Virginia

Number and Number Sense

• 6.1 The student will represent relationships between quantities using ratios, and will use appropriate notations, such as a/b, a to b, and a:b.

  • 6.1.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

    • 6.1.a.1 Represent a relationship between two quantities using ratios.

      • Write a ratio (6-R.1)
      • Write a ratio: word problems (6-R.3)

    • 6.1.a.2 Represent a relationship in words that makes a comparison by using the notations a/b, a:b, and a to b.

      • Write a ratio (6-R.1)
      • Write a ratio: word problems (6-R.3)

    • 6.1.a.3 Create a relationship in words for a given ratio expressed symbolically.

• 6.2 The student will a) represent and determine equivalencies among fractions, mixed numbers, decimals, and percents; and b) compare and order positive rational numbers.

  • 6.2.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

    • 6.2.a.1 Represent ratios as fractions (proper or improper), mixed numbers, decimals, and/or percents. (a)

    • 6.2.a.2 Determine the decimal and percent equivalents for numbers written in fraction form (proper or improper) or as a mixed number, including repeating decimals. (a)

      • Convert between decimals and fractions (6-I.11)
      • Convert between decimals and mixed numbers (6-I.12)
      • Convert between percents, fractions, and decimals (6-S.4)

    • 6.2.a.3 Represent and determine equivalencies among decimals, percents, fractions (proper or improper), and mixed numbers that have denominators that are 12 or less or factors of 100. (a)

      • Equivalent fractions review (6-I.3)
      • Write fractions in lowest terms (6-I.4)
      • Convert between improper fractions and mixed numbers (6-I.8)

    • 6.2.a.4 Compare two percents using pictorial representations and symbols (<, ≤, ≥, >, =). (b)

      • Compare percents to each other and to fractions (6-S.6)

    • 6.2.a.5 Order no more than four positive rational numbers expressed as fractions (proper or improper), mixed numbers, decimals, and percents (decimals through thousandths, fractions with denominators of 12 or less or factors of 100). Ordering may be in ascending or descending order. (b)

      • Put decimal numbers in order (6-F.4)
      • Put a mix of decimals, fractions, and mixed numbers in order (6-I.14)

• 6.3 The student will a) identify and represent integers; b) compare and order integers; and c) identify and describe absolute value of integers.

  • 6.3.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

    • 6.3.a.1 Model integers, including models derived from practical situations. (a)

      • Understanding integers (6-M.1)

    • 6.3.a.2 Identify an integer represented by a point on a number line. (a)

      • Integers on number lines (6-M.2)
      • Graph integers on horizontal and vertical number lines (6-M.3)

    • 6.3.a.3 Compare and order integers using a number line. (b)

      • Compare integers (6-M.7)
      • Put integers in order (6-M.8)

    • 6.3.a.4 Compare integers, using mathematical symbols (<, ≤, >, ≥, =). (b)

      • Compare integers (6-M.7)

    • 6.3.a.5 Identify and describe the absolute value of an integer. (c)

      • Absolute value (6-M.6)

• 6.4 The student will recognize and represent patterns with whole number exponents and perfect squares.

  • 6.4.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

    • 6.4.a.1 Recognize and represent patterns with bases and exponents that are whole numbers.

      • Write multiplication expressions using exponents (6-D.1)

    • 6.4.a.2 Recognize and represent patterns of perfect squares not to exceed 20², by using grid paper, square tiles, tables, and calculators.

    • 6.4.a.3 Recognize powers of 10 with whole number exponents by examining patterns in place value.

Computation and Estimation

• 6.5 The student will a) multiply and divide fractions and mixed numbers; b) solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of fractions and mixed numbers; and c) solve multistep practical problems involving addition, subtraction, multiplication, and division of decimals.

