Circles

Chapter 3

Composite Circular Figures

Chapter 4

Speed, Density, and Compound Units

Textbook 6Scale Drawings, Proportional Measurement, and Geometric Design

4 chapters · 25 lessons

Chapter 1

Reading Scale Drawings

Chapter 2

Creating and Rescaling Drawings

Chapter 3

Length, Area, and Volume under Scaling

Chapter 4

Geometric Design under Constraints

Textbook 7Angle Relationships, Constructions, Cross-Sections, and Informal Geometry

4 chapters · 25 lessons

Chapter 1

Complementary, Supplementary, and Vertical Angles

Chapter 2

Parallel Lines, Transversals, and Polygon Angle Reasoning

Chapter 3

Compass-and-Straightedge Constructions

Chapter 4

Cross-Sections and Three-Dimensional Visualization

Textbook 8Theoretical Probability, Experimental Probability, and Simulations

4 chapters · 25 lessons

Chapter 1

Probability, Outcomes, Events, and Sample Spaces

Chapter 2

Theoretical and Experimental Probability

Chapter 3

Probability Simulations

Chapter 4

Simple Compound Events

Textbook 9Sampling, Population Comparison, and Statistical Arguments

4 chapters · 25 lessons

Chapter 1

Populations, Samples, and Sampling Methods

Chapter 2

Bias, Survey Design, and Evidence Limits

Chapter 3

Comparing Populations with Samples

Chapter 4

Sampling Variability and Simulation

Textbook 10Percent Applications, Simple Interest, and Integrated Modeling

4 chapters · 25 lessons

Chapter 1

Markup, Discount, Tax, and Final Price

Chapter 2

Percent Increase, Decrease, and Error

Chapter 3

Full Simple-Interest Calculations

Chapter 4

Full Grade 7 Modeling Challenges

Every competency, covered.

The 195 competencies set for Grade 7 Mathematics, and every one has lessons behind it.

Rational Numbers and Signed Operations

51 competencies

Formulate a mathematical problem from a real situation involving rational quantities, proportions, equations, inequalities, probability, geometry, statistics, or finance.
Define variables and units.
Identify constraints, assumptions, and relevant information.
Determine whether a proportional model is appropriate.
Represent a situation using equations, inequalities, proportions, ratio tables, graphs, diagrams, simulations, or statistical displays.
Solve the mathematical model accurately.
Interpret the constant of proportionality, coefficients, inequality bounds, probability values, or statistical summaries in context.
Interpret intermediate and final results with units.
Validate a model using observed data, substitution, alternative representations, estimation, or limiting cases.
Evaluate whether a model’s assumptions remain valid across the stated range.
Revise a model when the relationship is nonproportional, a constraint is missing, a sample is biased, or a simulation is invalid.
Compare two possible models and explain which better fits the situation and evidence.
State the practical limitations of a simplified linear or proportional model.
Explain a rational number as a number expressible as a ratio of two integers with a nonzero denominator.
Represent rational numbers as fractions, terminating decimals, repeating decimals, or integers.
Convert among fractions, decimals, and percentages, including signed values.
Locate positive and negative rational numbers on a number line.
Compare and order rational numbers in different forms.
Interpret absolute value as distance from zero for rational numbers.
Determine the distance between two rational numbers on a number line.
Add integers fluently.
Subtract integers fluently.
Multiply and divide integers fluently.
Add and subtract signed fractions.
Multiply and divide signed fractions.
Add and subtract signed decimals.
Multiply and divide signed decimals.
Apply sign rules correctly across rational-number operations.
Evaluate numerical expressions containing several rational-number operations.
Apply the order of operations to signed rational-number expressions.
Use properties of operations to rewrite and simplify rational-number calculations.
Estimate rational-number sums, differences, products, and quotients.
Interpret signed rational-number results in contexts involving change, position, temperature, elevation, balance, or directional quantity.
Explain why a repeating decimal may represent a rational number.
Check rational-number calculations using inverse operations, estimation, or equivalent representations.
Distinguish rational numbers from numerical forms whose status has not yet been established, without formally introducing irrational numbers.
Construct a logical mathematical argument from definitions, properties, equations, diagrams, and calculations.
Distinguish an example supporting a claim from a general argument establishing the claim.
Use a counterexample to disprove a universal claim.
Explain why one counterexample is sufficient to disprove an “always” statement.
Make a conjecture from patterns or examples.
Test a conjecture using additional cases, boundary cases, and counterexamples.
Use rational-number properties to justify transformations.
Justify each equality-preserving or order-preserving step in an equation or inequality.
Identify assumptions in a proportional, geometric, statistical, probability, or financial argument.
Critique a mathematical solution for validity, clarity, completeness, and efficiency.
Identify circular reasoning, unsupported generalization, or reliance on a misleading diagram at an introductory level.
Compare arithmetic, algebraic, graphical, and proportional methods.
Use estimation, bounds, substitution, units, and inverse operations to validate results.
Explain why a valid procedure applies beyond one example.
Communicate an informal proof or general mathematical argument in a logically ordered form.

