WEEKS (WHEN WILL YOU TEACH THE Web viewanalogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract,

Algebra IIA , B, & CModified: August 4, 2014

Algebra 2 A1st Semester

Timeline:1st Semester

1.5 Weeks

Vocabulary:

Arithmetic MeanArithmetic SequenceArithmetic SeriesCommon DifferenceCommon RatioExplicit FormulaGeometric MeanRecursive FormulaSequenceSeriesTermSummation NotationInfinite Series

Unit 1: Sequences and Series

New State Standards:

A-SSE Seeing Structure in Expressions1. Interpret expressions that represent a quantity in terms of its context.★

a. Interpret parts of an expression, such as terms, factors, and coefficients.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.★

F-IF Interpreting Functions3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★

F-BF Building Functions1. Write a function that describes a relationship between two quantities.★

a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★

F-LE Linear, Quadratic and Exponential Models2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

College Readiness:Range (13-15) Basic Operations & Applications: Perform one-operation computation with whole numbers and decimalsRange (13-15) Basic Operations & Applications: Solve problems in one or two steps using whole numbersRange (16-19) Basic Operations & Applications: Solve routine one-step arithmetic problems (using whole numbers, fractions, and decimals) such as single step percentRange (16-19) Basic Operations & Applications: Solve some routine two-step arithmetic problemsRange (20-23) Numbers: Concepts & Properties: Exhibit knowledge of elementary number concepts including rounding, the ordering of decimals, pattern identification, absolute value, primes, andgreatest common factorRange (33-36) Numbers: Concepts & Properties: Draw conclusions based on number concepts, algebraic properties, and/or relationships between expressions and numbersRange (13-15) Expressions, Equations, & Inequalities: Exhibit knowledge of basic expressions (e.g., identify an expression for a total as b + g)Range (13-15) Expressions, Equations, & Inequalities: Solve equations in the form x + a = b, where a and b are whole numbers or decimals

Activities:

Concepts and Skills:Students will: Generate a sequence Use a recursive formula Identify an arithmetic

sequence Use an arithmetic mean Identify a geometric

sequence Use a geometric mean Write and evaluate

arithmetic and geometric series

Write a series in summation notation

Find the sum of a finite arithmetic series

Find the sum of a finite geometric series

Find the sum of an infinite arithmetic series

Find the sum of an infinite geometric series

Unit Learning Targets

I can use the fundamental counting principle to count the number of ways an event can happen.

I can use permutations

Resources:

Algebra II textbook: Larson, Boswell, Kanold, Stiff

CH: 8 Internet Research Infinite Algebra 1 Infinite Geometry Infinite Algebra 2 Edmodo Quality Core Infinite Algebra II Discovery Education www.Khanacademy.org brightstorm.com/math/

algebra-2/

Strategies: Graphing calculator

applications Exit slips ADP activities Enrichment activities ACT/SAT integration Real World Applications Vocabulary Builders

http://www.Khanacademy.org/


Range (16-19) Expressions, Equations, & Inequalities: Solve one-step equations having integer or decimal answersRange (20-23) Expressions, Equations, & Inequalities: Evaluate algebraic expressions by substituting integers for unknown quantitiesRange (20-23) Expressions, Equations, & Inequalities: Solve routine first-degree equationsRange (28-32) Expressions, Equations, & Inequalities: Manipulate expressions and equationsRange (33-36) Expressions, Equations, & Inequalities: Write expressions that require planning and/or manipulating to accurately model a situation

NCTM:Numbers & Operations

use number-theory arguments to justify relationships involving whole numbers. judge the reasonableness of numerical computations and their results

Algebra generalize patterns using explicitly defined and recursively defined functions understand relations and functions and select, convert flexibly among, and use various

representations for them use a variety of symbolic representations, including recursive and parametric equations, for

functions and relations use symbolic algebra to represent and explain mathematical relationships use symbolic expressions, including iterative and recursive forms, to represent relationships

arising from various contexts

Quality CoreA1A: Identify properties of real numbers and use them and the correct order of operations to simplify expressionsA1D: Solve single-step and multistep equations and inequalities in one variableA1J: Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid conclusionsB1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsH2A: Find the nth term of an arithmetic or geometric sequenceH2B: Find the position of a given term of an arithmetic or geometric sequenceH2C: Find sums of a finite arithmetic or geometric seriesH2D: Use sequences and series to solve real-world problemsH2E: Use sigma notation to express sums

to count the number of ways an event can happen.

I can combination to count the number of ways can happen.

I can use the binomial theorem to expand a binomial that is raised to power.

I can find the theoretical and experimental probabilities.

I can find geometric probabilities.

I can find probabilities of unions and intersections of two events.

I can use the compliments to find the probability of


an event. I can find the

probability of independent events.

I can find the probability of dependent events.

I can find the binomial probabilities and analyze binomial distributions.

I can test the hypothesis.

I can calculate probabilities using normal distributions.

I can use normal distributions to approximate binomial distributions.

I can find expected values of collections of outcomes.



1 weeksLinear Inequalities

1.5 weeksSystem of Inequalities/ Linear Programming

Vocabulary:InequalitiesExpressionsEquationsSubstitutionOne step equationsTwo step equationsLike termsTermsFactorsCoefficientsDegreeDistributiveClosureCommutativeAssociativeIdentitiesInverse PropertiesLeading CoefficientLiteral EquationConstraintsLinear ProgrammingObjective FunctionFeasible RegionVerticesOptimizationMaximumMinimumCompound InequalityAbsolute ValueIntersectionInterval NotationSet Notation

Unit 2: InequalitiesUnit 3: Systems of Inequalities and Linear ProgrammingNew State Standards:A-SSE Seeing Structure in ExpressionsInterpret the structure of expressions1. Interpret expressions that represent a quantity in terms of its context.

a) Interpret parts of an expression, such as terms, factors, and coefficients.

A-CED Creating EquationsCreate equations that describe numbers or relationships1. Create equations and inequalities in one variable and use them to solve problems3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.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-REI Reasoning with Equations and InequalitiesUnderstand solving equations as a process of reasoning and explain the reasoning1. 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.

Solve equations and inequalities in one variable3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Represent and solve equations and inequalities graphically11. 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.★

College Readiness:(Range 13-15) Basic Operations and Applications: Perform one-operation computation withwhole numbers and decimals(Range 13-15) Basic Operations and Applications: Solve problems in one or two stepsusing whole numbers(Range 16-19) Basic Operations and Applications: Solve some routine two-step arithmetic problems(Range 30-36) Numbers: Concepts & Properties: Draw conclusions based on number concepts, algebraic properties, and/or relationships between expressions and numbers(Range 20-23) Numbers: Concepts & Properties: Exhibit knowledge of elementary number concepts including rounding, the ordering of decimals, pattern identification, absolute value, primes, and greatest common factor(Range 13-15) Expressions, Equations, & Inequalities: Exhibit knowledge of basic expressions (e.g., identify an expression for a total as b + g)(Range 13-15) Expressions, Equations, & Inequalities: Solve equations in the form x + a = b, where a and b are whole numbers or decimals(Range 16-19) Expressions, Equations, & Inequalities: Substitute whole numbers for unknown

Activities:

Concepts and Skills:Students will: Find distance and midpoint Solve equations and

inequalities for word problems

Write and solve equations and inequalities for word problems

Simplify expressions and interpret their parts

Identify and use algebraic properties of equality

Solve literal equations (for a specified variable)

Solve system of equation and inequalities using various methods (including graphing)

Solve linear programming problems

Solve absolute value inequalities

Multiple forms for linear equations including point slope

Learning Targets I can evaluate

algebraic expressions.

I can simplify algebraic expressions by combining like terms.

