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10/8/2014 Instruction: Algebra 2 Common Core Template

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Chicopee Public Schools

High School Mathematics

ALGEBRA 2 7-30-2012

STANDARDS FOR MATHEMATICAL PRACTICE

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

STANDARDS FOR MATHEMATICAL CONTENT

CRITICAL AREAS

1. Relate arithmetic of rational expressions to arithmetic of rational numbers.

2. Expand understandings of functions and graphing to include trigonometric functions.

3. Synthesize and generalize functions and extend understanding of exponential functions to logarithmic functions.

4. Relate data display and summary statistics to probability and explore a variety of data collection methods.

DOMAINS

1. Number and Quantity (N)

2. Algebra (A)

3. Functions (F)

4. Statistics and Probability (SP)

NUMBER AND QUANTITY - THE COMPLEX NUMBER SYSTEM

Perform arithmetic operations with complex numbers.

Standard Description / Objectives (UNPACKED) Resources/Strategies Assessments

N-CN.1 Know there is a complex number i such that = -1, andevery complex number has the form a + bi with a and b real.

McDougal-Littell Chapter 4.6 – PerformOperations with Complex Numbers pp. 275

Formativeassessment: Daily

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Ø Perform arithmetic operations with complex numbers.

Ø Understand number systems.

Ø Understand how mathematical ideas build on one another.

Ø

– 283 / Taught in third term (February);pacing is 2 days. Relate the real andimaginary parts of a complex number to thereal (horizontal) and imaginary (vertical)axes. Ask students what kinds of numberthey used when they were young children,and what new kinds of numbers theylearned as they moved through school.Explain that each time they learned aboutnew kinds of numbers, such as fractions,negative numbers, and irrational numbers,they were able to solve equations theycould not solve before. Complex numbersare just one more step In that process.

Homework quiz, oronline quiz available atclasszone.com

Summativeassessment: Chaptertest.

N-CN.2 Use the relation = –1 and the commutative, associative,and distributive properties to add, subtract, and multiplycomplex numbers.

Ø Perform arithmetic operations with complex numbers.

Ø Understand number systems.

Ø Understand how mathematical ideas build on one another.

McDougal-Littell Chapter 4.6 – PerformOperations with Complex Numbers pp. 275– 283 / Taught in third term (February);pacing is 2 days. To add or subtractcomplex numbers, add or subtract their realparts and their imaginary parts separately(associative & commutative properties). Tomultiply complex numbers, use thedistributive property or the FOIL method. Todivide complex numbers, multiply thenumerator and denominator by the complexconjugate of the denominator.

Formativeassessment: DailyHomework quiz, oronline quiz available atclasszone.com


Use complex numbers in polynomial identities and equations.

Standard Description /Objectives (UNPACKED) Resources/Strategies Assessments

N-CN.7 Solve quadratic equations with real coefficients that havecomplex solutions.

Ø Use complex numbers in polynomial identities andequations.

Ø Analyze situations using algebraic symbols;

Ø Use models to represent relationships.

Ø Solve quadratic equations by factoring, finding square roots,and completing the square.

McDougal-Littell Chapter 4.3 – 4.5, 4.7 –Solve quadratic equations by factoring,finding square roots, and completing thesquare. pp. 252 – 274 / Taught in third term(January); pacing is 6 days. Solve x2 + bx +c = 0 by factoring, solve ax2 + bx + c = 0 byfactoring, solve quadratic functions byfinding square roots, and completing thesquare.



N-CN.8 (+) Extend polynomial identities to the complex numbers. Forexample, rewrite x2 + 4 as (x + 2i)(x – 2i).


Ø Analyze situations using algebraic symbols.


McDougal-Littell Chapter 4.3 – 4.5, 4.7, 4.8,5.4 – Special factoring patterns. Taught inthird term (January); pacing is inclusive ofthe 6 days from previous standard and willinclude 1 – 2 additional days extending tothe complex numbers. Recognizingtrinomials by their special factoring patterns(identities) here: Solve (a+b) 2 = a 2 + 2ab +b 2 page 253

(a+b)(c+d) = ac + ad + bc + bd page 252

a 2 - b 2 = (a+b)(a-b) (Difference of squares)page 253

a 3 b 3 = (a b)(a 2 ab + b 2) (Sum andDifference of Cubes) page 354

x 2 + (a+b)x + ab = (x + a)(x + b) page 252

if ax 2 + bx + c = 0 then x = ( -b (b 2 -4ac) ) / 2a (Quadratic Formula) page 292



N.CN.9 (+) Know the Fundamental Theorem of Algebra; show that it istrue for quadratic polynomials.

McDougal-Littell Chapter 5.7 – Applying theFundamental Theorem of Algebra. pp. 379 -386 Taught in third term (March); pacing is

Formativeassessment: DailyHomework quiz, or




Ø Determine the number of possible solutions of a polynomialfunction.


2 days. Have students determine thepossible number of positive, negative, andimaginary zeros of a polynomial function.Include the corollary to the FToA as well asComplex & Irrational Conjugates theorems,and Descartes’ Rule of Signs.

online quiz available atclasszone.com


NUMBER AND QUANTITY - VECTOR AND MATRIX QUANTITIES

Represent and model with vector quantities.


N-VM.1 (+) Recognize vector quantities as having both magnitude anddirection. Represent vector quantities by directed linesegments, and use appropriate symbols for vectors and theirmagnitudes (e.g., v, |v|, || v||, v).

Ø Describe, represent, and find the magnitude and direction ofvectors

Ø Represent and model with vector quantities.

McDougal-Littell Geometry Chapter 7.4 –Translations & Vectors pp. 421 - 428;McDougal-Littell Geometry Chapter 9.7Vectors pp. 573 - 579. Glencoe McGraw-Hill- Chapter 8.7 Vectors & Computer Lab pp.593 - 601.

LTF Module: “Applications of Vectors”

LTF Module: “Particle Motion Along aHorizontal Line (Mod 7)”

Needs to be added to first term (October);pacing is 2 - 3 days. Students usegeometric sketches, scale drawings, tables,verbal descriptions, and algebraic equationsto investigate vectors.

