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Mathematics Common Core State Standards High School Algebra I and II COMMON CORE CLUSTER STANDARDS College and Career Readiness Cluster Standards for Mathematics 1. Interpret the structure of expressions. COMMON CORE STATE STANDARD WHAT DOES THIS STANDARD MEAN THAT A STUDENT MUST KNOW AND BE ABLE TO DO? Algebra I and II Linear, quadratic, and exponential expressions are the focus in Algebra I and integer exponents extended to rational exponents – square and cubed roots. Algebra II extends to polynomial and rational expressions. Seeing Structure in Expressions 1. Interpret expressions that represent a quantity Students identify the different parts of the expression and explain in terms of its context. their meaning within the context of the problem. a. Interpret parts of an expression, such as terms, factors, and coefficients. Students decompose expressions and make sense of the multiple b. Interpret complicated expressions by factors and terms by explaining the meaning of the individual parts. viewing one or more of their parts as a single entity. For example, interpret P(1 Students classify polynomials by degree, term, even, or odd. +r)ᴺ as the product of P and a factor not depending on P. Students factor polynomials of degree two and higher. Students identify parts of exponential decay and growth. Students use the appropriate models for simple compound interest and continuously compounded interest. 2. Use the structure of an expression to identify ways to rewrite it. Students rewrite algebraic expressions in different equivalent forms such as factoring or combining like terms. Students rewrite between the three different forms of linear equations. Students rewrite a quadratic in factored form. Students rewrite an inverse variation function in terms of its constant. 1 Reformatted Saddlebrook Preparatory School Students add, subtract, multiply, and divide to write a simpler form of a rational expression. 2. Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeroes of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. c. Use the properties of exponents to transform expressions for exponential functions. Students write expressions in equivalent forms by factoring to find the zeroes of a quadratic function and explain the meaning of the zeroes. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. Students use the formula to solve real world problems such as calculating the height of a tree after n years given the initial height of the tree and the rate the tree grows each year. Students write expressions in equivalent forms by completing the square to convey the vertex form to find the maximum or minimum value of a quadratic function, and explain the meaning of the vertex. Students identify parts of exponential decay and growth. Students use properties of exponents (power of a power, product of powers, power of a product and rational exponents) to write equivalent form of an exponential function to explain the approximate rate of growth or decay. Students use the geometric series formula to solve a problem. 2 Reformatted Saddlebrook Preparatory School Mathematics Common Core State Standards High School Algebra I and II COMMON CORE CLUSTER STANDARDS College and Career Readiness Cluster Standards for Mathematics 1. Perform arithmetic operations on polynomials. 2. Understand the relationship between zeroes and factors of polynomials to perform multi-digit arithmetic. COMMON CORE STATE STANDARD WHAT DOES THIS STANDARD MEAN THAT A STUDENT MUST KNOW AND BE ABLE TO DO? Algebra I and II Linear and quadratic polynomial expressions are the expectation in Algebra 1 and beyond quadratic polynomial expressions is the expectation for Algebra II. Arithmetic with Polynomials and Rational Expressions 1. Understand that polynomials form a system Students understand the definition of a polynomial. analogous to the integers, namely, they are closed under the operations of addition, Students combine like terms and understand closure. subtraction, and multiplication; add, subtract, and multiply polynomials. Students add, subtract, and multiply polynomials and understand how closure applies under these operations. Students apply the Remainder Theorem. 2. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a is a factor of p(x). 3. Identify zeroes of polynomials when suitable factorization are available, and use the zeroes to construct a rough graph of the function defined by the polynomial. Students understand that a is a root of a polynomial function if and only if x – a is a factor of the function. Students use synthetic division to find the remainder or a polynomial. Students find the zeroes of a polynomial when the polynomial is factored. Students use the zeroes of a function to sketch a graph of the function. 3 Reformatted Saddlebrook Preparatory School 3. Use polynomial identities to solve problems. 4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x² + y²)² = (x² - y²)² + (2xy)² can be used to generate Pythagorean triples. Students calculate polynomial identities that include but are not limited to the product of the sum and difference of two terms, the difference of two squares, the sum and difference of two cubes, and the square of a binomial. Students prove polynomial identities by showing steps and providing reasons. 5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)ᴺ in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. 4. Rewrite rational expressions. 6. Rewrite simple rational expressions in different forms: write a(x) / b(x) in the form q(x) + r(x) / b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for more complicated examples, a computer algebra system. 7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Students illustrate how polynomial identities are used to determine numerical relationships. Students demonstrate calculations for Pascal’s Triangle to determine the coefficients of the binomial expansion. Students use the Binomial Theorem to find the nth term in the expansion of a binomial to a positive power. Students rewrite rational expressions, a(x)/b(x), in the form q(x) + r(x)/b(x) by using factoring, long division, or synthetic division. Students simplify rational expressions by adding, subtracting, multiplying, or dividing. Students understand that rational expressions are closed under addition, subtraction, multiplication, and division (by a nonzero expression). 