I can solve linear equations

I can use linear

Resources:



algebra-2/





quantities to evaluate expressions(Range 16-19) Expressions, Equations, & Inequalities: Solve one-step equations having integer or decimal answers(Range 16-19) Expressions, Equations, & Inequalities: Combine like terms (e.g., 2x + 5x)(Range 20-23) Expressions, Equations, & Inequalities: Evaluate algebraic expressions by substituting integers for unknown quantities(Range 20-23) Expressions, Equations, & Inequalities: Add and subtract simple algebraic expressions(Range 20-23) Expressions, Equations, & Inequalities: Solve routine first-degree equations(Range 20-23) Expressions, Equations, & Inequalities: Perform straightforward word-to-symbol translations(Range 24-27) Expressions, Equations, & Inequalities: Solve real-world problems using first degreeEquations(Range 24-27) Expressions, Equations, & Inequalities: Write expressions, equations, or inequalities with a single variable for common pre-algebra settings (e.g., rate and distance problems and problems that can be solved by using proportions)(Range 24-27) Expressions, Equations, & Inequalities: Solve first-degree inequalities that do not require reversing the inequality sign(Range 28-32) Expressions, Equations, & Inequalities: Manipulate expressions and equations(Range 28-32) Expressions, Equations, & Inequalities: Write expressions, equations, and inequalities for common algebra settings(Range 28-32) Expressions, Equations, & Inequalities: Solve linear inequalities that require reversing the inequality sign(Range 33-36) Expressions, Equations, & Inequalities: Write expressions that require planning and/or manipulating to accurately model a situation(Range 33-36) Expressions, Equations, & Inequalities: Write equations and inequalities that require planning, manipulating, and/or solving

NCTM:Number and Operations:

Develop a deeper understanding of very large and very small numbers and of various representations of them;

Use number-theory arguments to justify relationships involving whole numbers.Algebra:

understand the meaning of equivalent forms of expressions, equations, inequalities, and relations;

write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases;

Use symbolic algebra to represent and explain mathematical relationships judge the meaning, utility, and reasonableness of the results of symbol manipulations,

including those carried out by technology. Draw reasonable conclusions about a situation being modeled

Quality Core:A1D: Solve single-step and multistep equations and inequalities in one variableA1F: Write linear equations in standard form and slope-intercept form when given two points, a point and the slope, or the graph of the equationA1G: Graph a linear equation using a table of values, x- and y-intercepts, or slope-intercept formA1H: Find the distance and midpoint between two points in the coordinate planeA1J: Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid

equations to solve real world problems.

I can rewrite equations with more than one variable

I can rewrite common formulas

I can solve simple inequalities

I can solve compound inequalities

I can solve absolute value equations and inequalities.

I can use absolute value equations and inequalities to solve real world problems.


conclusionsB1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsD1A: Solve linear inequalities containing absolute valueD1B: Solve compound inequalities containing “and” and “or” and graph the solution setD2A: Graph a system of linear inequalities in two variables with and without technology to find the solution set to the systemD2B: Solve linear programming problems by finding maximum and minimum values of a function over a region defined by linear inequalities


1.5 Weeks

Vocabulary:Augmented MatrixDeterminantEqual MatricesMatrixMatrix AdditionMatrix ElementMatrix EquationMatrix MultiplicationRow OperationsScalar MultiplicationVariable MatrixZero MatrixSquare MatrixInverse Matrix

Unit 4: Matrices

New State Standards:N-VM Vector & Matrix Quantities6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.8. (+) Add, subtract, and multiply matrices of appropriate dimensions.9. (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.10. (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

A-REI Reasoning with Equations & Inequalities1. 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.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.8. (+) Represent a system of linear equations as a single matrix equation in a vector variable.9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

College Readiness:(Range 13-15) Basic Operations & Applications: Perform one-operation computation with whole numbers and decimals

Activities:

Concepts and Skills:Students will: Write the dimensions of a

matrix Identify a matrix element Use identity and inverse

matrices Subtract matrices Determine equal matrices Find unknown matrix

elements Use scalar products Solve matrix equations

with scalars Multiply matrices Determine if matrix

multiplication is defined Verify matrix inverses Evaluate determinant of

2X2 matrix Find an inverse matrix Solve a matrix equation Evaluate determinant of

3X3 matrix Use technology to solve

matrix problems

Resources:



algebra-2/





(Range 13-15) Basic Operations & Applications: Solve problems in one or two steps using whole numbers(Range 16-19) Basic Operations & Applications: Solve routine one-step arithmetic problems (using whole numbers, fractions, and decimals) such as single step percent(Range 16-19) Basic Operations & Applications: Solve some routine two-step arithmetic problems(Range 33-36) Numbers: Concepts &Properties: Draw conclusions based on number concepts, algebraic properties, and/or relationships between expressions and numbers(Range 13-15) Numbers: Concepts &Properties: Perform one-operation computation with whole numbers and decimals(Range 13-15) Expressions, Equations, &Inequalities: Exhibit knowledge of basic expressions (e.g., identify an expression for a total as b + g)(Range 13-15) Expressions, Equations, &Inequalities: Solve equations in the form x + a = b, where a and b are whole numbers or decimals(Range 16-19) Expressions, Equations, &Inequalities: Solve one-step equations having integer or decimal answers(Range 20-23) Expressions, Equations, &Inequalities: Add and subtract simple algebraic expressions(Range 20-23) Expressions, Equations, &Inequalities: Solve routine first-degree equations(Range 28-32) Expressions, Equations, &Inequalities: Manipulate expressions and equations(Range 28-32) Expressions, Equations, &Inequalities: Find solutions to systems of linear equations


understand vectors and matrices as systems that have some of the properties of the real-number system

judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities

develop an understanding of properties of, and representations for, the addition and multiplication of vectors and matrices

develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases

judge the reasonableness of numerical computations and their resultsAlgebra:

understand relations and functions and select, convert flexibly among, and use various representations for them

interpret representations of functions of two variables understand the meaning of equivalent forms of expressions, equations, inequalities, and

relations write equivalent forms of equations, inequalities, and systems of equations and solve them

with fluency—mentally or with paper and pencil in simple cases and using technology in all cases

judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.

Quality Core:A1A: Identify properties of real numbers and use them and the correct order of operations to simplify expressionsA1D: Solve single-step and multistep equations and inequalities in one variable

Write a system as a matrix equation

Solve a system of two equations

Solve a system of three equations

Use Cramer’s Rule Write an augmented matrix Write a system from and

augmented matrix


A1E: Solve systems of two linear equations using various methods, including elimination, substitution, and graphingB1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsD1C: Solve algebraically a system containing three variablesF1A: Evaluate and simplify polynomial expressions and equationsI1A: Add, subtract, and multiply matricesI1B: Use addition, subtraction, and multiplication of matrices to solve real-world problemsI1C: Calculate the determinant of 2 × 2 and 3 × 3 matricesI1D: Find the inverse of a 2 × 2 matrixI1E: Solve systems of equations by using inverses of matrices and determinantsI1F: Use technology to perform operations on matrices, find determinants, and find inverses


2.5 weeks

Vocabulary:Quadratic FunctionFactorGCFMonomialBinomialTrinomialPerfect Square TrinomialDifference of Squares

Unit 5: Factoring Quadratics and Complex Numbers

New State Standards:A-SSE Seeing Structure in ExpressionsInterpret the structure of expressions1. Interpret expressions that represent a quantity in terms of its context.★

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.2. 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).

College Readiness:(Range 16-19) Number Concepts and Properties: Recognize one-digit factors of a number(Range 24-27) Number Concepts and Properties: Work with numerical factors(Range 28-32) Number Concepts and Properties: Apply number properties involving prime factorization(Range 28-32) Number Concepts and Properties: Apply number properties involving even/odd numbers and factors/multiples(Range 20-23) Expressions, Equations, and Inequalities: Multiply two binomials*(Range 24-27) Expressions, Equations, and Inequalities: Factor simple quadratics (e.g., the difference of squares and perfect square trinomials) *(Range 28-32) Expressions, Equations, and Inequalities: Manipulate expressions and equations

NCTM:Algebra:

Understand the meaning of equivalent forms of expressions Use symbolic algebra to represent and explain mathematical relationships

Quality Core:B1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsC1A: Identify complex numbers and write their conjugatesC1C: Simplify quotients of complex numbersF1B: Factor polynomials using a variety of methods (e.g., factor theorem, synthetic division, long

Activities:

Concepts and Skills:Students will: Define parts of a quadratic

function Find greatest common

factors Factor quadratics Include graphical

representations of factoring

Students will: Solve by factoring Solve by finding square

roots Solving by graphing Simplify radicals using i Perform operations on

complex numbers (including rationalize the denominator)

Factor using imaginary numbersFinding complex solutions

Resources:



algebra-2/





division, sums and differences of cubes, grouping)


Unit 6: Solving Quadratic Equations Activities: Resources:


2 week

Vocabulary:Quadratic FunctionFactorGCFMonomialBinomialTrinomialPerfect Square TrinomialDifference of SquaresComplete the squareComplex numberQuadratic FormulaDiscriminantZerosRadicalsSquare rootsStandard formConjugate

New State Standards:N-CN Complex Number System1. 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.2. Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.7. Solve quadratic equations with real coefficients that have complex solutions8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.