Formativeassessment: DailyHomework quiz, oronline quiz available atglencoe.com


N-VM.3 (+) Solve problems involving velocity and other quantities thatcan be represented by vectors.

Ø Use real world problems to represent and model with vectorquantities.

McDougal-Littell Geometry Chapter 7.4 –Translations & Vectors pp. 421 - 428;McDougal-Littell Geometry Chapter 9.7Vectors pp. 573 - 579. Glencoe McGraw-Hill- Chapter 8.7 Vectors & Computer Lab pp.593 - 601.

LTF Module: “Applications of Vectors”


Needs to be added to first term (October);pacing is 2 - 3 days.

Have students use the plethora of real worldproblems in the aforementioned books. Thecomputer lab requires the use of a ruler andprotractor.

Formativeassessment: DailyHomework quiz, oronline quiz available atglencoe.com


Perform operations on matrices and use matrices in applications.

N-VM.6 (+) Use matrices to represent and manipulate data, e.g., torepresent payoffs or incidence relationships in a network.

Ø Perform operations on matrices

Ø Use matrices in applications

Ø Analyze situations using algebraic symbols.

Ø Use models to understand relationships

McDougal-Littell Chapter 3.5 – 3.6 –Perform Basic Matrix Operations. pp. 187 –199 / Taught in second term (November);pacing is 3 days.

LTF Module: “Solving Systems of LinearEquations”

LTF Module: “Solving Systems of LinearInequalities(Mod2)”

Students will learn how to performoperations with matrices so they canorganize, manipulate and solverelationships in models. Graphingcalculators can simplify the calculations forreal world problems.





N-VM.8 (+) Add, subtract, and multiply matrices of appropriatedimensions.

Ø Perform operations on matrices

McDougal-Littell Chapter 3.5 – 3.6 –Perform Basic Matrix Operations. pp. 187 –199 / Taught in second term (November);pacing is 3 days. Students will learn how toperform operations with matrices so theycan organize, manipulate and solverelationships in models.



N-VM.12 (+) Work with 2 ´ 2 matrices as transformations of the plane,and interpret the absolute value of the determinant in terms ofarea.

Ø Be able to express plane transformations in algebraic andmatrix form.

Ø Use matrices to represent figures in a coordinate plane.

Ø Use matrices to represent transformations on figures in acoordinate plane.

Ø Use matrices to solve area problems

McDougal-Littell Chapter 3.7 – EvaluateDeterminants & Apply Cramer’s Rule; Usematrices to solve area problems. pp. 187 –199 / Taught in second term (November);pacing is 3 days. Students will learn how toperform operations with matrices so theycan organize, manipulate and solverelationships in models.

Glencoe McGraw-Hill - Chapter 9.6Dilations pp. 660 - 667 & Computer Lab -Transformations using Matrices pp. 668 -669. Needs to be added to second term(November); pacing is 2 days. Graphingcalculators are needed for computer labactivity. McDougal-Littell – Skills Review –The Coordinate Plane; Transformations pp.987 – 988 / Needs to be added in secondterm (November); pacing is 3 days.Supplemental material: Chapter 9 Matrices& Transformations; pacing is 2 days

LTF Module: “Solving Systems of LinearEquations”

LTF Module: “Solving Systems of LinearInequalities(Mod 2)”



ALGEBRA - SEEING STRUCTURE IN EXPRESSIONS

Interpret the structure of expressions .


A-SSE.1 Interpret expressions that represent a quantity in terms of itscontext. «

Ø Be able understand and explain meanings of operations asto their use in a word problem or algebraic expression.

McDougal-Littell Chapter 1.2 Evaluate &Simplify algebraic expressions. pp. 10 - 17

LTF Module “Related Rates – Area andVolume Applications (A2) (Mod 12)”

LTF Module “Literal Equations – GeometricFormulas (Mod 12)”

Taught in first term (September); pacing is 2days. Linear, quadratic, and exponentialexpressions are the focus in Algebra I, andinteger exponents are extended to rationalexponents (only those with square or cubedroots). In Algebra II, the expectation is toextend to polynomial and rationalexpressions.



A-SSE.1.a Interpret parts of an expression, such as terms, factors, andcoefficients.

Ø Identify the different parts of the expression and explaintheir meaning within the context of a problem.

McDougal-Littell - Chapter 1.2 Evaluate &Simplify algebraic expressions. pp. 10 - 17

LTF Module “Transformations in NumericalData (Mod 11)”

Taught in first term (September); pacing is 2days.





A-SSE.1.b Interpret complicated expressions by viewing one or more oftheir parts as a single entity . For example, interpret P(1 + r)nas the product of P and a factor not depending on P.

Ø Decompose expressions and make sense of the multiplefactors and terms by explaining the meaning of the individualparts.

McDougal-Littell - Chapter 7.1 ExponentialGrowth Functions (Extended). pp. 481 &484 Taught in first term (September); pacingis 1 day. Students could find the balance ofan account using Example 5 along withguided practice 6. Homework could includeproblems 37 & 38 on page 484.



A-SSE.2 Use the structure of an expression to identify ways to rewriteit. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing itas a difference of squares that can be factored as (x2 – y2)(x2

+ y2).

Ø Rewrite algebraic expressions in different equivalent formssuch as factoring or combining like terms.

o Use factoring techniques such as common factors,grouping, the difference of two squares, the sum or differenceof two cubes, or a combination of methods to factorcompletely.

o Simplify expressions including combining like terms, usingthe distributive property and other operations withpolynomials.

McDougal-Littell - Chapter 4.3 & 4.4Factoring pp. 252 - 265. Taught in secondterm (January); pacing is 3 days.Recognizing different forms and specialfactoring patterns as well as solvingquadratic equations, area and real life wordproblems should be the study strategy.

LTF Module “Transformations in NumericalData (Mod 11)”

Taught in third term (January); pacing is 3days.