Proportional Relationships, Tables, Graphs, and Equations

45 competencies

Formulate a mathematical problem from a real situation involving rational quantities, proportions, equations, inequalities, probability, geometry, statistics, or finance.
Define variables and units.
Identify constraints, assumptions, and relevant information.
Determine whether a proportional model is appropriate.
Represent a situation using equations, inequalities, proportions, ratio tables, graphs, diagrams, simulations, or statistical displays.
Solve the mathematical model accurately.
Interpret the constant of proportionality, coefficients, inequality bounds, probability values, or statistical summaries in context.
Interpret intermediate and final results with units.
Validate a model using observed data, substitution, alternative representations, estimation, or limiting cases.
Evaluate whether a model’s assumptions remain valid across the stated range.
Revise a model when the relationship is nonproportional, a constraint is missing, a sample is biased, or a simulation is invalid.
Compare two possible models and explain which better fits the situation and evidence.
State the practical limitations of a simplified linear or proportional model.
Determine whether two quantities have a proportional relationship.
Explain that a proportional relationship has a constant multiplicative comparison.
Calculate the constant of proportionality from a table.
Calculate the constant of proportionality from a verbal description or pair of corresponding values.
Interpret the constant of proportionality as a unit rate.
Write an equation of the form y = kx for a proportional relationship.
Use a proportional equation to determine an unknown value.
Construct a table from a proportional equation.
Graph a proportional relationship in the coordinate plane.
Explain why the graph of a proportional relationship passes through the origin.
Identify the point (1, k) as representing the unit rate in a proportional graph where the scales permit.
Determine whether a graph represents a proportional relationship.
Compare two proportional relationships represented in tables, graphs, equations, or verbal descriptions.
Solve proportions using equivalent ratios, unit rates, equations, or cross-products with explanation.
Distinguish proportional relationships from additive, fixed-fee, curved, or otherwise nonproportional relationships.
Apply proportional reasoning to scale, rates, percentages, measurement, and financial contexts.
State and evaluate assumptions required by a proportional model.
Construct a logical mathematical argument from definitions, properties, equations, diagrams, and calculations.
Distinguish an example supporting a claim from a general argument establishing the claim.
Use a counterexample to disprove a universal claim.
Explain why one counterexample is sufficient to disprove an “always” statement.
Make a conjecture from patterns or examples.
Test a conjecture using additional cases, boundary cases, and counterexamples.
Use rational-number properties to justify transformations.
Justify each equality-preserving or order-preserving step in an equation or inequality.
Identify assumptions in a proportional, geometric, statistical, probability, or financial argument.
Critique a mathematical solution for validity, clarity, completeness, and efficiency.
Identify circular reasoning, unsupported generalization, or reliance on a misleading diagram at an introductory level.
Compare arithmetic, algebraic, graphical, and proportional methods.
Use estimation, bounds, substitution, units, and inverse operations to validate results.
Explain why a valid procedure applies beyond one example.
Communicate an informal proof or general mathematical argument in a logically ordered form.

