Algebra Cluster: Interpret the structure of expressions. A.SSE.1: Interpret expressions that represent a quantity in terms of its context (Modeling standard). a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity For example, interpret P(1+r) n as the product of P and a factor not depending on P. A.SSE.2: Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). A.SSE.1 Cluster: Interpret the structure of expressions. MP: Reason abstractively and quantitatively Knowledge: Define and recognize parts of an expression, such as terms, factors, and coefficients Reasoning: Interpret parts of an expression, such as terms, factors, and coefficients in terms of the context Reasoning: Interpret complicated expressions, in terms of the context, by viewing one or more of their parts as a single entity A.SSE.2 Cluster: Interpret the structure of expressions. Knowledge: Identify ways to rewrite expressions, such as difference of squares, factoring out a common monomial, and regrouping Knowledge: Identify various structures of expressions Reasoning: Use the structure of an expression to identify ways to rewrite it Reasoning: Classify expressions by structure and develop strategies to assist in classification 30
Cluster: Write expressions in equivalent forms to solve problems. A.SSE.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 zeros 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. For example the expression 1.15t can be rewritten as (1.151/12)12t 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. A.SSE.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. For example, calculate mortgage payments. A.SSE.3 Cluster: Write expressions in equivalent forms to solve problems. MP: Reason abstractively and quantitatively Knowledge: Explain the connection between the factored form of a quadratic expression and the zeros of the function it defines Knowledge: Explain the connection between the completed square form of a quadratic expression and the maximum or minimum value of the function it defines Knowledge: Explain the properties of the quantity represented by the quadratic expression Reasoning: Explain the properties of the quantity or quantities represented by the transformed exponential expression Reasoning: Complete the square on a quadratic expression to produce an equivalent form of an expression Reasoning: Choose and produce an equivalent form of a quadratic expression to reveal and explain properties of the quantity represented by the original expression Reasoning: Use the properties of exponents to transform simple expressions for exponential functions Demonstration: Choose and produce an equivalent form of an exponential expression to reveal and explain properties of the quantity represented by the original expression Demonstration: Choose and produce an equivalent form of a quadratic expression to reveal and explain properties of the quantity represented by the original expression Demonstration: Factor a quadratic expression to produce an equivalent form of the original expression 31
A.SSE.4 Cluster: Write expressions in equivalent forms to solve problems. MP: Reason abstractly and quantitatively Knowledge: Find the first term in a geometric sequence given at least two other terms Knowledge: Define a geometric series as a series with a constant ratio between successive terms Demonstration: Use the formula S + a (1-rn)/(1-r) to solve problems Demonstration: Derive a formula [i.e., equivalent to the formula S + a (1-rn)/(1-r)] for the sum of a finite geometric series (when the common ratio is not 1) Cluster: Perform arithmetic operations on polynomials. A.APR.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. A.APR.1 Cluster: Perform arithmetic operations on polynomials. Knowledge: Define closure Knowledge: Identify that the sum, difference, or product of two polynomials will always be a polynomial, which means that polynomials are closed under the operations of addition, subtraction, and multiplication Demonstration: Apply arithmetic operations of addition, subtraction, and multiplication to polynomials 32
Cluster: Understand the relationship between zeros and factors of polynomials. A.APR.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). A.APR.3: Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. A.APR.2 Cluster: Understand the relationship between zeros and factors of polynomials. Knowledge: Define the remainder theorem for polynomial division and divide polynomials Reasoning: Given a polynomial p(x) and a number a, divide p(x) by (x a) to find p(a), then apply the remainder theorem and conclude that p(x) is divisible by x a, if and only if p(a) = 0 A.APR.3 Cluster: Understand the relationship between zeros and factors of polynomials. Knowledge: Factor polynomials using any available method Demonstration: Create a sign chart for a polynomial f(x) using the polynomial s x-intercepts and testing the domain intervals for which f(x) greater than and less than zero Demonstration: Use the x-intercepts of a polynomial function and the sign chart to construct a rough graph of the function Cluster: Use polynomial identities to solve problems. A.APR.4: Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x 2 + y 2 )2 = (x 2 y 2) 2 + (2xy) 2 can be used to generate Pythagorean triples. A.APR.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. A.APR.4 Cluster: Use polynomial identities to solve problems. MP: Reason abstractively and quantitatively Knowledge: Explain that an identity shows a relationship between two quantities or expressions, that is true for all values of the variables, over a specified set Reasoning: Prove polynomial identities Demonstration: Use polynomial identities to describe numerical relationships 33
A.APR.5 Cluster: Use polynomial identities to solve problems. Knowledge: Define the Binomial Theorem and compute combinations Knowledge: Apply the Binomial Theorem to expand (x+y)n, when n is a positive integer and x and y are any number, rather than expanding by multiplying Reasoning: Explain the connection between Pascal s Triangle and the determination of the coefficients in the expansion of (x+y)n, when n is a positive integer and x and y are any number Cluster: Rewrite rational expressions. A.APR.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 the more complicated examples, a computer algebra system. A.APR.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. A.APR.6 Cluster: Rewrite rational expressions. Demonstration: Use inspection to 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). Demonstration: Use long division to 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). Demonstration: Use a computer algebra system to rewrite complicated 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). A.APR.7 Cluster: Rewrite rational expressions. MP: Reason abstractly and quantitatively Knowledge: Add, subtract, multiply, and divide rational expressions Demonstration: Informally verify that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression 34
Cluster: Create equations that describe numbers or relationships. A.CED.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. A.CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A.CED.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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. A.CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. A.CED.1 Cluster: Create equations that describe numbers or relationships. Knowledge: Describe the relationships between the quantities in the problem (for example, how the quantities are changing or growing with respect to each other); express these relationships using mathematical operations to create an appropriate equation or inequality to solve Knowledge: Describe the relationships between the quantities in the problem (for example, how the quantities are changing or growing with respect to each other); express these relationships using mathematical operations to create an appropriate equation or inequality to solve Reasoning: Use all available types of functions to create such equations, including root functions, but constrain to simple cases Reasoning: Compare and contrast problems that can be solved by different types of equations Reasoning: Compare and contrast problems that can be solved by different types of equations (linear, exponential Demonstration: Solve linear and exponential equations in one variable Demonstration: Solve inequalities in one variable Demonstration: Solve all available types of equations and inequalities, including root equations and inequalities, in one variable Demonstration: Create equations and inequalities in one variable and use them to solve problems Demonstration: Create equations and inequalities in one variable to model real-world situations Demonstration: Create equations (linear, exponential) and inequalities in one variable and use them to solve problems 35
A.CED.2 Cluster: Create equations that describe numbers or relationships. MP: Reason abstractly and quantitatively Knowledge: Identify the quantities in a mathematical problem or realworld situation that should be represented by distinct variables and describe what quantities the variables represent Reasoning: Graph one or more created equation on coordinate axes with appropriate labels and scales Reasoning: Justify which quantities in a mathematical problem or realworld situation are dependent and independent of one another and which operations represent those relationships Reasoning: Determine appropriate units for the labels and scale of a graph depicting the relationship between equations created in two or more variables Demonstration: Create at least two equations in two or more variables to represent relationships between quantities A.CED.3 Cluster: Create equations that describe numbers or relationships. MP: Reason abstractly and quantitatively Knowledge: Recognize when a modeling context involves constraints Reasoning: Interpret solutions as viable or nonviable options in a modeling context Reasoning: Determine when a problem should be represented by equations, inequalities, systems of equations and/or inequalities Reasoning: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities A.CED.4 Cluster: Create equations that describe numbers or relationships. MP: Reason abstractly and quantitatively Knowledge: Define a quantity of interest to mean any numerical or algebraic quantity (e.g., 2(a/b)=d in which 2 is the quantity of interest showing that d must be even; (πr 2 h/3)=vcone and πr 2 h=vcylinder showing that Vcylinder=3*Vcone ) Reasoning: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., π r 2 can be re-written as (π r)r which makes the form of this expression resemble Bh. The quantity of interest could also be (a + b)n = an b 0 + a(n - 1) b 1 +... + a 0 b n). 36
Cluster: Understand solving equations as a process of reasoning and explain the reasoning. A.REI.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. A.REI.2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. A.REI.1 Cluster: Understand solving equations as a process of reasoning and explain the reasoning. Knowledge: Know that solving an equation means that the equation remains balanced during each step Knowledge: Recall the properties of equality Reasoning: Explain why, when solving equations, it is assumed that the original equation is equal Reasoning: Determine if an equation has a solution Reasoning: Choose an appropriate method for solving the equation Reasoning: Justify solution(s) to equations by explaining each step in solving a simple equation using the properties of equality, beginning with the assumption that the original equation is equal Reasoning: Construct a mathematically viable argument justifying a given, or self-generated, solution method A.REI.2 Cluster: Understand solving equations as a process of reasoning and explain the reasoning. MP: Reason abstractly and quantitatively Knowledge: Determine the domain of a rational function Knowledge: Determine the domain of a radical function Demonstration: Solve radical equations in one variable Demonstration: Solve rational equations in one variable Demonstration: Give examples showing how extraneous solutions may arise when solving rational and radical equations 37
