Standards Detail
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Subject Area - CC.2:
Mathematics
Mathematics
- Standard Area - CC.2.2: Algebraic Concepts
- Grade Level - CC.2.2.PREK: GRADE Pre-Kindergarten
Standard - CC.2.2.PREK.A.1
Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
- Standard Area - CC.2.2: Algebraic Concepts
- Grade Level - CC.2.2.K: GRADE K
Standard - CC.2.2.K.A.1
Extend concepts of putting together and taking apart to add and subtract within 10.
- Standard Area - CC.2.2: Algebraic Concepts
- Grade Level - CC.2.2.1: GRADE 1
Standard - CC.2.2.1.A.1
Represent and solve problems involving addition and subtraction within 20.
Standard - CC.2.2.1.A.2
Understand and apply properties of operations and the relationship between addition and subtraction.
- Standard Area - CC.2.2: Algebraic Concepts
- Grade Level - CC.2.2.2: GRADE 2
Standard - CC.2.2.2.A.1
Represent and solve problems involving addition and subtraction within 100.
Standard - CC.2.2.2.A.2
Use mental strategies to add and subtract within 20.
Standard - CC.2.2.2.A.3
Work with equal groups of objects to gain foundations for multiplication.
- Standard Area - CC.2.2: Algebraic Concepts
- Grade Level - CC.2.2.3: GRADE 3
Standard - CC.2.2.3.A.1
Represent and solve problems involving multiplication and division.
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Assessment Anchor - M03.B-O.1 Represent and solve problems involving multiplication and division.
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Anchor Descriptor - M03.B-O.1.1 Understand various meanings of multiplication and division.
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Eligible Content - M03.B-O.1.1.1 Interpret and/or describe products of whole numbers (up to and including 10 × 10). Example 1: Interpret 35 as the total number of objects in 5 groups, each containing 7 objects. Example 2: Describe a context in which a total number of objects can be expressed as 5 × 7.
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Alternate Eligible Content - M03.B-O.1.1.1a Use amodel in a multiplicationsituation.
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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Eligible Content - M03.B-O.1.1.2 Interpret and/or describe whole-number quotients of whole numbers (limit dividends through 50 and limit divisors and quotients through 10). Example 1: Interpret 48 ÷ 8 as the number of objects in each share when 48 objects are partitioned equally into 8 shares, or as a number of shares when 48 objects are partitioned into equal shares of 8 objects each. Example 2: Describe a context in which a number of shares or a number of groups can be expressed as 48 ÷ 8.Anchor Descriptor - M03.B-O.1.2 Solve mathematical and realworld problems using multiplication and division, including determining the missing number in a multiplication and/or division equation.Eligible Content - M03.B-O.1.2.1 Use multiplication (up to and including 10 × 10) and/or division (limit dividends through 50 and limit divisors and quotients through 10) to solve word problems in situations involving equal groups, arrays, and/or measurement quantities.Eligible Content - M03.B-O.1.2.2 Determine the unknown whole number in a multiplication (up to and including 10 × 10) or division (limit dividends through 50 and limit divisors and quotients through 10) equation relating three whole numbers. Example: Determine the unknown number that makes an equation true.Standard - CC.2.2.3.A.2
Understand properties of multiplication and the relationship between multiplication and division.
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Assessment Anchor - M03.B-O.2 Understand properties of multiplication and the relationship between multiplication and division.
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Anchor Descriptor - M03.B-O.2.1 Use properties to simplify and solve multiplication problems.
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Eligible Content - M03.B-O.2.1.1 Apply the commutative property of multiplication (not identification or definition of the property).Eligible Content - M03.B-O.2.1.2 Apply the associative property of multiplication (not identification or definition of the property).Anchor Descriptor - M03.B-O.2.2 Relate division to a missingnumber multiplication equation.Eligible Content - M03.B-O.2.2.1 Interpret and/or model division as a multiplication equation with an unknown factor. Example: Find 32 ÷ 8 by solving 8 × ? = 32.Standard - CC.2.2.3.A.3
Demonstrate multiplication and division fluency.
Standard - CC.2.2.3.A.4Solve problems involving the four operations, and identify and explain patterns in arithmetic.
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Assessment Anchor - M03.B-O.3 Solve problems involving the four operations, and identify and explain patterns in arithmetic.
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Anchor Descriptor - M03.B-O.3.1 Use operations, patterns, and estimation strategies to solve problems (may include word problems).
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Eligible Content - M03.B-O.3.1.1 Solve two-step word problems using the four operations (expressions are not explicitly stated). Limit to problems with whole numbers and having whole-number answers.
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Alternate Eligible Content - M03.B-O.3.1.1a Solve a 1-step real-world problem involving numbers under 10 using addition or subtraction.
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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Eligible Content - M03.B-O.3.1.2 Represent two-step word problems using equations with a symbol standing for the unknown quantity. Limit to problems with whole numbers and having whole-number answers.Eligible Content - M03.B-O.3.1.3 Assess the reasonableness of answers. Limit problems posed with whole numbers and having whole-number answers.Eligible Content - M03.B-O.3.1.4 Solve two-step equations using order of operations (equation is explicitly stated with no grouping symbols).Eligible Content - M03.B-O.3.1.5 Identify arithmetic patterns (including patterns in the addition table or multiplication table) and/or explain them using properties of operations. Example 1: Observe that 4 times a number is always even. Example 2: Explain why 6 times a number can be decomposed into three equal addends.
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Alternate Eligible Content - M03.B-O.3.1.5a Identify a mathematical pattern in a real-world problem.
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. -
Alternate Eligible Content - M03.B-O.3.1.5b Identifythe 3 next terms in a mathematical pattern (increasing by 2, 5 or 10).
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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Eligible Content - M03.B-O.3.1.6 Create or match a story to a given combination of symbols (+, –, ×, ÷, <, >, and =) and numbers.Eligible Content - M03.B-O.3.1.7 Identify the missing symbol (+, –, ×, ÷, <, >, and =) that makes a number sentence true.
- Standard Area - CC.2.2: Algebraic Concepts
- Grade Level - CC.2.2.4: GRADE 4
Standard - CC.2.2.4.A.1Represent and solve problems involving the four operations.
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Assessment Anchor - M04.B-O.1 Use the four operations with whole numbers to solve problems.
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Anchor Descriptor - M04.B-O.1.1 Use numbers and symbols to model the concepts of expressions and equations.
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Eligible Content - M04.B-O.1.1.1 Interpret a multiplication equation as a comparison. Represent verbal statements of multiplicative comparisons as multiplication equations. Example 1: Interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Example 2: Know that the statement 24 is 3 times as many as 8 can be represented by the equation 24 = 3 × 8 or 24 = 8 × 3.Eligible Content - M04.B-O.1.1.2 Multiply or divide to solve word problems involving multiplicative comparison, distinguishing multiplicative comparison from additive comparison. Example: Know that 3 × 4 can be used to represent that Student A has 4 objects and Student B has 3 times as many objects not just 3 more objects.
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Alternate Eligible Content - M04.B-O.1.1.2a Useamodel to solve a real-world multiplication problem.
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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Eligible Content - M04.B-O.1.1.3 Solve multi-step word problems posed with whole numbers using the four operations. Answers will be either whole numbers or have remainders that must be interpreted yielding a final answer that is a whole number. Represent these problems using equations with a symbol or letter standing for the unknown quantity.
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Alternate Eligible Content - M04.B-O.1.1.3a Solve a real-world problem with one or more steps using addition or subtraction.
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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Eligible Content - M04.B-O.1.1.4 Identify the missing symbol (+, –, ×, ÷, =, <, and >) that makes a number sentence true (single-digit divisor only).
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Standard - CC.2.2.4.A.2Develop and/or apply number theory concepts to find factors and multiples.
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Assessment Anchor - M04.B-O.2 Gain familiarity with factors and multiples.
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Anchor Descriptor - M04.B-O.2.1 Develop and apply number theory concepts to represent numbers in various ways.
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Eligible Content - M04.B-O.2.1.1 Find all factor pairs for a whole number in the interval 1 through 100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the interval 1 through 100 is a multiple of a given onedigit number. Determine whether a given whole number in the interval 1 through 100 is prime or composite.
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Alternate Eligible Content - M04.B-O.2.1.1a Identify the multiples of 5 to 100 and 10 to 100 (e.g., count money).
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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Standard - CC.2.2.4.A.4Generate and analyze patterns using one rule.
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Assessment Anchor - M04.B-O.3 Generate and analyze patterns.
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Anchor Descriptor - M04.B-O.3.1 Recognize, describe, extend, create, and replicate a variety of patterns.
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Eligible Content - M04.B-O.3.1.1 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. Example 1: Given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms alternate between odd and even numbers. Example 2: Given the rule “increase the number of sides by 1” and starting with a triangle, observe that the tops of the shapes alternate between a side and a vertex.
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Alternate Eligible Content - M04.B-O.3.1.1a Extend a pattern when shown a model and told the rule.
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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Eligible Content - M04.B-O.3.1.2 Determine the missing elements in a function table (limit to +, –, or × and to whole numbers or money).Eligible Content - M04.B-O.3.1.3 Determine the rule for a function given a table (limit to +, –, or × and to whole numbers).
- Standard Area - CC.2.2: Algebraic Concepts
- Grade Level - CC.2.2.5: GRADE 5
Standard - CC.2.2.5.A.1Interpret and evaluate numerical expressions using order of operations.
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Assessment Anchor - M05.B-O.1 Write and interpret numerical expressions.
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Anchor Descriptor - M05.B-O.1.1 Analyze and complete calculations by applying the order of operations.
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Eligible Content - M05.B-O.1.1.1 Use multiple grouping symbols (parentheses, brackets, or braces) in numerical expressions and evaluate expressions containing these symbols.Eligible Content - M05.B-O.1.1.2 Write simple expressions that model calculations with numbers and interpret numerical expressions without evaluating them. Example 1: Express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Example 2: Recognize that 3 × (18,932 + 921) is three times as large as 18,932 + 921 without having to calculate the indicated sum or product.
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Standard - CC.2.2.5.A.4Analyze patterns and relationships using two rules.
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Assessment Anchor - M05.B-O.2 Analyze patterns and relationships.
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Anchor Descriptor - M05.B-O.2.1 Create, extend, and analyze patterns.
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Eligible Content - M05.B-O.2.1.1 Generate two numerical patterns using two given rules. Example: Given the rule “add 3” and the starting number 0 and given the rule “add 6” and the starting number 0, generate terms in the resulting sequences.
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Alternate Eligible Content - M05.B-O.2.1.1a Identify and extend numeric patterns.
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. -
Alternate Eligible Content - M05.B-O.2.1.1b Generate a pattern that follows 1 or more rules provided.
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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Eligible Content - M05.B-O.2.1.2 Identify apparent relationships between corresponding terms of two patterns with the same starting numbers that follow different rules. Example: Given two patterns in which the first pattern follows the rule “add 8” and the second pattern follows the rule “add 2,” observe that the terms in the first pattern are 4 times the size of the terms in the second pattern.
- Standard Area - CC.2.2: Algebraic Concepts
- Grade Level - CC.2.2.6: GRADE 6
Standard - CC.2.2.6.B.1Apply and extend previous understandings of arithmetic to algebraic expressions.
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Assessment Anchor - M06.B-E.1 Apply and extend previous understandings of arithmetic to numerical and algebraic expressions.
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Anchor Descriptor - M06.B-E.1.1 Identify, write, and evaluate numerical and algebraic expressions.
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Eligible Content - M06.B-E.1.1.1 Write and evaluate numerical expressions involving whole-number exponents.Eligible Content - M06.B-E.1.1.2 Write algebraic expressions from verbal descriptions. Example: Express the description “five less than twice a number” as 2y – 5.Eligible Content - M06.B-E.1.1.3 Identify parts of an expression using mathematical terms (e.g., sum, term, product, factor, quotient, coefficient, quantity). Example: Describe the expression 2(8 + 7) as a product of two factors.Eligible Content - M06.B-E.1.1.4 Evaluate expressions at specific values of their variables, including expressions that arise from formulas used in real-world problems. Example: Evaluate the expression b^2 – 5 when b = 4.Eligible Content - M06.B-E.1.1.5 Apply the properties of operations to generate equivalent expressions. Example 1: Apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x. Example 2: Apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y). Example 3: Apply properties of operations to y + y + y to produce the equivalent expression 3y.Standard - CC.2.2.6.B.2
Understand the process of solving a one-variable equation or inequality and apply to real-world and mathematical problems.
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Assessment Anchor - M06.B-E.2 Interpret and solve one-variable equations and inequalities.
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Anchor Descriptor - M06.B-E.2.1 Create, solve, and interpret onevariable equations or inequalities in real-world and mathematical problems.
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Eligible Content - M06.B-E.2.1.1 Use substitution to determine whether a given number in a specified set makes an equation or inequality true.Eligible Content - M06.B-E.2.1.2 Write algebraic expressions to represent real-world or mathematical problems.
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Alternate Eligible Content - M06.B-E.2.1.2a Select an algebraic expression involvingaddition or subtraction of whole numbers to solve a 1-step real-world problem.
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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Eligible Content - M06.B-E.2.1.3 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all non-negative rational numbers.
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Alternate Eligible Content - M06.B-E.2.1.3a Use a 1-step algebraic expression to solve a real-world problem involving addition or subtraction of whole numbers.
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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Eligible Content - M06.B-E.2.1.4 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem and/or represent solutions of such inequalities on number lines.
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Standard - CC.2.2.6.B.3Represent and analyze quantitative relationships between dependent and independent variables.
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Assessment Anchor - M06.B-E.3 Represent and analyze quantitative relationships between dependent and independent variables.
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Anchor Descriptor - M06.B-E.3.1 Use variables to represent two quantities in a real-world problem that change in relationship to one another.
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Eligible Content - M06.B-E.3.1.1 Write an equation to express the relationship between the dependent and independent variables. Example: In a problem involving motion at a constant speed of 65 units, write the equation d = 65t to represent the relationship between distance and time.
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Alternate Eligible Content - M06.B-E.3.1.1a Identify the relationship between two variables in an 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.
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Eligible Content - M06.B-E.3.1.2 Analyze the relationship between the dependent and independent variables using graphs and tables and/or relate these to an equation.
- Standard Area - CC.2.2: Algebraic Concepts
- Grade Level - CC.2.2.7: GRADE 7
Standard - CC.2.2.7.B.1Apply properties of operations to generate equivalent expressions.
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Assessment Anchor - M07.B-E.1 Represent expressions in equivalent forms.
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Anchor Descriptor - M07.B-E.1.1 Use properties of operations to generate equivalent expressions.
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Eligible Content - M07.B-E.1.1.1 Apply properties of operations to add, subtract, factor, and expand linear expressions with rational coefficients. Example 1: The expression 1/2 • (x + 6) is equivalent to 1/2 • x + 3. Example 2: The expression 5.3 – y + 4.2 is equivalent to 9.5 – y (or –y + 9.5). Example 3: The expression 4w – 10 is equivalent to 2(2w – 5).
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Standard - CC.2.2.7.B.3Model and solve real-world and mathematical problems by using and connecting numerical, algebraic, and/or graphical representations.
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Assessment Anchor - A1.1.1
Operations with Real Numbers and Expressions
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Anchor Descriptor - A1.1.1.1 Represent and/or use numbers in equivalent forms (e.g., integers, fractions, decimals, percents, square roots, and exponents).
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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).
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Anchor Descriptor - A1.1.1.2 Apply number theory concepts to show relationships between real numbers in problem solving settings.
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Eligible Content - A1.1.1.2.1 Find the Greatest Common Factor (GCF) and/or the Least Common Multiple (LCM) for sets of monomials.
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Anchor Descriptor - A1.1.1.3 Use exponents, roots and/or absolute value to solve problems.
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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).
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Anchor Descriptor - A1.1.1.4 Use estimation strategies in problem-solving situations.
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Eligible Content - A1.1.1.4.1 Use estimation to solve problems.
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Anchor Descriptor - A1.1.1.5 Simplify expressions involving polynomials.
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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.
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Assessment Anchor - M07.B-E.2 Solve real-world and mathematical problems using numerical and algebraic expressions, equations, and inequalities.
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Anchor Descriptor - M07.B-E.2.1 Solve multi-step real-world and mathematical problems posed with positive and negative rational numbers.
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Eligible Content - M07.B-E.2.1.1 Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate. Example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50 an hour (or 1.1 × $25 = $27.50).
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Anchor Descriptor - M07.B-E.2.2 Use variables to represent quantities in a real-world or mathematical problem and construct simple equations and inequalities to solve problems.
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Eligible Content - M07.B-E.2.2.1 Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Example: The perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
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Alternate Eligible Content - M07.B-E.2.2.1a Select an algebraic expression (equations or inequalities) using addition or subtraction of fractions, decimals, or positive/negative integers to solve a 1-step real-world problem.
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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Eligible Content - M07.B-E.2.2.2 Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers, and graph the solution set of the inequality. Example: A salesperson is paid $50 per week plus $3 per sale. This week she wants her pay to be at least $100. Write an inequality for the number of sales the salesperson needs to make and describe the solutions.
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Anchor Descriptor - M07.B-E.2.3 Determine the reasonableness of the answer(s) in problemsolving situations.
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Eligible Content - M07.B-E.2.3.1 Determine the reasonableness of answer(s) or interpret the solution(s) in the context of the problem. Example: If you want to place a towel bar that is 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
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Alternate Eligible Content - M07.B-E.2.3.1a Identify a reasonable solution in the context of a problem using the four basic operations and numbers under 20.
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 Area - CC.2.2: Algebraic Concepts
- Grade Level - CC.2.2.8: GRADE 8
Standard - CC.2.2.8.B.1Apply concepts of radicals and integer exponents to generate equivalent expressions.
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Assessment Anchor - A1.1.1
Operations with Real Numbers and Expressions
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Anchor Descriptor - A1.1.1.1 Represent and/or use numbers in equivalent forms (e.g., integers, fractions, decimals, percents, square roots, and exponents).
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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).
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Anchor Descriptor - A1.1.1.2 Apply number theory concepts to show relationships between real numbers in problem solving settings.
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Eligible Content - A1.1.1.2.1 Find the Greatest Common Factor (GCF) and/or the Least Common Multiple (LCM) for sets of monomials.
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Anchor Descriptor - A1.1.1.3 Use exponents, roots and/or absolute value to solve problems.
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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).
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Anchor Descriptor - A1.1.1.4 Use estimation strategies in problem-solving situations.
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Eligible Content - A1.1.1.4.1 Use estimation to solve problems.
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Anchor Descriptor - A1.1.1.5 Simplify expressions involving polynomials.
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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.
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Assessment Anchor - M08.B-E.1 Demonstrate an understanding of expressions and equations with radicals and integer exponents.
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Anchor Descriptor - M08.B-E.1.1 Represent and use expressions and equations to solve problems involving radicals and integer exponents.
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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.
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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.
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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.2Understand the connections between proportional relationships, lines, and linear equations.
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Assessment Anchor - A1.2.1
Functions
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Anchor Descriptor - A1.2.1.1 Analyze and/or use patterns or relations.
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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).
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Anchor Descriptor - A1.2.1.2 Interpret and/or use linear functions and their equations, graphs or tables.
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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).
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Assessment Anchor - M08.B-E.2 Understand the connections between proportional relationships, lines, and linear equations.
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Anchor Descriptor - M08.B-E.2.1 Analyze and describe linear relationships between two variables, using slope.
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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.
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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.
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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.
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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.
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Standard - CC.2.2.8.B.3Analyze and solve linear equations and pairs of simultaneous linear equations.
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Assessment Anchor - A1.1.2
Linear Equations
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Anchor Descriptor - A1.1.2.1 Write, solve and/or graph linear equations and inequalities using various methods.
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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).
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Anchor Descriptor - A1.1.2.2 Write, solve and/or graph systems of linear equations using various methods.
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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).
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Assessment Anchor - M08.B-E.3 Analyze and solve linear equations and pairs of simultaneous linear equations.
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Anchor Descriptor - M08.B-E.3.1 Write, solve, graph, and interpret linear equations in one or two variables, using various methods.
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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).
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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.
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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.
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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.
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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.
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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.1Define, evaluate, and compare functions.
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Assessment Anchor - A1.1.2
Linear Equations
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Anchor Descriptor - A1.1.2.1 Write, solve and/or graph linear equations and inequalities using various methods.
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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).
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Anchor Descriptor - A1.1.2.2 Write, solve and/or graph systems of linear equations using various methods.
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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).
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Assessment Anchor - A1.2.1
Functions
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Anchor Descriptor - A1.2.1.1 Analyze and/or use patterns or relations.
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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).
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Anchor Descriptor - A1.2.1.2 Interpret and/or use linear functions and their equations, graphs or tables.
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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).
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Assessment Anchor - M08.B-F.1 Analyze and interpret functions.
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Anchor Descriptor - M08.B-F.1.1 Define, evaluate, and compare functions displayed algebraically, graphically, or numerically in tables or by verbal descriptions.
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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.2Use concepts of functions to model relationships between quantities.
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Assessment Anchor - A1.1.2
Linear Equations
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Anchor Descriptor - A1.1.2.1 Write, solve and/or graph linear equations and inequalities using various methods.
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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).
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Anchor Descriptor - A1.1.2.2 Write, solve and/or graph systems of linear equations using various methods.
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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).
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Assessment Anchor - A1.2.1
Functions
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Anchor Descriptor - A1.2.1.1 Analyze and/or use patterns or relations.
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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).
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Anchor Descriptor - A1.2.1.2 Interpret and/or use linear functions and their equations, graphs or tables.
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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).
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Assessment Anchor - A1.2.2
Coordinate Geometry
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Anchor Descriptor - A1.2.2.1 Describe, compute and/or use the rate of change (slope) of a line.
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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.-
Anchor Descriptor - A1.2.2.2 Analyze and/or interpret data on a scatter plot.
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Eligible Content - A1.2.2.2.1 Draw, find and/or write an equation for a line of best fit for a scatter plot.
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Assessment Anchor - M08.B-F.2 Use functions to model relationships between quantities.
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Anchor Descriptor - M08.B-F.2.1 Represent or interpret functional relationships between quantities using tables, graphs, and descriptions.
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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.
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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.
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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.
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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.
- Standard Area - CC.2.2: Algebraic Concepts
- Grade Level - CC.2.2.HS: GRADE High School
Standard - CC.2.2.HS.C.1Use the concept and notation of functions to interpret and apply them in terms of their context.
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Assessment Anchor - A1.2.1
Functions
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Anchor Descriptor - A1.2.1.1 Analyze and/or use patterns or relations.
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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).
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Anchor Descriptor - A1.2.1.2 Interpret and/or use linear functions and their equations, graphs or tables.
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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).
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Assessment Anchor - A1.2.2
Coordinate Geometry
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Anchor Descriptor - A1.2.2.1 Describe, compute and/or use the rate of change (slope) of a line.
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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.-
Anchor Descriptor - A1.2.2.2 Analyze and/or interpret data on a scatter plot.
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Eligible Content - A1.2.2.2.1 Draw, find and/or write an equation for a line of best fit for a scatter plot.
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Assessment Anchor - A2.2.1
Patterns, Relations, and Functions
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Anchor Descriptor - A2.2.1.1 Analyze and/or use patterns or relations.
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Eligible Content - A2.2.1.1.1 Analyze a set of data for the existence of a pattern and represent the pattern with a rule algebraically and/or graphically.Eligible Content - A2.2.1.1.2
Identify and/or extend a pattern as either an arithmetic or geometric sequence (e.g., given a geometric sequence, find the 20th term).
Eligible Content - A2.2.1.1.3 Determine the domain, range or inverse of a relation.Eligible Content - A2.2.1.1.4 Identify and/or determine the characteristics of an exponential, quadratic, or polynomial function (e.g., intervals of increasing/decreasing, intercepts, zeros, and asymptotes).-
Assessment Anchor - G.2.2
Measurements of Two-Dimensional Shapes and Figures
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Anchor Descriptor - G.2.2.1 Use and/or compare measurements of angles.
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Eligible Content - G.2.2.1.1 Use properties of angles formed by intersecting lines to find the measures of missing angles.Eligible Content - G.2.2.1.2 Use properties of angles formed when two parallel lines are cut by a transversal to find the measures of missing angles.
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