SECTION 210:15-3-57. Standard Two: Algebraic Reasoning and Algebra

(a)   Statement of the standard. Students will focus on number and operations to develop fluency with an importance of understanding numbers, ways of representing numbers, relationships among numbers, relationships among number systems, and meanings of operations and how they relate to one another. Students will place an emphasis on the development of estimation to determine the reasonableness of solutions and answers and to compute efficiently and proficiently.

(b)   Standard Two objectives for Pre-Kindergarten. The following objectives apply for students in Pre-Kindergarten:

(1)   Recognize, duplicate, and extend patterns.

(A)   Objective 1. Sort and group up to 5 objects into a set based upon characteristics such as color, size, and shape and explain verbally what the objects have in common.

(B)   Objective 2. Recognize, duplicate, and extend repeating patterns involving manipulatives, sound, movement, and other contexts.

(c)   Standard Two objectives for Kindergarten. The following objectives apply for students in Kindergarten:

(1)   Duplicate patterns in a variety of contexts.

(A)   Objective 1. Sort and group up to 10 objects into a set based upon characteristics such as color, size, and shape. Explain verbally what the objects have in common.

(B)   Objective 2. Recognize, duplicate, complete, and extend repeating, shrinking and growing patterns involving shape, color, size, objects, sounds, movement, and other contexts.

(d)   Standard Two objectives for Grade 1. The following objectives apply for students in Grade 1:

(1)   Identify patterns found in real-world and mathematical situations.

(A)   Objective 1. Identify, create, complete, and extend repeating, growing, and shrinking patterns with quantity, numbers, or shapes in a variety of real-world and mathematical contexts.

(e)   Standard Two objectives for Grade 2. The following objectives apply for students in Grade 2:

(1)   Describe the relationship found in patterns to solve real-world and mathematical problems.

(A)   Objective 1. Represent, create, describe, complete, and extend growing and shrinking patterns with quantity and numbers in a variety of real-world and mathematical contexts.

(B)   Objective 2. Represent and describe repeating patterns involving shapes in a variety of contexts.

(2)   Use number sentences involving unknowns to represent and solve real-world and mathematical problems.

(A)   Objective 1. Use objects and number lines to represent number sentences.

(B)   Objective 2. Generate real-world situations to represent number sentences and vice versa.

(C)   Objective 3. Apply commutative and identity properties and number sense to find values for unknowns that make number sentences involving addition and subtraction true or false.

(f)   Standard Two objectives for Grade 3. The following objectives apply for students in Grade 3:

(1)   Describe and create representations of numerical and geometric patterns.

(A)   Objective 1. Create, describe, and extend patterns involving addition, subtraction, or multiplication to solve problems in a variety of contexts.

(B)   Objective 2. Describe the rule (single operation) for a pattern from an input/output table or function machine involving addition, subtraction, or multiplication.

(C)   Objective 3. Explore and develop visual representations of growing geometric patterns and construct the next steps.

(2)   Use number sentences involving multiplication and unknowns to represent and solve real-world and mathematical problems.

(A)   Objective 1. Find unknowns represented by symbols in arithmetic problems by solving one-step open sentences (equations) and other problems involving addition, subtraction, and multiplication. Generate real-world situations to represent number sentences.

(B)   Objective 2. Recognize, represent, and apply the number properties (commutative, identity, and associative properties of addition and multiplication) using models and manipulatives to solve problems.

(g)   Standard Two objectives for Grade 4. The following objectives apply for students in Grade 4:

(1)   Use multiple representations of patterns to solve real-world and mathematical problems.

(A)   Objective 1. Create an input/output chart or table to represent or extend a numerical pattern.

(B)   Objective 2. Describe the single operation rule for a pattern from an input/output table or function machine involving any operation of a whole number.

(C)   Objective 3. Create growth patterns involving geometric shapes and define the single operation rule of the pattern.

(2)   Use multiplication and division with unknowns to create number sentences representing a given problem situation.

(A)   Objective 1. Use number sense, properties, of multiplication and the relationship between multiplication and division to solve problems and find values for the unknowns represented by letters and symbols that make number sentences true.

(B)   Objective 2. Solve for unknowns in problems by solving open sentences (equations) and other problems involving addition, subtraction, multiplication, or division with whole numbers. Use real-world situations to represent number sentences and vice versa.

(h)   Standard Two objectives for Grade 5. The following objectives apply for students in Grade 5:

(1)   Describe and graph patterns of change created through numerical patterns.

(A)   Objective 1. Use tables and rules of up to two operations to describe patterns of change and make predictions and generalizations about real-world and mathematical problems.

(B)   Objective 2. Use a rule or table to represent ordered pairs of whole numbers and graph these ordered pairs on a coordinate plane, identifying the origin and axes in relation to the coordinates.

(2)   Understand and interpret expressions, equations, and inequalities involving variables and whole numbers, and use them to represent and evaluate real-world and mathematical problems.

(A)   Objective 1. Generate equivalent numerical expressions and solve problems involving whole numbers by applying the commutative, associative, and distributive properties and order of operations (no exponents).

(B)   Objective 2. Determine whether an equation or inequality involving a variable is true or false for a given value of the variable.

(C)   Objective 3. Evaluate expressions involving variables when values for the variables are given.

(i)   Standard Two objectives for Grade 6. The following objectives apply for students in Grade 6:

(1)   Recognize and represent relationships between varying quantities; translate from one representation to another; use patterns, tables, graphs, and rules to solve real-world and mathematical problems.

(A)   Objective 1. Plot integer- and rational-valued (limited to halves and fourths) ordered pairs as coordinates in all four quadrants and recognize the reflective relationships among coordinates that differ only by their signs.

(B)   Objective 2. Represent relationships between two varying quantities involving no more than two operations with rules, graphs, and tables; translate between any two of these representations.

(C)   Objective 3. Use and evaluate variables in expressions, equations, and inequalities that arise from various contexts, including determining when or if, for a given value of the variable, an equation or inequality involving a variable is true or false.

(2)   Use properties of arithmetic to generate equivalent numerical expressions and evaluate expressions involving positive rational numbers.

(A)   Objective 1. Generate equivalent numerical expressions and evaluate expressions involving positive rational numbers by applying the commutative, associative, and distributive properties and order of operations to solve real-world and mathematical problems.

(3)   Use equations and inequalities to represent real-world and mathematical problems and use the idea of maintaining equality to solve equations. Interpret solutions in the original context.

(A)   Objective 1. Represent real-world or mathematical situations using expressions, equations, and inequalities involving variables and rational numbers.

(B)   Objective 2. Use number sense and properties of operations and equality to solve real-world and mathematical problems involving equations in the form x + p = q and px = q are nonnegative rational numbers. Graph the solution on a number line, interpret the solution in the original context, and assess the reasonableness of the solution.

(j)   Standard Two objectives for Grade 7. The following objectives apply for students in Grade 7:

(1)   Understand the concept of proportionality in real-world and mathematical situations, and distinguish between proportional and other relationships.

(A)   Objective 1. Describe that the relationship between two variables, x and y, is proportional if it can be expressed in the form y/x = k or y = kx; distinguish proportional relationships from other relationships, including inversely proportional relationship (xy = k or y = k/x).

(B)   Objective 2. Recognize that the graph of a proportional relationship is a line through the origin and the coordinate (1, r), where both r and the slope are the unit rate (constant of proportionality, k).

(2)   recognize proportional relationships in real-world and mathematical situations; represent these and other relationships with tables, verbal descriptions, symbols, and graphs; solve problems involving proportional relationships and interpret results in the original context.

(A)   Objective 1. Represent proportional relationships with tables, verbal descriptions, symbols, and graphs; translate from one representation to another. Determine and compare the unit rate (constant of proportionality, slope, or rate of change) given any of these representations.

(B)   Objective 2. Solve multi-stop problems involving proportional relationships involving distance-time, percent increase or decrease, discounts, tips, unit pricing, similar figures, and other real-world and mathematical situations.

(C)   Objective 3. Use proportional reasoning to solve real-world and mathematical problems involving ratios.

(D)   Objective 4. Use proportional reasoning to assess the reasonableness of solutions.

(3)   Represent and solve linear equations and inequalities.

(A)   Objective 1. Write and solve problems leading to linear equations with one variable in the form px + q = r and (x + q) = r, where p, q, and r are rational numbers.

(B)   Objective 2. Represent, write, solve, and graph problems leading to linear inequalities with one variable in the form x + p > q and x + p < q, where p and q are nonnegative numbers.

(C)   Objective 3. Represent real-world or mathematical situations using equations and inequalities involving variables and rational numbers.

(4)   Use order of operations and properties of operations to generate equivalent numerical and algebraic expressions containing rational numbers and grouping symbols; evaluate such expressions.

(A)   Objective 1. Use properties of operations (limited to associative, commutative, and distributive) to generate equivalent numerical and algebraic expressions containing rational numbers, grouping symbols, and whole number exponents.

(B)   Objective 2. Apply understanding of order of operations and grouping symbols when using calculators and other technologies.

(k)   Standard Two objectives for Pre-Algebra. The following objectives apply for students in Pre-Algebra:

(1)   Understand the concept of function in real-world and mathematical situations, and distinguish between linear and nonlinear functions.

(A)   Objective 1. Recognize that a function is a relationship between an independent variable and a dependent variable in which the value of the independent variable determines the value of the dependent variable.

(B)   Objective 2. Use linear functions to represent and explain real-world and mathematical situations.

(C)   Objective 3. Identify a function as linear if it can be expressed in the form y = mx + b or if its graph is a straight line.

(2)   Recognize linear functions in real-world and mathematical situations; represent linear functions and other functions with tables, verbal descriptions, symbols, and graphs; solve problems involving linear functions and interpret results in the original context.

(A)   Objective 1. Represent linear functions with tables, verbal descriptions, symbols, and graphs; translate from one representation to another.

(B)   Objective 2. Identify, describe, and analyze linear relationships between two variables.

(C)   Objective 3. Identify graphical properties of linear functions including slope and intercepts. Know that the slope equals the rate of change, and that the y-intercept is zero when the function represents a proportional relationship.

(D)   Objective 4. Predict the effect on the graph of a linear function when the slope or y-intercept changes. Use appropriate tools to examine these effects.

(E)   Objective 5. Solve problems involving linear functions and interpret results in the original context.

(3)   Generate equivalent numerical and algebraic expressions and use algebraic expressions and use algebraic properties to evaluate expressions.

(A)   Objective 1. Use substitution to simplify and evaluate algebraic expressions.

(B)   Objective 2. Justify steps in generating equivalent expressions by identifying the properties used, including the properties of operations (associative, commutative, and distributive laws) and the order of operations, including grouping symbols.

(4)   Represent real-world and mathematical problems using equations and inequalities involving linear expressions. Solve and graph equations and inequalities symbolically and graphically. Interpret solutions in the original context.

(A)   Objective 1. Illustrate, write, and solve mathematical and real-world problems using linear equations with one variable with one solution, infinitely many solutions, or no solutions. Interpret solutions in the original context.

(B)   Objective 2. Represent, write, solve, and graph problems leading to linear inequalities with one variable in the form px + q > r and px + q < r, where p, q, and r are rational numbers.

(C)   Objective 3. Represent real-world situations using equations and inequalities involving one variable.

(l)   Standard Two objectives for Algebra 1. The following objectives apply for students in Algebra 1:

(1)   Represent and solve mathematical and real-world problems using linear equations, absolute value equations, and systems of equations; interpret solutions in the original context.

(A)   Objective 1. Use knowledge of solving equations with rational values to represent and solve mathematical and real-world problems (e.g. angle measures, geometric formulas, science, or statistics) and interpret the solutions in the original context.

(B)   Objective 2. Solve absolute value equations and interpret the solutions in the original context.

(C)   Objective 3. Analyze and solve real-world and mathematical problems involving systems of linear equations with a maximum of two variables by graphing (may include graphing calculator or other appropriate technology), substitution, and elimination. Interpret the solutions in the original context.

(2)   Represent and solve real-world and mathematical problems using linear inequalities, compound inequalities and systems of linear inequalities; interpret solutions in the original context.

(A)   Objective 1. Represent relationships in various contexts with linear inequalities; solve the resulting inequalities, graph on a coordinate plane, and interpret the solutions.

(B)   Objective 2. Represent relationships in various contexts with compound and absolute value inequalities, graph on a coordinate plane, and interpret the solutions.

(C)   Objective 3. Solve systems of linear inequalities with a maximum of two variables; graph and interpret the solutions on a coordinate plane.

(3)   Generate equivalent algebraic expressions and use algebraic properties to evaluate expressions and arithmetic and geometric sequences.

(A)   Objective 1. Solve equations involving several variables for one variable in terms of the others.

(B)   Objective 2. Simplify polynomial expressions by adding, subtracting, or multiplying.

(C)   Objective 3. Factor common monomial factors from polynomial expressions and factor quadratic expressions with a leading coefficient of 1.

(D)   Objective 4. Evaluate linear, absolute value, rational, and radical expressions. Include applying a nonstandard operation such as a ʘ b = 2a + b.

(E)   Objective 5. Recognize that arithmetic sequences are linear using equations, tables, graphs, and verbal descriptions. Use the pattern, find the next term.

(F)   Objective 6. Recognize that geometric sequences are exponential using equations, tables, graphs, and verbal descriptions. Given the formula (x) = (r)^x, find the next term and define the meaning of within the context of the problem.

(4)   Analyze mathematical change involving linear equations in real-world and mathematical problems.

(A)   Objective 1. Calculate and interpret slope and the x- and y-intercepts of a line using a graph, an equation, two points, or a set of data points to solve real-world and mathematical problems.

(B)   Objective 2. Solve mathematical and real-world problems involving lines that are parallel, perpendicular, horizontal, or vertical.

(C)   Objective 3. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line.

(D)   Objective 4. Translate between a graph and a situation described qualitatively.

(m)   Standard Two objectives for Algebra 2. The following objectives apply for students in Algebra 2:

(1)   Represent and solve mathematical and real-world problems using nonlinear equations and systems of linear equations; interpret the solutions in the original context.

(A)   Objective 1. Represent real-world or mathematic problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist.

(B)   Objective 2. Represent real-world or mathematical problems using exponential equations, such as compound interest, depreciation, and population growth, and solve these equations graphically (including graphing calculator or other appropriate technology) or algebraically.

(C)   Objective 3. Solve one-variable rational equations and check for extraneous solutions.

(D)   Objective 4. Solve polynomial equations with real roots using various methods and tools that may include factoring, polynomial division, synthetic division, graphing calculators or other appropriate technology.

(E)   Objective 5. Solve square root equations with one variable and check for extraneous solutions.

(F)   Objective 6. Solve common and natural logarithmic equations using the properties of logarithms.

(G)   Objective 7. Solve real-world and mathematical problems that can be modeled using arithmetic or finite geometric sequences or series given the n^th terms and sum formulas. Graphing calculators or other appropriate technology may be used.

(H)   Objective 8. Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology).

(I)   Objective 9. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology.

(2)   Represent and analyze mathematical situations and structures using algebraic symbols using various strategies to write equivalent forms of expressions.

(A)   Objective 1. Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies.

(B)   Objective 2. Add, subtract, multiply, divide, and simplify polynomial and rational expressions.

(C)   Objective 3. Recognize that a quadratic function has different equivalent representations [(x) = ax + bx + c, (x) = a(x - h) + k, and f(x) = (x - h)(x - k)]. Identify and use the representation that is most appropriate to solve real-world and mathematical problems.

(D)   Objective 4. Rewrite expressions involving radicals and rational exponents using the properties of exponents.

[Source: Added at 33 Ok Reg 1324, eff 9-11-16]