Mathematics Grade 7

Program of Studies
Standards, Benchmarks, and Indicators

Standards
The standards describe what students should know or be able to do before they enroll in the subsequent high school credit-bearing courses of Algebra or Geometry. The ten standards for mathematics at grade 7 are separated into two sets -- content standards and process standards. The five content standards--numerical reasoning, algebra, geometry, measurement, and data analysis and probability--describe the specific mathematical content taught in the course. The five process standards--problem solving, reasoning, communication, connections, and real-world representations--address the mathematical processes that are developed and reinforced in each of the content standards.

Benchmarks
Benchmarks clarify each standard at each specific grade level and provide a statement of grade level expectations for student achievement using age-appropriate materials. Benchmarks are organized under the five standards and mark progress toward the achievement of those standards.  Each benchmark is annotated with its correlation to the Virginia SOLs.

Indicators
Indicators state specific student expectations and behaviors that demonstrate the student's learning. Indicators state the skills and knowledge that, taken together, mark the accomplishment of a benchmark. Enrichment indicators state the skills and knowledge that demonstrate enhanced and extended learning related to the benchmark.

Note: Clicking on Benchmark X.X.X will take you to a representative problem for that benchmark.

Standard 1
    Solve problems involving numbers and number operations.

Benchmark 7.1.1
Students represent rational numbers in equivalent forms and graph real numbers on a number line. (SOL 7.1 and 8.6)
Indicators
1. Represent rational numbers as fractions, decimals, and percents using hands-on materials, diagrams, and symbols.

2. Select and use an appropriate form of a rational number to solve a problem.

3. Order and graph rational and irrational numbers on a number line.

4. Identify a whole number between 0 and 400 as a perfect square or find the two consecutive integers between which the number's square root lies.

Enrichment Indicators
5. Convert repeating decimals to common fraction form.

6. Illustrate the density of rational numbers.

7. Represent numbers in bases other than base ten, including binary and hexadecimal numbers.

Benchmark 7.1.2
Students simplify and evaluate numerical and variable expressions involving rational numbers, scientific notation, and exponents. (SOL 7.1, 7.3, 8.2, and 8.5)
Indicators
1. Perform operations with rational numbers.

2. Use order of operations to simplify numerical expressions.

3. Convert numbers written in scientific notation into standard notation.

4. Convert numbers written in standard notation into scientific notation.

5. Use laws of exponents to simplify expressions with positive and negative exponents.

6. Distinguish between numerical and variable expressions.

7. Distinguish between simplifying numerical expressions and evaluating variable expressions.

8. Evaluate variable expressions by substitution.

9. Use key terms related to exponential expressions:

  • base
  • numerical coefficient
  • standard notation
  • exponent
  • scientific notation
  • calculator notation
10. Use technologies to perform numerical calculations.
Enrichment Indicators
11. Derive laws of exponents, including fractional exponents.

12. Perform operations in number bases other than base ten.

13. Perform operations on sets.

14. Simplify radical expressions.

Benchmark 7.1.3
Students explain and apply properties of operations with real numbers and classify numbers into subsets of the real number system. (SOL 7.4 and 8.3)
Indicators
1. Explain orally and in writing properties of operations with real numbers:
  • associative properties for addition and multiplication
  • commutative properties for addition and multiplication
  • distributive property
  • identity elements for addition and multiplication
  • inverse elements for addition and multiplication
  • closure properties for addition and multiplication
    • 2. Use properties of operations to justify the process of simplifying numerical and variable expressions.

    3. Represent relationships among subsets of the set of real numbers using diagrams.

    4. Compare and contrast subsets of the real numbers:

  • natural numbers
  • whole numbers
  • irrational numbers
  • integers
  • rational numbers
  • real numbers
    • Enrichment Indicators
    5. Apply group properties to operation tables and examine mathematical systems.

    6. Define and compare sets of numbers, including complex numbers.

    7. Define and apply union, intersection, difference, complement, universal set, cardinal number, ordinal number, set builder notation, and quantification notation.

    Benchmark 7.1.4
    Students solve problems involving number theory and scientific notation.(SOL 7.2)
    Indicators
    1. Distinguish between factors and terms.

