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Subject Area - CC.2:
Mathematics
  • Standard Area - CC.2.2: Algebraic Concepts
  • Grade Level - CC.2.2.8: GRADE 8
Standard - CC.2.2.8.B.1

Apply concepts of radicals and integer exponents to generate equivalent expressions.

Eligible Content:
5 Total
PASA DLM
Essential Elements (EE)
Assessments
Materials & Resources
  • Standard Area - CC.2.2: Algebraic Concepts
  • Assessment Anchor - A1.1.1:

    Operations with Real Numbers and Expressions

    • Eligible Content - A1.1.1.1.1 Compare and/or order any real numbers (rational and irrational may be mixed).
    • Eligible Content - A1.1.1.1.2

      Simplify square roots (e.g., √24 = 2√6).
      • Eligible Content - A1.1.1.2.1 Find the Greatest Common Factor (GCF) and/or the Least Common Multiple (LCM) for sets of monomials.
        • Eligible Content - A1.1.1.3.1 Simplify/evaluate expressions involving properties/laws of exponents, roots and/or absolute value to solve problems (exponents should be integers from -10 to 10).
          • Eligible Content - A1.1.1.4.1 Use estimation to solve problems.
            • Eligible Content - A1.1.1.5.1 Add, subtract and/or multiply polynomial expressions (express answers in simplest form – nothing larger than a binomial multiplied by a trinomial).
            • Eligible Content - A1.1.1.5.2

              Factor algebraic expressions, including difference of squares and trinomials (trinomials limited to the form ax2+bx+c where a is equal to 1 after factoring out all monomial factors).
            • Eligible Content - A1.1.1.5.3 Simplify/reduce a rational algebraic expression.
              • Anchor Descriptor - M08.B-E.1.1 Represent and use expressions and equations to solve problems involving radicals and integer exponents.
                • Eligible Content - M08.B-E.1.1.1 Apply one or more properties of integer exponents to generate equivalent numerical expressions without a calculator (with final answers expressed in exponential form with positive exponents). Properties will be provided. Example: 3^12 x 3^-15 = 3^-3 = 1/(3^3)
                • Eligible Content - M08.B-E.1.1.2 Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Evaluate square roots of perfect squares (up to and including 12^2) and cube roots of perfect cubes (up to and including 5^3) without a calculator. Example: If x^2 = 25 then x = ±√25.
                  • Alternate Eligible Content - M08.B-E.1.1.2a Identify the meaning of an exponent (limited to exponents of 2 and 3).
                    Alternate Eligible Content is designed for students assessed using the PA Alternate System of Assessment (PASA). Essentialized Example resources assist teachers in designing instruction that reduces content complexity while maintaining alignment to academic standards.
                • Eligible Content - M08.B-E.1.1.3 Estimate very large or very small quantities by using numbers expressed in the form of a single digit times an integer power of 10 and express how many times larger or smaller one number is than another. Example: Estimate the population of the United States as 3 × 10^8 and the population of the world as 7 × 10^9 and determine that the world population is more than 20 times larger than the United States’ population.
                • Eligible Content - M08.B-E.1.1.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Express answers in scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology (e.g., interpret 4.7EE9 displayed on a calculator as 4.7 × 10^9).
                Standard - CC.2.2.8.B.2

                Understand the connections between proportional relationships, lines, and linear equations.

                Eligible Content:
                5 Total
                PASA DLM
                Essential Elements (EE)
                Assessments
                Materials & Resources
                • Standard Area - CC.2.2: Algebraic Concepts
                • Assessment Anchor - A1.2.1:

                  Functions

                  • Eligible Content - A1.2.1.1.1 Analyze a set of data for the existence of a pattern and represent the pattern algebraically and/or graphically.
                  • Eligible Content - A1.2.1.1.2 Determine if a relation is a function given a set of points or a graph.
                  • Eligible Content - A1.2.1.1.3 Identify the domain or range of a relation (may be presented as ordered pairs, a graph, or a table).
                    • Eligible Content - A1.2.1.2.1 Create, interpret and/or use the equation, graph or table of a linear function.
                    • Eligible Content - A1.2.1.2.2 Translate from one representation of a linear function to another (graph, table and equation).
                      • Anchor Descriptor - M08.B-E.2.1 Analyze and describe linear relationships between two variables, using slope.
                        • Eligible Content - M08.B-E.2.1.1 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: Compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
                          • Alternate Eligible Content - M08.B-E.2.1.1a Compare two proportional relationships shown in graph form.
                            Alternate Eligible Content is designed for students assessed using the PA Alternate System of Assessment (PASA). Essentialized Example resources assist teachers in designing instruction that reduces content complexity while maintaining alignment to academic standards.
                        • Eligible Content - M08.B-E.2.1.2 Use similar right triangles to show and explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane.
                        • Eligible Content - M08.B-E.2.1.3 Derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
                          • Alternate Eligible Content - M08.B-E.2.1.3a Identify the slopeand y-intercept of a line on a graph.
                            Alternate Eligible Content is designed for students assessed using the PA Alternate System of Assessment (PASA). Essentialized Example resources assist teachers in designing instruction that reduces content complexity while maintaining alignment to academic standards.
                          Standard - CC.2.2.8.B.3

                          Analyze and solve linear equations and pairs of simultaneous linear equations.

