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Domain: Reasoning with Equations and Inequalities A -RE I Cluster: Understand solving equations as a process of reasoning and explain the reasoning Mathematical Content Standard: 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Featured Mathematical Practice: Standard Clarification/Example: High School: Algebra Correlated WA Standard: Pacing: Assuming an equation has a solution, construct a convincing argument that justifies each step in the solution process. Justifications may include the associative, commutative, and division properties, combining like terms, multiplication by 1, etc. Task Analysis: Level: Understand properties such as associative, commutative, etc. Solve a multi-step equation Justify their reasoning for each step in the process of solving the equation Vocabulary: Prior Equation Variable Solve Justify Explicit Associative Commutative Introductory Sample Assessment Item: Domain: Cluster: Level: Reasoning with Equations and Inequalities A -RE I Understand solving equations as a process of reasoning and explain the reasoning High School: Algebra Mathematical Content Standard: 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Featured Mathematical Practice: Standard Clarification/Example: Pacing: Solve simple rational and radical equations in one variable and provide examples of how extraneous solutions arise. Task Analysis: Correlated WA Standard: Know the difference between a rational and a radical equation is Understand what an extraneous solution is Solve simple rational and radical equations in one variable Interpret the meaning of the derived solution and relate it to the context of the problem in order to find possible extraneous solutions Vocabulary: Prior Radical Rational Explicit Extraneous solutions Introductory Sample Assessment Item: Domain: Cluster: Level: Reasoning with Equations and Inequalities A -REI Solve equations and inequalities in one variable High School: Algebra Mathematical Content Standard: 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Featured Mathematical Practice: Standard Clarification/Example: Correlated WA Standard: Pacing: Solve linear equations in one variable, including coefficients represented by letters. Solve linear inequalities in one variable, including coefficients represented by letters. Task Analysis: Solve a one-step, two-step, and multi-step equations and inequalities Solve a formula for the indicated variable Understand that you can solve for a particular variable in terms of other unknowns (coefficients as letters rather than numbers) Understand the constraints letter coefficients have Vocabulary: Prior Coefficient Variable Explicit Introductory Constraints Sample Assessment Item: Domain: Cluster: Level: Reasoning with Equations and Inequalities A -REI Solve equations and inequalities in one variable High School: Algebra Mathematical Content Standard: 4. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. Featured Mathematical Practice: Standard Clarification/Example: Transform a quadratic equation written in standard form by completing the square. (x – =q Derive the quadratic formula by completing the square on the standard form of a quadratic equation. Task Analysis: Recognize what a quadratic equation is Organize the quadratic equation into standard form Apply the rules of completing the square Derive the quadratic formula by completing the square on the standard form of a quadratic equation. p)2 Vocabulary: Prior Standard form Quadratic formula Explicit Completing the square Introductory Correlated WA Standard: Pacing: Sample Assessment Item: Domain: Cluster: Level: Reasoning with Equations and Inequalities A -REI Solve equations and inequalities in one variable High School: Algebra Mathematical Content Standard: 4. Solve quadratic equations in one variable. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Featured Mathematical Practice: Standard Clarification/Example: Pacing: Solve quadratic equations in one variable by simple inspection, taking the square root, factoring, and completing the square. Understand why taking the square root of both sides of an equation yields two solutions. Use the quadratic formula to solve any quadratic equation, recognizing the formula produces all complex solutions. Write the solutions in the form a ± bi, where a and b are real numbers. Explain how complex solutions affect the graph of a quadratic equation. Task Analysis: Correlated WA Standard: Recognize a quadratic equation Determine the appropriate solving method Recognize the effect square root has on the number of possible real solutions Understand the consequence of taking the square root of a negative number Solve quadratic equations involving imaginary numbers Explain how complex solutions affect the graph of a quadratic equation Vocabulary: Prior Square root Completing the square Quadratic formula Factoring Solving Explicit Complex numbers Introductory Sample Assessment Item: Domain: Cluster: Level: Reasoning with Equations and Inequalities A -REI Solve systems of equations High School: Algebra Mathematical Content Standard: 5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Featured Mathematical Practice: Standard Clarification/Example: Pacing: Solve systems of equations using the elimination method (sometimes called linear combinations). Solve a system of equations by substitution (solving for one variable in the first equation and substitution it into the second equation). Task Analysis: Correlated WA Standard: Recognize a system of equations Solve the system of equations by elimination Solve the system of equations by substitution Prove that given a system of equations in two variables has the same solution using the elimination method and the substitution method Vocabulary: Prior System Explicit Elimination Substitution Introductory Sample Assessment Item: Domain: Cluster: Level: Reasoning with Equations and Inequalities A -REI Solve systems of equations High School: Algebra Mathematical Content Standard: 6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Featured Mathematical Practice: Standard Clarification/Example: Pacing: Solve systems of equations using graphs. Task Analysis: Correlated WA Standard: Create a graph with appropriate labels and scale Graph a linear equation Graph two linear equations on the same graph Determine the point of intersection Describe what the point of intersection represents within the context of the problem Vocabulary: Prior Graph Labels Scale Explicit Point of intersection Introductory Sample Assessment Item: Domain: Cluster: Level: Reasoning with Equations and Inequalities A -REI Solve systems of equations High School: Algebra Mathematical Content Standard: 7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2+ y2=3. Featured Mathematical Practice: Standard Clarification/Example: Pacing: Solve a system containing a linear equation and a quadratic equation in two variables (conic sections possible) graphically and symbolically. Task Analysis: Correlated WA Standard: Create a graph with appropriate labels and scale Graph a linear equation Graph a quadratic equation in two variables (including circles) Find the points of intersection Describe what the point of intersection represents within the context of the problem Solve a linear equation for one variable Apply the substitution method of solving a system by substituting the linear expression into the quadratic equation Simplify the new quadratic equation Solve the quadratic equation in one variable Apply the solution(s) back into an original equation and solve for the other variable Describe the solution within the context of the problem Vocabulary: Prior Explicit Conic sections Introductory Sample Assessment Item: Domain: Cluster: Level: Reasoning with Equations and Inequalities A -REI Solve systems of equations High School: Algebra Mathematical Content Standard: 8. (+) Represent a system of linear equations as a single matrix equation in a vector variable. Featured Mathematical Practice: Standard Clarification/Example: Pacing: Write a system of linear equations as a single matrix equation. Task Analysis: Correlated WA Standard: Vocabulary: Construct a matrix from a given system of equations Prior Understand the idea of the number of equations to be as least as many as the number of unknown Use row operations to solve the matrix Explicit Understand when a matrix is in reduced row echelon form Determine the values of the unknowns from the reduced row echelon form Matrix Reduced row echelon form Introductory Sample Assessment Item: Domain: Cluster: Level: Reasoning with Equations and Inequalities A -REI Solve systems of equations High School: Algebra Mathematical Content Standard: 9. (+) 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). Featured Mathematical Practice: Standard Clarification/Example: Pacing: Find the inverse of the coefficient matrix in the equation, if it exits. Use the inverse of the coefficient matrix to solve the system. Use technology for matrices with dimensions 3 by 3 or greater. Task Analysis: Correlated WA Standard: Find the dimension of matrices. Understand when matrices can be multiplied. Understand that matrix multiplication is not commutative. Understand the concept of an identity matrix. Understand why multiplication by the inverse of the coefficient matrix yields a solution to the system (if it exists). Use the inverse of the coefficient matrix to solve the system. Vocabulary: Prior Inverse Matrix System Commutative Explicit Determinant Identity matrix Introductory Sample Assessment Item: Domain: Cluster: Level: Reasoning with Equations and Inequalities A -REI Represent and solve equations and inequalities graphically High School: Algebra Mathematical Content Standard: 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Featured Mathematical Practice: Standard Clarification/Example: Pacing: Understand that all solutions to an equation in two variables are contained on the graph of that equation. Task Analysis: Correlated WA Standard: Understand that the graph represents the infinite number of solutions to the equation Explain what the solutions are within the context of the problem Vocabulary: Prior Infinite Explicit Introductory Sample Assessment Item: Domain: Cluster: Level: Reasoning with Equations and Inequalities A -REI Represent and solve equations and inequalities graphically High School: Algebra Mathematical Content Standard: 11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.* Featured Mathematical Practice: Standard Clarification/Example: Pacing: Explain why the intersection of y = f(x) and y = g(x) is the solution of f(x) = g(x) for any combination of linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Task Analysis: Correlated WA Standard: Use technology to graph the equations and determine their point of intersection, Use tables of values, or Use successive approximations that become closer and closer to the actual value. Understand how a particular input of x gives you the same output of y within the system of equations Vocabulary: Prior Function System of equations Explicit Introductory Sample Assessment Item: Domain: Cluster: Level: Reasoning with Equations and Inequalities A -REI Represent and solve equations and inequalities graphically High School: Algebra Mathematical Content Standard: 12. Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Featured Mathematical Practice: Standard Clarification/Example: Correlated WA Standard: Pacing: Graph the solutions to a linear inequality in two variables as a half-plane, excluding the boundary for noninclusive inequalities. Graph the solution set to a system of linear inequalities in two variables as the intersection of their corresponding half-planes. Task Analysis: Graph a linear inequality Understand that the plane is the solution set (shaded region) Understand if the boundary is open or closed (solid or dotted line) Graph a system of linear inequalities Identify the solution to the system as the overlapping half-planes Vocabulary: Prior Linear inequality System Explicit Planes Open/closed boundary Introductory Sample Assessment Item: