2 Map.d… · Web viewTimeline: 1st 9 Weeks. 2 weeks. Not for Accel. Vocabulary: Natural. Whole. Integers. Rational. Irrational. Real. Fraction. Percent. Addition. Subtraction

Algebra 2 Curriculum MapModified: July, 2013

Timeline:1st 9 Weeks

2 weeksNot for Accel

Vocabulary:NaturalWholeIntegersRationalIrrationalRealFractionPercentAdditionSubtractionMultiplicationDivisionNumeratorDenominatorCommon denominatorGCFLCMExpressionEquationPrime

Unit A: Basic Operations and Applications

New State Standards: Number and Quantity(The Real Number System)HSN-RN.BUse properties of rational and irrational numbers.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.

College Readiness: (Range 13-15) Basic Operations and Applications: Solve problems in one or two steps using whole numbers(Range 16-19) Basic Operations and Applications: Solve some routine two-step arithmetic problems (Range 16-19) Basic Operations and Applications: Solve routine one-step arithmetic problems (using whole numbers, fractions, and decimals) such as single step (Range 13-15) Numbers: Concepts &Properties: Recognize equivalent fractions and fractions in lowest terms (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: Order fractions(Range 24-27) Numbers: Concepts &Properties: Work with numerical factors

NCTM: Number and Operations:

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

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;

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

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: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 problems

Activities:Rational and Irrational Numbers 2

Concepts and Skills:Students will: Classify and order

numbers Represent and use numbers

in a variety of equivalent forms, such as integers, percents, fractions, decimals and rationals

Describe relationships among fractions, decimals and percents

Describe effects of arithmetic operations

Classify numbers as rational or irrational and order them

Add, subtract, multiply and divide whole numbers, fractions, decimals, integers, and rationals

Use prime factorization to find the greatest common factor(GCF) and the least common multiple (LCM) of numbers

Simplify expressions and equations with parentheses, exponents, multiplication, division, addition and subtraction

Add, subtract, multiply and divide with integers, decimals, fractions, percents, proper and improper fractions

Apply the order of operations to rational numbers including integers, decimal fractions, percents, and proper and improper fractions

Resources:

Strategies:



3.5 weeks2 weeks

Vocabulary:Standard PositionInitial SideTerminal SideCoterminal AnglesUnit CircleCosineSineTangentSecantCosecantCotangentCentral AngleReference AngleRadianDegreesRatioAngleOppositeAdjacentRight TriangleSpecial Right TrianglesHypotenuseArc LengthProportion

Unit 1: Basic Trigonometry

New State Standards: Functions(Trigonometric Functions)HSF-TF.AExtend the domain of trigonometric functions using the unit circle.1. 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.

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 aright 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:


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 angles

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

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

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

Resources:

Strategies:


G3C: Measure angles in standard position using degree or radian measure and convert a measure from one unit to the other

Timeline:1stSemester

1.5 weeks2 weeks

Vocabulary:Sine FunctionSine CurveCosine FunctionCosine CurveTangent FunctionTangent CurveAmplitudePeriodMaximumMinimumPhase ShiftsTransformationHorizontal ShiftVertical ShiftDomainRange

Unit 2: Trigonometric Functions

New State Standards: Functions(Trigonometric Functions)HSF-TF.AExtend the domain of trigonometric functions using the unit circle.4. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric

functions.

Functions(Trigonometric Functions)HSF-TF.BModel periodic phenomena with trigonometric functions.5. Choose trigonometric functions to model periodic phenomena with specified amplitude,

frequency, and midline.

Functions(Trigonometric Functions)HSF-TF.CProve and apply trigonometric identities.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: (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 aright 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

Activities:Ferris Wheel (FAL)

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

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

Resources:

Strategies:


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

and their compositionsAlgebra:


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


1.5 weeks

Vocabulary:Quadratic FunctionFactorGCFMonomialBinomialTrinomialPerfect Square TrinomialDifference of Squares

Unit 3: Factoring Quadratics

New State Standards: Algebra( Seeing Structure in Expressions)HSA-SSE.AInterpret the structure of expressions.1. 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 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

Activities:

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

function Find greatest common

factors Factor quadratics Include graphical

representations of factoring

Resources:

Strategies:


NCTM: Algebra:

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

Quality Core:A1C: Factor trinomials in the form ax^2 + bx + c 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 division, sums and differences of cubes, grouping)


3 weeks

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

Unit 4: Solving Quadratic Equations

New State Standards: Number and Quantity (The Complex Number System)HSN-CN.APerform arithmetic operations with complex numbers.1. 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.

Number and Quantity (The Complex Number System)HSN-CN.CUse complex numbers in polynomial identities and equations.7. Solve quadratic equations with real coefficients that have complex solutions

8. (+) 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.

Algebra(Seeing Structure in Expressions)HSA-SSE.AInterpret the structure of expressions.1. Interpret expressions that represent a quantity in terms of its context.*



Activities:Solving Quadratic Equations: Cutting Corners FAL

Concepts and Skills: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 numbers

Finding complex solutions Solve perfect square

trinomial equations 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

Resources:

Strategies:



+ y2).

Algebra(Seeing Structure in Expressions)HSA-SSE.BWrite expressions in equivalent forms to solve problems.3. Choose and produce an equivalent form of an expression to reveal and explain properties of

the quantity represented by the expression.*a. Factor a quadratic expression to reveal the 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.

Algebra(Creating Equations)HSA-CED.ACreate equations that describe numbers or relationships.2. Create equations in two or more variables to represent relationships between quantities;

graph equations on coordinate axes with labels and scales.

Algebra(Reasoning with Equations and Inequalities)HSA-REI.AUnderstand solving equations as a process of reasoning and explain the reasoning.1. Explain each step in solving a simple equation as following from the equality of numbers

asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Algebra(Reasoning with Equations and Inequalities)HSA-REI.BSolve equations and inequalities in one variable.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.

Algebra(Reasoning with Equations and Inequalities)HSA-REI.DRepresent and solve equations and inequalities graphically.10. Understand that the graph of an equation in two variables is the set of all its solutions

plotted in the coordinate plane, often forming a curve (which could be a line).

Find the discriminant Use the discriminant to

determine number and type of solutions


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 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 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 problemsC1A: Identify complex numbers and write their conjugatesC1B: Add, subtract, and multiply complex numbersC1C: Simplify quotients of 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)

Timeline:2nd 9 Weeks

2.5 weeks

Vocabulary:Quadratic FunctionParabolaFactorMonomialBinomialTrinomialZerosStandard FormVertex FormIntercept Formx-intercepty-interceptVertexDomain

Unit 5: Properties and Graphs of Quadratics

New State Standards: Number and Quantity(Quantities)HSN-Q.AReason quantitatively and use units to solve problems.1. 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.

Number and Quantity(The Complex Number System)HSN-CN.CUse complex numbers in polynomial identities and equations.7. Solve quadratic equations with real coefficients that have complex solutions


Algebra

Activities:Functions in Everyday Situations FAL

Forming Quadratics FAL

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

Resources:

Strategies:


Range (Seeing Structure in Expressions)HSA-SSE.AInterpret the structure of expressions.1. Interpret expressions that represent a quantity in terms of its context.*




+ y2).


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


Algebra(Arithmetic with Polynomials and Rational Expressions)HSA-APR.BUnderstand the relationship between zeros and factors of polynomials.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.





Algebra(Reasoning with Equations and Inequalities)HSA-REI.BSolve equations and inequalities in one variable.

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


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.



Functions(Interpreting Functions)HSF-IF.AUnderstand the concept of a function and use function notation.1. 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).

Functions(Interpreting Functions)HSF-IF.BInterpret functions that arise in applications in terms of the 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.

Functions(Interpreting Functions)HSF-IF.CAnalyze functions using different representations.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 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 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 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 technology


E2C: 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)

Timeline:2nd 9 Weeks

4 weeks

Vocabulary:ConjugatesFactor TheoremFundamental Theorem of AlgebraImaginary Root TheoremIrrational Root TheoremMultiplicityPolynomial FunctionRational Root TheoremRemainder TheoremStandard form of PolynomialLong DivisionSynthetic DivisionDegreeDegree of PolynomialDifference of CubesMultiple ZeroPolynomialRelative MaximumRelative MinimumSum of CubesEnd BehaviorIncreasingDecreasing

Unit 6: Polynomials and Polynomial Functions

New State Standards: Number and Quantity (The Complex Number System)HSN-CN.APerform arithmetic operations with complex numbers.1. 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.