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.

2. 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).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.

A-CED Creating Equations2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A-REI Reasoning with Equations & Inequalities1. 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.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 x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

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).

College Readiness:(Range 13-15) Basic Operations and Applications: Perform one-operation computation withwhole numbers and decimals(Range 13-15) Basic Operations and Applications: Solve problems in one or two stepsusing whole numbers(Range 16-19) Basic Operations and Applications: Solve some routine two-step arithmetic problems

Concepts and Skills:Students will: Complete the square to get

vertex form Solve by completing the

square Derive the quadratic

formula Find complex solutions Find x-intercepts where

leading coefficient not equal to 1

Rewrite quadratics in vertex from

Use the Quadratic Formula Find the discriminant Use the discriminant to

determine number and type of solutions

Unit Learning TargetsStudents will be able to:

I can solve quadratic equations for integer roots using the Zero-Product Property

I can solve quadratic equations for fractional roots using the Zero-Product



algebra-2/





(Range 16-19) Numbers: Concepts & Properties: Recognize one-digit factors of a number(Range 20-23) Numbers: Concepts & Properties: Exhibit knowledge of elementary number concepts including rounding, the ordering of decimals, pattern identification, absolute value, primes, and greatest common factor(Range 24-27) Numbers: Concepts & Properties: Work with numerical factors(Range 24-27) Numbers: Concepts & Properties: Work with squares and square roots of numbers(Range 24-27) Numbers: Concepts & Properties: Exhibit some knowledge of the complex numbers †(Range 28-32) Numbers: Concepts & Properties: Apply number properties involving prime factorization(Range 28-32) Numbers: Concepts & Properties: Apply number properties involving even/odd numbers and factors/multiples(Range 30-36) Numbers: Concepts & Properties: Draw conclusions based on number concepts, algebraic properties, and/or relationships between expressions and numbers(Range 20-23) Numbers: Concepts & Properties: Exhibit knowledge of elementary number concepts including rounding, the ordering of decimals, pattern identification, absolute value, primes, and greatest common factor(Range 13-15) Expressions, Equations, & Inequalities: Exhibit knowledge of basic expressions (e.g., identify an expression for a total as b + g)(Range 13-15) Expressions, Equations, & Inequalities: Solve equations in the form x + a = b, where a and b are whole numbers or decimals (Range 16-19) Expressions, Equations, & Inequalities: Solve one-step equations having integer or decimal answers(Range 16-19) Expressions, Equations, & Inequalities: Combine like terms (e.g., 2x + 5x)(Range 20-23) Expressions, Equations, & Inequalities: Evaluate algebraic expressions by substituting integers for unknown quantities(Range 20-23) Expressions, Equations, & Inequalities: Add and subtract simple algebraic expressions(Range 20-23) Expressions, Equations, & Inequalities: Solve routine first-degree equations(Range 24-27) Expressions, Equations, & Inequalities: Identify solutions to simple quadraticequations(Range 24-27) Expressions, Equations, & Inequalities: Factor simple quadratics (e.g., the difference of squares and perfect square trinomials) *(Range 28-32) Expressions, Equations, & Inequalities: Manipulate expressions and equations(Range 28-32) Expressions, Equations, & Inequalities: Solve quadratic equations(Range 28-32) Functions : Evaluate quadratic functions, expressed in function notation, at integer values

NCTM:Algebra:

Judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology. Use symbolic algebra to represent and explain mathematical relationships

Number and Operations: Compare and contrast the properties of numbers and number systems, including the rational

and real numbers, and understand complex numbers as solutions to quadratic equations that do not have real solutions

Quality Core:A1C: Factor trinomials in the form ax2 + bx + cA1D: Solve single-step and multistep equations and inequalities in one variableA1J: Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid conclusions

Property I can solve

quadratic equations by first factoring out a GCF

I can write quadratic equations in standard form

I can use the Square Root Property to solve an equation

I can solve quadratic equations by completing the square

I can use the discriminant to determine the number of real roots

I can solve quadratic equations using the Quadratic Formula,


B1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsC1B: Add, subtract, and multiply complex numbersE1A: Solve quadratic equations and inequalities using various techniques, including completing the square and using the quadratic formulaE1B: Use the discriminant to determine the number and type of roots for a given quadratic equationE1C: Solve quadratic equations with complex number solutionsE1D: Solve quadratic systems graphically and algebraically with and without technologyF1B: Factor polynomials using a variety of methods (e.g., factor theorem, synthetic division, long division, sums and differences of cubes, grouping)


2 weeks

Vocabulary:Quadratic FunctionParabolaFactorMonomialBinomialTrinomialZerosStandard FormVertex FormIntercept Formx-intercepty-interceptVertexDomainRange

Unit 7: Properties and Graphs of Quadratics

New State Standards:N-Q Quantities1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

N-CN Complex Number System7. Solve quadratic equations with real coefficients that have complex solutions9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.


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. 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.

A-APR Arithmetic with Polynomials & Rational Expressions3. 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-CED Creating Equations2. Create equations in two or more variables to represent relationships between quantities; graph

Activities:

Concepts and Skills:Students will: Graph quadratic functions

in vertex, intercept, and standard form

Given any quadratic equation, write equation in other forms

Graph using transformations

Determine characteristics based on form including shape, direction, vertex, symmetry, and intercepts

Write a quadratic based on zeros

Describe the domain and range of quadratic

Understand the concepts are similar given x = y^2

Graph quadratic inequality Graph system of quadratic

inequalities

Learning Targets I can graph

quadratic functions

Resources:


CH: 8 Internet Research Infinite Algebra 1 Infinite Geometry Infinite Algebra 2 Edmodo Quality Core Infinite Algebra II Discovery Education www.Khanacademy.org brightstorm.com/math/algebra-2/

Strategies: Graphing calculator applications Exit slips ADP activities Enrichment activities ACT/SAT integration Real World Applications Vocabulary Builders



equations on coordinate axes with labels and scales.

A-REI Reasoning with Equations & Inequalities1. 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.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 x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

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).F-IF Interpreting Functions1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).4. 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.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

a. 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.

9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

College Readiness:(Range 13-15) Basic Operations and Applications: Perform one-operation computation withwhole numbers and decimals(Range 13-15) Basic Operations and Applications: Solve problems in one or two stepsusing whole numbers(Range 16-19) Basic Operations and Applications: Solve some routine two-step arithmetic problems(Range 16-19) Numbers: Concepts & Properties: Recognize one-digit factors of a number(Range 20-23) Numbers: Concepts & Properties: Exhibit knowledge of elementary number concepts including rounding, the ordering of decimals, pattern identification, absolute value, primes, and greatest common factor(Range 24-27) Numbers: Concepts & Properties: Work with numerical factors(Range 24-27) Numbers: Concepts & Properties: Work with squares and square roots of numbers(Range 24-27) Numbers: Concepts & Properties: Exhibit some knowledge of the complex numbers †(Range 28-32) Numbers: Concepts & Properties: Apply number properties involving prime