Expressions with Rational Coefficients and Multi-Step Equations

47 competencies

Identify terms, constants, variables, and rational coefficients in an expression.
Translate verbal statements into expressions containing rational coefficients.
Evaluate expressions containing positive and negative rational values.
Apply the distributive property with positive and negative rational coefficients.
Combine like terms.
Explain why only like terms may be combined through common variable structure.
Factor a common numerical or rational coefficient from an expression.
Determine whether two expressions are equivalent.
Use properties and counterexamples to justify or reject a claim about equivalent expressions.
Solve multi-step addition and subtraction equations.
Solve multi-step multiplication and division equations.
Solve equations containing parentheses using the distributive property.
Solve equations with positive or negative rational coefficients.
Solve equations whose solutions are rational numbers.
Check an equation solution by substitution.
Write a multi-step equation representing a real situation.
Interpret each term and coefficient in a contextual equation.
Compare algebraic and arithmetic solution methods.
Identify the first equality-breaking step in an incorrect equation solution.
Formulate a mathematical problem from a real situation involving rational quantities, proportions, equations, inequalities, probability, geometry, statistics, or finance.
Define variables and units.
Identify constraints, assumptions, and relevant information.
Determine whether a proportional model is appropriate.
Represent a situation using equations, inequalities, proportions, ratio tables, graphs, diagrams, simulations, or statistical displays.
Solve the mathematical model accurately.
Interpret the constant of proportionality, coefficients, inequality bounds, probability values, or statistical summaries in context.
Interpret intermediate and final results with units.
Validate a model using observed data, substitution, alternative representations, estimation, or limiting cases.
Evaluate whether a model’s assumptions remain valid across the stated range.
Revise a model when the relationship is nonproportional, a constraint is missing, a sample is biased, or a simulation is invalid.
Compare two possible models and explain which better fits the situation and evidence.
State the practical limitations of a simplified linear or proportional model.
Construct a logical mathematical argument from definitions, properties, equations, diagrams, and calculations.
Distinguish an example supporting a claim from a general argument establishing the claim.
Use a counterexample to disprove a universal claim.
Explain why one counterexample is sufficient to disprove an “always” statement.
Make a conjecture from patterns or examples.
Test a conjecture using additional cases, boundary cases, and counterexamples.
Use rational-number properties to justify transformations.
Justify each equality-preserving or order-preserving step in an equation or inequality.
Identify assumptions in a proportional, geometric, statistical, probability, or financial argument.
Critique a mathematical solution for validity, clarity, completeness, and efficiency.
Identify circular reasoning, unsupported generalization, or reliance on a misleading diagram at an introductory level.
Compare arithmetic, algebraic, graphical, and proportional methods.
Use estimation, bounds, substitution, units, and inverse operations to validate results.
Explain why a valid procedure applies beyond one example.
Communicate an informal proof or general mathematical argument in a logically ordered form.