Number and Quantity (The Complex Number System)HSN-CN.CUse complex numbers in polynomial identities and equations.7. Solve quadratic equations with real coefficients that have complex solutions

8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).






+ y2).

Algebra

Activities:Representing PolynomialsFAL

Manipulating Polynomials FAL

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 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

Resources:

Strategies:


(Seeing Structure in Expressions)HSA-SSE.BWrite expressions in equivalent forms to solve problems.3. Choose and produce an equivalent form of an expression to reveal and explain properties of

the quantity represented by the expression.★a. Factor a quadratic expression to reveal the zeros of the function it defines.b. Complete the square in a quadratic expression to reveal the maximum or


Algebra(Arithmetic with Polynomials and Rational Expressions)HSA-APR.APerform arithmetic operations on polynomials.1. Understand that polynomials form a system analogous to the integers, namely, they are

closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Algebra(Arithmetic with Polynomials and Rational Expressions)HSA-APR.BUnderstand the relationship between zeros and factors of 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.

Algebra(Arithmetic with Polynomials and Rational Expressions)HSA-APR.CUse polynomial identities to solve problems.4. Prove polynomial identities and use them to describe numerical relationships. For example,

the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.

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.



Algebra(Reasoning with Equations and Inequalities)HSA-REI.A

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)


Understand solving equations as a process of reasoning and explain the reasoning.1. Explain each step in solving a simple equation as following from the equality of numbers






2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.



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.*



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.

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: 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: 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)F2A: 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 equations


F2D: Use technology to graph a polynomial function and approximate the zeros, minimum, and maximum; determine domain and range of the polynomial function

Timeline:3rd 9 Weeks

1 week

Vocabulary:Parent FunctionAmplitudeMaximumMinimumTransformationHorizontal ShiftVertical ShiftHorizontal StretchVertical StretchDomainRangeExponential

Unit 7: Transforming Functions

New State Standards: Number and Quantity(Quantities)HSN-Q.AReason quantitatively and use units to solve problems.3. Choose a level of accuracy appropriate to limitations on measurement when reporting

quantities.







Functions(Building Functions)HSF-BF.BBuild new functions from existing functions.3. Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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

Activities:

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

Resources:

Strategies:


(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


2.5 weeks3 weeks

Vocabulary:CenterCircleConic SectionCo-verticesDirectrixEllipseFocus of ParabolaFocus of EllipseFocus of HyperbolaHyperbolaMajor AxisMinor AxisRadiusStandard Form of CircleTransverse AxisVertices of EllipseVertices of Hyperbola

Unit 8: Conics


quantities.





Activities:

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

Resources:

Strategies:













Functions(Building Functions)HSF-BF.B

ellipse Graph 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


Build new functions from existing functions.3. Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for


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 single-step 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(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 with fluency—mentally or with paper and pencil in simple cases and using technology in all cases

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 functions


E2A: 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 graphE3A: Identify conic sections (e.g., parabola, circle, ellipse, hyperbola) from their equations in standard formE3B: Graph circles and parabolas and their translations from given equations or characteristics with and without technologyE3C: Determine characteristics of circles and parabolas from their equations and graphsE3D: Identify and write equations for circles and parabolas from given characteristics and graphs


2 weeks

Vocabulary:Like RadicalsNth RootPrincipal RootRadical EquationRadical FunctionRadicandRational ExponentRationalize DenominatorSquare Root EquationSquare Root FunctionExtraneous Solution

Unit 9: Radical Functions and Rational Exponents


quantities.

Number and Quantity(The Real Number System)HSN-RN.AExtend the properties of exponents to rational exponents.1. 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.

2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Number and Quantity(The Real Number System)HSN-RN.BUse properties of rational and irrational numbers.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.


the quantity represented by the expression.*

Activities:

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

expressions Multiply radicals Divide radicals Rationalize the

denominator Add and subtract radical

expressions Multiply binomial radical

expressions 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 Solve quadratic equations

that lead to extraneous solutions

Resources:

Strategies:




2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.










