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Skills available for Colorado fifth-grade math standards

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5.1 Number Sense, Properties, and Operations

5.1.1 The decimal number system describes place value patterns and relationships that are repeated in large and small numbers and forms the foundation for efficient algorithms
- 5.1.1.a Explain that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
  - Convert between standard and expanded form (5-A.1)
  - Place value (5-A.2)
  - Place values in decimal numbers (5-G.5)
  - 5.1.1.a.i Explain patterns in the number of zeros of the product when multiplying a number by powers of 10.
  - 5.1.1.a.ii Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.
    - Multiply a decimal by a power of ten (5-I.3)
    - Decimal division patterns over increasing place values (5-J.2)
  - 5.1.1.a.iii Use whole-number exponents to denote powers of 10.
- 5.1.1.b Read, write, and compare decimals to thousandths.
  - 5.1.1.b.i Read and write decimals to thousandths using base-ten numerals, number names, and expanded form.
  - 5.1.1.b.ii Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
- 5.1.1.c Use place value understanding to round decimals to any place.
  - Round decimals (5-G.10)
  - Estimate sums and differences of decimals using rounding (5-H.9)
- 5.1.1.d Convert like measurement units within a given measurement system.
  - 5.1.1.d.i Convert among different-sized standard measurement units within a given measurement system.
  - 5.1.1.d.ii Use measurement conversions in solving multi-step, real world problems.
5.1.2 Formulate, represent, and use algorithms with multi-digit whole numbers and decimals with flexibility, accuracy, and efficiency
- 5.1.2.a Fluently multiply multi-digit whole numbers using the standard algorithm.
- 5.1.2.b Find whole-number quotients of whole numbers.
  - 5.1.2.b.i Use strategies based on place value, the properties of operations, and/or the relationship between multiplication and division.
  - 5.1.2.b.ii Illustrate and explain calculations by using equations, rectangular arrays, and/or area models.
- 5.1.2.c Add, subtract, multiply, and divide decimals to hundredths.
  - 5.1.2.c.i Use concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
  - 5.1.2.c.ii Relate strategies to a written method and explain the reasoning used.
- 5.1.2.d Write and interpret numerical expressions.
  - 5.1.2.d.i Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
    - Evaluate numerical expressions (5-O.5)
  - 5.1.2.d.ii Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.
5.1.3 Formulate, represent, and use algorithms to add and subtract fractions with flexibility, accuracy, and efficiency
- 5.1.3.a Use equivalent fractions as a strategy to add and subtract fractions.
  - 5.1.3.a.i Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.
  - 5.1.3.a.ii Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions with like denominators.
  - 5.1.3.a.iii Solve word problems involving addition and subtraction of fractions referring to the same whole.
5.1.4 The concepts of multiplication and division can be applied to multiply and divide fractions
- 5.1.4.a Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b).
  - Fractions review (5-K.1)
  - Divide fractions by whole numbers (5-N.8)
- 5.1.4.b Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.
  - Divide 2-digit and 3-digit numbers by 2-digit numbers: word problems (5-D.15)
  - Fractions of a whole: word problems (5-K.2)
- 5.1.4.c Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. In general, (a/b) × (c/d) = ac/bd.
- 5.1.4.d Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths.
  - 5.1.4.d.i Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
- 5.1.4.e Interpret multiplication as scaling (resizing), by:
  - 5.1.4.e.i Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
  - 5.1.4.e.ii Apply the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.
    - Multiply two fractions using models (5-M.20)
- 5.1.4.f Solve real world problems involving multiplication of fractions and mixed numbers.
- 5.1.4.g Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.
  - Divide unit fractions by whole numbers (5-N.2)
  - Divide fractions by whole numbers (5-N.8)
- 5.1.4.h Interpret division of a whole number by a unit fraction, and compute such quotients.
  - Divide whole numbers by unit fractions (5-N.4)
  - Divide whole numbers by fractions (5-N.9)
- 5.1.4.i Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions.
  - Divide whole numbers by unit fractions using models (5-N.3)
  - Divide fractions and mixed numbers: word problems (5-N.12)