Write expressions in equivalent forms to solve problems.


A-SSE.4 Derive the formula for the sum of a finite geometric series(when the common ratio is not 1), and use the formula tosolve problems. For example, calculate mortgage payments. «

Ø Develop the formula for the sum of a finite geometric serieswhen the ratio is not 1.

Ø Use the formula to solve real world problems such ascalculating the height of a tree after n years given the initialheight of the tree and the rate the tree grows each year.Calculate mortgage payments.

McDougal-Littell - Chapter 12.1 - 12.3Define & Use Sequences and Series,Analyze Geometric Sequences & Series pp.794 - 819. Topic needs to be added to thefourth term (June); pacing is 5 days.

Khan Academy: ”Algebra II: BinomialExpansions, Geometric Series Sum”

LTF Module: “Arithmetic Sequences”

It is important to balance conceptualunderstanding and procedural fluency inwork with equivalent expressions. Forexample, ask students to think about anexercise program such as running 5 milesthe first week, and increasing the distanceby 3 miles per week. They will learn to usesequences & series to represent thenumber of miles run each week and thetotal numbers of miles run. Also, considerextending finite geometric series to infinitegeometric series for enrichment.



ALGEBRA - ARITHMETIC WITH POLYNOMIALS AND RATIONAL EXPRESSIONS

Perform arithmetic operations on polynomials.


A-APR.1 Understand that polynomials form a system analogous to theintegers, namely, they are closed under the operations ofaddition, subtraction, and multiplication; add, subtract, andmultiply polynomials.

Ø Understand the definition of a polynomial.

Ø Understand the concepts of combining like terms andclosure.

McDougal-Littell - Chapter 5.2 & 5.3Evaluate & Graph Polynomial Functions;Add, Subtract, and Multiply Polynomials pp.337 - 352. Taught in third term (March);pacing is 3 days. Recognizing differentforms and special factoring patterns as wellas solving quadratic equations, area andreal life word problems should be the studystrategy.





Ø Add, subtract, and multiply polynomials and understandhow closure applies under these operations

A-APR.MA.1.a Divide polynomials.

Ø Divide polynomials by understanding polynomial longdivision

McDougal-Littell - Chapter 5.5 Apply theRemainder & Factor theorem pp. 362 - 369.Taught in third term (March); pacing is 2days. Use polynomial long division andsynthetic division so that you can factorpolynomials and solve real life problemsthat are non-linear. Choose one problemfrom exercises and use both polynomiallong division and synthetic division to showhow the two processes are related.



Understand the relationship between zeros and factors of polynomials.

A-APR.2 Know and apply the Remainder Theorem: For a polynomialp(x) and a number a, the remainder on division by x – a isp(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

Ø Understand and apply the Remainder Theorem.

Ø Understand that a is a root of a polynomial function if andonly if x-a is a factor of the function.

McDougal-Littell - Chapter 5.5 Apply theRemainder & Factor theorem pp. 362 - 369.Taught in third term (March); pacing is 2days. Use polynomial long division andsynthetic division so that you can factorpolynomials and solve real life problemsthat are non-linear. Choose one problemfrom exercises and use both polynomiallong division and synthetic division to showhow the two processes are related.



A-APR.3 Identify zeros of polynomials when suitable factorizations areavailable, and use the zeros to construct a rough graph of thefunction defined by the polynomial.

Ø Find the zeros of a polynomial when the polynomial isfactored.

Ø Use the zeros of a function to sketch a graph of thefunction.

McDougal-Littell - Chapter 5.6 Find RationalZeros pp. 370 - 377. Taught in third term(March); pacing is 2 days. Students shouldlearn a way to list all possible rational zerosof a polynomial function. Select thereasonable values from the list and usethem to graph the function. Stress that agraph shows only the reasonable values tocheck for rational zeros and that thesevalues may not be actual zeros.



Use polynomial identities to solve problems.

A-APR.4 Prove polynomial identities and use them to describenumerical relationships. For example, the polynomial identity(x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generatePythagorean triples.

Ø Understand that polynomial identities include but are notlimited to the product of the sum and difference of two terms,the difference of two squares, the sum and difference of twocubes, the square of a binomial, etc .

Ø Prove polynomial identities by showing steps and providingreasons.

Ø Illustrate how polynomial identities are used to determinenumerical relationships such as 252 = (20+5)2 = 20 2 + 2*20*5+ 52

McDougal-Littell - Chapter 5.3 Add,Subtract, and Multiply Polynomials pp. 337 -352. Taught in third term (March); pacing is1 day added to this topic to extend it toproving polynomial identities. Recognizingdifferent forms and special factoringpatterns as well as solving quadraticequations, area and real life word problemsshould be the study strategy.

(a+b) 2 = a 2 + 2ab + b 2

(a+b)(c+d) = ac + ad + bc + bd

a 2 - b 2 = (a+b)(a-b) (Difference ofsquares)

a 3 b 3 = (a b)(a 2 ab + b 2) (Sum andDifference of Cubes)

x 2 + (a+b)x + AB = (x + a)(x + b)

if ax 2 + bx + c = 0 then x = ( -b (b 2 -4ac) ) / 2a (Quadratic Formula)



A-APR.5 (+) Know and apply the Binomial Theorem for the expansionof (x + y)n in powers of x and y for a positive integer n, wherex and y are any numbers, with coefficients determined forexample by Pascal’s Triangle. (The Binomial Theorem can beproved by mathematical induction or by a combinatorialargument.)

McDougal-Littell - Chapter 10.2Combinations & the Binomial Theorem pp.690 - 697. Topic needs to be added to thefourth term (June); pacing is 1 day.

Khan Academy: ”Precalculus”




Ø For small values of n, use Pascal’s Triangle to determinethe coefficients of the binomial expansion.

Ø Use the Binomial Theorem to find the nth term in theexpansion of a binomial to a positive power.

LTF Module “Combinations (Mod 6)” Summativeassessment: Chaptertest.

Rewrite rational expressions.