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 single-step 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 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 +

NCTM: Numbers & Operations

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


representations of themd. compare and contrast the properties of numbers and number systems, including the


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 resultsAlgebra:


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 problemsG1B: 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 Week

Vocabulary:Direct VariationInverse VariationJoint VariationInverseComposition of FunctionsHorizontal Line Test

Unit 10: Function Operations


quantities.


closed under the operations of addition, subtraction, and multiplication; add, subtract, and

Activities:

Concepts and Skills:Students will: Model inverse variation Identify and solve

problems using direct, joint, and inverse variation

Find an inverse from a graph or table

Decide if a function has an inverse

Verify inverse by composition

Resources:

Strategies:


multiply polynomials.

Functions(Building Functions)HSF-BF.BBuild new functions from existing functions.4. 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.






Functions(Interpreting Functions)HSF-IF.CAnalyze functions using different representations.8. Write a function defined by an expression in different but equivalent forms to reveal and

explain different properties of the function.


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 problemsC1D: Perform operations on functions, including function composition, and determine domain and range for each of the given functionsF1A: Evaluate and simplify polynomial expressions and equationsG1A: Solve mathematical and real-world rational equation problems (e.g., work or rate problems)


2 weeks

Vocabulary:Rational ExpressionSimplest FormComplex FractionRational Equations

Unit 11: Rational Functions


quantities.







+ y2).

Activities:

Concepts and Skills:Students will: Simplify rational

expressions Add, Subtract, Multiply,

and Divide rational expressions

Find Least Common Multiples

Solve rational equations

Resources:

Strategies:



the quantity represented by the expression.*



Algebra(Arithmetic with Polynomials and Rational Expressions)HSA-APR.DRewrite rational expressions.

6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.






Functions


(Interpreting Functions)HSF-IF.CAnalyze functions using different representations.8. Write a function defined by an expression in different but equivalent forms to reveal and

explain different properties of the function.


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 integer


values

NCTM: Algebra:




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)G1A: Solve mathematical and real-world rational equation problems (e.g., work or rate problems)

Timeline:4th 9 Weeks

2 Weeks

Vocabulary:Augmented MatrixDeterminantEqual MatricesMatrixMatrix AdditionMatrix ElementMatrix EquationMatrix MultiplicationRow OperationsScalar Multiplication

Unit 12: Matrices

New State Standards: Number and Quantity(Vector and Matrix Quantities)HSN-VM.CPerform operations on matrices and use matrices in applications.6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence

relationships in a network.

7. (+) 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

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 Multiply matrices Determine if matrix

multiplication is defined

Resources:

Strategies:


Variable MatrixZero MatrixSquare MatrixInverse Matrix

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.



Algebra(Reasoning with Equations and Inequalities)HSA-REI.CSolve systems of equations.5. Prove that, given a system of two equations in two variables, replacing one equation by the

sum of that equation and a multiple of the other produces a system with the same solutions.

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(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

Evaluate determinant of 2X2 matrix

Find an inverse matrix Solve a matrix equation Evaluate determinant of

3X3 matrix Use technology to solve

matrix problems Write a system as a matrix

equation Solve a system of two

equations Solve a system of three

equations Write an augmented matrix Write a system from and

augmented matrix


(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:



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 variableA1E: 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 weeks

Unit 13: Equations, Inequalities & Systems

New State Standards: Algebra

Activities:Solving Linear Equations in Two Variables FAL

Resources:

Strategies:


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

(Seeing Structure in Expressions)HSA-SSE.AInterpret the structure of expressions.1. Interpret expressions that represent a quantity in terms of its context.*

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

Algebra(Creating Equations)HSA-CED.ACreate equations that describe numbers or relationships.1. Create equations and inequalities in one variable and use them to solve problems

3. 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.



Algebra(Reasoning with Equations and Inequalities)HSA-REI.BSolve equations and inequalities in one variable.3. Solve linear equations and inequalities in one variable, including equations with coefficients

represented by letters.

Algebra(Reasoning with Equations and Inequalities)HSA-REI.DRepresent and solve equations and inequalities graphically.11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y

= g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.*

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

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

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


(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 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


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:



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 variableA1E: Solve systems of two linear 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 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 setD1C: Solve algebraically a system containing three variablesD2A: 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:AsymptoteChange of Base FormulaCommon LogarithmExponential EquationExponential FunctionLogarithmLogarithmic EquationLogarithmic FunctionNatural Logarithmic Function

Unit 14: Exponential and Logarithmic Functions

New State Standards: Number and Quantity(Quantities)HSN-Q.AReason quantitatively and use units to solve problems.1. 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.