5.2 Patterns, Functions, and Algebraic Structures

5.2.1 Number patterns are based on operations and relationships
- 5.2.1.a Generate two numerical patterns using given rules.
- 5.2.1.b Identify apparent relationships between corresponding terms.
  - Multiplication input/output tables: find the rule (5-C.24)
- 5.2.1.c Form ordered pairs consisting of corresponding terms from the two patterns, and graphs the ordered pairs on a coordinate plane.
  - Objects on a coordinate plane (5-T.2)
- 5.2.1.d Explain informally relationships between corresponding terms in the patterns.
- 5.2.1.e Use patterns to solve problems including those involving saving and checking accounts.
  - Use a rule to complete a number sequence (5-S.1)
- 5.2.1.f Explain, extend, and use patterns and relationships in solving problems, including those involving saving and checking accounts such as understanding that spending more means saving less.

5.3 Data Analysis, Statistics, and Probability

5.3.1 Visual displays are used to interpret data
- 5.3.1.a Represent and interpret data.
  - 5.3.1.a.i Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8).
    - Create line plots (5-V.10)
    - Create and interpret line plots with fractions (5-V.11)
  - 5.3.1.a.ii Use operations on fractions for this grade to solve problems involving information presented in line plots.
    - Interpret line plots (5-V.9)

5.4 Shape, Dimension, and Geometric Relationships

5.4.1 Properties of multiplication and addition provide the foundation for volume an attribute of solids.
- 5.4.1.a Model and justify the formula for volume of rectangular prisms.
  - 5.4.1.a.i Model the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes.
    - Volume of cubes and rectangular prisms (5-DD.15)
  - 5.4.1.a.ii Show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base.
    - Volume of cubes and rectangular prisms (5-DD.15)
  - 5.4.1.a.iii Represent threefold whole-number products as volumes to represent the associative property of multiplication.
    - Volume of cubes and rectangular prisms (5-DD.15)
- 5.4.1.b Find volume of rectangular prisms using a variety of methods and use these techniques to solve real world and mathematical problems.
  - 5.4.1.b.i Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
    - Volume of rectangular prisms made of unit cubes (5-DD.13)
  - 5.4.1.b.ii Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths.
    - Volume of cubes and rectangular prisms (5-DD.15)
  - 5.4.1.b.iii Use the additive nature of volume to find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts.
    - Volume of rectangular prisms made of unit cubes (5-DD.13)
5.4.2 Geometric figures can be described by their attributes and specific locations in the plane
- 5.4.2.a Graph points on the coordinate plane to solve real-world and mathematical problems.
  - Objects on a coordinate plane (5-T.2)
  - Graph points on a coordinate plane (5-T.3)
- 5.4.2.b Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
  - Graph points on a coordinate plane (5-T.3)
  - Coordinate planes as maps (5-T.7)
- 5.4.2.c Classify two-dimensional figures into categories based on their properties.
  - 5.4.2.c.i Explain that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.
    - Classify quadrilaterals (5-AA.9)
  - 5.4.2.c.ii Classify two-dimensional figures in a hierarchy based on properties.

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5.1 Number Sense, Properties, and Operations

5.1.1 The decimal number system describes place value patterns and relationships that are repeated in large and small numbers and forms the foundation for efficient algorithms

5.1.1.a Explain that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

5.1.1.a.i Explain patterns in the number of zeros of the product when multiplying a number by powers of 10.

5.1.1.a.ii Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.

5.1.1.a.iii Use whole-number exponents to denote powers of 10.

5.1.1.b Read, write, and compare decimals to thousandths.

5.1.1.b.i Read and write decimals to thousandths using base-ten numerals, number names, and expanded form.

5.1.1.b.ii Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

5.1.1.c Use place value understanding to round decimals to any place.

5.1.1.d Convert like measurement units within a given measurement system.

5.1.1.d.i Convert among different-sized standard measurement units within a given measurement system.

5.1.1.d.ii Use measurement conversions in solving multi-step, real world problems.

5.1.2 Formulate, represent, and use algorithms with multi-digit whole numbers and decimals with flexibility, accuracy, and efficiency

5.1.2.a Fluently multiply multi-digit whole numbers using the standard algorithm.

5.1.2.b Find whole-number quotients of whole numbers.

5.1.2.b.i Use strategies based on place value, the properties of operations, and/or the relationship between multiplication and division.

5.1.2.b.ii Illustrate and explain calculations by using equations, rectangular arrays, and/or area models.

5.1.2.c Add, subtract, multiply, and divide decimals to hundredths.

5.1.2.c.i Use concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

5.1.2.c.ii Relate strategies to a written method and explain the reasoning used.

5.1.2.d Write and interpret numerical expressions.

5.1.2.d.i Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

5.1.2.d.ii Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

5.1.3 Formulate, represent, and use algorithms to add and subtract fractions with flexibility, accuracy, and efficiency

5.1.3.a Use equivalent fractions as a strategy to add and subtract fractions.

5.1.3.a.i Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.

5.1.3.a.ii Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions with like denominators.

5.1.3.a.iii Solve word problems involving addition and subtraction of fractions referring to the same whole.

5.1.4 The concepts of multiplication and division can be applied to multiply and divide fractions

5.1.4.a Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b).

5.1.4.b Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.

5.1.4.c Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. In general, (a/b) × (c/d) = ac/bd.

5.1.4.d Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths.

5.1.4.d.i Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

5.1.4.e Interpret multiplication as scaling (resizing), by:

5.1.4.e.i Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.

5.1.4.e.ii Apply the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.

5.1.4.f Solve real world problems involving multiplication of fractions and mixed numbers.

5.1.4.g Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.

5.1.4.h Interpret division of a whole number by a unit fraction, and compute such quotients.

5.1.4.i Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions.

5.2 Patterns, Functions, and Algebraic Structures

5.2.1 Number patterns are based on operations and relationships

5.2.1.a Generate two numerical patterns using given rules.

5.2.1.b Identify apparent relationships between corresponding terms.

5.2.1.c Form ordered pairs consisting of corresponding terms from the two patterns, and graphs the ordered pairs on a coordinate plane.

5.2.1.d Explain informally relationships between corresponding terms in the patterns.

5.2.1.e Use patterns to solve problems including those involving saving and checking accounts.

5.2.1.f Explain, extend, and use patterns and relationships in solving problems, including those involving saving and checking accounts such as understanding that spending more means saving less.

5.3 Data Analysis, Statistics, and Probability

5.3.1 Visual displays are used to interpret data

5.3.1.a Represent and interpret data.

5.3.1.a.i Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8).

5.3.1.a.ii Use operations on fractions for this grade to solve problems involving information presented in line plots.

5.4 Shape, Dimension, and Geometric Relationships

5.4.1 Properties of multiplication and addition provide the foundation for volume an attribute of solids.

5.4.1.a Model and justify the formula for volume of rectangular prisms.

5.4.1.a.i Model the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes.

5.4.1.a.ii Show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base.

5.4.1.a.iii Represent threefold whole-number products as volumes to represent the associative property of multiplication.

5.4.1.b Find volume of rectangular prisms using a variety of methods and use these techniques to solve real world and mathematical problems.

5.4.1.b.i Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

5.4.1.b.ii Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths.

5.4.1.b.iii Use the additive nature of volume to find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts.

5.4.2 Geometric figures can be described by their attributes and specific locations in the plane

5.4.2.a Graph points on the coordinate plane to solve real-world and mathematical problems.

5.4.2.b Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

5.4.2.c Classify two-dimensional figures into categories based on their properties.

5.4.2.c.i Explain that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.

5.4.2.c.ii Classify two-dimensional figures in a hierarchy based on properties.