4 Reformatted Saddlebrook Preparatory School COMMON CORE CLUSTER STANDARDS College and Career Readiness Cluster Standards for Mathematics 1. Create equations that describe numbers or relationships. COMMON CORE STATE STANDARD Algebra I and II Linear and quadratic polynomial expressions are the expectation in Algebra 1 and beyond quadratic polynomial expressions is the expectation for Algebra II. Creating Equations 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. 4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. WHAT DOES THIS STANDARD MEAN THAT A STUDENT MUST KNOW AND BE ABLE TO DO? Students create linear, quadratic, rational and exponential equations and inequalities in one variable and use them in a contextual situation to solve problems. Students create linear and quadratic equations from a set of given points. Students create equations in two or more variables to represent relationships between quantities. Students graph equations in two variables on a coordinate plane and label the axes and scales. Students write and use a system of equations and/or inequalities to solve real world problems recognizing that equations and inequalities represent the constraints of the problem. Students solve multi-variable formulas or literal equations for a specific variable. 5 Reformatted Saddlebrook Preparatory School Mathematics Common Core State Standards High School Algebra I and II COMMON CORE CLUSTER STANDARDS College and Career Readiness Cluster Standards for Mathematics COMMON CORE STATE STANDARD WHAT DOES THIS STANDARD MEAN THAT A STUDENT MUST KNOW AND BE ABLE TO DO? Algebra I students build on experiences graphing and solving systems of linear equations. Algebra I and II Algebra I students focus on and master A.REI.1 for linear equations and be able to extend and apply their reasoning to other types of equations in future courses. Students solve exponential equations with logarithms in Algebra II and extend to simple rational and radical equations. Reasoning with Equations and Inequalities 1. Understand solving equations as a 1. Explain each step in solving a simple equation Students construct a convincing argument that justifies each step in process of reasoning and explain the as following from the equality of numbers the solution process (assuming an equation has a solution). reasoning. asserted at the previous step, starting from the Justifications may include the associative, commutative, and assumption that the original equation has a division properties, combining terms, multiplication by the identity. solution. Construct a viable argument to justify a solution method. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 2. Solve equations and inequalities in one variable. 3.Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Students solve simple rational and radical equations in one variable, and identify and explain why they arise. Students solve linear equations in one variable, including coefficients represented by letters. 6 Reformatted Saddlebrook Preparatory School 4a. Use the method of completing the square to transform any quadratic equations in x into an equation of the form (x – p)² = q that has the same solutions. Derive the quadratic formula from this form. 4b. 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 plus/minus b for real numbers a and b. Students transform quadratic equations written in standard form to an equation in vertex form by completing the square. Students demonstrate deriving the quadratic formula by completing the square on the standard form of a quadratic equation. Students solve quadratic equations in one variable by simple inspection taking the square root, factoring, and completing the square. Students understand why taking the square root of both sides of an equation yields two solutions. Students use the quadratic formula to solve any quadratic equation, recognizing the formula produces all complex solutions. Students explain how complex solutions affect the graph of a quadratic equation. Students solve systems of equations using the elimination method. 3. Solve systems of equations. 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. Students solve a system of equations by substitution. Students solve a system of equations using matrices. Students solve systems of equations using graphs. 6. Solve systems of linear equations exactly and approximately (e.g. with graphs), focusing on pairs of linear equations in two variables. 7 Reformatted Saddlebrook Preparatory School 7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Students solve a system containing a linear equation and a quadratic equation in two variables graphically and symbolically. 8. (+) Represent a system of linear equations as a single matrix equation in a vector variable. Students write a system of linear equations as a single matrix equation. 9. (+) Students find the inverse of a matrix if it exists and use it to solve systems of linear equations. Students find the inverse of the coefficient matrix in the equation if it exists. Students use the inverse of the coefficient matrix to solve the system. Students use technology for matrices with dimensions 3 by 3 or greater. 4. Represent and solve equations and inequalities graphically. 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). Students demonstrate that all solutions to an equation in two variables are contained on the graph of that equation. 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. Students explain why the interaction 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. Students graph the solutions to a linear inequality in two variables as a half-plane, excluding the boundary for non-inclusive inequalities. Students graph the solutions set to a system of linear inequalities in two variables as the intersection of their corresponding half-planes. 8 Reformatted Saddlebrook Preparatory School 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 halfplanes. Students graph half-plane two variable linear inequality solutions and graph a solution set in two variables of the intersection of the half-planes. 9 Reformatted Saddlebrook Preparatory School