using a table I can graph to

find the zeros of a quadratic function

I can analyze graphs of functions


factorization(Range 28-32) Numbers: Concepts & Properties: Apply number properties involving even/odd numbers and factors/multiples(Range 30-36) Numbers: Concepts & Properties: Draw conclusions based on number concepts, algebraic properties, and/or relationships between expressions and numbers(Range 20-23) Numbers: Concepts & Properties: Exhibit knowledge of elementary number concepts including rounding, the ordering of decimals, pattern identification, absolute value, primes, and greatest common factor(Range 13-15) Expressions, Equations, & Inequalities: Exhibit knowledge of basic expressions (e.g., identify an expression for a total as b + g)(Range 13-15) Expressions, Equations, & Inequalities: Solve equations in the form x + a = b, where a and b are whole numbers or decimals (Range 16-19) Expressions, Equations, & Inequalities: Solve one-step equations having integer or decimal answers(Range 16-19) Expressions, Equations, & Inequalities: Combine like terms (e.g., 2x + 5x)(Range 20-23) Expressions, Equations, & Inequalities: Evaluate algebraic expressions by substituting integers for unknown quantities(Range 20-23) Expressions, Equations, & Inequalities: Add and subtract simple algebraic expressions(Range 20-23) Expressions, Equations, & Inequalities: Solve routine first-degree equations(Range 24-27) Expressions, Equations, & Inequalities: Identify solutions to simple quadraticequations(Range 24-27) Expressions, Equations, & Inequalities: Factor simple quadratics (e.g., the difference of squares and perfect square trinomials) *(Range 28-32) Expressions, Equations, & Inequalities: Manipulate expressions and equations(Range 28-32) Expressions, Equations, & Inequalities: Solve quadratic equations(Range 20-23) Graphical Representations : Locate points in the coordinate plane(Range 28-32) Graphical Representations : Interpret and use information from graphs in the coordinate plane(Range 28-32) Graphical Representations : Recognize special characteristics of parabolas and circles (e.g., the vertex of a parabola and the center or radius of a circle)†(Range 33-36) Graphical Representations : Identify characteristics of graphs based on a set of conditions or on a general equation such as y = ax² + c(Range 33-36) Graphical Representations : Solve problems integrating multiple algebraic and/or geometric concepts(Range 33-36) Graphical Representations : Analyze and draw conclusions based on information from graphs in the coordinate plane(Range 28-32) Functions : Evaluate quadratic functions, expressed in function notation, at integer values

NCTM:Algebra:


Understand relations and functions and select, convert flexibly among, and use various representations for them

analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior

understand the meaning of equivalent forms of expressions, equations, inequalities, and relations

write equivalent forms of equations, inequalities, and systems of equations and solve them





Quality Core:A1B: Multiply monomials and binomialsA1C: Factor trinomials in the form ax^2 + bx + cA1D: Solve single-step and multistep equations and inequalities in one variableA1J: Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid conclusionsB1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsC1B: Add, subtract, and multiply complex numbersE1A: Solve quadratic equations and inequalities using various techniques, including completing the square and using the quadratic formulaE1B: Use the discriminant to determine the number and type of roots for a given quadratic equationE1C: Solve quadratic equations with complex number solutionsE1D: Solve quadratic systems graphically and algebraically with and without technologyE2A: Determine the domain and range of a quadratic function; graph the function with and without technologyE2C: Graph a system of quadratic inequalities with and without technology to find the solution set to the systemF1B: Factor polynomials using a variety of methods (e.g., factor theorem, synthetic division, long division, sums and differences of cubes, grouping)

Algebra 2B2nd SemesterTimeline:2nd Trimester

1.5 weeks

Vocabulary:ConjugatesFactor TheoremFundamental Theorem of AlgebraImaginary Root TheoremIrrational Root TheoremMultiplicityPolynomial Function

Unit 8: Polynomials and Polynomial Functions

New State Standards:N-CN Complex Number System1. 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.2. Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.7. Solve quadratic equations with real coefficients that have complex solutions8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.


a. Interpret parts of an expression, such as terms, factors, and coefficients.

Activities:

Concepts and Skills:Students will: Classify polynomials Write a polynomial in

standard form write a polynomial in

factored form Find zeros of polynomial

functions Write a polynomial

function from its zeros Find and use the

multiplicity of a zero Perform polynomial long

division

Resources:



Strategies: Graphing calculator applications



Rational Root TheoremRemainder TheoremStandard form of PolynomialLong DivisionSynthetic DivisionDegreeDegree of PolynomialDifference of CubesMultiple ZeroPolynomialRelative MaximumRelative MinimumSum of CubesEnd BehaviorIncreasingDecreasing

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-APR Arithmetic with Polynomials & Rational Expressions1. 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.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 zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.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.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-CED Creating Equations2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A-REI Reasoning with Equations & Inequalities1. 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.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).

F-IF Interpreting Functions1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.4. 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.★6. 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.★7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★

b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

Use synthetic division Check factors Evaluate polynomials

using synthetic division Solve polynomial

equations by factoring Factor sum/difference of

cubes Solve a polynomial

equation Factor by using a quadratic

pattern Solve higher-degree

polynomial equations Find rational roots Use Rational Root

Theorem Find irrational roots Find imaginary roots Write polynomials

equations from roots Use Fundamental Theorem

of Algebra Use zero’s and end

behavior to create a graphical representation of a polynomial

Identify key features of polynomial graphs (including domain, range, roots, relative max, relative min, increasing and decreasing intervals, positive or negative, symmetries, and end behavior)

Exit slips ADP activities Enrichment activities ACT/SAT integration Real World Applications Vocabulary Builders



College Readiness:(Range 13-15) Basic Operations and Applications: Perform one-operation computation withwhole numbers and decimals(Range 13-15) Basic Operations and Applications: Solve problems in one or two stepsusing whole numbers(Range 16-19) Basic Operations and Applications: Solve some routine two-step arithmetic problems(Range 16-19) Numbers: Concepts & Properties: Recognize one-digit factors of a number(Range 20-23) Numbers: Concepts & Properties: Exhibit knowledge of elementary number concepts including rounding, the ordering of decimals, pattern identification, absolute value, primes, and greatest common factor(Range 24-27) Numbers: Concepts & Properties: Find and use the least common multiple(Range 24-27) Numbers: Concepts & Properties: Work with numerical factors(Range 24-27) Numbers: Concepts & Properties: Work with squares and square roots of numbers(Range 24-27) Numbers: Concepts & Properties: Work with cubes and cube roots of numbers*(Range 24-27) Numbers: Concepts & Properties: Exhibit some knowledge of the complex numbers †(Range 28-32) Numbers: Concepts & Properties: Apply number properties involving prime factorization(Range 28-32) Numbers: Concepts & Properties: Apply number properties involving even/odd numbers and factors/multiples(Range 28-32) Numbers: Concepts & Properties: Apply rules of exponents(Range 28-32) Numbers: Concepts & Properties: Multiply two complex numbers†(Range 33-36) Numbers: Concepts & Properties: Draw conclusions based on number concepts, algebraic properties, and/or relationships between expressions and numbers(Range 33-36) Numbers: Concepts & Properties: Apply properties of complex numbers(Range 13-15) Expressions, Equations, & Inequalities: Exhibit knowledge of basic expressions (e.g., identify an expression for a total as b + g)(Range 13-15) Expressions, Equations, & Inequalities: Solve equations in the form x + a = b, where a and b are whole numbers or decimals(Range 16-19) Expressions, Equations, & Inequalities: Solve one-step equations having integer or decimal answers(Range 16-19) Expressions, Equations, & Inequalities: Combine like terms (e.g., 2x + 5x)(Range 20-23) Expressions, Equations, & Inequalities: Evaluate algebraic expressions by substituting integers for unknown quantities(Range 20-23) Expressions, Equations, & Inequalities: Add and subtract simple algebraic expressions(Range 24-27) Expressions, Equations, & Inequalities: Identify solutions to simple quadraticequations(Range 24-27) Expressions, Equations, & Inequalities: Add, subtract, and multiply polynomials * (Range 24-27) Expressions, Equations, & Inequalities: Factor simple quadratics (e.g., the difference of squares and perfect square trinomials) *(Range 28-32) Expressions, Equations, & Inequalities: Manipulate expressions and equations(Range 28-32) Expressions, Equations, & Inequalities: Solve quadratic equations(Range 28-32) Functions : Evaluate quadratic functions, expressed in function notation, at integer values(Range 20-23) Graphical Representations : Locate points in the coordinate plane(Range 28-32) Graphical Representations : Interpret and use information from graphs in the coordinate plane(Range 33-36) Graphical Representations : Identify characteristics of graphs based on a set of conditions or on a general equation such as y = ax² + c


