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CCRS StandardStandard IDEvidence of Student AttainmentTeacher VocabularyKnowledgeSkillsUnderstandingResources1. Know there is a complex number i such that i2 = 1, and every complex number has the form a + bi with a and b real.The Complex Number SystemPerform arithmetic operations with complex numbers.Number/QuantityNCN.1Students:Given an equation where x2 is less than zero,
l Explain by repeated reasoning from square roots in the positive numbers what conditions a solution must satisfy, how defining a number i by the equation i2 = 1 would satisfy those conditions, and extend the real numbers to a set called the complex numbers,
l Explain how adding and/or multiplying i by any real number results in a complex number and is real when the multiplier is zero.l Complex numberStudents know:
l Which manipulations of radicals produce equivalent forms, for example, "8 + "18 `" "26 but 2"2 + 3"2 = 5"2,
l That the extension of the real numbers which allows equations such as x2 = 1 to have solutions is known as the complex numbers and the defining feature of the complex numbers is a number i, such that i2 = 1.Students are able to:
l Perform manipulations of radicals, including those involving square roots of negative numbers, to produce a variety of forms, for example, "(8) = i"(8) = 2i"(2).Students understand that:
l When quadratic equations do not have real solutions, the number system must be extended so that solutions exist, and the extension must maintain properties of arithmetic in the real numbers. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54289" \t "_blank" ALEX Resources2. Use the relation i2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.The Complex Number SystemPerform arithmetic operations with complex numbers.Number/QuantityNCN.2Students:
l Produce equivalent expressions in the form a + bi, where a and b are real for combinations of complex numbers by using addition, subtraction, and multiplication and justify that these expressions are equivalent through the use of properties of operations and equality (Tables 3 and 4). Students know:
l Combinations of operations on complex number that produce equivalent expressions,
l Properties of operations and equality that verify this equivalence.Students are able to:
l Perform arithmetic manipulations on complex numbers to produce equivalent expressions.Students understand that:
l Complex number calculations follow the same rules of arithmetic as combining real numbers and algebraic expressions. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54290" \t "_blank" ALEX Resources3. Solve quadratic equations with real coefficients that have complex solutions.The Complex Number SystemUse complex numbers in polynomial identities and equations. (Polynomials with real coefficients.)Number/QuantityNCN.7Students:Given a contextual situation in which a quadratic solution is necessary find all solutions real or complex.l complex solutionStudents know:
l strategies for solving quadratic equationsStudents are able to:
l apply the quadratic equation
l provide solutions in complex formStudents understand that:
l all quadratic equations have two solutions: real or imaginary
l some contextual situations are better suited to quadratic solutionsClick below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54292" \t "_blank" ALEX Resources4. (+) Extend polynomial identities to the complex numbers. Example: Rewrite x2 + 4 as (x + 2i)(x 2i).The Complex Number SystemUse complex numbers in polynomial identities and equations. (Polynomials with real coefficients.)Number/QuantityNCN.8Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54293" \t "_blank" ALEX Resources5. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.The Complex Number SystemUse complex numbers in polynomial identities and equations. (Polynomials with real coefficients.)Number/QuantityNCN.9Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54295" \t "_blank" ALEX Resources6. Interpret expressions that represent a quantity in terms of its context.&
Interpret parts of an expression, such as terms, factors, and coefficients.
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.Seeing Structure in ExpressionsInterpret the structure of expressions. (Polynomial and rational.)AlgebraASSE.1Students:Given a contextual situation and an expression that does model it,
l Connect each part of the expression to the corresponding piece of the situation,
l Interpret parts of the expression such as terms, factors, and coefficients.l Terms
l Factors
l CoefficientsStudents know:
l Interpretations of parts of algebraic expressions such as terms, factors, and coefficients.Students are able to:
l Produce mathematical expressions that model given contexts,
l Provide a context that a given mathematical expression accurately fits,
l Explain the reasoning for selecting a particular algebraic expression by connecting the quantities in the expression to the physical situation that produced them, (e.g., the formula for the area of a trapezoid can be explained as the average of the two bases multiplied by height).