  • 6.5.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

    • 6.5.a.1 Demonstrate/model multiplication and division of fractions (proper or improper) and mixed numbers using multiple representations. (a)

      • Multiply two fractions using models (6-K.5)
      • Divide whole numbers by unit fractions using models (6-L.1)
      • Divide whole numbers and fractions using models (6)
      • Divide fractions using models (6)
      • Divide fractions and mixed numbers using models (6)
      • Divide fractions and mixed numbers using models: word problems (6)

    • 6.5.a.2 Multiply and divide fractions (proper or improper) and mixed numbers. Answers are expressed in simplest form. (a)

      • Multiply two fractions (6-K.6)
      • Multiply three or more fractions and whole numbers (6-K.10)
      • Multiply mixed numbers and whole numbers (6-K.12)
      • Multiply mixed numbers (6-K.13)
      • Multiply three or more mixed numbers, fractions, and/or whole numbers (6-K.15)
      • Divide whole numbers and unit fractions (6-L.3)
      • Divide fractions (6-L.6)
      • Divide fractions and mixed numbers (6-L.9)

    • 6.5.a.3 Solve single-step and multistep practical problems that involve addition and subtraction with fractions (proper or improper) and mixed numbers, with and without regrouping, that include like and unlike denominators of 12 or less. Answers are expressed in simplest form. (b)

      • Add and subtract fractions with like denominators: word problems (6-J.2)
      • Add and subtract fractions with unlike denominators: word problems (6-J.4)
      • Add and subtract mixed numbers: word problems (6-J.7)
      • Maps with fractional distances (6-J.9)

    • 6.5.a.4 Solve single-step and multistep practical problems that involve multiplication and division with fractions (proper or improper) and mixed numbers that include denominators of 12 or less. Answers are expressed in simplest form. (b)

      • Multiply fractions: word problems (6-K.7)
      • Multiply mixed numbers: word problems (6-K.14)
      • Divide fractions by whole numbers in recipes (6-L.4)
      • Divide fractions and mixed numbers: word problems (6-L.10)

    • 6.5.a.5 Solve multistep practical problems involving addition, subtraction, multiplication and division with decimals. Divisors are limited to a three-digit number, with decimal divisors limited to hundredths. (c)

      • Maps with decimal distances (6-G.5)

• 6.6 The student will a) add, subtract, multiply, and divide integers; b) solve practical problems involving operations with integers; and c) simplify numerical expressions involving integers.

  • 6.6.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

    • 6.6.a.1 Model addition, subtraction, multiplication and division of integers using pictorial representations or concrete manipulatives. (a)

      • Add integers using counters (6-N.1)
      • Subtract integers using counters (6-N.4)

    • 6.6.a.2 Add, subtract, multiply, and divide two integers. (a)

      • Add integers (6-N.3)
      • Subtract integers (6-N.6)
      • Add and subtract integers: find the sign (6-N.7)
      • Add and subtract integers: input/output tables (6-N.9)
      • Multiply integers: find the sign (6-N.12)
      • Multiply integers (6-N.13)
      • Divide integers: find the sign (6-N.14)
      • Divide integers (6-N.15)
      • Add, subtract, multiply, or divide two integers (6-O.12)

    • 6.6.a.3 Solve practical problems involving addition, subtraction, multiplication, and division with integers. (b)

      • Add, subtract, multiply, or divide two whole numbers: word problems (6-O.2)

    • 6.6.a.4 Use the order of operations and apply the properties of real numbers to simplify numerical expressions involving more than two integers. Expressions should not include braces { } or brackets [ ], but may contain absolute value bars ||. Simplification will be limited to three operations, which may include simplifying a whole number raised to an exponent of 1, 2 or 3. (c)

      • Add three or more integers (6-N.10)
      • Evaluate numerical expressions involving integers (6-O.13)

Measurement and Geometry

• 6.7 The student will a) derive π (pi); b) solve problems, including practical problems, involving circumference and area of a circle; and c) solve problems, including practical problems, involving area and perimeter of triangles and rectangles.

  • 6.7.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

    • 6.7.a.1 Derive an approximation for pi (3.14 or 22/7) by gathering data and comparing the circumference to the diameter of various circles, using concrete materials or computer models. (a)

    • 6.7.a.2 Solve problems, including practical problems, involving circumference and area of a circle when given the length of the diameter or radius. (b)

      • Circumference of circles (6)
      • Circles: word problems (6)
      • Area between two circles (6)
      • Area of circles (6)

    • 6.7.a.3 Solve problems, including practical problems, involving area and perimeter of triangles and rectangles. (c)

      • Area of rectangles and squares (6-FF.2)
      • Area of triangles (6-FF.6)
      • Area between two rectangles (6-FF.13)
      • Area between two triangles (6-FF.14)
      • Rectangles: relationship between perimeter and area (6-FF.15)

• 6.8 The student will a) identify the components of the coordinate plane; and b) identify the coordinates of a point and graph ordered pairs in a coordinate plane.

  • 6.8.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

    • 6.8.a.1 Identify and label the axes, origin, and quadrants of a coordinate plane. (a)

    • 6.8.a.2 Identify the quadrant or the axis on which a point is positioned by examining the coordinates (ordered pair) of the point. Ordered pairs will be limited to coordinates expressed as integers. (a)

      • Quadrants (6-X.4)

    • 6.8.a.3 Graph ordered pairs in the four quadrants and on the axes of a coordinate plane. Ordered pairs will be limited to coordinates expressed as integers. (b)

      • Graph points on a coordinate plane (6-X.3)

    • 6.8.a.4 Identify ordered pairs represented by points in the four quadrants and on the axes of the coordinate plane. Ordered pairs will be limited to coordinates expressed as integers. (b)

      • Objects on a coordinate plane (6-X.2)

    • 6.8.a.5 Relate the coordinates of a point to the distance from each axis and relate the coordinates of a single point to another point on the same horizontal or vertical line. Ordered pairs will be limited to coordinates expressed as integers. (b)

    • 6.8.a.6 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to determine the length of a side joining points with the same first coordinate or the same second coordinate. Ordered pairs will be limited to coordinates expressed as integers. Apply these techniques in the context of solving practical and mathematical problems. (b)

      • Distance between two points (6-X.7)
      • Graph triangles and quadrilaterals (6-CC.8)

• 6.9 The student will determine congruence of segments, angles, and polygons.

  • 6.9.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

    • 6.9.a.1 Identify regular polygons.

      • Identify and classify polygons (6-CC.1)

    • 6.9.a.2 Draw lines of symmetry to divide regular polygons into two congruent parts.

      • Line symmetry (6-DD.1)

    • 6.9.a.3 Determine the congruence of segments, angles, and polygons given their properties.

      • Similar and congruent figures (6-DD.5)

    • 6.9.a.4 Determine whether polygons are congruent or noncongruent according to the measures of their sides and angles.

      • Similar and congruent figures (6-DD.5)

Probability and Statistics

• 6.10 The student, given a practical situation, will a) represent data in a circle graph; b) make observations and inferences about data represented in a circle graph; and c) compare circle graphs with the same data represented in bar graphs, pictographs, and line plots.

  • 6.10.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

    • 6.10.a.1 Collect, organize and represent data in a circle graph. The number of data values should be limited to allow for comparisons that have denominators of 12 or less or those that are factors of 100 (e.g., in a class of 20 students, 7 choose apples as a favorite fruit, so the comparison is 7 out of 20, 7/20, or 35%). (a)

    • 6.10.a.2 Make observations and inferences about data represented in a circle graph. (b)

      • Interpret circle graphs (6-GG.16)

    • 6.10.a.3 Compare data represented in a circle graph with the same data represented in bar graphs, pictographs, and line plots. (c)

• 6.11 The student will a) represent the mean of a data set graphically as the balance point; and b) determine the effect on measures of center when a single value of a data set is added, removed, or changed.

  • 6.11.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

    • 6.11.a.1 Represent the mean of a set of data graphically as the balance point represented in a line plot. (a)

    • 6.11.a.2 Determine the effect on measures of center when a single value of a data set is added, removed, or changed. (b)

      • Changes in mean, median, mode, and range (7-DD.4)

Patterns, Functions, and Algebra

• 6.12 The student will a) represent a proportional relationship between two quantities, including those arising from practical situations; b) determine the unit rate of a proportional relationship and use it to find a missing value in a ratio table; c) determine whether a proportional relationship exists between two quantities; and d) make connections between and among representations of a proportional relationship between two quantities using verbal descriptions, ratio tables, and graphs.