    2. Distinguish between factors and multiples.

    3. Distinguish between prime and composite numbers.

    4. Use prime factorizations to determine the least common multiple of a set of numbers.

    5. Use prime factorizations to determine the greatest common factor of a set of numbers.

    6. Apply factors and multiples to solve problems.

    7. Apply scientific notation to solve problems involving very large and very small numbers.

    Benchmark 7.1.5
    Students solve proportional reasoning problems including percent applications and scale drawings.
    (SOL 7.5, 7.6, 7.7, 8.1, and 8.4)
    Indicators
    1. Distinguish between proportional and nonproportional relationships.

    2. Use proportional and algebraic processes to solve proportional reasoning problems.

    3. Solve problems involving sales tax, discount, tip, and simple interest.

    4. Solve problems involving compound interest.

    5. Solve problems involving percent of increase/decrease.

    6. Determine the size of the image or preimage of a scale drawing.

    Enrichment Indicators
    7. Solve problems requiring proportional reasoning, including problems involving binomials in the numerator and compound unit conversion.
    Standard 2
        Solve problems involving algebra.
    Benchmark 7.2.1
    Students solve multistep equations in one variable and translate and solve related word problems. (SOL 7.23, 7.24, 7.25, and 8.16)
    Indicators
    1. Represent and demonstrate steps in solving equations using hands-on materials, diagrams, and algebraic sentences.

    2. Demonstrate understanding of additive and multiplicative properties of equality.

    3. Demonstrate understanding and application of the distributive property.

    4. Isolate the variable into one member of the equation.

    5. Check the solution by using substitution and simplification.

    6. Translate word problems into algebraic equations and solve.

    7. Use key terms related to equations:

  • variable
  • constant
  • expression
  • coefficient
  • term
    • 8. Solve literal equations for a given variable.
    Enrichment Indicators
    9. Solve nonlinear equations with rational roots.
    Benchmark 7.2.2
    Students solve and graph multistep inequalities in one variable. (SOL 7.25)
    Indicators
    1. Represent and demonstrate steps in solving inequalities using hands-on materials, diagrams, and algebraic sentences.

    2. Check the solution set by using substitution and simplification.

    3. Draw a graph on a number line to represent the solution set.

    Benchmark 7.2.3
    Students represent linear functions using tables, graphs, and equations.(SOL 7.22, 7.26, 8.17, 8.18,
    and 8.19)
    Indicators
    1. Graph ordered pairs that represent a function on a coordinate plane.

    2. Write an equation that represents a function from a table of values.

    3. Write a table of values that represents the graph of a function.

    4. Write a table of values that represents the equation of a function.

    5. Graph functions using graphing technologies.

    6. Use key terms related to functions:

  • domain
  • dependent variable
  • range
  • independent variable
    • 7. Recognize and apply the slope and intercept in representing an equation in slope-intercept form.
    Enrichment Indicators
    8. Solve linear systems of equations in two or more variables.

    9. Recognize the equations and graphs of absolute value, greatest integer, and exponential functions.

    10. Recognize the equations of conic sections.

    11. Manipulate equations to make horizontal and vertical shifts in their graphs.

    12. Sketch trigonometric functions.

    13. Find the nth term of an arithmetic or geometric sequence.

    14. Find the sum of the terms of an arithmetic or geometric sequence.

    Standard 3
        Construct and analyze geometric figures.
    Benchmark 7.3.1
    Students solve problems involving geometric transformations.(SOL 8.9)
    Indicators
    1. Demonstrate reflections, translations, and rotations.

    2. Illustrate a translation as the product of two reflections.

    3. Dilate a geometric figure by a positive scale factor.

    4. Use key terms related to transformations:

  • image
  • preimage
  • rotation
  • reflection
  • translation
  • dilation
    • 5. Identify real-world applications of transformations.
    Enrichment Indicators
    6. Transform geometric figures using matrices.
    Benchmark 7.3.2
    Students solve problems involving angle relationships. (SOL 7.22, 8.7, and 8.15)
    Indicators
    1. Recognize and describe pairs of angles formed by two lines cut by a transversal.
  • corresponding angles
  • alternate interior angles
  • alternate exterior angles
  • supplementary angles
  • vertical angles
    • 2. Explain angle relationships when a pair of parallel lines is cut by a transversal.