((a + b)/2) hStudents understand that:
l Physical situations can be represented by algebraic expressions which combine numbers from the context, variables representing unknown quantities, and operations indicated by the context,
l Different but equivalent algebraic expressions can be formed by approaching the context from a different perspective. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54297" \t "_blank" ALEX Resources7. Use the structure of an expression to identify ways to rewrite it. For example, see x4 y4 as (x2)2 (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 y2)(x2 + y2).Seeing Structure in ExpressionsInterpret the structure of expressions. (Polynomial and rational.)AlgebraASSE.2Students:
l Make sense of algebraic expressions by identifying structures within the expression which allow them to rewrite it in useful ways. Students know:
l Algebraic properties (including those in Tables 3, 4, and 5),
l When one form of an algebraic expression is more useful than an equivalent form of that same expression. Students are able to:
l Use algebraic properties to produce equivalent forms of the same expression by recognizing underlying mathematical structures.Students understand that:
l Generating simpler, but equivalent, algebraic expressions facilitates the investigation of more complex algebraic expressions. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54301" \t "_blank" ALEX Resources8. 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.&Seeing Structure in ExpressionsWrite expressions in equivalent forms to solve problems.AlgebraASSE.4Students:
l Present and defend the derivation of the formula for the sum of a finite geometric series. One approach:S = a + ar + ar2 + ar3 + . . . + arnrS = ar + ar2 + ar3 + . . . + arn+1rS  S = arn+1  a = a(rn+1  1) S = a(rn+1  1)/(r  1),
l Recognize geometric series which exist in problem situations and use the sum formula to simplify and solve problems.l Geometric seriesStudents know:
l Characteristics of a geometric series,
l Techniques for performing algebraic manipulations and justifications for the equivalence of the resulting expressions (including Tables 3, 4, and 5).Students are able to:
l Identify the regularity that exists in a series as being that which defines it as a geometric series.
l Accurately perform the procedures involved in using geometric series to solve contextual problems,
l Explain with mathematical reasoning why each step in the derivation of the formula for the sum of a finite geometric series is legitimate, including explaining why the formula does not hold for a common ratio of 1.Students understand that:
l When each term of a geometric series is multiplied by a value, the result is a new geometric series,
l When many problems exist with the same mathematical structure, formulas are useful generalizations for efficient solution of problems, (e.g., mortgage payment calculation with geometric series).Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54304" \t "_blank" ALEX Resources9. 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.Arithmetic with Polynomials and Rational ExpressionsPerform arithmetic operations on polynomials. (Beyond quadratic.)AlgebraAAPR.1Students:
l Use the repeated reasoning from prior knowledge of properties of arithmetic on integers to progress consistently to rules for arithmetic on polynomials,
l Accurately perform combinations of operations on various polynomials.l Polynomials
l ClosureStudents know:
l Corresponding rules of arithmetic of integers, specifically what it means for the integers to be closed under addition, subtraction, and multiplication, and not under division,
l Procedures for performing addition, subtraction, and multiplication on polynomials.Students are able to:
l Communicate the connection between the rules for arithmetic on integers and the corresponding rules for arithmetic on polynomials,
l Accurately perform combinations of operations on various polynomials.Students understand that:
l There is an operational connection between the arithmetic on integers and the arithmetic on polynomials. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54307" \t "_blank" ALEX Resources10. 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).Arithmetic with Polynomials and Rational ExpressionsUnderstand the relationship between zeros and factors of polynomials.AlgebraAAPR.2Students:Given a polynomial p(x):
l Identify when (x  a) is a factor of the given polynomial p(x),
l Identify when (x  a)is not a factor, then the remainder when p(x) is divided by (x  a) is p(a). l If and only if
l Remainder theoremStudents know:
l Procedures for dividing a polynomial p(x) by a linear polynomial (x  a), (e.g., long division and synthetic division). Students are able to:
l Accurately perform procedures for dividing a polynomial p(x) by a linear polynomial (x  a),
l Evaluate a polynomial p(x) for any value of x. Students understand that:
l There is a structural relationship between the value of a in (x  a), as well as the remainder when p(x) is divided by (x  a). Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54309" \t "_blank" ALEX Resources11. 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.Arithmetic with Polynomials and Rational ExpressionsUnderstand the relationship between zeros and factors of polynomials.AlgebraAAPR.3Students:Given any polynomial,
l Analyze and determine if suitable factorizations exist,
l Use the root determined by these factorizations to construct a graph of the given polynomials. The graph should include all real roots,
l Utilize techniques such as plotting points between and outside roots or use technology to find the general shape of the graph. l Zeros of polynomial
l Factorization Students know:
l When a factorization of a polynomial reveals a root of that polynomial,
l When a rearrangement of the terms of a polynomial expression can reveal a recognizable factorable form of the polynomial,
l Relationships of roots to points on the graph of the polynomial. Students are able to:
l Use techniques for factoring polynomials. Students understand that:
l Important features of the graph can be revealed by inputting values between the identified roots of the given polynomial.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54310" \t "_blank" ALEX Resources12. Prove polynomial identities and use them to describe numerical relationships. Example: The polynomial identity (x2 + y2)2 = (x2 y2)2 + (2xy)2 can be used to generate Pythagorean triples.Arithmetic with Polynomials and Rational ExpressionsUse polynomial identities to solve problems.AlgebraAAPR.4Students:
l Use properties of operations on polynomials to justify identities such as:
(a+b)2 = a2 + 2ab + b2
(a+b)(c+d) = ac + ad + bc + bd
a2  b2 = (a+b)(ab)
x2 + (a+b)x + ab = (x + a)(x + b)
(x2 + y2)2 = (x2  y2)2 + (2xy)2,
l Use these identities to describe numerical relationships (e.g., identity 3 can be used to mentally compute 79 x 81, or identity 5 can be used as a generator for Pythagorean triples). l Polynomial identityStudents know:
l Distributive Property of multiplication over addition.Students are able to:
l Accurately perform algebraic manipulations on polynomial expressions. Students understand that:
l Reasoning with abstract polynomial expressions reveals the underlying structure of the Real Number System,
l Justification of generalizations is necessary before using these generalizations in applied settings.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54312" \t "_blank" ALEX Resources13. (+) 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 Pascals Triangle.1 (1The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)Arithmetic with Polynomials and Rational ExpressionsUse polynomial identities to solve problems.AlgebraAAPR.5Students:
l Expand (x + y)n for sequential cases n = 1,2,3... that indicate a pattern, note the regularity in the pattern of exponents and coefficients, and generalize the expansion to the Binomial Theorem by, for example, connecting the pattern of Pascal's Triangle to the pattern of coefficients in the binomial expansion or through combinatory,
l Use the patterns present in the Binomial Theorem to expand binomials and identify coefficients of particular terms. l Binomial Theorem
l Pascal's Triangle
l CombinatoryStudents know:
l Distributive Property of multiplication over addition for polynomials,
l The generation pattern for Pascal's Triangle and which Binomial Expansion term has coefficients corresponding to each row.Students are able to:
l Accurately perform algebraic manipulations on polynomial expressions,
l Generate rows of Pascal's Triangle. Students understand that:
l Regularities noted in one part of mathematics may also be seen in very different areas of mathematics, (i.e., Pascal's Triangle from counting procedures and the Binomial Theorem) and these regularities are useful in computing or manipulating mathematical expressions.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54314" \t "_blank" ALEX Resources14. 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.Arithmetic with Polynomials and Rational ExpressionsRewrite rational expressions. (Linear and quadratic denominators.)AlgebraAAPR.6Students: Given a rational expression in the form a(x)/b(x),
l Rewrite rational expressions of the form q(x) + r(x)/b(x) with the degree of r(x) less than the degree of b(x),choosing the most appropriate technique from inspection when b(x)is a common factor of the terms of a(x), long division for other examples and a computer algebra system for more complicated examples. l Rational expression
l Degree of polynomial
l InspectionStudents know:
l Techniques for long division of polynomials,
l Techniques for utilizing a computer algebra system.Students are able to:
l Accurately perform polynomial long division,
l Efficiently and accurately use a computer algebra system to divide polynomials. Students understand that:
l The role of the remainder in polynomial division is analogous to that of the remainder in whole number division,
l Different forms of rational expressions are useful to reveal important features of the expression.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54316" \t "_blank" ALEX Resources15. (+) 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.Arithmetic with Polynomials and Rational ExpressionsRewrite rational expressions. (Linear and quadratic denominators.)AlgebraAAPR.7Students:
l Will explain with mathematical reasoning why adding, subtracting, multiplying, or dividing a rational expression by a nonzero rational expression must yield a rational expression. Alternatively, the sum, product, difference, or quotient of rational expression must be rational.
Given two rational expressions,
l Accurately produce a single rational expression that is the sum, product. difference, or quotient (nonzero divisor) of the two rational expressions.l Rational expression
l Closed under an operationStudents know:
l Techniques for performing the operations on polynomials. Students are able to:
l Accurately perform addition, subtraction, multiplication, and division of rational expressions. Students understand that:
l They can communicate a mathematical justification for all four operations on rational expressions being closed,
l The structure of mathematics (closed under the four operations) present in the system of rational numbers is also present in the system of rational expressions.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54317" \t "_blank" ALEX Resources16. 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. Creating Equations*Create equations that describe numbers or relationships. (Equations using all available types of expressions, including simple root functions.)AlgebraACED.1Students:Given a contextual situation that may include linear, quadratic, exponential, or rational functional relationships in one variable,
l Model the relationship with equations or inequalities and solve the problem presented in the contextual situation for the given variable.