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


Activities:

Concepts and Skills:Students will: Graph exponential function Translate exponential

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

Resources:

Strategies:





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%.

Algebra(Creating Equations)HSA-CED.ACreate equations that describe numbers or relationships.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.






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

Solve exponential equations

Solve logarithmic equations

Use logarithmic properties to solve an equation


input x. The graph of f is the graph of the equation y = f(x).



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.*


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.


a. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.


Functions(Building Functions)HSF-BF.ABuild a function that models a relationship between two quantities.1. 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.


Functions(Building Functions)HSF-BF.BBuild new functions from existing functions.3. Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for


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.

Functions(Linear, Quadratic, and Exponential Models)HSF-LE.AConstruct and compare linear, quadratic, and exponential models and solve problems.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.

Functions(Linear, Quadratic, and Exponential Models)HSF-LE.BInterpret expressions for functions in terms of the situation they model.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 expression is 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 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

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 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:


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;



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 graph


G2A: Graph exponential and logarithmic functions with and without technologyG2B: Convert exponential equations to logarithmic form and logarithmic equations to exponential form


1 week

Vocabulary:Arithmetic MeanArithmetic SequenceArithmetic SeriesCommon DifferenceCommon RatioExplicit FormulaGeometric MeanRecursive FormulaSequenceSeriesTermSummation NotationInfinite Series

Unit 15: Sequences and Series

New State Standards: Algebra(Seeing Structure in Expressions)HSA-SSE.AInterpret the structure of expressions.1. Interpret expressions that represent a quantity in terms of its context.*

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

Algebra(Seeing Structure in Expressions)HSA-SSE.BInterpret the structure of expressions.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.*

Functions(Interpreting Functions)HSF-IF.AUnderstand the concept of a function and use function notation.3. 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.



Functions(Building Functions)HSF-BF.ABuild a function that models a relationship between two quantities.1. 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.*

Functions

Activities:

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

sequence Identify a geometric

sequence 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

Resources:

Strategies:


(Linear, Quadratic, and Exponential Models)HSF-LE.AConstruct and compare linear, quadratic, and exponential models and solve problems.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).

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 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 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 decimalsRange (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


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


2 weeks

Vocabulary:Binomial ProbabilityBox and Whisker PlotConditional ProbabilityCumulative ProbabilityInterquartile RangeMeasures of Central TendencyMeasures of VariationNormal DistributionOutlierPercentileProbability DistributionQuartilesSampleSample SpaceStandard DeviationStandard Normal CurveZ-Score

Unit 16: Data Analysis and Probability

New State Standards: Statistics and Probability(Interpreting Categorical and Quantitative Data)HSS-ID.ASummarize, represent, and interpret data on a single count or measurement variable.4. Use the mean and standard deviation of a dataset to fit it to a normal distribution and to

estimate population percentages. Recognize that there are datasets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

Statistics and Probability(Making Inferences and Justifying Conclusions)HSS-IC.AUnderstand and evaluate random processes underlying statistical experiments.1. Understand statistics as a process for making inferences about population parameters based

on a random 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?

Statistics and Probability(Making Inferences and Justifying Conclusions)HSS-IC.BMake inferences and justify conclusions from sample surveys, experiments, and observational studies.3. Recognize the purposes of and differences among sample surveys, experiments, and

observational studies; explain how randomization relates to each.

4. Use data from a sample survey to estimate population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

Activities:

Concepts and Skills:Students will: Make a frequency table Calculate probability

distributions Find conditional

probability Construct tree diagrams Compute Combinations

and Permutations Find measures of central

tendency 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

Resources:

Strategies:


5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

6. Evaluate reports based on data.

Statistics and Probability(Using Probability to Make Decisions)HSS-MD.BUse probability to evaluate outcomes of decisions.6. (+) 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, medical testing, 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

Documents

2 Map.d… · Web viewTimeline: 1st 9 Weeks. 2 weeks. Not for Accel. Vocabulary: Natural. Whole. Integers. Rational. Irrational. Real. Fraction. Percent. Addition. Subtraction