A-APR.6 Rewrite simple rational expressions in different forms; writea(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 thedegree of b(x), using inspection, long division, or, for the morecomplicated examples, a computer algebra system.

Ø Rewrite rational expressions, in the form byusing factoring, long division, or synthetic division. Use acomputer algebra system for complicated examples to assistwith building a broader conceptual understanding.

McDougal-Littell - Chapter 8.1 - 8.6 RationalFunctions pp. 548 - 611 Taught in fourthterm (June); pacing is 12 days andextending it to include A computer algebrasystem such as “Geogebra ” can be usedfor complicated examples.



A-APR.7 (+) Understand that rational expressions form a systemanalogous to the rational numbers, closed under addition,subtraction, multiplication, and division by a nonzero rationalexpression; add, subtract, multiply, and divide rationalexpressions.

Ø Simplify rational expressions by adding, subtracting,multiplying, or dividing.

Ø Understand that rational expressions are closed underaddition, subtraction, multiplication, and division (by a nonzeroexpression).

McDougal-Littell - Chapter 8.1 - 8.6 RationalFunctions pp. 548 - 611 Taught in fourthterm (June); pacing is 12 days andextending it to include A computer algebrasystem such as “Geogebra ” can be usedfor complicated examples.



ALGEBRA - CREATING EQUATIONS

Create equations that describe numbers or relationships.


A-CED.1 Create equations and inequalities in one variable and usethem to solve problems. Include equations arising from linearand quadratic functions, and simple rational and exponentialfunctions. «

Ø Create linear, quadratic, rational and exponential equationsand inequalities in one variable and use them in a contextualsituation to solve problems.

McDougal-Littell - Chapter 1.3 -1.7 LinearEquations & Inequalities pp. 18 - 59

McDougal-Littell - Chapter 4.5 SolveQuadratic Equations pp. 266 - 274

McDougal-Littell - Chapter 7.5 - 7.7 SolveExponential & Logarithmic Equations pp.507 - 536

McDougal-Littell - Chapter 8.6 SolveRational Equations & Rational Inequalitiespp. 589 - 600

Taught in multiple terms and is a compositeof several chapters. Should be included asthe year progresses and can be used at theend-of-the-year review for final exam.

LTF Module : “Adaptation of 2003 AB3 (Mod5)”


LTF Module: “Motion Problems UsingExponential and Natural LogarithmicFunctions (Mod 7)”

LTF Module: “Population Growth (Mod 10)”

LTF Module: “Transformations in NumericalData (Mod 11)”





A-CED.2 Create equations in two or more variables to representrelationships between quantities; graph equations oncoordinate axes with labels and scales. «

Ø Create equations in two or more variables to representrelationships between quantities.

Ø Graph equations in two variables on a coordinate plane andlabel the axes and scales.

McDougal-Littell - Chapter 3.1 - 3.3 LinearSystems pp. 150 - 173 Taught in secondterm (November); pacing is 3 days.

LTF Module: “Solving Systems of LinearInequalities (Mod 2)”

LTF Module: “CHARACTERISTICS OFDISCONTINUOUS PIECEWISEFUNCTION (Mod 1)”

LTF Module: “Accumulation with aQuadratic Function (Mod 5)”

LTF Module: “Accumulation (PC) (Mod 5)”

LTF Module: “Writing Functions – FindingExtrema (Mod 9)”


LTF Module: “Related Rates – Area andVolume Applications (A2) (Mod 12)”



A-CED.3 Represent constraints by equations or inequalities, and bysystems of equations and/or inequalities, and interpretsolutions as viable or non-viable options in a modelingcontext. For example, represent inequalities describingnutritional and cost constraints on combinations of differentfoods. «

Ø Write and use a system of equations and/or inequalities tosolve a real world problem. Recognize that the equations andinequalities represent the constraints of the problem. Use theObjective Equation and the Corner Principle to determine thesolution to the problem. (Linear Programming)

McDougal-Littell - Chapter 3.3 Extension -Use Linear Programming pp. 174 - 176Taught in second term (November); pacingis 3 days.


LTF Module: “Piecewise Functions (Mod 3)“



LTF Module: “Writing Functions – FindingExtrema”



A-CED.4 Rearrange formulas to highlight a quantity of interest, usingthe same reasoning as in solving equations. For example,rearrange Ohm’s lawV = IR to highlight resistance R

Ø Solve multi-variable formulas or literal equations, for aspecific variable..

McDougal-Littell - Chapter 1.4 – RewriteFormulas and Equations pp. 26 – 33 Taughtin first term (September) Ask students towrite a list of steps to follow in solvingequations using exercises 18 - 32 andproblems 35, 38, and 39 on pp. 31 - 32.

Many formulas used in physics andelectronics involve radicals. Ask studentswho are taking or have taken these coursesto bring in some of these formulas andexplain to the class how the formulas areused.



LTF Module: “Literal Equations – GeometricFormulas (Mod 12)”



ALGEBRA - REASONING WITH EQUATIONS AND INEQUALITIES

Understand solving equations as a process of reasoning and explain the reasoning.




A-REI.2 Solve simple rational and radical equations in one variable,and give examples showing how extraneous solutions mayarise.

Ø Solve simple rational and radical equations in one variableand provide examples of how extraneous solutions arise.

McDougal-Littell - Chapter 6.6 – SolveRadical Equations pp. 452 – 464. Taught infourth term (May). Pacing is two days. Manyformulas used in physics and electronicsinvolve radicals. Ask students who aretaking or have taken these courses to bringin some of these formulas and explain to theclass how the formulas are used.



Represent and solve equations and inequalities graphically.

A-REI.11 Explain why the x-coordinates of the points where the graphsof the equations y = f(x) and y = g(x) intersect are thesolutions of the equation f(x) = g(x); find the solutionsapproximately, 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 logarithmicfunctions. «

Ø Explain why the intersection of y = f(x) and y = g(x) is thesolution of f(x) = g(x) for any combination of linear, polynomial,rational, absolute value, exponential, and logarithmicfunctions. Find the solution(s) by:

o Using technology to graph the equations and determine theirpoint of intersection,

o Using tables of values, or

o Using successive approximations that become closer andcloser to the actual value.

