Curriculum / Global Curriculum / Grade 5

Grade 5 Mathematics.

10textbooks

250lessons

167competencies

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The whole year, lesson by lesson.

Textbook 1Whole Numbers to Millions and Negative Number Lines

4 chapters · 25 lessons

Chapter 1

Place Value through Millions

Chapter 2

Comparing, Ordering, Rounding, and Estimating Large Numbers

Chapter 3

Multi-Step Whole-Number Operations

Chapter 4

Negative Numbers as Positions

Textbook 2Unlike-Denominator Fraction Addition and Subtraction

4 chapters · 25 lessons

Chapter 1

Common Fractional Units

Chapter 2

Adding Fractions with Unlike Denominators

Chapter 3

Subtracting Fractions and Mixed Numbers

Chapter 4

Multi-Step Fraction Addition and Subtraction

Textbook 3Fraction Multiplication and Introductory Fraction Division

4 chapters · 25 lessons

Chapter 1

Whole Numbers Multiplied by Fractions

Chapter 2

Multiplying a Fraction by a Fraction

Chapter 3

Fraction Multiplication in Measurement and Area

Chapter 4

Introductory Fraction Division

Textbook 4Decimal Place Value, Addition, and Subtraction

4 chapters · 25 lessons

Chapter 1

Decimal Structure, Equivalence, and Number Lines

Chapter 2

Rounding and Estimating Decimals

Chapter 3

Adding Decimals

Chapter 4

Subtracting Decimals

Textbook 5Decimal Multiplication and Division

4 chapters · 25 lessons

Chapter 1

Multiplying a Decimal by a Whole Number

Chapter 2

Multiplying Two Decimals

Chapter 3

Dividing a Decimal by a Whole Number

Chapter 4

Dividing by a Decimal

Textbook 6Patterns, Tables, Ordered Pairs, and the First-Quadrant Coordinate Plane

4 chapters · 25 lessons

Chapter 1

Related Numerical Patterns and Tables

Chapter 2

Numerical Expressions and Operation Structure

Chapter 3

Ordered Pairs and the First Quadrant

Chapter 4

Graphing Rules and Coordinate Models

Textbook 7Shape Hierarchies and Areas of More Shapes

4 chapters · 25 lessons

Chapter 1

Triangle and Quadrilateral Hierarchies

Chapter 2

Area through Decomposition and Rearrangement

Chapter 3

Composite Rectilinear Area

Chapter 4

Geometry on the Coordinate Plane

Textbook 8Volume, Metric Conversion, and Applied Measurement

4 chapters · 25 lessons

Chapter 1

Volume as Unit-Cube Filling

Chapter 2

Equal Layers and Rectangular Arrangements

Chapter 3

Metric Conversions with Fractions and Decimals

Chapter 4

Multi-Step Applied Measurement

Textbook 9Line Graphs, Fractional and Decimal Data, and the Mean

4 chapters · 25 lessons

Chapter 1

Constructing and Reading Line Graphs

Chapter 2

Trends, Comparison, and Fractional or Decimal Data

Chapter 3

Mean as Fair Share

Chapter 4

Choosing and Evaluating Data Representations

Textbook 10Unit Pricing, Budgets, Spend–Save Mathematics, and Integrated Modeling

4 chapters · 25 lessons

Chapter 1

Unit Pricing and Package Comparison

Chapter 2

Budgets and Spend–Save Allocations

Chapter 3

Financial Tables, Repeated Periods, and Multi-Step Problems

Chapter 4

Full Grade 5 Mathematical Modeling Challenges

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.

See how a lesson works

Every competency, covered.

The 167 competencies set for Grade 5 Mathematics, and every one has lessons behind it.

Whole Numbers to Millions and Negative Number Lines

44 competencies

Identify the real quantities and relationships needed to formulate a mathematical problem.
Distinguish relevant information from distracting or unnecessary information.
State simple assumptions required by a model.
Represent a real situation using numerical expressions, tables, diagrams, coordinate points, graphs, measurement relationships, or financial calculations.
Use a table to organize how one quantity changes with another.
Solve the mathematical representation accurately.
Interpret intermediate and final results in the original context.
Check the result using estimation, units, benchmarks, graph behavior, or an alternative model.
Determine whether a model’s output is reasonable and usable.
Revise a model when assumptions, operations, units, or representations do not fit.
Explain one limitation of a simplified model.
Read and write whole numbers through millions.
Explain the value of each digit in a whole number through the millions place.
Represent large numbers in standard, expanded, word, and place-value forms.
Compose and decompose large whole numbers flexibly.
Compare and order whole numbers through millions.
Round whole numbers to a stated place.
Estimate sums, differences, products, and quotients involving large whole numbers.
Apply whole-number operations accurately in multi-step contexts.
Select an efficient mental, visual, or written whole-number strategy.
Use estimation and inverse operations to check whole-number calculations.
Interpret zero as a reference point separating positive and negative positions on a number line.
Locate negative whole numbers on horizontal and vertical number lines.
Compare and order positive and negative whole numbers using their positions.
Identify pairs of numbers located the same distance from zero on opposite sides.
Interpret negative numbers in contexts such as temperature, elevation, depth, and change relative to a reference level.
Explain why a number farther left on a number line is less, including when both numbers are negative.
Determine distance between a number and zero using number-line counting without formal absolute-value notation.
Avoid applying unsupported negative-number operation rules before they have been formally developed.
Identify relevant quantities, relationships, constraints, assumptions, and unknowns in a multi-step problem.
Select and justify a strategy from several available representations or procedures.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Break a complex problem into manageable intermediate questions.
Explain the meaning of each intermediate result.
Estimate before or after calculating.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.

Unlike-Denominator Fraction Addition and Subtraction

36 competencies

Use equivalent fractions to express unlike fractional parts in a common unit.
Identify a suitable common denominator in supported cases.
Add fractions with unlike denominators.
Subtract fractions with unlike denominators.
Explain unlike-denominator operations through fraction strips, area models, number lines, or bar models.
Add and subtract mixed numbers in supported contexts.
Regroup one whole as fractional parts when subtracting mixed numbers.
Simplify a fraction result when an evident common factor is available, without requiring advanced factorization.
Estimate fraction sums and differences using benchmarks.
Solve multi-step problems involving unlike-denominator fraction addition and subtraction.
Identify the real quantities and relationships needed to formulate a mathematical problem.
Distinguish relevant information from distracting or unnecessary information.
State simple assumptions required by a model.
Represent a real situation using numerical expressions, tables, diagrams, coordinate points, graphs, measurement relationships, or financial calculations.
Use a table to organize how one quantity changes with another.
Solve the mathematical representation accurately.
Interpret intermediate and final results in the original context.
Check the result using estimation, units, benchmarks, graph behavior, or an alternative model.
Determine whether a model’s output is reasonable and usable.
Revise a model when assumptions, operations, units, or representations do not fit.
Explain one limitation of a simplified model.
Identify relevant quantities, relationships, constraints, assumptions, and unknowns in a multi-step problem.
Select and justify a strategy from several available representations or procedures.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Break a complex problem into manageable intermediate questions.
Explain the meaning of each intermediate result.
Estimate before or after calculating.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.

Fraction Multiplication and Introductory Fraction Division

38 competencies

Interpret multiplication of a fraction by a whole number as repeated addition or equal groups of a fractional quantity.
Calculate a whole number multiplied by a fraction.
Interpret multiplication of two fractions as finding part of a part.
Represent fraction-by-fraction multiplication using area models.
Multiply two proper fractions in supported cases.
Multiply a fraction by a mixed number or mixed number by a fraction in carefully supported cases using decomposition.
Explain fraction multiplication as scaling.
Predict whether multiplying by a fraction less than one will make a positive quantity smaller.
Estimate a fraction product using benchmarks and whole-number reasoning.
Divide a unit fraction by a whole number using visual models.
Divide a whole number by a unit fraction using measurement or grouping models.
Explain the meaning of the quotient in introductory fraction-division situations.
Check introductory fraction division through multiplication and visual reconstruction.
Solve multi-step fraction problems while preserving the meaning of the whole and the unit.
Identify the real quantities and relationships needed to formulate a mathematical problem.
Distinguish relevant information from distracting or unnecessary information.
State simple assumptions required by a model.
Represent a real situation using numerical expressions, tables, diagrams, coordinate points, graphs, measurement relationships, or financial calculations.
Use a table to organize how one quantity changes with another.
Solve the mathematical representation accurately.
Interpret intermediate and final results in the original context.
Check the result using estimation, units, benchmarks, graph behavior, or an alternative model.
Determine whether a model’s output is reasonable and usable.
Revise a model when assumptions, operations, units, or representations do not fit.
Explain one limitation of a simplified model.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Break a complex problem into manageable intermediate questions.
Explain the meaning of each intermediate result.
Estimate before or after calculating.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.

Decimal Place Value, Addition, and Subtraction

26 competencies

Read and write decimals used in Grade 5 calculations.
Explain decimal values through ones, tenths, hundredths, and supported smaller place-value units where required.
Represent decimals using base-ten models, grids, number lines, money, and measurement.
Compare and order decimals using place value.
Round decimals to a stated whole-number or decimal place.
Estimate decimal sums, differences, products, and quotients.
Add decimals by aligning like place values.
Subtract decimals by aligning like place values.
Regroup across the decimal point using place-value reasoning.
Explain why decimal alignment is based on place value rather than the right edge of the numerals.
Solve multi-step problems involving all four decimal operations.
Select appropriate precision for a decimal answer according to the problem context.
Select and justify a strategy from several available representations or procedures.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Break a complex problem into manageable intermediate questions.
Explain the meaning of each intermediate result.
Estimate before or after calculating.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.

Decimal Multiplication and Division

36 competencies

Multiply a decimal by a whole number.
Multiply two decimals in supported cases.
Use an area model or place-value reasoning to explain decimal multiplication.
Determine a reasonable decimal-point position in a product using estimation.
Divide a decimal by a whole number.
Divide a whole number by a decimal in supported measurement contexts.
Divide a decimal by a decimal in supported terminating cases.
Use equivalent scaling to transform a decimal divisor into a whole-number divisor in supported cases.
Interpret a decimal quotient according to the context.
Check decimal division through multiplication and estimation.
Solve multi-step problems involving all four decimal operations.
Select appropriate precision for a decimal answer according to the problem context.
Identify the real quantities and relationships needed to formulate a mathematical problem.
Distinguish relevant information from distracting or unnecessary information.
State simple assumptions required by a model.
Represent a real situation using numerical expressions, tables, diagrams, coordinate points, graphs, measurement relationships, or financial calculations.
Use a table to organize how one quantity changes with another.
Solve the mathematical representation accurately.
Interpret intermediate and final results in the original context.
Check the result using estimation, units, benchmarks, graph behavior, or an alternative model.
Determine whether a model’s output is reasonable and usable.
Revise a model when assumptions, operations, units, or representations do not fit.
Explain one limitation of a simplified model.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Break a complex problem into manageable intermediate questions.
Explain the meaning of each intermediate result.
Estimate before or after calculating.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.