    3. Solve problems involving parallel lines cut by a transversal.

    4. Solve problems involving complementary and supplementary angles.

    5. Develop a procedure for determining the sum of the measures of the angles of any polygon.

    6. Develop a procedure for determining the measure of each angle of a regular polygon.

    7. Develop a procedure for determining the number of diagonals of any polygon.

    Enrichment Indicators
    8. Compare and contrast Euclidean and non-Euclidean geometries.

    9. Prove geometric theorems.

    Benchmark 7.3.3
    Students solve problems involving congruence and similarity. (SOL 7.12)
    Indicators
    1. Recognize and describe attributes of congruent and similar figures.

    2. Write relationships between the sides of congruent and similar figures as proportions.

    3. Use proportions to find measures of sides of similar figures.

    Enrichment Indicators
    4. Solve problems involving trigonometric ratios including using the unit circle.

    5. Apply triangle congruence theorems to solve problems.

    6. Apply triangle similarity theorems to solve problems.

    Benchmark 7.3.4
    Students classify and illustrate two- and three-dimensional figures. (SOL 7.10, 7.11, 7.14, and 8.10)
    Indicators
    1. Illustrate triangles and describe their attributes orally and in writing:
  • scalene
  • isosceles
  • equilateral
  • acute
  • right
  • obtuse
    • 2. Illustrate quadrilaterals and describe their attributes orally and in writing:
  • square
  • rhombus
  • parallelogram
  • rectangle
  • trapezoid
    • 3. Illustrate polygons and describe their attributes orally and in writing:
  • triangle
  • quadrilateral
  • pentagon
  • hexagon
  • heptagon
  • octagon
  • nonagon
  • decagon
    • 4. Draw nets for:
  • prisms
  • cylinders
  • cones
  • pyramids
    • 5. Inscribe equilateral triangles, squares, and hexagons in circles.
    Enrichment Indicators
    6. Construct congruent figures, bisectors, perpendiculars, and parallels.

    7. Inscribe a regular pentagon in a circle.

    Standard 4
        Solve problems involving measurement of geometric figures.
    Benchmark 7.4.1
    Students solve problems involving one-, two-, and three-dimensional measurements. (SOL 7.8, 7.9, 7.13, 8.8, and 8.11)
    Indicators
    1. Recognize and describe how the Pythagorean Theorem applies to right triangles.

    2. Use the Pythagorean Theorem to determine the measure of any side of a right triangle.

    3. Define key words related to measurement :

  • perimeter
  • area
  • volume
    • 4. Solve problems involving the area of two-dimensional figures.

    5. Construct three-dimensional models given the top, side, and front views.

    6. Solve problems involving the surface area and volume of a prism, cylinder, cone, or pyramid.

    Enrichment Indicators
    7. Derive the Pythagorean property in three different ways.

    8. Use Heron's Formula to determine the area of a triangle.

    9. Devise procedures for determining the surface area and volume of prisms, cylinders, pyramids, and cones.

    Standard 5
        Collect, organize, describe, and analyze data.
    Benchmark 7.5.1
    Students collect, organize, and display data. (SOL 7.20 and 8.14)
    Indicators
    1. Choose methods for collecting data, including sampling techniques and technologies.

    2. Organize data using matrices and charts.

    3. Determine the most appropriate type of graph to display data.

    4. Display data using frequency distributions, histograms, line plots, stem-and-leaf plots, box-and-whisker plots, bar graphs, circle graphs, line graphs, and scatterplots.

    5. Present data using graphing technologies.

    Benchmark 7.5.2
    Students analyze data for patterns and trends. (SOL 7.19)
    Indicators
    1. Recognize and identify patterns and trends in data sets.

    2. Determine values for the extremes, quartiles, and ranges of data sets.

    3. Choose the best measure of central tendency and measure of variation to solve a problem.

    4. Determine whether positive, negative, or no correlation exists among data points on a scatterplot.

    5. Identify gaps, outliers, and clusters in data sets.

    Enrichment Indicators
    6. Solve problems involving standard deviation.
    Benchmark 7.5.3
    Students interpret data to make inferences and predictions. (SOL 7.21 and 8.13)
    Indicators
    1. Make inferences based on data analysis.

    2. Make comparisons and predictions using information in statistical graphs.

    3. Write an interpretation of the data including the significance of gaps, outliers, and clusters.

    4. Summarize results and draw conclusions from data analysis.

    Enrichment Indicators

    5. Interpret normal distributions of selected data by using graphing technologies.

    Benchmark 7.5.4
    Students solve probability problems with simple and compound events. (SOL 7.15, 7.16, 7.17, and 8.12)
    Indicators
    1. Represent a sample space using a list, chart, or tree diagram.

    2. Distinguish between dependent and independent events.

    3. Demonstrate ways to solve problems involving dependent or independent events.

    4. Express probability as a ratio, decimal, or percent.

    5. Investigate and explain the difference between experimental and theoretical probability.

    6. Make predictions based on experimental and theoretical probability.

    Enrichment Indicators

    7. Make connections between probability and geometric concepts.

    8. Use set notation to illustrate and explain probability relationships.

    Benchmark 7.5.5
    Students solve problems using counting principles. (SOL 7.18)
    Indicators
    1. Represent the number of possible arrangements of objects using tree diagrams and the fundamental counting principle.

    2. Determine the number of combinations of a group of objects.

    3. Determine the number of permutations of a group of objects.

    Enrichment Indicators
    4. Derive formulas for permutations and combinations.

    5. Make connections between counting principles and related geometric concepts.

    6. Determine the expected outcome of an event.

    Standard 6
        Model real-world situations.
    Benchmark 7.6.1
    Students recognize and apply mathematical models to real-world situations.
    Indicators
    1. Cite examples of mathematics use in real-world settings.

    2. Describe the relationship between a mathematical model and a real-world situation.

    3. Provide multiple solutions to a real-world problem.

    4. Apply one solution path to multiple problems.

    5. Construct and use mathematical models to represent real-world situations.

    Standard 7
        Apply a variety of strategies to solve problems both within and outside mathematics.
    Benchmark 7.7.1
    Students use appropriate strategies to solve problems both within and outside mathematics.
    Indicators
    1. Formulate problems from a variety of mathematical situations.

    2. Formulate problems from a variety of situations in other disciplines.

    3. Explain, illustrate, and evaluate various strategies for solving problems.

    4. Apply a specific strategy to a variety of problems.

    5. Choose strategies appropriate to the problems.

    6. Evaluate the appropriateness of the strategies for the problems.

    Standard 8
        Communicate mathematically.
    Benchmark 7.8.1
    Students read, understand, and communicate mathematical ideas both orally and in writing.
    Indicators
    1. Use the symbols and vocabulary of mathematics to represent and describe mathematical ideas, generalizations, and relationships.

    2. Ask questions pertinent to concepts and procedures written in textbooks and other mathematics references.

    3. Write directions to accomplish a mathematical task.

    4. Restate and explain both orally and in writing mathematics concepts and procedures read in textbooks or other mathematics references.

    5. Explain both orally and in writing the mathematical meaning of information read in textbooks and other mathematics references.

    6. Write mathematical analyses of information from a variety of sources.

    7. Create webs, charts, graphs, and diagrams to communicate mathematical ideas.

    8. Use technologies to communicate mathematical understanding.

    Standard 9
        Reason mathematically.
    Benchmark 7.9.1
    Students reason numerically and spatially and justify reasoning in a variety of mathematical situations.
    Indicators
    1. Evaluate reasonableness of numeric information in data sets.

    2. Evaluate reasonableness of numeric results from solving problems and make necessary corrections.

    3. Represent numeric and spatial relationships using hands-on materials and diagrams.

    4. Translate numeric understanding of concepts and procedures into algebraic representation.

    5. Translate geometric reasoning into algebraic representation.

    6. Construct and defend valid mathematical assertions.

    Standard 10
        Make conceptual connections within and outside mathematics.
    Benchmark 7.10.1
    Students use a variety of representations to illustrate mathematical concepts, skills, and processes.
    Indicators
    1. Use algebraic, geometric, and physical models to represent numeric procedures.

    2. Use algebraic models to represent geometric relationships.

    3. Use geometric models to represent algebraic relationships.

    4. Make connections and apply mathematical models and processes to situations in other disciplines.

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