(Range 33-36) Graphical Representations : Analyze and draw conclusions based on information from graphs in the coordinate plane(Range 20-23) Functions: Evaluate quadratic functions, expressed in function notation, at integer values(Range 24-27) Functions: Evaluate polynomial functions, expressed in function notation, at integervalues


Compare and contrast the properties of numbers and number systems, including the rational and real numbers, and understand complex numbers as solutions to quadratic equations that do not have real solutions

Judge the reasonableness of numerical computations and their resultsAlgebra

understand relations and functions and select, convert flexibly among, and use various representations for them



write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases

Quality Core:A1B: Multiply monomials and binomialsA1C: Factor trinomials in the form ax2 + bx + cA1D: Solve single-step and multistep equations and inequalities in one variableA1J: Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid conclusionsB1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsE1A: Solve quadratic equations and inequalities using various techniques, including completing the square and using the quadratic formulaE1D: Solve quadratic systems graphically and algebraically with and without technologyF1A: Determine the number and type of rational zeros for a polynomial functionF1B: Factor polynomials using a variety of methods (e.g., factor theorem, synthetic division, long division, sums and differences of cubes, grouping)F1A: Evaluate and simplify polynomial expressions and equationsF2A: Determine the number and type of rational zeros for a polynomial functionF2B: Find all rational zeros of a polynomial functionF2C: Recognize the connection among zeros of a polynomial function, x-intercepts, factors of polynomials, and solutions of polynomial equationsF2D: Use technology to graph a polynomial function and approximate the zeros, minimum, and maximum; determine domain and range of the polynomial function

Timeline:2nd Semester1 week

Unit 9: Transforming FunctionsNew State Standards:N-Q Quantities

Activities: Resources:



Vocabulary:Parent FunctionAmplitudeMaximumMinimumTransformationHorizontal ShiftVertical ShiftHorizontal StretchVertical StretchDomainRangeExponential

3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

A-REI Reasoning with Equations & Inequalities10. 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).

F-IF Interpreting Functions1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★

F-BF Building Functions3. 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.

College Readiness:(Range 20-23) Graphical Representation: Locate points in the coordinate plane(Range 33-36) Graphical Representation: Identify characteristics of graphs based on a set of conditions or on a general equation such as y = ax² + c(Range 33-36) Graphical Representation: Analyze and draw conclusions based on information from graphs in the coordinate plane

NCTM:Algebra:

understand and perform transformations such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on more-complicated symbolic expressions

understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions

Quality Core:B1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsC1D: Perform operations on functions, including function composition, and determine domain and range for each of the given functionsE2A: Determine the domain and range of a quadratic function; graph the function with and without technologyE2B: Use transformations (e.g., translation, reflection) to draw the graph of a relation and determine a relation that fits a graph

Concepts and Skills:Students will: Graph main parent

functions Apply transformations to

parent functions Determine domain/ range

for parent functions Determine domain/ range

for transformed functions Graphically represent

transformations Verbally describe

transformations Graph transformations

based on a graph of f(x) that is not a defined function

Write equation based on transformations on a defined function



Timeline:2nd Semester

Unit 10: Rational Functions


Activities:

Concepts and Skills:

Resources:




1.5 Weeks

Vocabulary:

Inverse VariationJoint VariationRational ExpressionSimplest FormComplex FractionRational EquationsComposition of FunctionsHorizontal Line Test

N-Q Quantities3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.



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-APR Arithmetic with Polynomials & Rational Expressions1. 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.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 system7. (+) 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

Students will: Model inverse

variation Identify and solve

problems using direct, joint, and inverse variation

Simplify rational expressions

Add, Subtract, Multiply, and Divide rational expressions

Find Least Common Multiples

Adding and subtracting rational expressions

Simplifying complex fractions

Solve rational equations

Find an inverse from a graph or table

Decide if a function has an inverse

Verify inverse by composition

Unit Learning TargetsStudents will be able to: Find the domain of

a rational function Graph rational

functions using asymptotes

Apply rational functions to real-world problems

Make predictions





expressions.

F-BF Building Functions4. Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function that has an inverse and write an expression for the inverse. For example, f(x) = 2 x3 or f(x) = (x+1)/(x-1) for x ≠ 1. b. (+) Verify by composition that one function is the inverse of another. c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain.

A-REI Reasoning with Equations & Inequalities1. 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 method2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise

F-IF Interpreting Functions1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or

using rational functions


by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

College Readiness:(Range 13-15) Basic Operations & Applications: Perform one-operation computation with whole numbers and decimals(Range 13-15) Basic Operations & Applications: Solve problems in one or two steps using whole numbers(Range 16-19) Basic Operations & Applications: Solve routine one-step arithmetic problems (using whole numbers, fractions, and decimals) such as single step percent(Range 16-19) Basic Operations & Applications: Solve some routine two-step arithmetic problems(Range 16-19) Numbers: Concepts & Properties: Recognize one-digit factors of a number(Range 20-23) Numbers: Concepts & Properties: Exhibit knowledge of elementary number concepts including rounding, the ordering of decimals, pattern identification, absolute value, primes, andgreatest common factor(Range 24-27) Numbers: Concepts & Properties: Find and use the least common multiple(Range 24-27) Numbers: Concepts & Properties: Work with numerical factors(Range 28-32) Numbers: Concepts & Properties: Apply number properties involving prime factorization(Range 28-32) Numbers: Concepts & Properties: Apply number properties involving even/odd numbers and factors/multiples(Range 28-32) Numbers: Concepts & Properties: Apply rules of exponents(Range 13-15) Expressions, Equations, & Inequalities: Exhibit knowledge of basic expressions


(e.g., identify an expression for a total as b + g)(Range 16-19) Expressions, Equations, & Inequalities: Combine like terms (e.g., 2x + 5x)(Range 20-23) Expressions, Equations, & Inequalities: Add and subtract simple algebraic expressions(Range 20-23) Expressions, Equations, & Inequalities: Solve routine first-degree equations(Range 20-23) Expressions, Equations, & Inequalities: Multiply two binomials*(Range 24-27) Expressions, Equations, & Inequalities: Identify solutions to simple quadratic equations(Range 24-27) Expressions, Equations, & Inequalities: Add, subtract, and multiply polynomials *(Range 24-27) Expressions, Equations, & Inequalities: Factor simple quadratics (e.g., the difference of squares and perfect square trinomials) *(Range 28-32) Expressions, Equations, & Inequalities: Manipulate expressions and equations(Range 28-32) Expressions, Equations, & Inequalities: Solve quadratic equations(Range 24-27) Functions: Evaluate polynomial functions, expressed in function notation, at integervalues

NCTM:Algebra:

understand relations and functions and select, convert flexibly among, and use various representations for them;

understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions;

understand the meaning of equivalent forms of expressions, equations, inequalities, and relations;


write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases;

draw reasonable conclusions about a situation being modeled.

Quality Core:A1A: Identify properties of real numbers and use them and the correct order of operations to simplify expressionsA1B: Multiply monomials and binomialsA1C: Factor trinomials in the form ax2 + bx + cB1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsE1A: Solve quadratic equations and inequalities using various techniques, including completing the square and using the quadraticformulaF1A: Evaluate and simplify polynomial expressions and equationsF1B: Factor polynomials using a variety of methods (e.g., factor theorem, synthetic division, long division, sums and differences of cubes, grouping)

Timeline:2nd Trimester

1 weeks

Unit 11: Conics

New State Standards:N-Q Quantities

Activities: Resources:


CH: 8


Vocabulary:

CenterCircleConic SectionCo-verticesDirectrixEllipseFocus of ParabolaFocus of EllipseFocus of HyperbolaHyperbolaMajor AxisMinor AxisRadiusStandard Form of CircleTransverse AxisVertices of EllipseVertices of Hperbola

3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

A-CED Creating Equations*2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A-REI Reasoning with Equations & Inequalities1. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.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).