McDougal-Littell - Chapter 3.1 – GraphSystems of Equations p. 159. GraphingCalculator Activity

Ask each student to work with a partner.One student can graph the equations, andthe other can find the solution using the“intersect” feature of the calculator.

McDougal - Littell - Chapter 4.9 – Graph &Solve Quadratic Inequalities p. 305 exercise58.

Use a graphing calculator to demonstratehow to graph an inequality in two variablesby using the graph style that shades theregion either above or below the graph ofthe equation that is entered. Explain whythe intersection of y = f(x) and y = g(x) is thesolution of f(x) = g(x).

McDougal - Littell - Chapter 5.4 Workshop –Using Tables & Graphs to Solve Functionspp. 360 - 361

McDougal - Littell - Chapter 6.6 Point ofIntersection for Radical Functions Example6 p. 455

McDougal - Littell - Chapter 7.6 Point ofIntersection for Exponential FunctionsExamples 2, 7 & Alternative Methods p. 523These methods allow students to visualizesolutions by using a table, approximatingpoints that get closer and closer to theactual value as well as using the “intersect”feature on a graphing calculator to show theintersection of the two functions.





FUNCTIONS - INTERPRETING FUNCTIONS

Interpret functions that arise in applications in terms of the context.


F-IF.4 For a function that models a relationship between twoquantities, interpret key features of graphs and tables in termsof the quantities, and sketch graphs showing key featuresgiven a verbal description of the relationship. Key features

include: intercepts; intervals where the function is increasing,decreasing, positive, or negative; relative maximums andminimums; symmetries; end behavior; and periodicity. «

Ø Given a function, identify key features in graphs and tables

McDougall-Littell - Chapter 2.1 – RepresentRelations and Functions pp. 72 – 81 Taughtin first term (October). Pacing is 1 day.







including: intercepts; intervals where the function isincreasing, decreasing, positive, or negative; relativemaximums and minimums; symmetries; end behavior; andperiodicity.

Ø Given the key features of a function, sketch the graph.

LTF Module: “AVERAGE RATE OFCHANGE“


LTF Module: “ANALYZING FUNCTIONALBEHAVIOR USING GRAPHICALDISPLAYS, PART 1(Mod 4)”

LTF Module: “ANALYZING FUNCTIONALBEHAVIOR USING GRAPHICALDISPLAYS, PART 2 (Mod 4)”

LTF Module: “Adaptation of 2003 AB3 (Mod5)”



LTF Module: “Reading the Graph of aVelocity Function (Mod 7)”

LTF Module: “Rational FunctionOptimization Problem (Mod 9)”

LTF Module: “Writing Functions – FindingExtrema (Mod 9)”



LTF Module: “Literal Equations (Mod 12)”

F-IF.5 Relate the domain of a function to its graph and, whereapplicable, to the quantitative relationship it describes. Forexample, if the function h gives the number of person-hoursit takes to assemble n engines in a factory, then the positiveintegers would be an appropriate domain for the function. «

Ø Given the graph of a function, determine the practicaldomain of the function as it relates to the numericalrelationship it describes.

McDougall-Littell - Chapter 2.1 – RepresentRelations and Functions pp. 72 – 81 Taughtin first term (October). Pacing is 1 day.





LTF Module: “Rational FunctionOptimization Problem (Mod 9)”




F-IF.6 Calculate and interpret the average rate of change of afunction (presented symbolically or as a table) over a specifiedinterval. Estimate the rate of change from a graph. «

Ø Calculate the average rate of change over a specifiedinterval of a function presented symbolically or in a table.

Ø Estimate the average rate of change over a specifiedinterval of a function from the function’s graph.

Ø Interpret, in context, the average rate of change of afunction over a specified interval.

McDougal-Littell - Chapter 2.2 – Find Slopeand Rate of Change pp. 82 – 88. Taught infirst term (October). Pacing is 1 day.

McDougal-Littell - Chapter 2.6 – DrawScatter Plots and Best-Fitting Lines pp. 112– 120 Taught in first term (October). Pacingis 2 days.



LTF Module: “ANALYZING FUNCTIONAL





BEHAVIOR USING GRAPHICALDISPLAYS, PART 1 (Mod 4)”

LTF Module: “ANALYZING FUNCTIONALBEHAVIOR USING GRAPHICALDISPLAYS, PART 2 (Mod 4)””



LTF Module: “Collecting Linear andQuadratic Data with Cardboard Tubes (Mod10)”


Analyze functions using different representations.


F-IF.7 Graph functions expressed symbolically and show keyfeatures of the graph, by hand in simple cases and usingtechnology for more complicated cases. «

Ø Graph functions expressed symbolically and show keyfeatures of the graph. Graph simple cases by hand, and usetechnology to show more complicated cases including thefunctions described in F-IF.7.b, F-IF.7.c, F-IF.7.e.

In Algebra 2, the expectation is to developmodels for more complex or sophisticatedsituations than in previous courses.

LTF Module: “Solving Systems of LinearInequalities”


LTF Module: “Piecewise Functions“

LTF Module: “ANALYZING FUNCTIONALBEHAVIOR USING GRAPHICALDISPLAYS, PART 1”




F-IF.7.b Graph square root, cube root, and piecewise-definedfunctions, including step functions and absolute valuefunctions. «

Ø Square root, cube root, and piecewise-defined functions,including step functions and absolute value functions.

McDougall-Littell - Chapter 2.7 – UseAbsolute Value Functions &Transformations; Model Real–LifeSituations with Equations pp. 140 – 149.Taught in first term (October). Pacing is 2days.

McDougall-Littell - Chapter 6.5 – GraphSquare Root and Cube Root Functionspp.446 – 451. Taught in fourth term (May).Pacing is 1 day.