Inequalities and Proportional Mathematical Modeling

51 competencies

Explain an inequality as a comparison describing a range of possible values.
Use <, >, ≤, and ≥ accurately.
Represent an inequality solution on a number line.
Interpret open and closed endpoints.
Solve one-step inequalities.
Solve multi-step inequalities involving addition and subtraction.
Solve supported multi-step inequalities involving multiplication or division.
Reverse the inequality direction when multiplying or dividing by a negative number and explain why through number-line or order reasoning.
Check values that satisfy or do not satisfy an inequality.
Write an inequality representing a constraint or range in a real situation.
Interpret inequality solutions in context, including whole-number restrictions where appropriate.
Distinguish an equation’s exact solution from an inequality’s solution set.
Formulate a mathematical problem from a real situation involving rational quantities, proportions, equations, inequalities, probability, geometry, statistics, or finance.
Define variables and units.
Identify constraints, assumptions, and relevant information.
Determine whether a proportional model is appropriate.
Represent a situation using equations, inequalities, proportions, ratio tables, graphs, diagrams, simulations, or statistical displays.
Solve the mathematical model accurately.
Interpret the constant of proportionality, coefficients, inequality bounds, probability values, or statistical summaries in context.
Interpret intermediate and final results with units.
Validate a model using observed data, substitution, alternative representations, estimation, or limiting cases.
Evaluate whether a model’s assumptions remain valid across the stated range.
Revise a model when the relationship is nonproportional, a constraint is missing, a sample is biased, or a simulation is invalid.
Compare two possible models and explain which better fits the situation and evidence.
State the practical limitations of a simplified linear or proportional model.
Determine whether two quantities have a proportional relationship.
Explain that a proportional relationship has a constant multiplicative comparison.
Calculate the constant of proportionality from a table.
Calculate the constant of proportionality from a verbal description or pair of corresponding values.
Interpret the constant of proportionality as a unit rate.
Write an equation of the form y = kx for a proportional relationship.
Use a proportional equation to determine an unknown value.
Construct a table from a proportional equation.
Graph a proportional relationship in the coordinate plane.
Explain why the graph of a proportional relationship passes through the origin.
Identify the point (1, k) as representing the unit rate in a proportional graph where the scales permit.
Determine whether a graph represents a proportional relationship.
Compare two proportional relationships represented in tables, graphs, equations, or verbal descriptions.
Solve proportions using equivalent ratios, unit rates, equations, or cross-products with explanation.
Distinguish proportional relationships from additive, fixed-fee, curved, or otherwise nonproportional relationships.
Apply proportional reasoning to scale, rates, percentages, measurement, and financial contexts.
State and evaluate assumptions required by a proportional model.
Explain why one counterexample is sufficient to disprove an “always” statement.
Justify each equality-preserving or order-preserving step in an equation or inequality.
Identify assumptions in a proportional, geometric, statistical, probability, or financial argument.
Critique a mathematical solution for validity, clarity, completeness, and efficiency.
Identify circular reasoning, unsupported generalization, or reliance on a misleading diagram at an introductory level.
Compare arithmetic, algebraic, graphical, and proportional methods.
Use estimation, bounds, substitution, units, and inverse operations to validate results.
Explain why a valid procedure applies beyond one example.
Communicate an informal proof or general mathematical argument in a logically ordered form.

Circles, Circumference, Area, and Compound Measures

38 competencies

Identify radius, diameter, chord, circumference, center, arc, and sector in a circle.
Explain the relationship between radius and diameter.
Interpret π as the constant ratio of circumference to diameter.
Calculate circle circumference from diameter.
Calculate circle circumference from radius.
Calculate circle area from radius.
Explain the circle-area formula through rearrangement or decomposition.
Determine a missing radius, diameter, circumference, or area in supported cases.
Calculate areas and perimeters of composite figures containing circles or semicircles.
Calculate circumference and area of circles.
Calculate perimeter and area of composite figures involving circular and polygonal parts.
Calculate average speed as distance divided by time.
Calculate distance from speed and time.
Calculate time from distance and speed.
Convert between compatible speed units in supported cases.
Calculate density as mass divided by volume.
Calculate mass or volume from density and the other known quantity.
Interpret compound-unit notation.
Compare compound measurements using common units.
Solve multi-step measurement problems involving scale, circles, speed, density, area, or volume.
Evaluate whether a compound-measure result has reasonable units and magnitude.
Determine whether two quantities have a proportional relationship.
Explain that a proportional relationship has a constant multiplicative comparison.
Calculate the constant of proportionality from a table.
Calculate the constant of proportionality from a verbal description or pair of corresponding values.
Interpret the constant of proportionality as a unit rate.
Write an equation of the form y = kx for a proportional relationship.
Use a proportional equation to determine an unknown value.
Construct a table from a proportional equation.
Graph a proportional relationship in the coordinate plane.
Explain why the graph of a proportional relationship passes through the origin.
Identify the point (1, k) as representing the unit rate in a proportional graph where the scales permit.
Determine whether a graph represents a proportional relationship.
Compare two proportional relationships represented in tables, graphs, equations, or verbal descriptions.
Solve proportions using equivalent ratios, unit rates, equations, or cross-products with explanation.
Distinguish proportional relationships from additive, fixed-fee, curved, or otherwise nonproportional relationships.
Apply proportional reasoning to scale, rates, percentages, measurement, and financial contexts.
State and evaluate assumptions required by a proportional model.