F-IF Interpreting Functions1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.4. 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.★7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

F-BF Building Functions3. 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.

College Readiness:

(Range 13-15) Basic Operations & Applications: Solve problems in one or two steps using whole numbers(Range 16-19) Basic Operations & Applications: Solve routine one-step arithmetic problems (using whole numbers, fractions, and decimals) such as singlestep percent(Range 16-19) Basic Operations & Applications: Solve some routine two-step arithmetic problems (Range 13-15) Numbers: Concepts &Properties: Recognize equivalent fractions and fractions in lowest terms(Range 16-19) Numbers: Concepts &Properties: Recognize one-digit factors of a number(Range 20-23) Numbers: Concepts &Properties: Exhibit knowledge of elementary number concepts including rounding, the ordering of decimals, pattern identification, absolute value, primes, andgreatest common factor(Range 24-27) Numbers: Concepts &Properties: Find and use the least common multiple

Concepts and Skills:Students will: Graph a Circle Graph an ellipse Graph a hyperbola Identify the graphs of

conic sections Use the definition of a

parabola Write the equation of a

parabola Identify focus and directrix

of a parabola Graph using the equation

of a parabola Write the equation of a

circle Use translations to write an

equation of a circle Find the center and radius

of a circle Graph circle using center

and radius Write the equation of an

ellipse Find the foci of ellipse Use the foci of ellipse to

graph Graph hyperbola Find the foci of hyperbola Write the equation of a

translated ellipse Write the equation of a

translated hyperbola Write the equation of a

translated parabola Identify translated conic

sections

Internet Research Infinite Algebra 1 Infinite Geometry Infinite Algebra 2 Edmodo Quality Core Infinite Algebra II Discovery Education www.Khanacademy.org brightstorm.com/math/algebra-2/




(Range 24-27) Numbers: Concepts &Properties: Work with numerical factors(Range 24-27) Numbers: Concepts &Properties: Work with squares and square roots of numbers(Range 24-27) Numbers: Concepts &Properties: Work problems involving positive integer exponents*(Range 24-27) Numbers: Concepts &Properties: Determine when an expression is undefined*(Range 28-32) Numbers: Concepts &Properties: Apply number properties involving prime factorization(Range 28-32) Numbers: Concepts &Properties: Apply number properties involving even/odd numbers and factors/multiples(Range 28-32) Numbers: Concepts &Properties: Solve Apply number properties involving positive/negative numbers(Range 33-36) Numbers: Concepts &Properties: Draw conclusions based on number concepts, algebraic properties, and/or relationships between expressions and numbers(Range 13-15) Expressions, Equations, & Inequalities: Exhibit knowledge of basic expressions (e.g., identify an expression for a total as b + g)(Range 13-15) Expressions, Equations, & Inequalities: Solve equations in the form x + a = b, where a and b are whole numbers or decimals(Range 16-19) Expressions, Equations, & Inequalities: Solve one-step equations having integer or decimal answers(Range 16-19) Expressions, Equations, & Inequalities: Combine like terms (e.g., 2x + 5x)(Range 20-23) Expressions, Equations, & Inequalities: Evaluate algebraic expressions by substituting integers for unknown quantities(Range 20-23) Expressions, Equations, & Inequalities: Add and subtract simple algebraic expressions(Range 24-27) Expressions, Equations, & Inequalities: Identify solutions to simple quadraticequations(Range 24-27) Expressions, Equations, & Inequalities: Add, subtract, and multiply polynomials * (Range 24-27) Expressions, Equations, & Inequalities: Factor simple quadratics (e.g., the difference of squares and perfect square trinomials) *(Range 28-32) Expressions, Equations, & Inequalities: Manipulate expressions and equations(Range 28-32) Expressions, Equations, & Inequalities: Solve quadratic equations(Range 28-32) Functions : Evaluate quadratic functions, expressed in function notation, at integer values(Range 20-23) Graphical Representations : Locate points in the coordinate plane(Range 28-32) Graphical Representations : Interpret and use information from graphs in the coordinate plane(Range 33-36) Graphical Representations : Identify characteristics of graphs based on a set of conditions or on a general equation such as y = ax² + c(Range 33-36) Graphical Representations : Analyze and draw conclusions based on information from graphs in the coordinate plane(Range 20-23) Functions: Evaluate quadratic functions, expressed in function notation, at integer values(Range 24-27) Functions: Evaluate polynomial functions, expressed in function notation, at integervalues

NCTM:Algebra:



Understand relations and functions and select, convert flexibly among, and use various representations for them


understand and perform transformations such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on more-complicated symbolic expressions

interpret representations of functions of two variables understand the meaning of equivalent forms of expressions, equations, inequalities, and

relations write equivalent forms of equations, inequalities, and systems of equations and solve them


identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships



Quality Core:B1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsC1D: Perform operations on functions, including function composition, and determine domain and range for each of the given functionsE2A: Determine the domain and range of a quadratic function; graph the function with and without technologyE2B: Use transformations (e.g., translation, reflection) to draw the graph of a relation and determine a relation that fits a graph


2 Weeks

Vocabulary:Like RadicalsNth RootPrincipal RootRadical EquationRadical

Unit 12: Radical Functions and Rational Exponents

New State Standards:N-Q Quantities3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

N-RN The Real Number System1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

Activities:

Concepts and Skills:Students will: Find all real roots Find roots Simplify radical

expressions Multiply radicals Divide radicals Rationalize the

denominator Add and subtract

radical expressions Multiply binomial

radical expressions

Resources:


CH: 9 Internet Research Infinite Algebra 1 Infinite Geometry Infinite Algebra 2 Edmodo Quality Core Infinite Algebra II Discovery Education www.khanacademy.org brightstorm.com/math/algebra-

2/

http://www.khanacademy.org/


FunctionRadicandRational ExponentRationalize DenominatorSquare Root EquationSquare Root FunctionExtraneous Solution

2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

A-SSE Seeing Structure in Expressions3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★

A-REI Reasoning with Equations & Inequalities1. 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.2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.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).

F-IF Interpreting Functions1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.4. For a function that models a relationship between two

Multiplying conjugates

Rationalize binomial radical denominators

Convert between radical and rational exponents

Simplify expressions with rational exponents

Simplify numbers with rational exponents

Write rational exponents in simplest form

Solve square root equations

Solve radical equations with rational exponents

Check for extraneous solutions

Solving equations with two rational exponents

Find the domain and range of a given radical or rational function

Create graphs of root functions

Solve quadratic equations that lead to extraneous solutions




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.★7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★

b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

College Readiness:(Range 13-15) Basic Operations & Applications: Perform one-operation computation with whole numbers and decimals(Range 13-15) Basic Operations & Applications: Perform Solve problems in one or two stepsusing whole numbers(Range 16-19) Basic Operations & Applications: Solve routine one-step arithmetic problems (using whole numbers, fractions, and decimals) such as singlestep percent(Range 16-19) Basic Operations & Applications: Solve some routine two-step arithmetic problems(Range 20-23) Numbers: Concepts & Properties: Exhibit knowledge of elementary number concepts including rounding, the ordering of decimals, pattern identification, absolute value, primes, andgreatest common factor(Range 24-27) Numbers: Concepts & Properties: Work with numerical factors(Range 24-27) Numbers: Concepts & Properties: Work with squares and square roots of numbers(Range 24-27) Numbers: Concepts & Properties: Work problems

Unit Learning TargetsStudents will be able to: Recognize types of

polynomial functions

Describe the end behavior of polynomial functions

Match functions with graphs

Find the zeros of linear and quadratic functions

Determine whether a number is a zero of a function

Use factoring and multiplicity to describe the graph of the function

Factor a higher-order polynomial

Approximate the local minimum or maximum

Find a quartic regression model

Choose a polynomial regression model

Find the domain by graphing

Find horizontal asymptotes

Find a hole in the graph

Work with geometric figures


involving positive integer exponents*(Range 24-27) Numbers: Concepts & Properties: Work with cubes and cube roots of numbers*(Range 28-32) Numbers: Concepts & Properties: Apply number properties involving prime factorization(Range 28-32) Numbers: Concepts & Properties: Apply number properties involving even/odd numbers and factors/multiples(Range 28-32) Numbers: Concepts & Properties: Apply number properties involving positive/negative numbers(Range 28-32) Numbers: Concepts & Properties: Apply rules of exponents(Range 33-36) Numbers: Concepts & Properties: Draw conclusions based on number concepts, algebraic properties, and/or relationships between expressions and numbers(Range 33-36) Numbers: Concepts & Properties: Apply rules of exponents(Range 28-32) Numbers: Concepts & Properties: Apply rules of exponents(Range 13-15) Expressions, Equations, & Inequalities: Solve equations in the form x + a = b,where a and b are whole numbers or decimals(Range 16-19) Expressions, Equations, & Inequalities: Substitute whole numbers for unknownquantities to evaluate expressions(Range 16-19) Expressions, Equations, & Inequalities: Solve one-step equations having integeror decimal answers(Range 16-19) Expressions, Equations, & Inequalities: Combine like terms (e.g., 2x + 5x)(Range 20-23) Expressions, Equations, & Inequalities: Evaluate algebraic expressions by substituting integers for unknown quantities(Range 20-23) Expressions, Equations, & Inequalities: Multiply two binomials*


(Range 28-32) Expressions, Equations, & Inequalities: Manipulate expressions and equations(Range 20-23) Graphical Representation: Locate points in the coordinate plane(Range 33-36) Graphical Representation: Identify characteristics of graphs based on a set of conditions or on a general equation such as y = ax² +


c. develop a deeper understanding of very large and very small numbers and of various representations of them

d. compare and contrast the properties of numbers and number systems, including the rational and real numbers, and understand complex numbers as solutions to quadratic equations that do not have real solutions

e. judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities

f. judge the reasonableness of numerical computations and their results

Algebra: understand relations and functions and select, convert

flexibly among, and use various representations for them understand and compare the properties of classes of

functions, including exponential, polynomial, rational, logarithmic, and periodic functions


use symbolic algebra to represent and explain mathematical relationships

judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried


out by technology identify essential quantitative relationships in a situation

and determine the class or classes of functions that might model the relationships

Quality Core:B1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsG1A: Solve mathematical and real-world rational equation problems (e.g., work or rate problems)G1B: Simplify radicals that have various indicesG1C: Use properties of roots and rational exponents to evaluate and simplify expressionsG1D: Add, subtract, multiply, and divide expressions containing radicalsG1E: Rationalize denominators containing radicals and find the simplest common denominatorG1F: Evaluate expressions and solve equations containing nth roots or rational exponentsG1G: Evaluate and solve radical equations given a formula for a real-world situationE2B: Use transformations (e.g., translation, reflection) to draw the graph of a relation and determine a relation that fits a graph


1.5 Weeks

Vocabulary:

AsymptoteChange of Base Formula

Unit 13: Exponential and Logarithmic Functions

New State Standards:N-Q Quantities1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Activities:

Concepts and Skills:Students will: Write exponential

functions Graph exponential

function Analyze functions Graph exponential

functions with varying parameters

Translate exponential

Resources:


CH: 8 Internet Research Infinite Algebra 1 Infinite Geometry Infinite Algebra 2 Edmodo Quality Core Infinite Algebra II Discovery Education www.khanacademy.org

http://www.khanacademy.org/


Common LogarithmExponential EquationExponential FunctionLogarithmLogarithmic EquationLogarithmic FunctionNatural Logarithmic Function



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

3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★

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-CED 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.

A-REI Reasoning with Equations & Inequalities1. 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.10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often

functions Evaluate e Convert between

exponentials and logarithmic forms

Evaluate logarithms Graph logarithmic

functions Translate

logarithmic functions

Identify properties of logarithms

Expand and simplify logarithms

Solve exponential equations

Solve logarithmic equations

Use logarithmic properties to solve an equation

Unit Learning Targets Students will be able to: I can graph

exponential growth function.

I can use exponential growth functions to model real-life situation.

I can graph exponential decay functions.

I can use exponential decay functions to model real-life situations.

I can use the number e as the

brightstorm.com/math/algebra-2/




forming a curve (which could be a line).

F-IF Interpreting Functions1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.4. 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.★5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★

e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

a. Use the properties of exponents to interpret expressions for

base of exponential functions.

I can use the natural base e in real-life situations.

I can evaluate logarithmic functions.

I can graph logarithmic functions.

I can use properties of logarithms to solve real-life problems.

I can solve exponential equations.

I can solve logarithmic functions.

I can model data with exponential functions.

I can model data with power functions.

I can evaluate and graph logistic growth functions.

I can use logistic growth functions to model real-life quantities.


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.


F-BF Building Functions1. Write a function that describes a relationship between two quantities.★

a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

3. 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.4. Find inverse functions.

c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.

5. Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

F-LE Linear, Quadratic, & Exponential Models*1. Distinguish between situations that can be modeled with linear functions and with exponential functions.

a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.


b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.4. 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.5. Interpret the parameters in a linear or exponential function in terms of a context.

College Readiness:(Range 13-15) Basic Operations & Applications: Perform one-operation computation with whole numbers and decimals(Range 13-15) Basic Operations & Applications: Solve problems in one or two steps using whole numbers(Range 16-19) Basic Operations & Applications: Solve routine one-step arithmetic problems (using whole numbers, fractions, and decimals) such as single step percent(Range 16-19) Basic Operations & Applications: Solve some routine two-step arithmetic problems(Range 13-15) Number Concepts & Properties: Recognize equivalent fractions and fractions in lowest terms(Range 24-27) Number Concepts & Properties: Work with squares and square roots of numbers(Range 24-27) Number Concepts & Properties: Work problems


involving positive integer exponents*(Range 24-27) Number Concepts & Properties: Work with cubes and cube roots of numbers*(Range 24-27) Number Concepts & Properties: Determine when an expressionis undefined*(Range 28-32) Number Concepts & Properties: Apply number properties involving even/odd numbers and factors/multiples(Range 28-32) Number Concepts & Properties: Apply rules of exponents(Range 33-36) Number Concepts & Properties: Draw conclusions based on number concepts, algebraic properties, and/or relationships between expressions and numbers(Range 33-36) Number Concepts & Properties: Exhibit knowledge of logarithms and geometric sequences(Range 16-19) Expressions, Equations, & Inequalities: Substitute whole numbers for unknown quantities to evaluate expressions(Range 16-19) Expressions, Equations, & Inequalities: Solve one-step equations having integeror decimal answers(Range 20-23) Expressions, Equations, & Inequalities: Evaluate algebraic expressions by substituting integers for unknown quantities(Range 20-23) Expressions, Equations, & Inequalities: Solve routine first-degree equations(Range 28-32) Expressions, Equations, & Inequalities: Manipulate expressions and equations(Range 28-32) Expressions, Equations, & Inequalities: Write expressions, equations, and inequalities for common algebra settings(Range 33-36) Expressions, Equations, & Inequalities: Write expressions that require planning and/or manipulating to accurately model a situation(Range 33-36) Expressions, Equations, & Inequalities: Write equations and inequalities that require planning, manipulating,


and/or solving(Range 13-15) Graphical Representation: Identify the location of a point with a positive coordinate on the number line(Range 16-19) Graphical Representation: Locate points on the number line and in the first quadrant(Range 20-23) Graphical Representation: Locate points in the coordinate plane(Range 28-32) Graphical Representation: Interpret and use information from graphs in the coordinate plane(Range 33-36) Graphical Representation: Identify characteristics of graphs based on a set of conditions or on a general equation such as y = ax² + c(Range 20-23) Graphical Representation: Solve problems integrating multiple algebraic and/or geometric concepts(Range 33-36) Graphical Representation: Analyze and draw conclusions based on information from graphs in the coordinate plane


develop a deeper understanding of very large and very small numbers and of various representations of them

use number-theory arguments to justify relationships involving whole numbers

judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities

develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases

Judge the reasonableness of numerical computations and their results.


Algebra: understand relations and functions and select, convert

flexibly among, and use various representations for them; analyze functions of one variable by investigating rates of

change, intercepts, zeros, asymptotes, and local and global behavior;

understand and perform transformations such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on more-complicated symbolic expressions;

understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions;

interpret representations of functions of two variables understand the meaning of equivalent forms of

expressions, equations, inequalities, and relations; write equivalent forms of equations, inequalities, and

systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases;

use symbolic algebra to represent and explain mathematical relationships;

use a variety of symbolic representations, including recursive and parametric equations, for functions and relations;

judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.

identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships;

use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various


contexts; Draw reasonable conclusions about a situation being

modeled. Approximate and interpret rates of change from graphical

and numerical data.

Quality Core:A1A: Identify properties of real numbers and use them and the correct order of operations to simplify expressionsA1J: Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid conclusionsB1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsC1D: Perform operations on functions, including function composition, and determine domain and range for each of the given functionsE2B: Use transformations (e.g., translation, reflection) to draw the graph of a relation and determine a relation that fits a graphG2A: Graph exponential and logarithmic functions with and without technologyG2B: Convert exponential equations to logarithmic form and logarithmic equations to exponential form


1 weeks

Vocabulary:Standard PositionInitial SideTerminal SideCoterminal AnglesUnit Circle

Unit 14: Basic Trigonometry

New State Standards:F-TF FunctionsTrigonometric Functions1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.3. Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number.

Activities:

Concepts and Skills:Students will: Convert between radians

and degrees Find coterminal angles Find reference angles Work with special right

triangles Construct the unit circle

using special right triangles

Resources:





CosineSineTangentSecantCosecantCotangentCentral AngleReference AngleRadianDegreesRatioAngleOppositeAdjacentRight TriangleSpecial Right TrianglesHypotenuseArc LengthProportion

College Readiness:(Range 20-23) Properties of Plane Figures: Exhibit knowledge of basic angle properties and special sums of angle measures (e.g., 90°, 180°, and 360°)(Range 24-27) Properties of Plane Figures: Use several angle properties to find an unknown angle measure(Range 28-32) Properties of Plane Figures: Apply properties of 30°-60°-90°, 45°-45°-90°, similar, and congruent triangles(Range 28-32) Properties of Plane Figures: Use the Pythagorean theorem(Range 24-27) Functions: Express the sine, cosine, and tangent of an angle in a right triangle as a ratio of given side lengths(Range 28-32) Functions: Apply basic trigonometric ratios to solve right-triangle problems(Range 33-36) Functions: Exhibit knowledge of unit circle trigonometry

NCTM:Geometry:

use trigonometric relationships to determine lengths and angle measuresAlgebra:

Understand the meaning of equivalent forms of expressions Use symbolic algebra to represent and explain mathematical relationships Draw reasonable conclusions about a situation being modeled

Quality Core:A1I: Use sine, cosine, and tangent ratios to find the sides or angles of right trianglesB1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsG3A: Use the law of cosines and the law of sines to find the lengths of sides and measures of angles of triangles in mathematical andreal-world problemsG3B: Use the unit-circle definition of the trigonometric functions and trigonometric relationships to find trigonometric values for general anglesG3C: Measure angles in standard position using degree or radian measure and convert a measure from one unit to the other

Measure an angle in standard position

Sketch an angle in standard position

Find the cosine and sine of an angle

Find exact value of sine and cosine

Find coordinates of points on the unit circle

Use radian measures for angles

Find the length of an arc of a circle

Use proportions Find cosine and sine of

radian measures Find all six trig values for

an angle Find all six trig values for

a point not on the unit circle



1.5 weeks

Vocabulary:Sine FunctionSine CurveCosine FunctionCosine CurveTangent FunctionTangent Curve

Unit 15: Trigonometric Functions

New State Standards:F-TF FunctionsTrigonometric Functions4. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.9. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

College Readiness:

Activities:

Concepts and Skills:Students will: Estimate sine, cosine, and

tangent values in radians and degrees

Find the period, amplitude, domain, and range of the sine and cosine curves

Sketch the graph of the sine and cosine curves

Graph sine from an equation

Resources:





AmplitudePeriodMaximumMinimumPhase ShiftsTransformationHorizontal ShiftVertical ShiftDomainRange

(Range 20-23) Properties of Plane Figures: Exhibit knowledge of basic angle properties and special sums of angle measures (e.g., 90°, 180°, and 360°)(Range 24-27) Properties of Plane Figures: Use several angle properties to find an unknown angle measure(Range 28-32) Properties of Plane Figures: Apply properties of 30°-60°-90°, 45°-45°-90°, similar, and congruent triangles(Range 28-32) Properties of Plane Figures: Use the Pythagorean theorem(Range 24-27) Functions: Express the sine, cosine, and tangent of an angle in a right triangle as a ratio of given side lengths(Range 28-32) Functions: Apply basic trigonometric ratios to solve right-triangle problems(Range 33-36) Functions: Exhibit knowledge of unit circle trigonometry(Range 33-36) Functions: Match graphs of basic trigonometric functions with their equations

NCTM:Measurement:

make decisions about units and scales that are appropriate for problem situations involving measurement

Geometry: use trigonometric relationships to determine lengths and angle measures Use various representations to help understand the effects of simple transformations and

their compositionsAlgebra:

Understand the meaning of equivalent forms of expressions Use symbolic algebra to represent and explain mathematical relationships Draw reasonable conclusions about a situation being modeled

Quality Core:B1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsG3D: Graph the sine and cosine functions with and without technologyG3E: Determine the domain and range of the sine and cosine functions, given a graphG3F: Find the period and amplitude of the sine and cosine functions, given a graphG2G: Use sine, cosine, and tangent functions, including their domains and ranges, periodic nature, and graphs, to interpret and analyze relations

Graph and find the domain and range of the tangent function

Identify phase shifts Graph translations Graph a combined

translation Write an equation based on

transformations of a graph and or points



1 Weeks

Vocabulary:Binomial ProbabilityBox and Whisker PlotConditional ProbabilityCumulative ProbabilityInterquartile RangeMeasures of Central TendencyMeasures of Variation

Unit 16: Data Analysis and Probability


S.ID Interpreting Categorical and Quantitative Data4. 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.

S.IC Making Inferences and Justifying Conclusions1. Understand statistics as a process for making inferences about population parameters based on arandom sample from that population.2. Decide if a specified model is consistent with results from a given data-generating process, e.g.,using simulation. For 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?3. Recognize the purposes of and differences among sample surveys, experiments, and observational

Activities:

Concepts and Skills:Students will:

Make a frequency table

Calculate probability distributions

Find conditional probability

Construct tree diagrams

Find measures of central tendency

Resources:





Normal DistributionOutlierPercentileProbability DistributionQuartilesSampleSample SpaceStandard DeviationStandard Normal CurveZ-Score

studies; explain how randomization relates to each.4. Use data from a sample survey to estimate a population mean or proportion; develop a margin oferror through the use of simulation models for random sampling.5. Use data from a randomized experiment to compare two treatments; use simulations to decide ifdifferences between parameters are significant.6. Evaluate reports based on data.

S.MD Using Probability to Make Decisions6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medicaltesting, pulling a hockey goalie at the end of a game).

College Readiness:

NCTM:

Quality Core:H1A: Use the fundamental counting principle to count the number of ways an event can happenH1B: Use counting techniques, like combinations and permutations, to solve problems (e.g., to calculate probabilities)H1C: Find the probability of mutually exclusive and nonmutually exclusive eventsH1D: Find the probability of independent and dependent eventsH1E: Use unions, intersections, and complements to find probabilitiesH1F: Solve problems involving conditional probability

Construct box-and-whisker plots

Construct and use percentiles

Identify an outlier Find standard

deviation of set of data

Find and interpret z-scores

Determine sample sizes

Work with binomial distributions

Work with standard normal curve and normal distributions


Documents

WEEKS (WHEN WILL YOU TEACH THE Web viewanalogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract,