Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adopting approaches and solutions in novel situations.

MAT-12.AR.07 Create equations and inequalities and use them to solve problems. Include equations arising from linear and quadratic equations and simple rational and exponential equations.

MAT-05.NO.NBT.07 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10. Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

MAT-06.AR.EE.01 Write, read, and evaluate numerical expressions, including expressions with whole number exponents and grouping symbols.

MAT-08.AR.EE.01 Explain the relationship between repeated multiplication and the properties of integer exponents. Apply a single exponent property to generate equivalent numeric and algebraic expressions that include numerical coefficients.

MAT-09.NO.01 Explain how the definition of rational exponents follows from extending the properties of integer exponents; rewrite simple expressions involving radicals and rational exponents using the properties of exponents.

MAT-09.NO.02 Perform basic operations on simple radical expressions to write a simplified equivalent expression.

MAT-09.AR.06 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, and exponential functions.

MAT-09.AR.04 Create linear and exponential equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with proper labels and scales.

MAT-09.AR.F.06 Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.

MAT-0.AR.F.08 Identify situations that can be modeled with linear, quadratic, and exponential functions. Justify the most appropriate model for a situation based on the rate of change over equal intervals. Include situations in which a quantity grows or decays.

MAT-12.NO.01 Rewrite complex expressions involving radicals and rational exponents using the properties of exponents.

MAT-12.NO.02 Perform basic operations on advanced radicals and simplify radicals to write equivalent expressions.

MAT-12.AR.07Create equations and inequalities and use them to solve problems. Include equations arising from linear and quadratic functions and simple rational and exponential functions.

MAT-12.AR.08 Create equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with appropriate labels and scales.

MAT-12.AR.F.03 Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.

MAT-12.AR.F.06 Apply the inverse relationship between exponents and logarithms to solve problems.

MAT-12.AR.F.15 Use properties of logarithms to express the solution to abct = d where a, c, and d are real numbers and b is a positive real number. Evaluate the logarithm using technology when appropriate.

Linear Equations

MAT-07.AR.EE.01 Apply the properties of operations as strategies to add, subtract, factor, and expand linear expressions involving variables, integers, and/or non-negative fractions and decimals with an emphasis on writing equivalent expressions.

MAT-08.AR.EE.03 Explain the characteristics of a linear relationship, including identifying the slope and yintercept in tables, graphs, equations, and descriptions.

MAT-08.AR.EE.04 Represent linear relationships using tables, graphs, equations, and descriptions when given a relationship in one of these forms.

MAT-08.AR.EE.05 Solve linear equations with rational number coefficients and variables on both sides, including equations that require using the distributive property and/or combining and collecting like terms. Interpret the number of solutions. Give examples of linear equations in one variable with one solution, many solutions, or no solutions.

MAT-09.AR.03 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, and exponential functions.

MAT-09.AR.04 Create linear and exponential equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with appropriate labels and scales.

MAT-09.AR.05 Justify each step in solving a linear equation that may or may not have a solution.

MAT-09.AR.06 Solve linear equations and inequalities (to include compound inequalities) in one variable.

MAT-09.AR.07 Solve a system of linear equations graphically and algebraically. Create and solve a system of linear equations in context and interpret the results.

MAT-12.AR.05 Add, subtract, multiply, and divide rational expressions. Understand that rational expressions form a system analogous to rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression.

MAT-12.AR.07Create equations and inequalities and use them to solve problems. Include equations arising from linear and quadratic functions and simple rational and exponential functions.

MAT-12.AR.08 Create equations in two or more variables to represent relationships between quantities.

Graph equations on coordinate axes with appropriate labels and scales.

MAT-12.AR.09 Represent constraints by equations or inequalities and by systems of equations and/or inequalities and interpret solutions as viable or non-viable options in a modeling context.

MAT-12.AR.12 Solve simple rational and radical equations in one variable and identify extraneous solutions.

MAT-12.AR.F.14 Write arithmetic and geometric sequences both recursively and with an explicit formula and convert between the two forms. Use sequences to model situations.

MAT-12.AR.15 Apply the Factor and Remainder Theorems to determine efficiently whether a liner expression is a factor of a polynomial expression.

MAT-12.AR.16 Using graphs, technology, tables, or successive approximations, show that the solution(s) to the equation f(x) = g(x) is the x-value(s) that result in the y-values of f(x) and g(x) being the same.

MAT-12.AR.17 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

MAT-12.AR.18 Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

MAT-12.NO.13 Apply the Fundamental Theorem of Algebra to find all roots of a polynomial equation and determine the nature (i.e., integer, rational, irrational, real, complex) of the roots.

MAT-12.AR.19 Solve a system of equations in three or more variables with matrices (using technology).

MAT-12.AR.F.23 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions and interpret them in context.

Inequalities

MAT-06.AR.EE.04 Describe the concept of a solution to an equation or an inequality. Determine whether a given number is a solution to an equation or an inequality.

MAT-06.AR.EE.06 Write a statement of inequality of the form x > c or the form x < c to represent a constraint or condition. Recognize that inequalities of the form x > c or the form x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

MAT-07.AR.EE.03 Write and solve one- and two-step inequalities where coefficients and solutions are integers and/or non-negative fractions and decimals, including authentic problems. Graph the solution set of the inequality and interpret it in the context of the problem.

MAT-08.AR.EE.07 Solve and graph inequalities in one variable with rational number coefficients and variables on both sides, including equations that require using the distributive property and/or combining like terms.

MAT-08.AR.EE.08 Graph linear inequalities in two variables on a coordinate plane. Interpret the possible solutions in the context of authentic problems.

MAT-09.AR.03 Create equations and inequalities in two variables and use them to solve problems. Include equations arising from linear, quadratic, and exponential functions.

MAT-09.AR.06 Solve linear equations and inequalities (to include compound inequalities) in one variable.

MAT-09.AR.07 Solve a system of linear equations graphically and algebraically. Create and solve a system of linear equations in context and interpret the results.

MAT-09.AR.08 Graph the solution set to a two-variable system of linear inequalities. Create and graph the solution set to a two-variable system of linear inequalities in context.

MAT-09.AR.09 Solve absolute value equations and inequalities in one or two variables.

MAT-12.AR.07Create equations and inequalities and use them to solve problems. Include equations arising from linear and quadratic functions and simple rational and exponential functions.

MAT-12.AR.09 Represent constraints by equations or inequalities and by systems of equations and/or inequalities and interpret solutions as viable or non-viable options in a modeling context.

Quadratic Equations

MAT-09.AR.10 Solve quadratic equations in one variable by inspection (e.g., for x2 = 49) taking square roots, the quadratic formula, and factoring, as appropriate to the initial form of the equation.

MAT-12.NO.09 Apply the Fundamental Theorem of Algebra to determine the number of zeros for polynomial functions. Find all solutions to a polynomial equation.

MAT-12.AR.07Create equations and inequalities and use them to solve problems. Include equations arising from linear and quadratic functions and simple rational and exponential functions.

MAT-12.AR.10 Derive the quadratic formula from the form 0 = ax2 + bx + c.

MAT-12.AR.11 Solve quadratic equations with real coefficients that have solutions of the form a + bi and a - bi.

MAT-12.AR.14 Identify zeros of polynomials when suitable factorizations are available. Use the zeros to construct a rough graph of the function defined by the polynomial.

MAT-12.AR.17 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.