LTF Module: CHARACTERISTICS OFDISCONTINUOUS PIECEWISEFUNCTION (Mod 1)”




F-IF.7.c Graph polynomial functions, identifying zeros when suitablefactorizations are available, and showing end behavior. «

Ø Polynomial functions, identifying zeros when factorable, andshowing end behavior.

McDougal-Littell - Chapter 5.6 – FindRational Zeros pp. 370 – 378 Taught in thirdterm (March). Pacing is 2 days.

McDougal-Littell - Chapter 5.8 – AnalyzeGraphs of Polynomial Functions pp. 387 –392 Taught in third term (March). Pacing is1 day.






F-IF.7.e Graph exponential and logarithmic functions, showingintercepts and end behavior, and trigonometric functions,showing period, midline, and amplitude. «

Ø Exponential and logarithmic functions, showing interceptsand end behavior.

Ø Trigonometric functions, showing period, midline, andamplitude.

McDougall-Littell - Chapter 7.1 – GraphExponential Growth Functions pp. 478 –485. Taught in fourth term (May). Pacing is1 day.

McDougall-Littell - Chapter 7.2 – GraphExponential Decay Functions pp. 486 – 491Taught in fourth term (May). Pacing is 1day.

McDougall-Littell - Chapter 7.3 – UseFunctions involving e pp. 492 – 498. Taughtin fourth term (May). Pacing is 1 day.

McDougall-Littell - Chapter 7.4 – EvaluateLogarithms & Graph Log Functions pp. 499– 506. Taught in fourth term (May). Pacingis 2 days.





F-IF.8 Write a function defined by an expression in different butequivalent forms to reveal and explain different properties ofthe function.

Ø Write a function in equivalent forms to show differentproperties of the function.

Ø Explain the different properties of a function that arerevealed by writing a function in equivalent forms.

McDougal-Littell - Chapter 5.8 – AnalyzeGraphs of Polynomial Functions pp. 387 –392 Taught in third term (March). Pacing is1 day.




LTF Module: “Literal Equations – GeometricFormulas (Mod 12)”

LTF Module: “Literal Equations (Mod 12)”



F-IF.MA.8.c Translate among different representations of functions andrelations: graphs, equations, point sets, and tables.

Ø Explain how different forms, graphs, equations, point sets,and tables can represent the same function or relation.

McDougall-Littell - Chapter 2.1 – RepresentRelations and Functions pp. 72 – 81

McDougall-Littell - Chapter 5.9 – WritePolynomial Functions & Models pp. 72 – 81




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

Ø Compare the key features of two functions represented indifferent ways. For example, compare the end behavior of twofunctions; one of which is represented graphically and theother is represented symbolically.

McDougall-Littell - Chapter 2.1 – RepresentRelations and Functions pp. 72 – 81








FUNCTIONS - BUILDING FUNCTIONS

Build a function that models a relationship between two quantities.




F-BF.1 Write a function that describes a relationship between twoquantities. «

Ø Compose functions.

Ø From context, write an explicit expression, define arecursive process, or describe the calculations needed tomodel a function between two quantities.

McDougal-Littell - Chapter 6.3 – PerformFunction Operations and Composition pp.428-437 Taught in fourth term (April) Pacingis 1 day.

McDougal-Littell - Chapter 8.1 – ModelInverse and Joint Variation pp. 551 – 557.Taught in fourth term (May). Pacing is 1day.

LTF Module: CHARACTERISTICS OFDISCONTINUOUS PIECEWISEFUNCTION




F-BF.1.b Combine standard function types using arithmetic operations.For example, build a function that models the temperature of acooling body by adding a constant function to a decayingexponential, and relate these functions to the model. «

Ø Combine standard function types, such as linear andexponential, using arithmetic operations.

McDougal-Littell - Chapter 2.5 – ModelDirect Variation pp. 551 – 557. Taught infirst term (October). Pacing is 1 day.

McDougal-Littell - Chapter 8.1 – ModelInverse and Joint Variation pp. 551 – 557.Taught in fourth term (May). Pacing is 1day.



Build new functions from existing functions.


F-BF.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 positiveand negative); find the value of k given the graphs.Experiment with cases and illustrate an explanation of theeffects on the graph using technology. Include recognizingeven and odd functions from their graphs and algebraicexpressions for them.

Ø Identify, through experimenting with technology, the effecton the graph of a function by replacing f(x) with f(x) + k, kf(x),f(kx), and f(x + k) for specific values of k (both positive andnegative).

Ø Given the graphs of the original function and atransformation, determine the value of (k).

Ø Recognize even and odd functions from their graphs andequations.

McDougal-Littell - Pre-Chapter 2.7 –Exploring Transformations p. 121 . Taughtin first term (October). Pacing is ½ hourprior to teaching Chapter 2.7. Students canwork individually or in pairs graphing y=|x|and y=|x|+k; y=|x-h| and y=a|x-h|+k to showtransformations.

McDougal-Littell - Chapter 2.7 – UseAbsolute Value Functions andTransformations pp. 123 – 129. Taught infirst term (October). Pacing is 2 days.

McDougal-Littell - Post-Chapter 2.7 – UsePiecewise Functions p. 130 . Taught in firstterm (October). Pacing is 1 day afterteaching Chapter 2.7.

McDougal-Littell - Chapter 14.3 – VerifyingTrigonometric Identities pp. 924 – 928.Would be an extension of the above lessonsusing problem #35 on page 928 to showodd and even functions. Pacing is 1 day.




F-BF.4 Find inverse functions.

Ø Understand what an inverse function is and be able to findthe inverse of a given function.

McDougal-Littell - Chapter 6.4 – UseInverse Functions pp. 437 – 445. Taught infourth term (April). Pacing is 2 days. Explainto students that an inverse function is afunction that “undoes” a function. Askstudents how you undo adding a number,multiplying a number, cubing a number, andso on. Then show them that they can findan inverse of a function by undoing theoperations that have been done to x, but inthe reverse order.



F-BF.4.a Solve an equation of the form f(x) = c for a simple function f McDougal-Littell - Chapter 6.4 – Use Formative



that has an inverse and write an expression for the inverse.For example, f(x) =2x3 or f(x) = (x + 1)/(x - 1) for x ?1.

Ø Solve a function for the dependent variable and write theinverse of a function by interchanging the values of thedependent and independent variables.

Inverse Functions pp. 437 – 445. Taught infourth term (April). Pacing is 2 days.

In Algebra 2, use transformations offunctions to find models as studentsconsider increasingly more complexsituations. It may help to solidify theconcepts by providing groups of studentsgrid paper and have them graph f(x) = 2x3 +1 and the line y = x on the same coordinateplane. Have them graph the inverse of f byreflecting its graph over the line y = x, andthen verify that the image is the graph of f-1

=

assessment: DailyHomework quiz, oronline quiz available atclasszone.com


FUNCTIONS - LINEAR, QUADRATIC, AND EXPONENTIAL MODELS

Construct and compare linear, quadratic, and exponential models and solve problems.


F-LE.4 For exponential models, express as a logarithm the solution toabct = d where a, c, and d are numbers and the base b is 2,10, or e; evaluate the logarithm using technology. «

Ø Express logarithms as solutions to exponential functionsusing bases 2, 10, and e.

Ø Use technology to evaluate a logarithm.

McDougal-Littell - Chapter 7.5 – ApplyProperties of Logarithms pp. 507 – 514.Taught in fourth term (May). Pacing is 1day. Graphing Logarithmic Functions on acalculator Activity p. 514

In Algebra 2, consider extending this unit toinclude the relationship between propertiesof logarithms and properties of exponents,such as the connection between theproperties of exponents and the basiclogarithm property that log xy = log x +log y.



FUNCTIONS - TRIGONOMETRIC FUNCTIONS

Extend the domain of trigonometric functions using the unit circle.


F-TF.1 Understand radian measure of an angle as the length of thearc on the unit circle subtended by the angle.

Ø Know that if the length of an arc subtended by an angle isthe same length as the radius of the circle, then the measureof the angle is 1 radian.

Ø Know that the graph of the function, f, is the graph of theequation y=f(x).

McDougal-Littell - Chapter 13.1 – UseTrigonometry with Right Triangles pp. 850 –858. Need to be added to the fourth term(June). Pacing is 1 day.

McDougal-Littell - Chapter 13.2 – DefineGeneral Angles and Use Radian Measurepp. 859 – 865. Need to be added to thefourth term (June). Pacing is 1 day.



F-TF.2 Explain how the unit circle in the coordinate plane enables theextension of trigonometric functions to all real numbers,interpreted as radian measures of angles traversedcounterclockwise around the unit circle.

Ø Explain how radian measures of angles rotatedcounterclockwise in a unit circle are in a one-to-onecorrespondence with the nonnegative real numbers, and thatangles rotated clockwise in a unit circle are in a one-to-onecorrespondence with the non-positive real numbers.

McDougal-Littell - Chapter 13.3 – EvaluateTrigonometric Functions of Any Angle pp.866 – 872. Need to be added to the fourthterm (June). Pacing is 2 days.



Model periodic phenomena with trigonometric functions.

F-TF.5 Choose trigonometric functions to model periodic phenomenawith specified amplitude, frequency, and midline. «

Ø Use sine and cosine to model periodic phenomena such as

McDougal-Littell - Chapter 13.3 – EvaluateTrigonometric Functions of Any Angle pp.866 – 872. Needs to be added to the fourthterm (June). Pacing is 2 days. Students

Formativeassessment: DailyHomework quiz, oronline quiz available at



the ocean’s tide or the rotation of a Ferris wheel.

Ø Given the amplitude; frequency; and midline in situations orgraphs, determine a trigonometric function used to model thesituation.

should model trigonometric functions usingproblem #37 p. 871 for the rotation of theFerris Wheel and using the Rock Climbingproblem in example 6 and answeringquestion 11 What If? Exercise 11 for themidline situation. Problem #2a & 2b is alsoexcellent for modeling a unicycle wheel.

McDougal-Littell - Chapter 14.1 – GraphSine, Cosine, & Tangent Functions pp. 908– 914. Needs to be added to the fourth term(June). Pacing is 2 days.

McDougal-Littell - Chapter 14.2 – Translate& Reflect Trigonometric Graphs pp. 915 –922. Needs to be added to the fourth term(June). Pacing is 2 days.

Example 3 page 916 Ferris Wheel problemfollowed by Extra Example 3 page 917.

McDougal-Littell - Chapter 14.4 – SolveTrigonometric Equations pp. 931 – 932.Need to be added to the fourth term (June).Pacing is 1 day Example 3 page 932followed by Guided Practice and then UsingAlternative Methods page 938 canemphasize that there are more than oneapproach to problem solving strategies. Thisexample allows students to see the periodicrhythm of the tides.

classzone.com


Prove and apply trigonometric identities.

F-TF.8 Prove the Pythagorean identity sin2(?) + cos2(?) = 1 and use itto find sin(?), cos(?), or tan(?) given sin(?), cos(?), or tan(?)and the quadrant.

Ø Use the unit circle to prove the Pythagorean identity sin2(?)+ cos2(?) = 1.

Ø Given the value of the sin(?) or cos(?), use the Pythagoreanidentity sin2(?) + cos2(?) = 1 to calculate other trigonometricratios.

McDougal-Littell - Chapter 14.3 – VerifyingTrigonometric Identities pp. 924 – 929.Need to be added to the fourth term (June).Pacing is 1 day.



STATISTICS AND PROBABILITY - INTERPRETING CATEGORICAL AND QUANTITATIVE DATA

Summarize, represent, and interpret data on a single count or measurement variable.


S-ID.4 Use the mean and standard deviation of a data set to fit it to anormal distribution and to estimate population percentages.Recognize that there are data sets for which such a procedureis not appropriate. Use calculators, spreadsheets, and tablesto estimate areas under the normal curve. «

Ø Identify data sets as approximately normal or not.

Ø Use the mean and standard deviation to fit it to a normaldistribution where appropriate.

Ø Use calculators, spreadsheets, and tables to estimate areasunder the normal curve.

Ø Interprets areas under a normal curve in context.

McDougal-Littell - Chapter 11.1 – FindMeasures of Central Tendency & Dispersionpp. 744 – 749. Needs to be added to thefourth term (June). Pacing is 1 day.Students should be familiar with this topicprior to Algebra 2. May only need to use agraphing calculator to calculate the mean,median, range, and standard deviation.Calculator Activity on page 750 practiceprior to Chapter 11.3 is a good focus.

McDougal-Littell - Chapter 11.3 – UseNormal Distributions pp. 757 – 762. Needsto be added to the fourth term (June).Pacing is 1 day.

While students may have heard of thenormal distribution, it is unlikely that they willhave prior experience using it to makespecific estimates. Build on students’understanding of data distributions to helpthem see how the normal distribution uses





area to make estimates of frequencies(which can be expressed as probabilities).Emphasize that only some data are welldescribed by a normal distribution.

STATISTICS AND PROBABILITY - MAKING INFERENCES AND JUSTIFYING CONCLUSIONS

Understand and evaluate random processes underlying statistical experiments.

S-IC.1 Understand statistics as a process for making inferences to bemade about population parameters based on a randomsample from that population. «

Ø Explain in context the difference between values describinga population and a sample.

McDougal-Littell - Chapter 10.6 – Constructand Interpret Binomial Distributions pp. 724– 730. Needs to be added to the fourth term(June). Pacing is 2 days. Use the Extensionto 11.3 - Approximate Binomial Distributions& Test hypotheses to understand statisticalinferences.





S-IC.2 Decide if a specified model is consistent with results from agiven data-generating process, e.g., using simulation. Forexample, a model says a spinning coin falls heads up withprobability 0.5. Would a result of 5 tails in a row cause you toquestion the model? «

Ø Explain how well and why a sample represents the variableof interest from a population.

Ø Demonstrate understanding of the different kinds ofsampling methods.

Ø Design simulations of random sampling: assign digits inappropriate proportions for events, carry out the simulationusing random number generators and random number tablesand explain the outcomes in context of the population and theknown proportions.

McDougal-Littell - Chapter 10.4 – FindProbabilities of Disjoint & OverlappingEvents pp. 707 – 714. Needs to be added tothe fourth term (June). Pacing is 1 day.

Example 4 page 709 can be paired with theAlternative Method on page 714 to decide ifthe simulated model is consistent to thegiven data in example 4. Simulation will helpthe students visualize the relationshipbetween experimental and theoreticalprobabilities.

LTF Module: “INTRODUCTORY ACTIVITYVIEWING AND DESCRIBING DATA”

Include comparing theoretical and empiricalresults to evaluate the effectiveness of atreatment.



Make inferences and justify conclusions from sample surveys, experiments and observational studies.

S-IC.3 Recognize the purposes of and differences among samplesurveys, experiments, and observational studies; explain howrandomization relates to each. «

Ø Identify situations as sample survey, experiment, orobservational study. Discuss the appropriateness of eachone’s use in contexts with limiting factors.

Ø Design or evaluate sample surveys, experiments andobservational studies with randomization. Discuss theimportance of randomization in these processes.

McDougal-Littell - Chapter 11.4 – Select &Draw Conclusions from Samples pp. 766 –772. Need to be added to the fourth term(June). Pacing is 2 days. Use the Extensionto 11.4 - Design Surveys and Experimentsto identify situations, limiting factors, bias,and randomization.

Focus on the variability of results fromexperiments—that is, focus on statistics asa way of dealing with, not eliminating,inherent randomness.



S-IC.4 Use data from a sample survey to estimate a population meanor proportion; develop a margin of error through the use ofsimulation models for random sampling. «

Ø Use sample means and sample proportions to estimatepopulation values.

Ø Conduct simulations of random sampling to gather samplemeans and sample proportions. Explain what the results meanabout variability in a population and use results to calculatemargins of error for these estimates.







In earlier grades, students are introduced todifferent ways of collecting data and usegraphical displays and summary statistics tomake comparisons. These ideas arerevisited with a focus on how the way inwhich data is collected determines thescope and nature of the conclusions thatcan be drawn from that data.

S-IC.5 Use data from a randomized experiment to compare twotreatments; use simulations to decide if differences betweenparameters are significant. «

Ø Evaluate effectiveness and differences in two treatmentsbased on data from randomized experiments. Explain incontext.

Ø Use simulations to generate data simulating application oftwo treatments. Use results to evaluate significance ofdifferences.


The concept of statistical significance isdeveloped informally through simulation asmeaning a result that is unlikely to haveoccurred solely as a result of randomselection in sampling or random assignmentin an experiment.



S-IC.6 Evaluate reports based on data. «

Ø Read and explain in context data from outside reports.





STATISTICS AND PROBABILITY - USING PROBABILITY TO MAKE DECISIONS

Use probability to evaluate outcomes of decisions.

S-MD.6 (+) Use probabilities to make fair decisions (e.g., drawing bylots, using a random number generator). «

Ø Make decisions based on expected values. Use expectedvalues to compare long term benefits of several situations.

McDougal-Littell - Chapter 10.3 Define andUse Probability pp. 698 - 706. Topic needsto be added to the fourth term (June);pacing is 1 day.

The ideas of statistical inference areintroduced. Evaluating the risks associatedwith conclusions drawn from sample data(i.e. incomplete information) requires anunderstanding of probability concepts.



S-MD.7 (+) Analyze decisions and strategies using probabilityconcepts (e.g., product testing, medical testing, pulling ahockey goalie at the end of a game). (Replacing the hockeygoalie with an extra skater.) «

Ø Explain in context decisions made based on expectedvalues.

McDougal-Littell - Chapter 10.3 Define andUse Probability pp. 698 - 706. Topic needsto be added to the fourth term (June);pacing is 1 day.



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