Patterns, Tables, Ordered Pairs, and the First-Quadrant Coordinate Plane

36 competencies

Identify the real quantities and relationships needed to formulate a mathematical problem.
Distinguish relevant information from distracting or unnecessary information.
State simple assumptions required by a model.
Represent a real situation using numerical expressions, tables, diagrams, coordinate points, graphs, measurement relationships, or financial calculations.
Use a table to organize how one quantity changes with another.
Solve the mathematical representation accurately.
Interpret intermediate and final results in the original context.
Check the result using estimation, units, benchmarks, graph behavior, or an alternative model.
Determine whether a model’s output is reasonable and usable.
Revise a model when assumptions, operations, units, or representations do not fit.
Explain one limitation of a simplified model.
Generate numerical patterns from one- and two-operation rules.
Organize input and output values in a table.
Identify relationships between corresponding values in two numerical patterns.
Generate two related numerical patterns from separate rules.
Identify features that arise from the relationship between two rules.
Represent a numerical calculation using a numerical expression containing parentheses where appropriate.
Interpret a numerical expression without necessarily evaluating it immediately.
Evaluate supported numerical expressions using a clearly stated operation order.
Generate an ordered pair from a table or rule.
Explain that the first coordinate gives horizontal position and the second gives vertical position.
Plot and identify ordered pairs in the first quadrant.
Generate and graph ordered pairs from a numerical rule.
Interpret a plotted point according to the quantities represented on the axes.
Compare two related rules using tables and coordinate graphs.
Identify the limits of a rule when its assumptions no longer hold.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.

Shape Hierarchies and Areas of More Shapes

49 competencies

Classify two-dimensional shapes using side, angle, parallel-line, perpendicular-line, and symmetry properties.
Organize triangles into side-based and angle-based classification hierarchies.
Organize quadrilaterals into an inclusive hierarchy.
Explain why every square is a rectangle and rhombus but not every rectangle or rhombus is a square.
Use defining properties rather than appearance to classify a figure.
Identify a counterexample to an incorrect geometric classification claim.
Decompose a nonrectangular figure into familiar pieces.
Rearrange pieces without changing total area.
Determine area of supported nonrectangular figures using unit squares, decomposition, or rearrangement.
Determine area of composite rectilinear figures.
Find missing side lengths in composite-area problems.
Compare figures with equal area and different structure.
Plot vertices of a figure in the first quadrant.
Identify horizontal and vertical side lengths from coordinate differences in the first quadrant.
Classify or analyze simple coordinate-plane figures.
Use symmetry and geometric properties to complete a coordinate-plane figure in supported cases.
Calculate areas of supported nonrectangular and composite figures through decomposition.
Identify the real quantities and relationships needed to formulate a mathematical problem.
Distinguish relevant information from distracting or unnecessary information.
State simple assumptions required by a model.
Represent a real situation using numerical expressions, tables, diagrams, coordinate points, graphs, measurement relationships, or financial calculations.
Use a table to organize how one quantity changes with another.
Solve the mathematical representation accurately.
Interpret intermediate and final results in the original context.
Check the result using estimation, units, benchmarks, graph behavior, or an alternative model.
Determine whether a model’s output is reasonable and usable.
Revise a model when assumptions, operations, units, or representations do not fit.
Explain one limitation of a simplified model.
Generate an ordered pair from a table or rule.
Explain that the first coordinate gives horizontal position and the second gives vertical position.
Plot and identify ordered pairs in the first quadrant.
Generate and graph ordered pairs from a numerical rule.
Interpret a plotted point according to the quantities represented on the axes.
Compare two related rules using tables and coordinate graphs.
Identify relevant quantities, relationships, constraints, assumptions, and unknowns in a multi-step problem.
Select and justify a strategy from several available representations or procedures.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Break a complex problem into manageable intermediate questions.
Explain the meaning of each intermediate result.
Estimate before or after calculating.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.

Volume, Metric Conversion, and Applied Measurement

44 competencies

Explain volume as the amount of three-dimensional space occupied.
Use a unit cube as a standard unit of volume.
Measure volume by counting nonoverlapping unit cubes that fill a solid.
Explain that complete volume measurement requires cubes filling all layers without gaps.
Determine the number of cubes in one layer.
Determine total volume from equal layers of unit cubes.
Use multiplication to calculate cube counts in supported rectangular arrangements without making the formal prism formula the sole method.
Find a missing number of layers, cubes per layer, or dimension from volume information.
Distinguish length, area, and volume and their units.
Calculate areas of supported nonrectangular and composite figures through decomposition.
Convert measurements within metric length units.
Convert measurements within metric mass units.
Convert measurements within metric capacity or liquid-volume units.
Convert units involving decimals and fractions in supported cases.
Select a common unit before combining or comparing measurements.
Solve multi-step measurement problems involving area, volume, elapsed time, and conversion.
Explain whether a measurement result is reasonable for the object or context.
Interpret precision shown on a measuring instrument without formal error analysis.
Identify the real quantities and relationships needed to formulate a mathematical problem.
Distinguish relevant information from distracting or unnecessary information.
State simple assumptions required by a model.
Represent a real situation using numerical expressions, tables, diagrams, coordinate points, graphs, measurement relationships, or financial calculations.
Use a table to organize how one quantity changes with another.
Solve the mathematical representation accurately.
Interpret intermediate and final results in the original context.
Check the result using estimation, units, benchmarks, graph behavior, or an alternative model.
Determine whether a model’s output is reasonable and usable.
Revise a model when assumptions, operations, units, or representations do not fit.
Explain one limitation of a simplified model.
Identify relevant quantities, relationships, constraints, assumptions, and unknowns in a multi-step problem.
Select and justify a strategy from several available representations or procedures.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Break a complex problem into manageable intermediate questions.
Explain the meaning of each intermediate result.
Estimate before or after calculating.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.