(Please Note: This standard must be taught in conjunction with the standard that follows).Student know:
l When the situation presented in a contextual problem is most accurately modeled by a linear, quadratic, exponential, or rational functional relationship. Students are able to:
l Write equations or inequalities in one variable that accurately model contextual situations. Students understand that:
l Features of a contextual problem can be used to create a mathematical model for that problem. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54319" \t "_blank" ALEX Resources17. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.Creating Equations*Create equations that describe numbers or relationships. (Equations using all available types of expressions, including simple root functions.)AlgebraACED.2Students:Given a contextual situation expressing a relationship between quantities with two or more variables,
l Model the relationship with equations and graph the relationship on coordinate axes with labels and scales.
(Please Note: This standard must be taught in conjunction with the preceding standard).Students know:
l When a particular two variable equation accurately models the situation presented in a contextual problem.Students are able to:
l Write equations in two variables that accurately model contextual situations,
l Graph equations involving two variables on coordinate axes with appropriate scales and labels. Students understand that:
l There are relationships among features of a contextual problem, a created mathematical model for that problem, and a graph of that relationship. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54320" \t "_blank" ALEX Resources18. 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. Example: Represent inequalities describing nutritional and cost constraints on combinations of different foods.Creating Equations*Create equations that describe numbers or relationships. (Equations using all available types of expressions, including simple root functions.)AlgebraACED.3Students:Given a contextual situation involving constraints,
l Write equations or inequalities or a system of equations or inequalities that model the situation and justify each part of the model in terms of the context,
l Solve the equation, inequalities or systems and interpret the solution in the original context including discarding solutions to the mathematical model that cannot fit the real world situation (e.g., distance cannot be negative),
l Solve a system by graphing the system on the same coordinate grid and determine the point(s) or region that satisfies all members of the system,
l Determine the point(s) of the region satisfying all members of the system that maximizes or minimizes the variable of interest in the case of a system of inequalities.l ConstraintStudents know:
l When a particular system of two variable equations or inequalities accurately models the situation presented in a contextual problem,
l Which points in the solution of a system of linear inequalities need to be tested to maximize or minimize the variable of interest. Students are able to:
l Graph equations and inequalities involving two variables on coordinate axes,
l Identify the region that satisfies both inequalities in a system,
l Identify the point(s) that maximizes or minimizes the variable of interest in a system of inequalities,
l Test a mathematical model using equations, inequalities, or a system against the constraints in the context and interpret the solution in this context.Students understand that:
l A symbolic representation of relevant features of a real world problem can provide for resolution of the problem and interpretation of the situation and solution,
l Representing a physical situation with a mathematical model requires consideration of the accuracy and limitations of the model. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54321" \t "_blank" ALEX Resources19. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: Rearrange Ohms law V = IR to highlight resistance R.Creating Equations*Create equations that describe numbers or relationships. (Equations using all available types of expressions, including simple root functions.)AlgebraACED.4Students:
l Rearrange formulas which arise in contextual situations to isolate variables that are of interest for particular problems. For example, if the electric company charges for power by the formula COST = 0.03 KWH + 15, a consumer may wish to determine how many kilowatt hours they may use to keep the cost under particular amounts, by considering KWH< (COST  15)/0.03 which would yield to keep the monthly cost under $75, they need to use less than 2000 KWH.Students know:
l Properties of equality and inequality (Tables 4 and 5).Students are able to:
l Accurately rearrange equations or inequalities to produce equivalent forms for use in resolving situations of interest.Students understand that:
l The structure of mathematics allows for the procedures used in working with equations to also be valid when rearranging formulas,
l The isolated variable in a formula is not always the unknown and rearranging the formula allows for sensemaking in problem solving.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54323" \t "_blank" ALEX Resources20. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.Reasoning with Equations & InequalitiesUnderstand solving equations as a process of reasoning and explain the reasoning. (Simple rational and radical.)AlgebraAREI.2Students:
l Solve problems involving rational and radical equations in one variable,
l Identify extraneous solutions to these equations if any,
l Produce examples of equations that would or would not have extraneous solutions and communicate the conditions that lead to the extraneous solutions. l Rational equations
l Radical equations
l Extraneous solutionsStudents know:
l Algebraic rules for manipulating rational and radical equations,
l Conditions under which a solution is considered extraneous.Students are able to:
l Accurately rearrange rational and radical equations to produce a set of values to test against the conditions of the original situation and equation, and determine whether or not the value is a solution,
l Explain with mathematical and reasoning from the context (when appropriate) why a particular solution is or is not extraneous.Students understand that:
l Values which arise from solving equations may not satisfy the original equation,
l Values which arise from solving the equations may not exist due to considerations in the context.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54326" \t "_blank" ALEX Resources21. Explain why the xcoordinates 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.&Reasoning with Equations & InequalitiesRepresent and solve equations and inequalities graphically. (Combine polynomial, rational, radical, absolute value, and exponential functions.)AlgebraAREI.11Students: Given two functions (linear, polynomial, rational, absolute value, exponential, and logarithmic) that intersect (e.g., y= 3x and y= 2x),  Graph each function and identify the intersection point(s),  Explain solutions for f(x) = g(x) as the xcoordinate of the points of intersection of the graphs, and explain solution paths (e.g., the values that make 3x = 2x true, are the xcoordinate intersection points of y=3x and y=2x,  Use technology, tables, and successive approximations to produce the graphs, as well as to determine the approximation of solutions.l Functions
l Successive approximations
l Linear functions
l Polynomial functions
l Rational functions
l Absolute value functions
l Exponential functions
l Logarithmic functionsStudents know:
l Defining characteristics of linear, polynomial, rational, absolute value, exponential, and logarithmic graphs,
l Methods to use technology, tables, and successive approximations to produce graphs and tables for linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Students are able to:
l Determine a solution or solutions of a system of two functions,
l Accurately use technology to produce graphs and tables for linear, polynomial, rational, absolute value, exponential, and logarithmic functions,
l Accurately use technology to approximate solutions on graphs.Students understand that:
l When two functions are equal, the x coordinate(s) of the intersection of those functions is the value that produces the same output (yvalue) for both functions,
l Technology is useful to quickly and accurately determine solutions and produce graphs of functions. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54328" \t "_blank" ALEX Resources22. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.&Interpreting FunctionsInterpret functions that arise in applications in terms of the context. (Emphasize selection of appropriate models.)FunctionsFIF.4Students: Given a function that models a relationship between two quantities,
l Produce the graph and table of the function and show the key features (intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity) that are appropriate for the function.
Given key features from verbal description of a relationship,
l Sketch a graph with the given key features. l Function
l Key features Students know:
l Key features of function graphs (i.e., intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity),
l Methods of modeling relationships with a graph or table.Students are able to:
l Accurately graph any relationship,
l Interpret key features of a graph. Students understand that:
l The relationship between two variables determines the key features that need to be used when interpreting and producing the graph.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54330" \t "_blank" ALEX Resources23. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: If the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.&Interpreting FunctionsInterpret functions that arise in applications in terms of the context. (Emphasize selection of appropriate models.)FunctionsFIF.5Students: Given a contextual situation that is functional,
l Model the situation with a graph and construct the graph based on the parameters given in the domain of the context. l Function Students know:
l Techniques for graphing functions,
l Techniques for determining the domain of a function from its context.Students are able to:
l Interpret the domain from the context,
l Produce a graph of a function based on the context given.Students understand that:
l Different contexts produce different domains and graphs,
l Function notation in itself may produce graph points which should not be in the graph as the domain is limited by the context.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54331" \t "_blank" ALEX Resources24. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.&Interpreting FunctionsInterpret functions that arise in applications in terms of the context. (Emphasize selection of appropriate models.)FunctionsFIF.6Students:Given an interval on a graph or table,
l Calculate the average rate of change within the interval.
Given a graph of contextual situation,
l Estimate the rate of change between intervals that are appropriate for the summary of the context. l Average rate of change Students know:
l Techniques for graphing,
l Techniques for finding a rate of change over an interval on a table or graph,
l Techniques for estimating a rate of change over an interval on a graph. Students are able to:
l Calculate rate of change over an interval on a table or graph,
l Estimate a rate of change over an interval on a graph. Students understand that:
l The average provides information on the overall changes within an interval, not the details within the interval (an average of the endpoints of an interval does not tell you the significant features within the interval). Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54333" \t "_blank" ALEX Resources25. Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.&
Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions.