Cluster: Solve equations and inequalities in one variable. A.REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. A.REI.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. b. Solve quadratic equations by inspection (e.g., for x 2 = 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. A.REI.3 Cluster: Solve equations and inequalities in one variable. MP: Reason abstractively and quantitatively Knowledge: Recall properties of equality Reasoning: Determine the effect that rational coefficients have on the inequality symbol and use this to find the solution set Demonstration: Solve multi-step equations in one variable Demonstration: Solve multi-step inequalities in one variable Demonstration: Solve equations and inequalities with coefficients represented by letters A.REI.4 Cluster: Solve equations and inequalities in one variable. Knowledge: Recognize when the quadratic formula gives complex solutions Reasoning: Derive the quadratic formula by completing the square on a quadratic equation in x Reasoning: Express complex solutions as a ± bi for real number solutions as a and b Reasoning: Determine appropriate strategies to solve problems involving quadratic equations, as appropriate to the initial form of the equation Demonstration: 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 Demonstration: Solve quadratic equations in one variable 38
Cluster: Solve systems of equations. A.REI.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. A.REI.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. A.REI.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 x 2 + y 2 = 3. A.REI.8: (+) Represent a system of linear equations as a single matric equation in a vector variable. A.REI.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 x 3 or greater). A.REI.5 Cluster: Solve systems of equations. Knowledge: Recognize and use properties of equality to maintain equivalent systems of equations Reasoning: Justify that replacing one equation in a two-equation system with the sum of that equation and a multiple of the other will yield the same solutions as the original system A.REI.6 Cluster: Solve systems of equations. Knowledge: Solve systems of linear equations by any method Reasoning: Justify the method used to solve systems of linear equations exactly and approximately focusing on pairs of linear equations in two variables Reasoning: Notes from Appendix A: Build on student experiences graphing and solving systems of linear equations from middle school to focus on justification of the methods used. Include cases where the two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution); connect to GPE 5 where it is taught in Geometry, which requires students to prove the slope criteria for parallel lines 39
A.REI.7 Cluster: Solve systems of equations. Knowledge: Transform a simple system consisting of a linear equation and a quadratic equation in two variables so that a solution can be found algebraically and graphically Reasoning: Explain the correspondence between the algebraic and graphical solutions to a simple system consisting of a linear equation and a quadratic equation in two variables A.REI.8 Cluster: Solve systems of equations. Reasoning: Solve systems of equations Demonstration: Write a system of linear equations in vector variable form Demonstration: Write a system of linear equations as a single matric equation A.REI.9 Cluster: Solve systems of equations. Reasoning: Find the inverse of a matrix Demonstration: Solve a system of linear equations using inverse matrices Demonstration: Solve a system of linear equations with three or more variables using technology 40
Cluster: Represent and solve equations and inequalities graphically. A.REI.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). A.REI.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. A.REI.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. A.REI.10 A.REI.11 A.REI.12 Cluster: Represent and solve equations and inequalities graphically. Knowledge: Recognize that the graphical representation of an equation in two variables is a curve, which may be a straight line Knowledge: Explain why each point on a curve is a solution to its equation Cluster: Represent and solve equations and inequalities graphically. Knowledge: Recognize and use function notation to represent linear and exponential equations Knowledge: Recognize that if (x 1, y 1 ) and (x 2, y 2 ) share the same location in the coordinate plane that x 1 = x 2 and y 1 = y 2 Knowledge: Recognize that f(x) = g(x) means that there may be particular inputs of f and g for which the outputs of f and g are equal Knowledge: Recognize and use function notation to represent linear, polynomial, rational, absolute value, exponential, and radical equations. Reasoning: Explain why the x-coordinates of the points where the graph of the equations y = f(x) and y = g(x) intersect are the solutions of the equations f(x) = g(x) Reasoning: Approximate/find the solution(s) using an appropriate method. For example, using technology to graph the functions, make tables of values or find successive approximations Cluster: Represent and solve equations and inequalities graphically. Knowledge: Identify characteristics of a linear inequality and a system of linear inequalities, such as: boundary line and shading, and determine the appropriate points to test and derive a solution set from Reasoning: Explain the meaning of the intersection of shaded regions in a system of linear inequalities Demonstration: Graph a line or boundary line and shade the appropriate region for a two-variable linear inequalityt Demonstration: Graph a system of linear inequalities and shade the appropriate overlapping region for a system of linear inequalities PRODUCT 41