Line Graphs, Fractional and Decimal Data, and the Mean

42 competencies

Identify situations involving change over ordered time or another continuous sequence.
Construct a line graph using ordered data values.
Select and label appropriate horizontal and vertical axes.
Select a consistent numerical scale.
Interpret individual points and line segments on a line graph.
Describe increases, decreases, constant intervals, peaks, low points, and overall trends.
Compare two related lines on one graph in supported cases.
Interpret data values containing fractions and decimals.
Translate data among tables, line plots, bar graphs, and line graphs.
Explain why connecting points is meaningful only when the horizontal axis has an ordered continuous interpretation.
Explain the mean as redistributing a total equally among all data values.
Calculate the mean of a small whole-number or supported decimal dataset.
Determine a missing data value from a known mean and total in supported cases.
Explain that the mean need not be one of the observed values.
Compare the mean with the mode, range, and visible data pattern informally.
Identify misleading scales, unsupported trend claims, inappropriate connected lines, or omitted data.
Identify the real quantities and relationships needed to formulate a mathematical problem.
Distinguish relevant information from distracting or unnecessary information.
State simple assumptions required by a model.
Represent a real situation using numerical expressions, tables, diagrams, coordinate points, graphs, measurement relationships, or financial calculations.
Use a table to organize how one quantity changes with another.
Solve the mathematical representation accurately.
Interpret intermediate and final results in the original context.
Check the result using estimation, units, benchmarks, graph behavior, or an alternative model.
Determine whether a model’s output is reasonable and usable.
Revise a model when assumptions, operations, units, or representations do not fit.
Explain one limitation of a simplified model.
Identify relevant quantities, relationships, constraints, assumptions, and unknowns in a multi-step problem.
Select and justify a strategy from several available representations or procedures.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Break a complex problem into manageable intermediate questions.
Explain the meaning of each intermediate result.
Estimate before or after calculating.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.

Unit Pricing, Budgets, Spend–Save Mathematics, and Integrated Modeling

71 competencies

Identify situations involving change over ordered time or another continuous sequence.
Construct a line graph using ordered data values.
Select and label appropriate horizontal and vertical axes.
Select a consistent numerical scale.
Interpret individual points and line segments on a line graph.
Describe increases, decreases, constant intervals, peaks, low points, and overall trends.
Compare two related lines on one graph in supported cases.
Interpret data values containing fractions and decimals.
Translate data among tables, line plots, bar graphs, and line graphs.
Explain why connecting points is meaningful only when the horizontal axis has an ordered continuous interpretation.
Divide a decimal by a whole number.
Divide a whole number by a decimal in supported measurement contexts.
Divide a decimal by a decimal in supported terminating cases.
Use equivalent scaling to transform a decimal divisor into a whole-number divisor in supported cases.
Interpret a decimal quotient according to the context.
Check decimal division through multiplication and estimation.
Solve multi-step problems involving all four decimal operations.
Select appropriate precision for a decimal answer according to the problem context.
Calculate total cost, change, and remaining money in multi-step purchase situations.
Calculate unit price using whole-number or supported decimal division.
Compare unit prices for equal or different package quantities.
Compare total cost and cost per unit as different mathematical measures.
Use multiplication to calculate the cost of a selected quantity from a unit price.
Construct a budget with income or available amount, spending categories, a planned saving amount, and remaining balance.
Calculate the amount available for spending after a stated saving amount is set aside.
Calculate the saving amount remaining after stated expenses.
Compare two mathematically possible spend–save allocations without prescribing which personal choice is morally or behaviorally correct.
Revise quantities or prices so a plan fits a stated budget.
Use tables or numerical expressions to model repeated spending or saving over several equal periods without interest.
Validate financial calculations using estimation, inverse operations, unit labels, and budget constraints.
Identify the real quantities and relationships needed to formulate a mathematical problem.
Distinguish relevant information from distracting or unnecessary information.
State simple assumptions required by a model.
Represent a real situation using numerical expressions, tables, diagrams, coordinate points, graphs, measurement relationships, or financial calculations.
Use a table to organize how one quantity changes with another.
Solve the mathematical representation accurately.
Interpret intermediate and final results in the original context.
Check the result using estimation, units, benchmarks, graph behavior, or an alternative model.
Determine whether a model’s output is reasonable and usable.
Revise a model when assumptions, operations, units, or representations do not fit.
Explain one limitation of a simplified model.
Generate numerical patterns from one- and two-operation rules.
Organize input and output values in a table.
Identify relationships between corresponding values in two numerical patterns.
Generate two related numerical patterns from separate rules.
Identify features that arise from the relationship between two rules.
Represent a numerical calculation using a numerical expression containing parentheses where appropriate.
Interpret a numerical expression without necessarily evaluating it immediately.
Evaluate supported numerical expressions using a clearly stated operation order.
Generate an ordered pair from a table or rule.
Explain that the first coordinate gives horizontal position and the second gives vertical position.
Plot and identify ordered pairs in the first quadrant.
Generate and graph ordered pairs from a numerical rule.
Interpret a plotted point according to the quantities represented on the axes.
Compare two related rules using tables and coordinate graphs.
Identify the limits of a rule when its assumptions no longer hold.
Identify relevant quantities, relationships, constraints, assumptions, and unknowns in a multi-step problem.
Select and justify a strategy from several available representations or procedures.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Break a complex problem into manageable intermediate questions.
Explain the meaning of each intermediate result.
Estimate before or after calculating.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.

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The whole year, lesson by lesson.