                          Eligible Content:
                          10 Total
                          PASA DLM
                          Essential Elements (EE)
                          Assessments
                          Materials & Resources
                          • Standard Area - CC.2.2: Algebraic Concepts
                          • Assessment Anchor - A1.1.2:

                            Linear Equations

                            • Eligible Content - A1.1.2.1.1 Write, solve and/or apply a linear equation (including problem situations).
                            • Eligible Content - A1.1.2.1.2 Use and/or identify an algebraic property to justify any step in an equation solving process (linear equations only).
                            • Eligible Content - A1.1.2.1.3 Interpret solutions to problems in the context of the problem situation (linear equations only).
                              • Eligible Content - A1.1.2.2.1 Write and/or solve a system of linear equations (including problem situations) using graphing, substitution and/or elimination (limit systems to 2 linear equations).
                              • Eligible Content - A1.1.2.2.2 Interpret solutions to problems in the context of the problem situation (systems of 2 linear equations only).
                                • Anchor Descriptor - M08.B-E.3.1 Write, solve, graph, and interpret linear equations in one or two variables, using various methods.
                                  • Eligible Content - M08.B-E.3.1.1 Write and identify linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
                                    • Alternate Eligible Content - M08.B-E.3.1.1a Select an algebraic equation using addition or subtraction to solve a 2-step real-world problem with one variable.
                                      Alternate Eligible Content is designed for students assessed using the PA Alternate System of Assessment (PASA). Essentialized Example resources assist teachers in designing instruction that reduces content complexity while maintaining alignment to academic standards.
                                  • Eligible Content - M08.B-E.3.1.2 Solve linear equations that have rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
                                    • Alternate Eligible Content - M08.B-E.3.1.2a Solve a 2-step real-world problem using an algebraic equation involving addition or subtraction and one variable.
                                      Alternate Eligible Content is designed for students assessed using the PA Alternate System of Assessment (PASA). Essentialized Example resources assist teachers in designing instruction that reduces content complexity while maintaining alignment to academic standards.
                                  • Eligible Content - M08.B-E.3.1.3 Interpret solutions to a system of two linear equations in two variables as points of intersection of their graphs because points of intersection satisfy both equations simultaneously.
                                  • Eligible Content - M08.B-E.3.1.4 Solve systems of two linear equations in two variables algebraically and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
                                  • Eligible Content - M08.B-E.3.1.5 Solve real-world and mathematical problems leading to two linear equations in two variables. Example: Given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
                                    • Alternate Eligible Content - M08.B-E.3.1.5a Graph a linear equation.
                                      Alternate Eligible Content is designed for students assessed using the PA Alternate System of Assessment (PASA). Essentialized Example resources assist teachers in designing instruction that reduces content complexity while maintaining alignment to academic standards.
                                    Standard - CC.2.2.8.C.1

                                    Define, evaluate, and compare functions.

                                    Eligible Content:
                                    11 Total
                                    PASA DLM
                                    Essential Elements (EE)
                                    Assessments
                                    Materials & Resources
                                    • Standard Area - CC.2.2: Algebraic Concepts
                                    • Assessment Anchor - A1.1.2:

                                      Linear Equations

                                      • Eligible Content - A1.1.2.1.1 Write, solve and/or apply a linear equation (including problem situations).
                                      • Eligible Content - A1.1.2.1.2 Use and/or identify an algebraic property to justify any step in an equation solving process (linear equations only).
                                      • Eligible Content - A1.1.2.1.3 Interpret solutions to problems in the context of the problem situation (linear equations only).
                                        • Eligible Content - A1.1.2.2.1 Write and/or solve a system of linear equations (including problem situations) using graphing, substitution and/or elimination (limit systems to 2 linear equations).
                                        • Eligible Content - A1.1.2.2.2 Interpret solutions to problems in the context of the problem situation (systems of 2 linear equations only).
                                        • Assessment Anchor - A1.2.1:

                                          Functions

                                          • Eligible Content - A1.2.1.1.1 Analyze a set of data for the existence of a pattern and represent the pattern algebraically and/or graphically.
                                          • Eligible Content - A1.2.1.1.2 Determine if a relation is a function given a set of points or a graph.
                                          • Eligible Content - A1.2.1.1.3 Identify the domain or range of a relation (may be presented as ordered pairs, a graph, or a table).
                                            • Eligible Content - A1.2.1.2.1 Create, interpret and/or use the equation, graph or table of a linear function.
                                            • Eligible Content - A1.2.1.2.2 Translate from one representation of a linear function to another (graph, table and equation).
                                              • Anchor Descriptor - M08.B-F.1.1 Define, evaluate, and compare functions displayed algebraically, graphically, or numerically in tables or by verbal descriptions.
                                                • Eligible Content - M08.B-F.1.1.1 Determine whether a relation is a function.
                                                • Eligible Content - M08.B-F.1.1.2 Compare properties of two functions, each represented in a different way (i.e., algebraically, graphically, numerically in tables, or by verbal descriptions). Example: Given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
                                                • Eligible Content - M08.B-F.1.1.3 Interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear.
                                                Standard - CC.2.2.8.C.2

                                                Use concepts of functions to model relationships between quantities.

                                                Eligible Content:
                                                14 Total
                                                PASA DLM
                                                Essential Elements (EE)
                                                Assessments
                                                Materials & Resources
                                                • Standard Area - CC.2.2: Algebraic Concepts
                                                • Assessment Anchor - A1.1.2:

                                                  Linear Equations

                                                  • Eligible Content - A1.1.2.1.1 Write, solve and/or apply a linear equation (including problem situations).
                                                  • Eligible Content - A1.1.2.1.2 Use and/or identify an algebraic property to justify any step in an equation solving process (linear equations only).
                                                  • Eligible Content - A1.1.2.1.3 Interpret solutions to problems in the context of the problem situation (linear equations only).
                                                    • Eligible Content - A1.1.2.2.1 Write and/or solve a system of linear equations (including problem situations) using graphing, substitution and/or elimination (limit systems to 2 linear equations).
                                                    • Eligible Content - A1.1.2.2.2 Interpret solutions to problems in the context of the problem situation (systems of 2 linear equations only).
                                                    • Assessment Anchor - A1.2.1:

                                                      Functions

                                                      • Eligible Content - A1.2.1.1.1 Analyze a set of data for the existence of a pattern and represent the pattern algebraically and/or graphically.
                                                      • Eligible Content - A1.2.1.1.2 Determine if a relation is a function given a set of points or a graph.
                                                      • Eligible Content - A1.2.1.1.3 Identify the domain or range of a relation (may be presented as ordered pairs, a graph, or a table).
                                                        • Eligible Content - A1.2.1.2.1 Create, interpret and/or use the equation, graph or table of a linear function.
                                                        • Eligible Content - A1.2.1.2.2 Translate from one representation of a linear function to another (graph, table and equation).
                                                        • Assessment Anchor - A1.2.2:

                                                          Coordinate Geometry

                                                          • Eligible Content - A1.2.2.1.1 Identify, describe and/or use constant rates of change.
                                                          • Eligible Content - A1.2.2.1.2 Apply the concept of linear rate of change (slope) to solve problems.
                                                          • Eligible Content - A1.2.2.1.3

                                                            Write or identify a linear equation when given

                                                            • the graph of the line
                                                            • 2 points on the line, or
                                                            • the slope and a point on a line,

                                                            (Linear equation may be in point-slope, standard and/or slope-intercept form).
                                                          • Eligible Content - A1.2.2.1.4 Determine the slope and/or y-intercept represented by a linear equation or graph.
                                                            • Eligible Content - A1.2.2.2.1 Draw, find and/or write an equation for a line of best fit for a scatter plot.
                                                              • Anchor Descriptor - M08.B-F.2.1 Represent or interpret functional relationships between quantities using tables, graphs, and descriptions.
                                                                • Eligible Content - M08.B-F.2.1.1 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values.
                                                                  • Alternate Eligible Content - M08.B-F.2.1.1a Determine the missing value in a graph showing a real-world linear relationship.
                                                                    Alternate Eligible Content is designed for students assessed using the PA Alternate System of Assessment (PASA). Essentialized Example resources assist teachers in designing instruction that reduces content complexity while maintaining alignment to academic standards.
                                                                • Eligible Content - M08.B-F.2.1.2 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch or determine a graph that exhibits the qualitative features of a function that has been described verbally.
                                                                  • Alternate Eligible Content - M08.B-F.2.1.2a Describe the relationship between two variables with a linear relationship displayed in graph form.
                                                                    Alternate Eligible Content is designed for students assessed using the PA Alternate System of Assessment (PASA). Essentialized Example resources assist teachers in designing instruction that reduces content complexity while maintaining alignment to academic standards.
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