Scale Drawings, Proportional Measurement, and Geometric Design

49 competencies

Interpret a scale drawing and identify drawing and actual dimensions.
Determine a scale factor between a drawing and an actual object.
Calculate actual dimensions from a scale drawing.
Calculate drawing dimensions from actual measurements.
Reproduce a drawing at a different scale.
Explain how lengths, areas, and volumes respond differently to scale factors at an introductory level.
Calculate circumference and area of circles.
Calculate perimeter and area of composite figures involving circular and polygonal parts.
Use scale factors to convert between drawing and actual measurements.
Explain why an area scale factor is the square of a length scale factor in supported cases.
Explain why a volume scale factor is the cube of a length scale factor in introductory supported cases.
Calculate average speed as distance divided by time.
Calculate distance from speed and time.
Calculate time from distance and speed.
Convert between compatible speed units in supported cases.
Calculate density as mass divided by volume.
Calculate mass or volume from density and the other known quantity.
Interpret compound-unit notation.
Compare compound measurements using common units.
Solve multi-step measurement problems involving scale, circles, speed, density, area, or volume.
Evaluate whether a compound-measure result has reasonable units and magnitude.
Formulate a mathematical problem from a real situation involving rational quantities, proportions, equations, inequalities, probability, geometry, statistics, or finance.
Define variables and units.
Identify constraints, assumptions, and relevant information.
Determine whether a proportional model is appropriate.
Represent a situation using equations, inequalities, proportions, ratio tables, graphs, diagrams, simulations, or statistical displays.
Solve the mathematical model accurately.
Interpret the constant of proportionality, coefficients, inequality bounds, probability values, or statistical summaries in context.
Interpret intermediate and final results with units.
Validate a model using observed data, substitution, alternative representations, estimation, or limiting cases.
Evaluate whether a model’s assumptions remain valid across the stated range.
Revise a model when the relationship is nonproportional, a constraint is missing, a sample is biased, or a simulation is invalid.
Compare two possible models and explain which better fits the situation and evidence.
State the practical limitations of a simplified linear or proportional model.
Construct a logical mathematical argument from definitions, properties, equations, diagrams, and calculations.
Distinguish an example supporting a claim from a general argument establishing the claim.
Use a counterexample to disprove a universal claim.
Explain why one counterexample is sufficient to disprove an “always” statement.
Make a conjecture from patterns or examples.
Test a conjecture using additional cases, boundary cases, and counterexamples.
Use rational-number properties to justify transformations.
Justify each equality-preserving or order-preserving step in an equation or inequality.
Identify assumptions in a proportional, geometric, statistical, probability, or financial argument.
Critique a mathematical solution for validity, clarity, completeness, and efficiency.
Identify circular reasoning, unsupported generalization, or reliance on a misleading diagram at an introductory level.
Compare arithmetic, algebraic, graphical, and proportional methods.
Use estimation, bounds, substitution, units, and inverse operations to validate results.
Explain why a valid procedure applies beyond one example.
Communicate an informal proof or general mathematical argument in a logically ordered form.

Angle Relationships, Constructions, Cross-Sections, and Informal Geometry

38 competencies

Solve multi-step addition and subtraction equations.
Solve multi-step multiplication and division equations.
Solve equations containing parentheses using the distributive property.
Solve equations with positive or negative rational coefficients.
Solve equations whose solutions are rational numbers.
Check an equation solution by substitution.
Write a multi-step equation representing a real situation.
Interpret each term and coefficient in a contextual equation.
Compare algebraic and arithmetic solution methods.
Identify the first equality-breaking step in an incorrect equation solution.
Identify complementary and supplementary angle relationships.
Identify and calculate vertical angles.
Calculate unknown angles formed by intersecting lines.
Analyze angle relationships involving parallel lines and a transversal at an informal level.
Use angle sums in triangles and quadrilaterals.
Construct perpendicular bisectors using compass and straightedge or a validated digital tool.
Construct angle bisectors.
Construct triangles from stated side and angle information in feasible cases.
Explain whether supplied measurements determine no triangle, one triangle, or more than one possible triangle informally.
Identify and draw cross-sections of prisms, pyramids, cylinders, cones, and supported solids.
Explain how cross-section shape depends on the direction and position of the slicing plane.
Use cross-sections to reason about three-dimensional structure.
Construct informal geometric arguments using definitions, diagrams, angle relationships, and counterexamples.
Construct a logical mathematical argument from definitions, properties, equations, diagrams, and calculations.
Distinguish an example supporting a claim from a general argument establishing the claim.
Use a counterexample to disprove a universal claim.
Explain why one counterexample is sufficient to disprove an “always” statement.
Make a conjecture from patterns or examples.
Test a conjecture using additional cases, boundary cases, and counterexamples.
Use rational-number properties to justify transformations.
Justify each equality-preserving or order-preserving step in an equation or inequality.
Identify assumptions in a proportional, geometric, statistical, probability, or financial argument.
Critique a mathematical solution for validity, clarity, completeness, and efficiency.
Identify circular reasoning, unsupported generalization, or reliance on a misleading diagram at an introductory level.
Compare arithmetic, algebraic, graphical, and proportional methods.
Use estimation, bounds, substitution, units, and inverse operations to validate results.
Explain why a valid procedure applies beyond one example.
Communicate an informal proof or general mathematical argument in a logically ordered form.

Theoretical Probability, Experimental Probability, and Simulations

46 competencies

Formulate a mathematical problem from a real situation involving rational quantities, proportions, equations, inequalities, probability, geometry, statistics, or finance.
Define variables and units.
Identify constraints, assumptions, and relevant information.
Determine whether a proportional model is appropriate.
Represent a situation using equations, inequalities, proportions, ratio tables, graphs, diagrams, simulations, or statistical displays.
Solve the mathematical model accurately.
Interpret the constant of proportionality, coefficients, inequality bounds, probability values, or statistical summaries in context.
Interpret intermediate and final results with units.
Validate a model using observed data, substitution, alternative representations, estimation, or limiting cases.
Evaluate whether a model’s assumptions remain valid across the stated range.
Revise a model when the relationship is nonproportional, a constraint is missing, a sample is biased, or a simulation is invalid.
Compare two possible models and explain which better fits the situation and evidence.
State the practical limitations of a simplified linear or proportional model.
Explain probability as a numerical measure between 0 and 1 inclusive.
Interpret probabilities of 0, 1, and values between them.
Express probability as a fraction, decimal, or percentage.
Define an outcome, event, and sample space.
List a sample space for a simple experiment.
Calculate theoretical probability when outcomes are equally likely.
Explain why theoretical probability depends on a justified model of possible outcomes.
Conduct or interpret a probability experiment.
Calculate experimental probability from relative frequency.
Compare theoretical and experimental probability.
Explain why experimental results vary across samples.
Explain why experimental probability often becomes more stable over many trials.
Design or use a simulation representing a simple probability situation.
Determine whether a simulation fairly represents the intended probabilities.
Use organized lists, tables, or tree diagrams for simple compound events.
Calculate probabilities of simple two-stage events with small sample spaces.
Calculate the probability of an event not occurring using the complement in simple cases.
Evaluate probability claims for incorrect assumptions about equally likely outcomes.
Construct a logical mathematical argument from definitions, properties, equations, diagrams, and calculations.
Distinguish an example supporting a claim from a general argument establishing the claim.
Use a counterexample to disprove a universal claim.
Explain why one counterexample is sufficient to disprove an “always” statement.
Make a conjecture from patterns or examples.
Test a conjecture using additional cases, boundary cases, and counterexamples.
Use rational-number properties to justify transformations.
Justify each equality-preserving or order-preserving step in an equation or inequality.
Identify assumptions in a proportional, geometric, statistical, probability, or financial argument.
Critique a mathematical solution for validity, clarity, completeness, and efficiency.
Identify circular reasoning, unsupported generalization, or reliance on a misleading diagram at an introductory level.
Compare arithmetic, algebraic, graphical, and proportional methods.
Use estimation, bounds, substitution, units, and inverse operations to validate results.
Explain why a valid procedure applies beyond one example.
Communicate an informal proof or general mathematical argument in a logically ordered form.

Sampling, Population Comparison, and Statistical Arguments

51 competencies

Formulate a mathematical problem from a real situation involving rational quantities, proportions, equations, inequalities, probability, geometry, statistics, or finance.
Define variables and units.
Identify constraints, assumptions, and relevant information.
Determine whether a proportional model is appropriate.
Represent a situation using equations, inequalities, proportions, ratio tables, graphs, diagrams, simulations, or statistical displays.
Solve the mathematical model accurately.
Interpret the constant of proportionality, coefficients, inequality bounds, probability values, or statistical summaries in context.
Interpret intermediate and final results with units.
Validate a model using observed data, substitution, alternative representations, estimation, or limiting cases.
Evaluate whether a model’s assumptions remain valid across the stated range.
Revise a model when the relationship is nonproportional, a constraint is missing, a sample is biased, or a simulation is invalid.
Compare two possible models and explain which better fits the situation and evidence.
State the practical limitations of a simplified linear or proportional model.
Conduct or interpret a probability experiment.
Calculate experimental probability from relative frequency.
Compare theoretical and experimental probability.
Explain why experimental results vary across samples.
Explain why experimental probability often becomes more stable over many trials.
Design or use a simulation representing a simple probability situation.
Determine whether a simulation fairly represents the intended probabilities.
Construct a logical mathematical argument from definitions, properties, equations, diagrams, and calculations.
Distinguish an example supporting a claim from a general argument establishing the claim.
Use a counterexample to disprove a universal claim.
Explain why one counterexample is sufficient to disprove an “always” statement.
Make a conjecture from patterns or examples.
Test a conjecture using additional cases, boundary cases, and counterexamples.
Use rational-number properties to justify transformations.
Justify each equality-preserving or order-preserving step in an equation or inequality.
Identify assumptions in a proportional, geometric, statistical, probability, or financial argument.
Critique a mathematical solution for validity, clarity, completeness, and efficiency.
Identify circular reasoning, unsupported generalization, or reliance on a misleading diagram at an introductory level.
Compare arithmetic, algebraic, graphical, and proportional methods.
Use estimation, bounds, substitution, units, and inverse operations to validate results.
Explain why a valid procedure applies beyond one example.
Communicate an informal proof or general mathematical argument in a logically ordered form.
Distinguish a population from a sample.
Explain why a sample may be used to learn about a population.
Identify simple random, convenience, voluntary-response, and systematic sampling methods at an introductory level.
Explain how selection method may introduce bias.
Identify undercoverage, nonresponse, and leading-question concerns in supported examples.
Determine whether a sample is reasonably representative of a stated population.
Use sample statistics to make a cautious estimate about a population.
Compare two populations informally using samples.
Compare measures of center between two samples.
Compare measures of spread between two samples.
Compare visual distributions using dot plots, histograms, or box plots.
Describe overlap and separation between two distributions.
Explain why different random samples from the same population may produce different statistics.
Use repeated samples or simulation to examine sampling variability.
Distinguish a claim supported by sample evidence from a claim that extends beyond the evidence.
Critique a survey or study for population definition, sample method, wording, size, and possible bias without conducting formal inference.

Percent Applications, Simple Interest, and Integrated Modeling

74 competencies

Solve multi-step addition and subtraction equations.
Solve multi-step multiplication and division equations.
Solve equations containing parentheses using the distributive property.
Solve equations with positive or negative rational coefficients.
Solve equations whose solutions are rational numbers.
Check an equation solution by substitution.
Write a multi-step equation representing a real situation.
Interpret each term and coefficient in a contextual equation.
Compare algebraic and arithmetic solution methods.
Identify the first equality-breaking step in an incorrect equation solution.
Calculate markup as a percentage of cost.
Calculate a selling price after markup.
Calculate discount amount and sale price.
Calculate tax, tip, service charge, or another percentage addition.
Calculate a final price involving markup, discount, tax, or a fixed charge in supported multi-step cases.
Calculate percent increase.
Calculate percent decrease.
Determine an original amount from a final amount and a single percentage increase or decrease in supported cases.
Compare percentage changes involving different original amounts.
Calculate simple interest using principal, annual or period rate, and time.
Use I = Prt in supported simple-interest calculations.
Calculate principal, rate, or time when the other simple-interest quantities are known in supported cases.
Calculate the final amount under simple interest.
Represent simple-interest growth using a table, graph, or proportional-style equation.
Compare two simple-interest arrangements mathematically.
Explain why simple-interest growth is additive rather than compound.
Solve multi-step financial problems involving budgets, markup, discount, tax, and simple interest.
Validate financial calculations using estimation, equations, percentages, and constraints.
Distinguish financial calculation from personal judgment about saving, borrowing, spending, or risk.
Formulate a mathematical problem from a real situation involving rational quantities, proportions, equations, inequalities, probability, geometry, statistics, or finance.
Define variables and units.
Identify constraints, assumptions, and relevant information.
Determine whether a proportional model is appropriate.
Represent a situation using equations, inequalities, proportions, ratio tables, graphs, diagrams, simulations, or statistical displays.
Solve the mathematical model accurately.
Interpret the constant of proportionality, coefficients, inequality bounds, probability values, or statistical summaries in context.
Interpret intermediate and final results with units.
Validate a model using observed data, substitution, alternative representations, estimation, or limiting cases.
Evaluate whether a model’s assumptions remain valid across the stated range.
Revise a model when the relationship is nonproportional, a constraint is missing, a sample is biased, or a simulation is invalid.
Compare two possible models and explain which better fits the situation and evidence.
State the practical limitations of a simplified linear or proportional model.
Determine whether two quantities have a proportional relationship.
Explain that a proportional relationship has a constant multiplicative comparison.
Calculate the constant of proportionality from a table.
Calculate the constant of proportionality from a verbal description or pair of corresponding values.
Interpret the constant of proportionality as a unit rate.
Write an equation of the form y = kx for a proportional relationship.
Use a proportional equation to determine an unknown value.
Construct a table from a proportional equation.
Graph a proportional relationship in the coordinate plane.
Explain why the graph of a proportional relationship passes through the origin.
Identify the point (1, k) as representing the unit rate in a proportional graph where the scales permit.
Determine whether a graph represents a proportional relationship.
Compare two proportional relationships represented in tables, graphs, equations, or verbal descriptions.
Solve proportions using equivalent ratios, unit rates, equations, or cross-products with explanation.
Distinguish proportional relationships from additive, fixed-fee, curved, or otherwise nonproportional relationships.
Apply proportional reasoning to scale, rates, percentages, measurement, and financial contexts.
State and evaluate assumptions required by a proportional model.
Construct a logical mathematical argument from definitions, properties, equations, diagrams, and calculations.
Distinguish an example supporting a claim from a general argument establishing the claim.
Use a counterexample to disprove a universal claim.
Explain why one counterexample is sufficient to disprove an “always” statement.
Make a conjecture from patterns or examples.
Test a conjecture using additional cases, boundary cases, and counterexamples.
Use rational-number properties to justify transformations.
Justify each equality-preserving or order-preserving step in an equation or inequality.
Identify assumptions in a proportional, geometric, statistical, probability, or financial argument.
Critique a mathematical solution for validity, clarity, completeness, and efficiency.
Identify circular reasoning, unsupported generalization, or reliance on a misleading diagram at an introductory level.
Compare arithmetic, algebraic, graphical, and proportional methods.
Use estimation, bounds, substitution, units, and inverse operations to validate results.
Explain why a valid procedure applies beyond one example.
Communicate an informal proof or general mathematical argument in a logically ordered form.