Textbook 1Whole Numbers to Millions and Negative Number Lines

4 chapters · 25 lessons

Chapter 1

Place Value through Millions

Chapter 2

Comparing, Ordering, Rounding, and Estimating Large Numbers

Chapter 3

Multi-Step Whole-Number Operations

Chapter 4

Negative Numbers as Positions

Textbook 2Unlike-Denominator Fraction Addition and Subtraction

4 chapters · 25 lessons

Chapter 1

Common Fractional Units

Chapter 2

Adding Fractions with Unlike Denominators

Chapter 3

Subtracting Fractions and Mixed Numbers

Chapter 4

Multi-Step Fraction Addition and Subtraction

Textbook 3Fraction Multiplication and Introductory Fraction Division

4 chapters · 25 lessons

Chapter 1

Whole Numbers Multiplied by Fractions

Chapter 2

Multiplying a Fraction by a Fraction

Chapter 3

Fraction Multiplication in Measurement and Area

Chapter 4

Introductory Fraction Division

Textbook 4Decimal Place Value, Addition, and Subtraction

4 chapters · 25 lessons

Chapter 1

Decimal Structure, Equivalence, and Number Lines

Chapter 2

Rounding and Estimating Decimals

Chapter 3

Adding Decimals

Chapter 4

Subtracting Decimals

Textbook 5Decimal Multiplication and Division

4 chapters · 25 lessons

Chapter 1

Multiplying a Decimal by a Whole Number

Chapter 2

Multiplying Two Decimals

Chapter 3

Dividing a Decimal by a Whole Number

Chapter 4

Dividing by a Decimal

Textbook 6Patterns, Tables, Ordered Pairs, and the First-Quadrant Coordinate Plane

4 chapters · 25 lessons

Chapter 1

Related Numerical Patterns and Tables

Chapter 2

Numerical Expressions and Operation Structure

Chapter 3

Ordered Pairs and the First Quadrant

Chapter 4

Graphing Rules and Coordinate Models

Textbook 7Shape Hierarchies and Areas of More Shapes

4 chapters · 25 lessons

Chapter 1

Triangle and Quadrilateral Hierarchies

Chapter 2

Area through Decomposition and Rearrangement

Chapter 3

Composite Rectilinear Area

Chapter 4

Geometry on the Coordinate Plane

Textbook 8Volume, Metric Conversion, and Applied Measurement

4 chapters · 25 lessons

Chapter 1

Volume as Unit-Cube Filling

Chapter 2

Equal Layers and Rectangular Arrangements

Chapter 3

Metric Conversions with Fractions and Decimals

Chapter 4

Multi-Step Applied Measurement

Textbook 9Line Graphs, Fractional and Decimal Data, and the Mean

4 chapters · 25 lessons

Chapter 1

Constructing and Reading Line Graphs

Chapter 2

Trends, Comparison, and Fractional or Decimal Data

Chapter 3

Mean as Fair Share

Chapter 4

Choosing and Evaluating Data Representations

Textbook 10Unit Pricing, Budgets, Spend–Save Mathematics, and Integrated Modeling

4 chapters · 25 lessons

Chapter 1

Unit Pricing and Package Comparison

Chapter 2

Budgets and Spend–Save Allocations

Chapter 3

Financial Tables, Repeated Periods, and Multi-Step Problems

Chapter 4

Full Grade 5 Mathematical Modeling Challenges

Every competency, covered.

The 167 competencies set for Grade 5 Mathematics, and every one has lessons behind it.

Whole Numbers to Millions and Negative Number Lines

44 competencies

Identify the real quantities and relationships needed to formulate a mathematical problem.
Distinguish relevant information from distracting or unnecessary information.
State simple assumptions required by a model.
Represent a real situation using numerical expressions, tables, diagrams, coordinate points, graphs, measurement relationships, or financial calculations.
Use a table to organize how one quantity changes with another.
Solve the mathematical representation accurately.
Interpret intermediate and final results in the original context.
Check the result using estimation, units, benchmarks, graph behavior, or an alternative model.
Determine whether a model’s output is reasonable and usable.
Revise a model when assumptions, operations, units, or representations do not fit.
Explain one limitation of a simplified model.
Read and write whole numbers through millions.
Explain the value of each digit in a whole number through the millions place.
Represent large numbers in standard, expanded, word, and place-value forms.
Compose and decompose large whole numbers flexibly.
Compare and order whole numbers through millions.
Round whole numbers to a stated place.
Estimate sums, differences, products, and quotients involving large whole numbers.
Apply whole-number operations accurately in multi-step contexts.
Select an efficient mental, visual, or written whole-number strategy.
Use estimation and inverse operations to check whole-number calculations.
Interpret zero as a reference point separating positive and negative positions on a number line.
Locate negative whole numbers on horizontal and vertical number lines.
Compare and order positive and negative whole numbers using their positions.
Identify pairs of numbers located the same distance from zero on opposite sides.
Interpret negative numbers in contexts such as temperature, elevation, depth, and change relative to a reference level.
Explain why a number farther left on a number line is less, including when both numbers are negative.
Determine distance between a number and zero using number-line counting without formal absolute-value notation.
Avoid applying unsupported negative-number operation rules before they have been formally developed.
Identify relevant quantities, relationships, constraints, assumptions, and unknowns in a multi-step problem.
Select and justify a strategy from several available representations or procedures.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Break a complex problem into manageable intermediate questions.
Explain the meaning of each intermediate result.
Estimate before or after calculating.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.

Unlike-Denominator Fraction Addition and Subtraction

36 competencies

Use equivalent fractions to express unlike fractional parts in a common unit.
Identify a suitable common denominator in supported cases.
Add fractions with unlike denominators.
Subtract fractions with unlike denominators.
Explain unlike-denominator operations through fraction strips, area models, number lines, or bar models.
Add and subtract mixed numbers in supported contexts.
Regroup one whole as fractional parts when subtracting mixed numbers.
Simplify a fraction result when an evident common factor is available, without requiring advanced factorization.
Estimate fraction sums and differences using benchmarks.
Solve multi-step problems involving unlike-denominator fraction addition and subtraction.
Identify the real quantities and relationships needed to formulate a mathematical problem.
Distinguish relevant information from distracting or unnecessary information.
State simple assumptions required by a model.
Represent a real situation using numerical expressions, tables, diagrams, coordinate points, graphs, measurement relationships, or financial calculations.
Use a table to organize how one quantity changes with another.
Solve the mathematical representation accurately.
Interpret intermediate and final results in the original context.
Check the result using estimation, units, benchmarks, graph behavior, or an alternative model.
Determine whether a model’s output is reasonable and usable.
Revise a model when assumptions, operations, units, or representations do not fit.
Explain one limitation of a simplified model.
Identify relevant quantities, relationships, constraints, assumptions, and unknowns in a multi-step problem.
Select and justify a strategy from several available representations or procedures.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Break a complex problem into manageable intermediate questions.
Explain the meaning of each intermediate result.
Estimate before or after calculating.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.

Fraction Multiplication and Introductory Fraction Division

38 competencies

Interpret multiplication of a fraction by a whole number as repeated addition or equal groups of a fractional quantity.
Calculate a whole number multiplied by a fraction.
Interpret multiplication of two fractions as finding part of a part.
Represent fraction-by-fraction multiplication using area models.
Multiply two proper fractions in supported cases.
Multiply a fraction by a mixed number or mixed number by a fraction in carefully supported cases using decomposition.
Explain fraction multiplication as scaling.
Predict whether multiplying by a fraction less than one will make a positive quantity smaller.
Estimate a fraction product using benchmarks and whole-number reasoning.
Divide a unit fraction by a whole number using visual models.
Divide a whole number by a unit fraction using measurement or grouping models.
Explain the meaning of the quotient in introductory fraction-division situations.
Check introductory fraction division through multiplication and visual reconstruction.
Solve multi-step fraction problems while preserving the meaning of the whole and the unit.
Identify the real quantities and relationships needed to formulate a mathematical problem.
Distinguish relevant information from distracting or unnecessary information.
State simple assumptions required by a model.
Represent a real situation using numerical expressions, tables, diagrams, coordinate points, graphs, measurement relationships, or financial calculations.
Use a table to organize how one quantity changes with another.
Solve the mathematical representation accurately.
Interpret intermediate and final results in the original context.
Check the result using estimation, units, benchmarks, graph behavior, or an alternative model.
Determine whether a model’s output is reasonable and usable.
Revise a model when assumptions, operations, units, or representations do not fit.
Explain one limitation of a simplified model.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Break a complex problem into manageable intermediate questions.
Explain the meaning of each intermediate result.
Estimate before or after calculating.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.

Decimal Place Value, Addition, and Subtraction

26 competencies

Read and write decimals used in Grade 5 calculations.
Explain decimal values through ones, tenths, hundredths, and supported smaller place-value units where required.
Represent decimals using base-ten models, grids, number lines, money, and measurement.
Compare and order decimals using place value.
Round decimals to a stated whole-number or decimal place.
Estimate decimal sums, differences, products, and quotients.
Add decimals by aligning like place values.
Subtract decimals by aligning like place values.
Regroup across the decimal point using place-value reasoning.
Explain why decimal alignment is based on place value rather than the right edge of the numerals.
Solve multi-step problems involving all four decimal operations.
Select appropriate precision for a decimal answer according to the problem context.
Select and justify a strategy from several available representations or procedures.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Break a complex problem into manageable intermediate questions.
Explain the meaning of each intermediate result.
Estimate before or after calculating.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.

Decimal Multiplication and Division

36 competencies

Multiply a decimal by a whole number.
Multiply two decimals in supported cases.
Use an area model or place-value reasoning to explain decimal multiplication.
Determine a reasonable decimal-point position in a product using estimation.
Divide a decimal by a whole number.
Divide a whole number by a decimal in supported measurement contexts.
Divide a decimal by a decimal in supported terminating cases.
Use equivalent scaling to transform a decimal divisor into a whole-number divisor in supported cases.
Interpret a decimal quotient according to the context.
Check decimal division through multiplication and estimation.
Solve multi-step problems involving all four decimal operations.
Select appropriate precision for a decimal answer according to the problem context.
Identify the real quantities and relationships needed to formulate a mathematical problem.
Distinguish relevant information from distracting or unnecessary information.
State simple assumptions required by a model.
Represent a real situation using numerical expressions, tables, diagrams, coordinate points, graphs, measurement relationships, or financial calculations.
Use a table to organize how one quantity changes with another.
Solve the mathematical representation accurately.
Interpret intermediate and final results in the original context.
Check the result using estimation, units, benchmarks, graph behavior, or an alternative model.
Determine whether a model’s output is reasonable and usable.
Revise a model when assumptions, operations, units, or representations do not fit.
Explain one limitation of a simplified model.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Break a complex problem into manageable intermediate questions.
Explain the meaning of each intermediate result.
Estimate before or after calculating.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.

Patterns, Tables, Ordered Pairs, and the First-Quadrant Coordinate Plane

36 competencies

Identify the real quantities and relationships needed to formulate a mathematical problem.
Distinguish relevant information from distracting or unnecessary information.
State simple assumptions required by a model.
Represent a real situation using numerical expressions, tables, diagrams, coordinate points, graphs, measurement relationships, or financial calculations.
Use a table to organize how one quantity changes with another.
Solve the mathematical representation accurately.
Interpret intermediate and final results in the original context.
Check the result using estimation, units, benchmarks, graph behavior, or an alternative model.
Determine whether a model’s output is reasonable and usable.
Revise a model when assumptions, operations, units, or representations do not fit.
Explain one limitation of a simplified model.
Generate numerical patterns from one- and two-operation rules.
Organize input and output values in a table.
Identify relationships between corresponding values in two numerical patterns.
Generate two related numerical patterns from separate rules.
Identify features that arise from the relationship between two rules.
Represent a numerical calculation using a numerical expression containing parentheses where appropriate.
Interpret a numerical expression without necessarily evaluating it immediately.
Evaluate supported numerical expressions using a clearly stated operation order.
Generate an ordered pair from a table or rule.
Explain that the first coordinate gives horizontal position and the second gives vertical position.
Plot and identify ordered pairs in the first quadrant.
Generate and graph ordered pairs from a numerical rule.
Interpret a plotted point according to the quantities represented on the axes.
Compare two related rules using tables and coordinate graphs.
Identify the limits of a rule when its assumptions no longer hold.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.

Shape Hierarchies and Areas of More Shapes

49 competencies

Classify two-dimensional shapes using side, angle, parallel-line, perpendicular-line, and symmetry properties.
Organize triangles into side-based and angle-based classification hierarchies.
Organize quadrilaterals into an inclusive hierarchy.
Explain why every square is a rectangle and rhombus but not every rectangle or rhombus is a square.
Use defining properties rather than appearance to classify a figure.
Identify a counterexample to an incorrect geometric classification claim.
Decompose a nonrectangular figure into familiar pieces.
Rearrange pieces without changing total area.
Determine area of supported nonrectangular figures using unit squares, decomposition, or rearrangement.
Determine area of composite rectilinear figures.
Find missing side lengths in composite-area problems.
Compare figures with equal area and different structure.
Plot vertices of a figure in the first quadrant.
Identify horizontal and vertical side lengths from coordinate differences in the first quadrant.
Classify or analyze simple coordinate-plane figures.
Use symmetry and geometric properties to complete a coordinate-plane figure in supported cases.
Calculate areas of supported nonrectangular and composite figures through decomposition.
Identify the real quantities and relationships needed to formulate a mathematical problem.
Distinguish relevant information from distracting or unnecessary information.
State simple assumptions required by a model.
Represent a real situation using numerical expressions, tables, diagrams, coordinate points, graphs, measurement relationships, or financial calculations.
Use a table to organize how one quantity changes with another.
Solve the mathematical representation accurately.
Interpret intermediate and final results in the original context.
Check the result using estimation, units, benchmarks, graph behavior, or an alternative model.
Determine whether a model’s output is reasonable and usable.
Revise a model when assumptions, operations, units, or representations do not fit.
Explain one limitation of a simplified model.
Generate an ordered pair from a table or rule.
Explain that the first coordinate gives horizontal position and the second gives vertical position.
Plot and identify ordered pairs in the first quadrant.
Generate and graph ordered pairs from a numerical rule.
Interpret a plotted point according to the quantities represented on the axes.
Compare two related rules using tables and coordinate graphs.
Identify relevant quantities, relationships, constraints, assumptions, and unknowns in a multi-step problem.
Select and justify a strategy from several available representations or procedures.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Break a complex problem into manageable intermediate questions.
Explain the meaning of each intermediate result.
Estimate before or after calculating.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.

Volume, Metric Conversion, and Applied Measurement

44 competencies

Explain volume as the amount of three-dimensional space occupied.
Use a unit cube as a standard unit of volume.
Measure volume by counting nonoverlapping unit cubes that fill a solid.
Explain that complete volume measurement requires cubes filling all layers without gaps.
Determine the number of cubes in one layer.
Determine total volume from equal layers of unit cubes.
Use multiplication to calculate cube counts in supported rectangular arrangements without making the formal prism formula the sole method.
Find a missing number of layers, cubes per layer, or dimension from volume information.
Distinguish length, area, and volume and their units.
Calculate areas of supported nonrectangular and composite figures through decomposition.
Convert measurements within metric length units.
Convert measurements within metric mass units.
Convert measurements within metric capacity or liquid-volume units.
Convert units involving decimals and fractions in supported cases.
Select a common unit before combining or comparing measurements.
Solve multi-step measurement problems involving area, volume, elapsed time, and conversion.
Explain whether a measurement result is reasonable for the object or context.
Interpret precision shown on a measuring instrument without formal error analysis.
Identify the real quantities and relationships needed to formulate a mathematical problem.
Distinguish relevant information from distracting or unnecessary information.
State simple assumptions required by a model.
Represent a real situation using numerical expressions, tables, diagrams, coordinate points, graphs, measurement relationships, or financial calculations.
Use a table to organize how one quantity changes with another.
Solve the mathematical representation accurately.
Interpret intermediate and final results in the original context.
Check the result using estimation, units, benchmarks, graph behavior, or an alternative model.
Determine whether a model’s output is reasonable and usable.
Revise a model when assumptions, operations, units, or representations do not fit.
Explain one limitation of a simplified model.
Identify relevant quantities, relationships, constraints, assumptions, and unknowns in a multi-step problem.
Select and justify a strategy from several available representations or procedures.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Break a complex problem into manageable intermediate questions.
Explain the meaning of each intermediate result.
Estimate before or after calculating.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.

Line Graphs, Fractional and Decimal Data, and the Mean

42 competencies

Identify situations involving change over ordered time or another continuous sequence.
Construct a line graph using ordered data values.
Select and label appropriate horizontal and vertical axes.
Select a consistent numerical scale.
Interpret individual points and line segments on a line graph.
Describe increases, decreases, constant intervals, peaks, low points, and overall trends.
Compare two related lines on one graph in supported cases.
Interpret data values containing fractions and decimals.
Translate data among tables, line plots, bar graphs, and line graphs.
Explain why connecting points is meaningful only when the horizontal axis has an ordered continuous interpretation.
Explain the mean as redistributing a total equally among all data values.
Calculate the mean of a small whole-number or supported decimal dataset.
Determine a missing data value from a known mean and total in supported cases.
Explain that the mean need not be one of the observed values.
Compare the mean with the mode, range, and visible data pattern informally.
Identify misleading scales, unsupported trend claims, inappropriate connected lines, or omitted data.
Identify the real quantities and relationships needed to formulate a mathematical problem.
Distinguish relevant information from distracting or unnecessary information.
State simple assumptions required by a model.
Represent a real situation using numerical expressions, tables, diagrams, coordinate points, graphs, measurement relationships, or financial calculations.
Use a table to organize how one quantity changes with another.
Solve the mathematical representation accurately.
Interpret intermediate and final results in the original context.
Check the result using estimation, units, benchmarks, graph behavior, or an alternative model.
Determine whether a model’s output is reasonable and usable.
Revise a model when assumptions, operations, units, or representations do not fit.
Explain one limitation of a simplified model.
Identify relevant quantities, relationships, constraints, assumptions, and unknowns in a multi-step problem.
Select and justify a strategy from several available representations or procedures.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Break a complex problem into manageable intermediate questions.
Explain the meaning of each intermediate result.
Estimate before or after calculating.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.

Unit Pricing, Budgets, Spend–Save Mathematics, and Integrated Modeling

71 competencies

Identify situations involving change over ordered time or another continuous sequence.
Construct a line graph using ordered data values.
Select and label appropriate horizontal and vertical axes.
Select a consistent numerical scale.
Interpret individual points and line segments on a line graph.
Describe increases, decreases, constant intervals, peaks, low points, and overall trends.
Compare two related lines on one graph in supported cases.
Interpret data values containing fractions and decimals.
Translate data among tables, line plots, bar graphs, and line graphs.
Explain why connecting points is meaningful only when the horizontal axis has an ordered continuous interpretation.
Divide a decimal by a whole number.
Divide a whole number by a decimal in supported measurement contexts.
Divide a decimal by a decimal in supported terminating cases.
Use equivalent scaling to transform a decimal divisor into a whole-number divisor in supported cases.
Interpret a decimal quotient according to the context.
Check decimal division through multiplication and estimation.
Solve multi-step problems involving all four decimal operations.
Select appropriate precision for a decimal answer according to the problem context.
Calculate total cost, change, and remaining money in multi-step purchase situations.
Calculate unit price using whole-number or supported decimal division.
Compare unit prices for equal or different package quantities.
Compare total cost and cost per unit as different mathematical measures.
Use multiplication to calculate the cost of a selected quantity from a unit price.
Construct a budget with income or available amount, spending categories, a planned saving amount, and remaining balance.
Calculate the amount available for spending after a stated saving amount is set aside.
Calculate the saving amount remaining after stated expenses.
Compare two mathematically possible spend–save allocations without prescribing which personal choice is morally or behaviorally correct.
Revise quantities or prices so a plan fits a stated budget.
Use tables or numerical expressions to model repeated spending or saving over several equal periods without interest.
Validate financial calculations using estimation, inverse operations, unit labels, and budget constraints.
Identify the real quantities and relationships needed to formulate a mathematical problem.
Distinguish relevant information from distracting or unnecessary information.
State simple assumptions required by a model.
Represent a real situation using numerical expressions, tables, diagrams, coordinate points, graphs, measurement relationships, or financial calculations.
Use a table to organize how one quantity changes with another.
Solve the mathematical representation accurately.
Interpret intermediate and final results in the original context.
Check the result using estimation, units, benchmarks, graph behavior, or an alternative model.
Determine whether a model’s output is reasonable and usable.
Revise a model when assumptions, operations, units, or representations do not fit.
Explain one limitation of a simplified model.
Generate numerical patterns from one- and two-operation rules.
Organize input and output values in a table.
Identify relationships between corresponding values in two numerical patterns.
Generate two related numerical patterns from separate rules.
Identify features that arise from the relationship between two rules.
Represent a numerical calculation using a numerical expression containing parentheses where appropriate.
Interpret a numerical expression without necessarily evaluating it immediately.
Evaluate supported numerical expressions using a clearly stated operation order.
Generate an ordered pair from a table or rule.
Explain that the first coordinate gives horizontal position and the second gives vertical position.
Plot and identify ordered pairs in the first quadrant.
Generate and graph ordered pairs from a numerical rule.
Interpret a plotted point according to the quantities represented on the axes.
Compare two related rules using tables and coordinate graphs.
Identify the limits of a rule when its assumptions no longer hold.
Identify relevant quantities, relationships, constraints, assumptions, and unknowns in a multi-step problem.
Select and justify a strategy from several available representations or procedures.
Use numerical expressions, tables, coordinate graphs, fraction models, area models, bar models, line graphs, or measurement diagrams.
Break a complex problem into manageable intermediate questions.
Explain the meaning of each intermediate result.
Estimate before or after calculating.
Use inverse operations, benchmarks, substitution, or alternative representations to validate a result.
Compare two valid solution strategies for accuracy, clarity, and efficiency.
Evaluate another person’s mathematical reasoning.
Identify the first incorrect step in a solution.
Explain why an error affects later results.
Revise an incorrect solution while preserving any valid reasoning.
Make and test conjectures about numerical, fractional, decimal, coordinate, geometric, or data patterns.
Use examples and counterexamples informally to support or challenge a claim.
Communicate a sustained mathematical explanation using accurate notation, units, labels, and logical order.