  • 6.12.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

    • 6.12.a.1 Make a table of equivalent ratios to represent a proportional relationship between two quantities, when given a ratio. (a)

    • 6.12.a.2 Make a table of equivalent ratios to represent a proportional relationship between two quantities, when given a practical situation. (a)

      • Equivalent ratios: word problems (6-R.8)

    • 6.12.a.3 Identify the unit rate of a proportional relationship represented by a table of values or a verbal description, including those represented in a practical situation. Unit rates are limited to positive values. (b)

      • Unit rates (6-R.9)
      • Ratios and rates: word problems (6-R.15)

    • 6.12.a.4 Determine a missing value in a ratio table that represents a proportional relationship between two quantities using a unit rate. Unit rates are limited to positive values. (b)

      • Ratio tables (6-R.7)

    • 6.12.a.5 Determine whether a proportional relationship exists between two quantities, when given a table of values or a verbal description, including those represented in a practical situation. Unit rates are limited to positive values. (c).

      • Do the ratios form a proportion? (6-R.16)
      • Scale drawings: word problems (6-R.22)

    • 6.12.a.6 Determine whether a proportional relationship exists between two quantities given a graph of ordered pairs. Unit rates are limited to positive values. (c)

    • 6.12.a.7 Make connections between and among multiple representations of the same proportional relationship using verbal descriptions, ratio tables, and graphs. Unit rates are limited to positive values. (d)

• 6.13 The student will solve one-step linear equations in one variable, including practical problems that require the solution of a one-step linear equation in one variable.

  • 6.13.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representation to

    • 6.13.a.1 Identify examples of the following algebraic vocabulary: equation, variable, expression, term, and coefficient.

      • Identify terms and coefficients (6-Y.8)
      • Which x satisfies an equation? (6-Z.2)

    • 6.13.a.2 Represent and solve one-step linear equations in one variable, using a variety of concrete materials such as colored chips, algebra tiles, or weights on a balance scale.

      • Model and solve equations using algebra tiles (6-Z.5)
      • Write and solve equations that represent diagrams (6-Z.7)

    • 6.13.a.3 Apply properties of real numbers and properties of equality to solve a one-step equation in one variable. Coefficients are limited to integers and unit fractions. Numeric terms are limited to integers.

      • Solve one-step equations with whole numbers (6-Z.10)
      • Solve one-step multiplication and division equations with decimals and fractions (6-Z.12)
      • Solve one-step equations involving integers (6-Z.18)

    • 6.13.a.4 Confirm solutions to one-step linear equations in one variable.

      • Does x satisfy an equation? (6-Z.1)

    • 6.13.a.5 Write verbal expressions and sentences as algebraic expressions and equations.

      • Write variable expressions: two operations (6-Y.2)
      • Write variable expressions: word problems (6-Y.3)
      • Write an equation from words (6-Z.3)

    • 6.13.a.6 Write algebraic expressions and equations as verbal expressions and sentences.

    • 6.13.a.7 Represent and solve a practical problem with a one-step linear equation in one variable.

      • Solve one-step equations: word problems (6-Z.16)

• 6.14 The student will a) represent a practical situation with a linear inequality in one variable; and b) solve one-step linear inequalities in one variable, involving addition or subtraction, and graph the solution on a number line.

  • 6.14.a The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representation to

    • 6.14.a.1 Given a verbal description, represent a practical situation with a one-variable linear inequality. (a)

    • 6.14.a.2 Apply properties of real numbers and the addition or subtraction property of inequality to solve a one-step linear inequality in one variable, and graph the solution on a number line. Numeric terms being added or subtracted from the variable are limited to integers. (b)

    • 6.14.a.3 Given the graph of a linear inequality with integers, represent the inequality two different ways (e.g., x < –5 or –5 > x) using symbols. (b)

      • Write inequalities from number lines (6-AA.3)

    • 6.14.a.4 Identify a numerical value(s) that is part of the solution set of a given inequality. (a, b)

      • Solutions to inequalities (6-AA.1)