Graph polynomial functions, identifying zeros when suitable factorizatwx v w x
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Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.Interpreting FunctionsAnalyze functions using different representations. (Focus on using key features to guide selection of appropriate type of model function.)FunctionsFIF.7Students:Given a symbolic representation of a function (including linear, quadratic, square root, cube root, piecewisedefined functions, polynomial, exponential, logarithmic, trigonometric, and (+) rational),
l Produce an accurate graph (by hand in simple cases and by technology in more complicated cases) and justify that the graph is an alternate representation of the symbolic function,
l Identify key features of the graph and connect these graphical features to the symbolic function, specifically for special functions:
quadratic or linear (intercepts, maxima, and minima),
square root, cube root, and piecewisedefined functions, including step functions and absolute value functions (descriptive features such as the values that are in the range of the function and those that are not),
polynomial (zeros when suitable factorizations are available, end behavior),
(+) rational (zeros and asymptotes when suitable factorizations are available, end behavior),
exponential and logarithmic (intercepts and end behavior),
trigonometric functions (period, midline, and amplitude). Students know:
l Techniques for graphing,
l Key features of graphs of functions. Students are able to:
l Identify the type of function from the symbolic representation,
l Manipulate expressions to reveal important features for identification in the function,
l Accurately graph any relationship.Students understand that:
l Key features are different depending on the function,
l Identifying key features of functions aid in graphing and interpreting the function.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54335" \t "_blank" ALEX Resources26. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.Interpreting FunctionsAnalyze functions using different representations. (Focus on using key features to guide selection of appropriate type of model function.)FunctionsFIF.8Students:Given a contextual situation containing a function defined by an expression,
l Use algebraic properties to rewrite the expression in a form that makes key features of the function easier to find,
l Manipulate a quadratic function by factoring and completing the square to show zeros, extreme values, and symmetry of the graph,
l Explain and justify the meaning of zeros, extreme values, and symmetry of the graph in terms of the contextual situation,
l Apply exponential properties to expressions and explain and justify the meaning in a contextual situation. l Zeros
l Extreme values
l Symmetry
l Exponential growth or decayStudents know:
l Techniques to factor and complete the square,
l Properties of exponential expressions,
l Algebraic properties of equality (Table 4).Students are able to:
l Accurately manipulate (e.g., factoring, completing the square) expressions using appropriate technique(s) to reveal key properties of a function.Students understand that:
l An expression may be written in various equivalent forms,
l Some forms of the expression are more beneficial for revealing key properties of the function. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54339" \t "_blank" ALEX Resources27. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.Interpreting FunctionsAnalyze functions using different representations. (Focus on using key features to guide selection of appropriate type of model function.)FunctionsFIF.9Students:Given two functions represented differently (algebraically, graphically, numerically in tables, or by verbal descriptions),
l Use key features to compare the functions,
l Explain and justify the similarities and differences of the functions.Students know:
l Techniques to find key features of functions when presented in different ways,
l Techniques to convert a function to a different form (algebraically, graphically, numerically in tables, or by verbal descriptions).Students are able to:
l Accurately determine which key features are most appropriate for comparing functions,
l Manipulate functions algebraically to reveal key functions,
l Convert a function to a different form (algebraically, graphically, numerically in tables, or by verbal descriptions) for the purpose of comparing it to another function.Students understand that:
l Functions can be written in different but equivalent ways (algebraically, graphically, numerically in tables, or by verbal descriptions),
l Different representations of functions may aid in comparing key features of the functions. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54340" \t "_blank" ALEX Resources28. Combine standard function types using arithmetic operations. Example: Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.Building FunctionsBuild a function that models a relationship between two quantities. (Include all types of functions studied.)FunctionsFBF.1BStudents:Given a contextual situation containing two quantities,
l Express a relationship between the quantities through an explicit expression using function notation, recursive process, or steps for calculation,
l Explain and justify how the expression or process models the relationship between the given quantities,
l Create a new function by using standard function types and arithmetic operations to combine the original functions to model the relationship of the given quantities, (+) standards not covered.l Explicit expression
l Recursive process
l Decaying exponentialStudents know:
l Techniques for expressing functional relationships (explicit expression, a recursive process, or steps for calculation) between two quantities,
l Techniques to combine functions using arithmetic operations.Students are able to:
l Accurately develop a model that shows the functional relationship between two quantities,
l Accurately create a new function through arithmetic operations of other functions,
l Present an argument to show how the function models the relationship between the quantities.Students understand that:
l Relationships can be modeled by several methods (e.g., explicit expression or recursive process),
l Arithmetic combinations of functions may be used to improve the fit of a model.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54343" \t "_blank" ALEX Resources29. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.Building FunctionsBuild new functions from existing functions. (Include simple radical, rational, and exponential functions; emphasize common effect of each transformation across function types.)FunctionsFBF.3Students:Given a function in algebraic form,
l Graph the function, f(x), conjecture how the graph of f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k(both positive and negative) will change from f(x), and test the conjectures,
l Describe how the graphs of the functions were affected (e.g., horizontal and vertical shifts, horizontal and vertical stretches, or reflections),
l Use technology to explain possible effects on the graph from adding or multiplying the input or output of a function by a constant value,
l Recognize if a function is even or odd.
