Mathematics Grade 8

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 8 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 8.1.1
Students simplify and evaluate expressions and solve problems involving rational numbers and percent. (SOL 8.1, 8.2, 8.4, 8.5, and 8.6)
Indicators
1. Apply the order of operations to simplify numeric expressions and simplify algebraic expressions for given replacement values of the variables.

2. Simplify and evaluate expressions by using the laws of exponents.

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

4. Identify a whole number between 0 and 200 as a perfect square or find the two consecutive integers between which the number's square root lies.

5. Solve practical problems involving whole numbers, integers, rationals, and percents including sales tax, discount, tip, simple interest, compound interest, and percent of increase/decrease.

6. Use proportions to solve scaling problems.

7. Use technologies to perform numerical calculations.

8. Convert a repeating decimal to its common fraction form.

Enrichment Indicators
9. Solve problems by using trigonometric ratios.
Benchmark 8.1.2
Students explain and apply properties of operations with real numbers and classify numbers into subsets of the real number system. (SOL 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. Describe and represent relationships among subsets of the set of real numbers.

    4. Compare and contrast subsets of the real numbers:

  • natural numbers
  • whole numbers
  • irrational numbers
  • integers
  • rational numbers
  • real numbers
    • Enrichment Indicators
    5. Illustrate the density of rational numbers.

    6. Investigate group properties in operation tables.

    7. Explore mathematical systems.

    Standard 2
        Solve problems involving algebra.
    Benchmark 8.2.1
    Students solve multistep equations and inequalities in one variable and translate and solve related word problems. (SOL 8.16)
    Indicators
    1. Represent the steps in solving equations and inequalities using algebraic sentences.

    2. Identify the properties of equality and properties of operations used.

    3. Solve and check the solution by using substitution and simplification.

    4. Represent solution sets graphically.

    5. Translate word problems into algebraic equations and inequalities and solve.

    6. Use key terms related to equations:

  • variable
  • constant
  • expression
  • coefficient
  • term
    • 7. Solve literal equations for a given variable.
    Enrichment Indicators
    8. Represent solutions to equations and inequalities by using set notation.

    9. Solve nonlinear equations with rational roots.

    Benchmark 8.2.2
    Students describe and represent relations using tables, graphs, equations, and word statements.
    (SOL 8.15, 8.17, 8.18, and 8.19)
    Indicators
    1. Graph linear equations in two variables on the coordinate plane.

    2. Investigate and describe functional relations.

    3. Describe and represent relations using tables, graphs, and algebraic sentences.

    4. Create and solve problems using proportions, formulas, and functions.

    5. Graph functions using graphing technologies.

    6. Use key terms related to functions:

  • domain
  • dependent variable
  • range
  • independent variable
    • 7. Describe relations using word statements.

    8. Recognize and represent linear equations in slope-intercept form.

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

    10. Recognize and make horizontal and vertical shifts in the graphs of functions.

    11. Sketch trigonometric functions.

    12. Solve linear systems of equations in two variables.

    Standard 3
        Construct and analyze geometric figures.
    Benchmark 8.3.1
    Students solve problems involving geometric transformations.(SOL 8.9)
    Indicators
    1. Reflect, translate, rotate, and dilate geometric figures on a coordinate system.

    2. Use key terms related to transformations:

  • image
  • preimage
  • rotation
  • reflection
  • translation
  • dilation
    • 3. Identify and illustrate applications of geometric transformations such as tiling, fabric design, art, and scaling.
    Enrichment Indicators
    4. Transform geometric figures by using matrices.
    Benchmark 8.3.2
    Students solve problems involving angle relationships. (SOL 8.7 and 8.15)
    Indicators
    1. Recognize, verify by measuring, and describe pairs of angles:
  • corresponding angles
  • alternate interior angles
  • vertical angles
  • supplementary angles
  • complementary angles
  • alternate exterior 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. Construct congruent figures, bisectors, perpendiculars, and parallels.
    Standard 4
        Measure geometric figures.
    Benchmark 8.4.1
    Students classify, illustrate, and solve problems involving two- and three-dimensional measurements. (SOL 8.8, 8.10, and 8.11)
    Indicators
    1. Describe, classify, and construct plane and solid figures including prisms, pyramids, cylinders, and cones.

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

    3. Verify and apply the Pythagorean Theorem in solving problems.

    4. Solve problems involving the surface area and volume of cones and pyramids.

    5. Devise valid procedures to determine the surface area and volume of prisms, cylinders, pyramids, and cones.

    Standard 5
        Collect, organize, describe, and analyze data.
    Benchmark 8.5.1
    Students collect, organize, and display data. (SOL 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 8.5.2
    Students analyze and interpret data. (SOL 8.13)
    Indicators
    1. Identify patterns and trends in data sets.

    2. Write an interpretation of the data including measures of central tendency, extremes, quartiles, ranges, and correlation.

    3. Describe the significance of gaps, outliers, and clusters of data sets.

    4. Make comparisons, predictions, inferences, and conclusions from information in statistical graphs.

    Enrichment Indicators
    5. Solve problems that involve using standard deviation.

    6. Interpret normal distributions of selected data sets by using graphing technologies.

    Benchmark 8.5.3
    Students solve probability and counting problems. (SOL 8.12)
    Indicators
    1. Represent the number of possible arrangements of objects using the fundamental counting principle, combinations, and permutations.

    2. Solve probability problems involving dependent and independent events.

    3. Analyze problem situations such as games of chance, board games, or grading scales and make predictions using knowledge of probability.

    Enrichment Indicators
    4. Explore probability relationships in Pascal's Triangle.

    5. Determine the expected outcome of an event.

    Standard 6
        Model real-world situations.
    Benchmark 8.6.1
    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 8.7.1
    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 8.8.1
    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 8.9.1
    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 8.10.1
    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.


    Return to FCPS Mathematics Programs of Studies page