Given the graph of a function and the graph of a translation, stretch, or reflection of that function,
l Determine the value which was used to shift, stretch, or reflect the graph,
l Recognize if a function is even or odd.l Even and odd functionsStudents know:
l Graphing techniques of functions,
l Methods of using technology to graph functions,
l Techniques to identify even and odd functions both algebraically and from a graph.Students are able to:
l Accurately graph functions,
l Check conjectures about how a parameter change in a function changes the graph and critique the reasoning of others about such shifts,
l Identify shifts, stretches, or reflections between graphs,
l Determine when a function is even or odd.Students understand that:
l Graphs of functions may be shifted, stretched, or reflected by adding or multiplying the input or output of a function by a constant value,
l Even and odd functions may be identified from a graph or algebraic form of a function.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54346" \t "_blank" ALEX Resources30. Solve an equation of the form f(x) = c for a simple function f that has an inverse, and write an expression for the inverse. Example: f(x) =2 x3 or f(x) = (x+1)/(x 1) for x `" 1.Building FunctionsBuild new functions from existing functions. (Include simple radical, rational, and exponential functions; emphasize common effect of each transformation across function types.)FunctionsFBF.4AStudents:Given an invertible function in algebraic form,
l Use algebraic properties to find the inverse of the given function, (+) standards not covered.l InverseStudents know:
l Algebraic properties of equality (Table 4).Students are able to:
l Accurately perform algebraic properties to find the inverse.Students understand that:
l The inverse of a function interchanges the input and output values from the original function,
l The inverse of a function must also be a function to exist and the domain may need to be restricted to make this occur.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54347" \t "_blank" ALEX Resources31. For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers, and the base b is 2, 10, or e; evaluate the logarithm using technology.Linear, Quadratic, & Exponential Models*Construct and compare linear, quadratic, and exponential models and solve problems. (Logarithms as solutions for exponentials.)FunctionsFLE.4Students: Given a contextual situation involving exponential growth or decay,
l Develop an exponential function which models the situation,
l Rewrite the exponential function as an equivalent logarithmic function,
l Use logarithmic properties to rearrange the logarithmic function, to isolate the variable, and use technology to find an approximation of the solution.l Exponential modelsStudents know:
l Methods for using exponential and logarithmic properties to solve equations,
l Techniques for rewriting algebraic expressions using properties of equality (Table 4).Students are able to:
l Accurately use logarithmic properties to rewrite and solve an exponential equation,
l Use technology to approximate a logarithm.Students understand that:
l Logarithmic and exponential functions are inverses of each other, and may be used interchangeably to aid in the solution of problems. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54350" \t "_blank" ALEX Resources32. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.Interpreting Categorical & Quantitative DataSummarize, represent, and interpret data on a single count or measurement variable.Statistics & ProbabilitySID.4Students:Given a data set,
l Find the mean and standard deviation of the set and use them to fit data to a normal distribution (when appropriate) to estimate population percentages,
l Estimate areas under the normal curve using calculators, spreadsheets, and standard normal distribution tables.l Normal distribution
l Population Percentages
l Normal curveStudents know:
l Techniques to find the mean and standard deviation of data sets,
l Techniques to use calculators, spreadsheets, and standard normal distribution tables to estimate areas under the normal curve.Students are able to:
l Accurately find the mean and standard deviation of data sets,
l Make reasonable estimates of population percentages from the normal distribution,
l Read and use normal distribution tables and use calculators and spreadsheets to accurately estimate the areas under the normal curve.Students understand that:Under appropriate conditions,
l The mean and standard deviation of a data set can be used to fit the data set to a normal distribution,
l Population percentages can be estimated by areas under the normal curve using calculators, spreadsheets, and standard normal distribution tables.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54352" \t "_blank" ALEX Resources33. Understand statistics as a process for making inferences about population parameters based on a random sample from that population.Making Inferences & Justifying ConclusionsUnderstand and evaluate random processes underlying statistical experiments.Statistics & ProbabilitySIC.1Students:Given a statistical question about a population,
l Describe and justify a data collection process that would result in a random sample from which inferences about the population can be drawn,
l Explain and justify their reasoning concerning data collection processes that do not allow generalizations (e.g., nonrandom samples) from the sample to the population. l Population parameters
l Random samplesStudents know:
l Techniques for selecting random samples from a population.Students are able to:
l Accurately compute the statistics needed,
l Recognize if a sample is random,
l Reach accurate conclusions regarding the population from the sample.Students understand that:
l Statistics generated from an appropriate sample are used to make inferences about the population.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54354" \t "_blank" ALEX Resources34. Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation. Example: A model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?Making Inferences & Justifying ConclusionsUnderstand and evaluate random processes underlying statistical experiments.Statistics & ProbabilitySIC.2Students:Given a datagenerating process,
l Identify a model that fits the data generating process,
l Find the theoretical probability or expected values generated by the model,
l Simulate the process and compare the results to theoretical probability or expected values. l Simulation
l Specified modelStudents know:
l Methods to determine if results from simulations are consistent with theoretical models. Students are able to:
l Accurately find the theoretical values from a model of a datagenerating process,
l Conduct a simulation that matches a given datagenerating process,
l Identify and communicate the differences and similarities between a theoretical model and the simulation results. Students understand that:
l A data generating process yields results that may or may not fit the underlying theoretical model, and statistical methods are used to determine the accuracy of the fit. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54355" \t "_blank" ALEX Resources35. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.Making Inferences & Justifying ConclusionsMake inferences and justify conclusions from sample surveys, experiments, and observational studies.Statistics & ProbabilitySIC.3Students:Given a scenario in which a statistical question needs to be investigated,
l Select an appropriate method of data collection (sample surveys, experiments, and observational studies) and justify the selection,
l Show randomization leads to more accurate inferences to the population.l Sample surveys
l Experiments
l Observational studies
l RandomizationStudents know:
l Key components of sample surveys, experiments, and observational studies,
l Procedures for selecting random samples.Students are able to:
l Use key characteristics of sample surveys, experiments, and observational studies to select the appropriate technique for a particular statistical investigation. Students understand that:
l Sample surveys, experiments, and observational studies may be used to make inferences made about the population,
l Randomization is used to reduce bias in statistical procedures. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54358" \t "_blank" ALEX Resources36. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.Making Inferences & Justifying ConclusionsMake inferences and justify conclusions from sample surveys, experiments, and observational studies.Statistics & ProbabilitySIC.4Students:Given a scenario in which a statistical question may be investigated by a sample survey,
l Use random sampling to gather survey data and use the sample mean or proportion to estimate the corresponding population values,
l Design a simulation for the situation, carry out the simulation, explain what the results mean about variability in population means or proportions, and use results to calculate margins of error for these estimates.l Sample survey
l Population mean
l Proportion
l Margin of error
l Random samplingStudents know:
l Techniques for conducting a sample survey,
l Techniques for conducting a simulation of a sample survey situation.Students are able to:
l Design and conduct sample surveys,
l Accurately calculate the point estimate of the population mean or proportion,
l Design, conduct, and use the results from a simulation model to develop a margin of error for a sample survey.Students understand that:
l Results of sample surveys are used to find an estimate of a population mean or proportion,
l Using sample values to estimate population values must be done with a consideration of the error in the estimate,
l Statistical analysis and data displays often reveal patterns in data or populations, enabling predictions. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54359" \t "_blank" ALEX Resources37. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.Making Inferences & Justifying ConclusionsMake inferences and justify conclusions from sample surveys, experiments, and observational studies.Statistics & ProbabilitySIC.5Students:Given a scenario in which a statistical question may be investigated by a randomized experiment,
l Design and conduct a randomized experiment to evaluate differences in two treatments based on data from randomized experiments,
l Interpret and explain the results in the context of the original scenario,
l Design, conduct, and use simulations to generate data simulating application of the two treatments,
l Use results of the simulation to evaluate significance of differences in the parameters of interest.l Randomized experiment
l SignificantStudents know:
l Techniques for conducting randomized experiments,
l Techniques for conducting simulations of randomized experiment situations.Students are able to:
l Design and conduct randomized experiments with two treatments,
l Draw conclusions from comparisons of the data of the randomized experiment,
l Design, conduct, and use the results from simulations of a randomized experiment situation to evaluate the significance of the identified differences.Students understand that:
l Differences of two treatments can be justified by a significant difference of parameters from a randomized experiment,
l Statistical analysis and data displays often reveal patterns in data or populations, enabling predictions. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54360" \t "_blank" ALEX Resources38. Evaluate reports based on data.Making Inferences & Justifying ConclusionsMake inferences and justify conclusions from sample surveys, experiments, and observational studies.Statistics & ProbabilitySIC.6Students:Given data based reports,
l Evaluate the data collection procedures and the interpretations of results in the context that yielded the data.Students know:
l Experimental designs that are appropriate for situations present in reports.Students are able to:
l Interpret the data and the results presented in a report.Students understand that:
l Statistical analysis and data displays often reveal patterns in data or populations, enabling predictions,
l Misleading data displays and biased sampling procedures can be used in published reports and must be carefully analyzed.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54361" \t "_blank" ALEX Resources39. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).Using Probability to Make DecisionsUse probability to evaluate outcomes of decisions. (Include more complex situations.)Statistics & ProbabilitySMD.6AClick below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54363" \t "_blank" ALEX Resources40. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).Using Probability to Make DecisionsUse probability to evaluate outcomes of decisions. (Include more complex situations.)Statistics & ProbabilitySMD.7AClick below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=54364" \t "_blank" ALEX Resources
HYPERLINK "http://alex.state.al.us/insight/lauderdaleco/standardprint/getstandards" \l "#" print
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