Course Name: Algebra 2/Math III Unit #1Unit Title:Polynomials

Enduring understanding (Big Idea): Students will understand how to factor polynomials in multiple forms, analyze polynomial functions and their graphs by identifying end behavior and roots, and graph polynomial functions.Essential Questions: 1. How are factors and roots of a polynomial equation related?2. How does solving and graphing higher-order polynomials relate to solving and graphing quadratics?BY THE END OF THIS UNIT:

Students will know…• How to factor, solve, and graph polynomial functions• Start and end behavior of polynomials• How to write polynomials functions from points

Vocabulary:Distributive Property, Closure, Like Terms, Polynomial, Term, Root, Solution, X-Intercept, Zero, Factor, Remainder Theorem , Rational Root Theorem, Pascal’s Triangle, rational expression, simplified form, excluded value, equivalent fractions, asymptote, Complex Fractions, Rational Expressions, Minimum, Maximum, Symmetry, End Behavior, Interval, Periodicity, Domain, Domain Restrictions, Integers, Start Behavior, End Behavior, Relative extremes, zeroes, Regression, Function

Unit Resources:See attached standard guides for additional resources.

Building a Cardboard Box of Greatest Volume: http://questgarden.com/101/08/6/100420161846/index.htm

Representing Polynomials MARS Task – Analysis of Graphs of Quadratics and Cubics Based On Intercepts, Relative Min/Max, and Transformations: http://map.mathshell.org/materials/download.php?fileid=1271

Mathematical Practices in Focus:1. Make sense of problems and persevere in solving.2. Reason abstractly.3. Model with Mathematics.4. Use appropriate tools strategically.5. Look for and make use of structure.

CCSS-M Included:A-APR.1, 2, 3, 4, 5, 6, 7F-IF.4, 5, 7cA-REI.10

Suggested Pacing:11 days (including 1 review, 1 test) – NOT including rationals

Students will be able to:• Add, subtract, and multiply polynomials (the concept should have already been taught – include fraction and variable coefficients in Math 3)• Geometric and real-world applications of adding, subtracting, and multiplying polynomials• Long Division of Polynomials• Use Synthetic Division to completely factor polynomials• Use Remainder Theorem to evaluate functions and prove that values are zeroes of functions• Use Rational Root Theorem to determine possible roots of functions (honors)• Find zeroes of higher-order polynomial functions graphically and from factors.• Solve cubic and quartic functions.• Write polynomial functions given the roots• Factor Sum/Difference of Cubes• Binomial Expansion•Recognize and explain the domain of the function of the original and simplified rational expression• Simplify Rational Expressions• Multiplication and division of rational expressions• Addition and Subtraction of rational expressions• Identify the intercepts, relative minimums and maximums, lines of symmetry, and start and end behavior of graphs (Polynomial and Rational)• Identify the intervals on which a graph increases and decreases (Polynomial and Rational)• Graph a function based on its key features (Polynomial and Rational)• Determine the domain of a function from its graph, including analysis of the start and end behavior (Polynomial and Rational)• Determine the key domain restrictions of functions, including division by 0 and even roots of negative numbers (Polynomial and Rational)• Analyze word problems to determine domain restrictions (i.e. negative numbers, non-integers).• Descartes Rule of Signs (HONORS)• x = y2 Parabolas (HONORS)• Write polynomial functions from their points in real-world situations and use them to predict future values (regression)

Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.PAGE 1

Course Name: Algebra 2/Math III Unit #1Unit Title:Polynomials

CORE CONTENTCluster Title: Polynomial OperationsStandard A-APR.1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.Concepts and Skills to Master:• Add, subtract, and multiply polynomials (the concept should have already been taught – include fraction and variable coefficients in Math 3)• Geometric and real-world applications of adding, subtracting, and multiplying polynomialsSUPPORTS FOR TEACHERS

Critical Background Knowledge: Rules of exponents Combining like terms Formulas for perimeter, area and volume of basic geometric shapes Fraction Operations

Academic Vocabulary:Distributive Property, Closure, Like Terms, Polynomial, Term

Suggested Instructional Strategies: Teach adding/subtracting polynomials, through real-

world and geometric examples and situations Teach multiplying polynomials through real-world

and geometric examples

NCDPI Unpacking:The Closure Property means that when adding, subtracting or multiplying polynomials, the sum, difference, or product is also a polynomial. Polynomials are not closed under division because in some cases the result is a rational expression.

Resources:Algebra 2 Textbook Correlation: 1-3

MARS Task on Manipulating Polynomials: http://map.mathshell.org/materials/lessons.php? taskid=437&subpage=concept

Sample Assessment TasksSkill-based task:1. (3x3 – 4x2 + 1) – (2x + 4)

2. (2x2 + 3) + 4(x – 2)2

3. Simplify:

a2−b2

a+b

Problem Task:1) The area of a rectangle can be represented by the expression (2x2 – 17x + 6), and the height can be represented by the expression (x – 6). What expression represents the volume?

2) The radius of a sphere can be represented by the binomial (x + 2). What expressions represent the volume and surface area?

3) Explain why (x + y)2 ≠ x2 + y2.

NCDPI Examples:Ex. If the radius of a circle is (5x – 2) kilometers, write an expression for the area of the circle.Ex. Explain why (4𝑥2 + 3)2 does not equal (16𝑥4 + 9).

Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.PAGE 2

Course Name: Algebra 2/Math III Unit #1Unit Title:Polynomials

CORE CONTENTCluster Title: Understand the Relationship Between Zeroes and Factors of PolynomialsStandard: A-APR.2 – Know and apply the remainder theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).Concepts and Skills to Master:• Long Division of Polynomials• Use Synthetic Division to completely factor polynomials• Use Remainder Theorem to evaluate functions and prove that values are zeroes of functions• Use Rational Root Theorem to determine possible roots of functions (honors)SUPPORTS FOR TEACHERSCritical Background Knowledge:-Solutions to polynomial functions occur when the function equals 0.- Factors of polynomials set equal to 0 determine solutions.- Evaluation FunctionsAcademic Vocabulary:Root, Solution, X-Intercept, Zero, Factor, Polynomial, Remainder Theorem Suggested Instructional Strategies:- Teach long division first (for HONORS), closely followed by synthetic division, highlighting that synthetic only works for a polynomial divided by a binomial.- Use the distributive property/properties of equality to show how to synthetically divide a polynomial where x has a coefficient > 1.- Have students use synthetic division to discover the remainder theorem.

NCDPI Unpacking:The Remainder Theorem states that if a polynomial, 𝑝(𝑥) is divided by a monomial, (𝑥 – 𝑐), the remainder is the same as if you evaluate the polynomial for 𝑐, i.e. calculate 𝑝(𝑐). If the remainder when dividing by (𝑥 – 𝑐) is 0, or 𝑝(𝑐) = 0, then (𝑥 – 𝑐) is a factor of the polynomial. If 𝑓(𝑎) = 0, then (𝑥 − 𝑎) is a factor of 𝑓(𝑥), which means that 𝑎 is a root of the function 𝑓(𝑥). This is known as the Factor Theorem.

Resources:Algebra 2 Textbook Correlation: 5-4, 5-5

Remainder Theorem Discovery (to be uploaded)

Sample Assessment TasksSkill-based task:

1. For f(x) = 3x3 + 4x2 – 12x + 3, evaluate:

a) f(0) b) f(6) c) f(-2) d) f(1/2)

using division.

2. Let p(x) = x5 – 3x4 + 8x2 – 9x + 30. Find p(-2). What does your answer tell you about the factors of p(x)?

Problem Task:1) The area of a rectangle can be represented by the expression 2x3 – 17x2 + 31x – 6. If the width is represented by x – 6, what expression represents the length?

NCDPI Examples:Ex. Given 𝑓 (𝑥) = 2𝑥2 + 6𝑥 − 20, determine whether −5 is a root of the function, then write the function in factored form.

Ex. Compare the process of synthetic division to the process of long division for dividing polynomials.

Ex. Assume that (𝑥 − 𝑐) is a factor of 𝑓, which means that 𝑓 is divisible by 𝑥 − 𝑐 . Explain why it must be true that 𝑓 (𝑐) = 0.

Ex. Assume we know that 𝑓 (𝑐) = 0. Explain why it must be true that (𝑥 − 𝑐) is a factor of 𝑓.

Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.PAGE 3

Course Name: Algebra 2/Math III Unit #1Unit Title:Polynomials

CORE CONTENTCluster Title: Interpreting FunctionsStandard A-APR.3.Identify zeroes of polynomials when suitable factorizations are available, and use the zeroes to construct a rough graph of the function defined by the polynomial.Concepts and Skills to Master:• Find zeroes of higher-order polynomial functions graphically and from factors.• Solve cubic and quartic functions.• Write polynomial functions given the roots.SUPPORTS FOR TEACHERSCritical Background Knowledge:• Factoring• Simplifying RadicalsAcademic Vocabulary:Zeroes, Roots, Solutions, Intercepts, FactorsSuggested Instructional Strategies:

Students learned about solving quadratics in Math 2, so Math 3 extends to solving higher-order polynomials. Begin with a brief warm-up review of solving quadratics, but incorporate solving quadratics into solving the higher-order polynomials for those that need review.

Relate the x-intercepts to the solutions = 0 for the equation, highlighting that y = 0 at the x-intercepts.

Relate synthetic division to the solutions of the factors = 0, and use the process to solve the higher-order polynomials.

Relate roots to polynomial functions “in reverse” using a discovery lesson to extend from quadratics.

Teach start and end behavior rules and use to graph polynomials by their zeroes.

For HONORS, extend to include completing the square and

introduce x= y2

parabolas

NCDPI Unpacking:Find the zeros of a polynomial when the polynomial is factored. Then use the zeros to sketch the graph.

Resources:Algebra 2 Textbook Correlation: 5-2, 5-3

1) Solving Higher-Order Equations Performance Task (see wiki, use after solving cubics, before solving higher-order, all rational solutions)

2) Writing Equations from Roots Discovery (it uses the roots of quadratics to compare to their factors to allow students to see the application to higher-order polynomials) (to be uploaded)

3)Representing and Solving Real-World Quadratics (should not be our main lesson, but is a good support): http://map.mathshell.org/materials/lessons.php? taskid=432&subpage=problem

Sample Assessment TasksSkill-based task:Solve and sketch rough graphs of the following equations:(Find 3 cubic/quartic)

Problem Task:

NCDPI Examples:Ex. For a certain polynomial function, 𝑥 = 3 is a zero with multiplicity two, 𝑥 = 1 is a zero with multiplicity three, and 𝑥 = −3 is a zero with multiplicity one. Write a possible equation for this function and sketch its graph.

CORE CONTENTCluster Title: Polynomial IdentitiesStandards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.PAGE 4

Course Name: Algebra 2/Math III Unit #1Unit Title:Polynomials

Standard A-APR.4. Prove polynomial identities and use them to describe numerical relationships. (For example, the polynomial identity (x2 + y2)2 = (x2 – y2) + (2xy)2 can be used to generate Pythagorean triples.)Concepts and Skills to Master:• Use the Rational Root Theorem to determine possible roots of polynomials (HONORS)• Factor Sum/Difference of Cubes• Use Synthetic Division to Completely Factor Higher-Order Polynomials• Given the roots of a polynomial, write the equation of the polynomialSUPPORTS FOR TEACHERSCritical Background Knowledge:• Factoring• Synthetic DivisionAcademic Vocabulary:Rational Root Theorem, Root, ZeroesSuggested Instructional Strategies:• Use synthetic division/factoring higher-orders to review factoring (maybe include factoring warm-up – they learned how to factor in Math 1 and 2).• Expand factoring knowledge to sums and differences of cubes (relate to difference of squares).• For HONORS, expand synthetic division to the rational root theorem to help them find the divisor.

NCDPI Unpacking:Prove polynomial identities algebraically by showing steps and providing reasons or explanations.

Resources:Algebra 2 Textbook Correlation: 5-2, 5-3,5-4, 5-5

Algebra Tiles Explanation of Sum/Difference of Cubes: http://www.mathnstuff.com/math/algebra/tt21.htm

Algebra II For Dummies Explanation of Synthetic Division/Remainder Theorem: http://books.google.com/books?id=VmPwZHw_DKwC&pg= PA118&lpg=PA118&dq=synthetic+division+tasks&source= bl&ots=7L88xbs6Kf&sig=o0eX-bIk4N0hIVhhVlLpSB1Rtcs&hl=en&sa=X& ei=3liQUY2LM4X28wS19ICwAQ&ved=0CDUQ6AEwATgK#v= onepage&q=synthetic%20division%20tasks&f=false

Sample Assessment TasksSkill-based task:1) Factor x3 – 3x2 + x – 3 if one factor is ____.2) Factor 8x3 + 125.

Problem Task:1) The volume of a cube can be represented by the expression x3 – 125. What expression represents the width of the cube?

NCDPI Examples:The following examples are meant to be investigated by students considering analogous problems, and trying special cases and simpler forms of the original problem in order to gain insight into its solution(s).

Ex. Is (2x – 3)2 - 64 equivalent to(2x – 11)(2x + 5)? Explain why or why not.

Ex. Jessie claims that (x + y)2 = x2 + 2xy + y2. Is he correct? Prove why or why not.

Ex. Prove x3 – y3 = (x – y)(x2 + xy + y2). Justify each step.

Ex. Solve the quadratic ax2 + bx + c = 0 Justifying each step. What was interesting about the result?

CORE CONTENTCluster Title: Interpreting FunctionsStandard A-APR.5. Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined by Pascal’s Triangle.Concepts and Skills to Master:

Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.PAGE 5

Course Name: Algebra 2/Math III Unit #1Unit Title:Polynomials

• Binomial ExpansionSUPPORTS FOR TEACHERSCritical Background Knowledge:• Multiplying Polynomials• Identifying PatternsAcademic Vocabulary:Pascal’s TriangleSuggested Instructional Strategies:Have students fill out Pascal’s Triangle for the first 12 rows. Then, they should multiply (x + y) to the 6th power manually until they discover the pattern. (Start the standard with the argumentation task so they discover the pattern.)

NCDPI Unpacking:The Binomial Theorem describes the algebraic expansion of powers of a binomial. There are patterns that develop with the coefficients and the variables when expanding binomials. Pascal’s triangle is a triangular array that identifies the coefficients of an expanded binomial. The numbers in Pascal’s triangle are also evaluations of combinations, nCr. The values of the combinations correspond with the coefficients of the expanded binomial, which indicates how many times that term will appear in the completely expanded form. This is a connection between Probability and Algebra that should be made explicit. For example, when squaring the binomial (𝑎 + 𝑏), note that the product 𝑎𝑏 occurs twice: (a + b)2 = a2 + ab + ab + b2 = a2 + 2ab + b2. Using combinatorics, the coefficient of the second term would be 2C1 = 2.

Resources:Algebra 2 Textbook Correlation: 5-7

Pyramid Power (Pearson Activities/Games/Puzzles 5-7, accessible through Pearson Success Net or wiki)

Binomial Theorem Enrichment (Pearson Enrichment 5-7, accessible through Pearson Success Net or wiki)

Sample Assessment TasksSkill-based task:1) Expand (x + y)5.2) What is the fourth term in (2x – y)6?

Problem Task:1) Textbook pg. 329 #24

Task: 1) Fill out the first 12 rows of Pascal’s Triangle.2) Distribute and write out the solutions to: (x + y)0, (x + y)1, (x + y)2, (x + y)3, (x + y)4, (x + y)5, and (x + y)6.3) Explain the relationship between Pascal’s Triangle and your solutions, and apply this relationship to compute (x + y)10.4) How do you think the answer would change if you computed (x + 5)10? Justify your answer.

NCDPI Examples:Ex. Explain how to generate a row of Pascal’s triangle.Ex. What are the coefficients of the expanded terms of (𝑎 + 𝑏)5?Ex. Using the binomial theorem, expand (𝑎 + 𝑏)5.Ex. Why are the coefficients of a binomial expansion equal to values of nCr?

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

Concepts and Skills to Master:• Recognize and explain the domain of the function of the original and simplified rational expressionStandards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.PAGE 6

Course Name: Algebra 2/Math III Unit #1Unit Title:Polynomials

• Simplify Rational Expressions

SUPPORTS FOR TEACHERSCritical Background Knowledge:• Simplifying numerical fractions• Factoring (GCF, difference of squares, trinomials)• Domain and excluded values

Academic Vocabulary:rational expression, simplified form, excluded value, equivalent fractions, asymptoteSuggested Instructional Strategies:

Stress that students need to factor first to simplify rationals, and they can only divide out factors in factored form.

Relate the domain restrictions of division by 0 to the graph (no point at that x-coordinate, asymptote).

Students should relate computational skills to real – world applications, see problem tasks.

NCDPI Unpacking:

Rewrite rational expressions,

a( x )b( x ) in the form

q ( x )+ r ( x )b ( x )

using long division, synthetic division or with expressions that pose difficulty by hand, use a computer algebra system such as the TI Inspire CAS or Ipad applications. When dividing a polynomial by a polynomial, the new form is the quotient plus the remainder divided by the divisor. This process should be connected to dividing with numbers. The quotient represents the number of times something will divide, plus the parts or pieces remaining. Know that the degree of the quotient is less than thedegree of the dividend. Connect division of polynomials to the remainder theorem when 𝑏(𝑥) is in the form (𝑥 − 𝑐).

Resources:Algebra 2 Textbook Correlation: 8-2, 8-3, 8-4

Jogging Rates (Real-World Application): http://www.google.com/url?sa=t&rct=j&q=&esrc=s&frm=1&source= web&cd=1&cad=rja&ved=0CC8QFjAA&url=http%3A%2F%2F www.gwinnett.k12.ga.us%2FPhoenixHS%2Fmath%2Fgrade09% 2Funit02%2F08-Task-Jogging_Rtnl%2520Expression_. pdf&ei=6FqQUZX_ FIam8QSz1YCACA&usg= AFQjCNFhRIckRWHle2yvRnmOw9L5IuKjBA

Graphing Rationals (Great progression from technology to by hand, highlights domain restrictions well): https://www.google.com/url?sa=t&rct=j&q=&esrc=s &frm=1&source=web&cd=3&ved=0CDkQFjAC&url =https%3A%2F%2Fwww.georgiastandards.org% 2FFrameworks%2FGSO%2520Frameworks% 2FAcc-Math-III-Unit-3-SE-Rational-Functions.pdf&ei=T1uQUYyFNZGy9gT3qYDICg &usg=AFQjCNF4MDd0XdfVJsY3YQezv0lAVohLlA

Sample Assessment TasksSkill-based task:

Given the rational expression ; state the domain and the simplified value.

Problem Task:

1) Given the rational expression ; state the domain and the simplified value. Explain why the domain of the simplified function must reflect the domain of the original function. Justify that a complicated rational function is equivalent to a simplified functions using graphs and tables.

2) Using a real-world situation discussed in class (STANDARD) or Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.PAGE 7

Course Name: Algebra 2/Math III Unit #1Unit Title:Polynomials

creating your own real-world situation (HONORS), use a rational expression to illustrate the situation. (For example, a class with x students is throwing a party that will cost $50. How much does each member of the class contribute?) Now, based on your situation, explain any limits on the domain that would exist based on the situation or real-world limitations. Is the rational expression the best mathematical means of describing your situation? Why or why not?

NCDPI Examples:

Ex. We know from arithmetic, that a fraction like

32710 indicates

the division of 327 by 10. The result can be expressed 32 R 7 or

as 32 +

710 . Use division of polynomials to show that

−x2+4 x+8x+1 can be written with an equivalent expression in

the form of q ( x )+r (x )

x+1 .

Ex. Divide. Write the answer in the form of quotient plus

remainder/divisor.

x4+3 xx2−4

Ex. Use a computer algebra system to rewrite the following rational expression in quotient and remainder form9 x3+9 x2−x+2

x+ 23

CORE CONTENTCluster Title: Rewrite Rational ExpressionsStandard A-APR.7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply and divide rational expressions.Concepts and Skills to Master:• Multiplication and division of rational expressions• Addition and Subtraction of rational expressionsSUPPORTS FOR TEACHERSCritical Background Knowledge:Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.PAGE 8

Course Name: Algebra 2/Math III Unit #1Unit Title:Polynomials

• Factoring• Simplifying, add, subtract, multiply and divide numerical fractions• Like and unlike denominatorsAcademic Vocabulary:Complex Fractions, Rational Expressions Suggested Instructional Strategies:• Teach multiplication and before addition and subtraction• Students should be encouraged to factor all numerators and denominators prior applying the operations, and stress that they cannot divide out factors unless the rationals are in factored form• Complex fractions should be selected based on difficulty – HONORS students should have more rigorous problems• Students should relate computational skills to real – world applications, see problem tasks.

NCDPI Unpacking:When performing any operation on a rational expression, the result is always another rational expression, which is the Closure Property for rational expressions. Compare this to the Closure Property for polynomials. Perform operations with rational expressions, division by nonzero rational expressions only.

Resources:Algebra 2 Textbook Correlation: 8-4, 8-5

Rational Equations Paideia (helps students solve and determine if solutions are realistic in a real-world setting): Linked to wiki

Pearson Enrichment 8-5 – The Superposition Principle (accessible through Pearson Success Net or wiki)

Sample Assessment TasksSkill-based task:

Given and Find

State all restrictions on the domain of the functions and the simplified expression.

NCDPI Examples:

A rectangle has an area of

x2+x−2x3

sq. ft. and a height of x2

x−1 ft. Express the width of the rectangle as a rational expression in terms of 𝑥.

Problem Task:

1) Given , students will explain the techniques and rationale for simplifying, including the domain and excluded values.

2) Real world problems - see textbook pg. 540 problems #37,45,46

3) Write and simplify a complex fraction to find the average rate of speed for a plane flight from North Carolina to California that travels 400 mph west and 520 east on the 3,200 mile flight. a) Why is the average speed NOT 460 mph? [(400 + 520)/2] Explain the difference between the rational expression and the basic formula for computing mean.b) What could account for the differences in speeds between the trip east and the trip west?

CORE CONTENTCluster Title: Interpret Functions That Arise in Applications Standard F-IF.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.Concepts and Skills to Master:• Identify the intercepts, relative minimums and maximums, lines of symmetry, and start and end behavior of graphs.• Identify the intervals on which a graph increases and decreases.• Graph a function based on its key features.SUPPORTS FOR TEACHERSCritical Background Knowledge:• Graph and identify points on the coordinate planeStandards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.PAGE 9

Course Name: Algebra 2/Math III Unit #1Unit Title:Polynomials

• Solve one-variable equationsAcademic Vocabulary:x-intercept, y-intercept, Minimum, Maximum, Symmetry, End Behavior, Interval, PeriodicitySuggested Instructional Strategies:• Ensure that students can identify intercepts, maximum points, minimum points, start and end behavior, symmetry, and periodicity early in the unit from actual graphs, then using technology• Use technology/graph paper to plot key features and functions.

NCDPI Unpacking:This standard should be revisited with every function your class is studying. Students should be able to move fluidly between graphs, tables, words, and symbols and understand the connections between the different representations. For example,when given a table and graph of a function that models a real-life situation, explain how the table relates to the graph and vise versa. Also explain the meaning of the characteristics of the graph and table in the context of the problem as follows:At the course three level, students should extend the previous course work with function types to focus on polynomial and trigonometric functions.• Polynomials – emphasis should be on the commonalities and differences between quadratics and power functions, relative max/mins.• Trigonometric functions –emphasis should be on periodicity, amplitude, frequency, and midline. (F-TF.5)

Note – This standard should be seen as related to F-IF.7 with the key difference being students can interpret from a graph or sketch graphs from a verbal description of key features.

Resources:Algebra 2 Textbook Correlation: Mainly 5-1 and 5-9 for this unit (polynomials), but the concept repeats itself throughout several units

1) Start and End Behavior Discovery (students use calculator to graph various functions with +/- leading coefficients and even and odd degrees to see relationships) (to be uploaded)

Sample Assessment TasksSkill-based task:On what interval does the graph of f(x) = x2 increase?

Problem Task:1) Given the graph of a function, identify the intercepts, minimum and maximum values, lines of symmetry, and end behavior. Using the basic functions learned in the previous objective, how is this function similar/different?

2) Find the y-intercepts of f(x) = 2x, f(x) = 0.5(2)x, and f(x) = 4(2)x? Based on the value of f(0) in each function, explain the rationale for the difference in y-intercepts. Use the pattern to form a hypothesis involving the y-intercept of an exponential function.

Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.PAGE 10

Course Name: Algebra 2/Math III Unit #1Unit Title:Polynomials

NCDPI Examples:Ex. Insert graph or equation of a polynomial function from a real-life situation.a. What are the x-intercepts and y-intercepts and explain them in the context of the problem?

b. Identify any maximums or minimums and explain their meaning in the context of the problem.

c. Describe the intervals of increase and decrease and explain them in the context of the problem.

F-IF.4 When given a verbal description of the relationship between two quantities, sketch a graph of the relationship, showing key features: Ex. Jaquan found that for a period of 7 days, the daily high was 85° and the low was 65°. He also noticed that the sunrise and sunset temperature was 75°. Sketch a graph showing the fluctuation temperatures during those 7 days.

CORE CONTENTCluster Title: Interpret Functions That Arise in Applications Standard F-IF.5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.Concepts and Skills to Master:• Determine the domain of a function from its graph, including analysis of the start and end behavior.• Determine the key domain restrictions of functions, including division by 0 and even roots of negative numbers.• Analyze word problems to determine domain restrictions (i.e. negative numbers, non-integers).

SUPPORTS FOR TEACHERSCritical Background Knowledge:• Ability to identify x values by looking at a graph

Academic Vocabulary:Domain, Domain Restrictions, Integers, Start Behavior, End Behavior

Suggested Instructional Strategies: Resources:Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.PAGE 11

Course Name: Algebra 2/Math III Unit #1Unit Title:Polynomials

• Compare the graphs of parent functions to their equations to determine any restrictions algebraically.• Students write word problems to satisfy various domain restrictions (positive numbers, non-negative numbers, integers, all real numbers).• Identify the range of functions based on their graphs and equations.

NCDPI Unpacking:Given a function, determine its domain. Describe the connections between the domain and the graph of the function. Know that the domain taken out of context is a theoretical domain and that the practical domain of a function is found based on a contextual situation given, and is the input values that make sense to the constraints of the problem context.

Algebra 2 Textbook Correlation: Introduced in 2-1, but additional resources can be found in the radical unit (Unit 6) and rationals unit (Unit 8)

Domain Discovery (discovers domains of parent functions): on wiki

Domains of Radicals and Rationals Discovery (highlights the domain restrictions of radicals and rationals) (to be uploaded)

Sample Assessment TasksSkill-based task:What are the domains of f(x) = x, f(x) = x2, f(x) = x3, f(x) = √x, and f(x) = 1/x? Explain the rationale for any restrictions (or lack of restrictions).

Problem Task:1) The summer before going to college, a student earned a promotion to shift supervisor at her job at Starbucks! The new position pays $10.20/hour. If the student’s paycheck was modeled by a function, what would be the domain and range? Explain your answer. Why is this domain and range different than other similar functions?

2) Create graphs of functions with the following domains: {All real numbers}, {All real numbers for x ≠ 0}, {All real numbers for x ≥ 0), {All real numbers for x < 0}, {All positive integers}, and {All integers}.1. How do your graphs accurately represent the required domains?2. What are the similarities and differences between your graphs? 3. For each graph, think of either a math function OR a real-world situation where the domain would apply. Explain why your math function OR real-world situation must have the domain you identified.

Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.PAGE 12

Course Name: Algebra 2/Math III Unit #1Unit Title:Polynomials

NCDPI Examples:Ex. A rocket is launched from 180 feet above the ground at time t = 0. The function that models this situation is given by h(t) = – 16t2

+ 96t + 180, where t is measured in seconds and h is height above the ground measured in feet.a. What is the theoretical domain for the function? How do you know this?b. What is the practical domain for t in this context? Explain.c. What is the height of the rocket two seconds after it was launched?d. What is the maximum value of the function and what does it mean in context?e. When is the rocket 100 feet above the ground?f. When is the rocket 250 feet above the ground?g. Why are there two answers to part f but only one practical answer for part e?h. What are the intercepts of this function? What do they mean in the context of this problem?i. What are the intervals of increase and decrease on the practical domain? What do they mean in the context of the problem?

CORE CONTENTCluster Title: Analyze Functions Using Different RepresentationsStandard F.IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (Specifically c – Graph polynomial functions, identifying zeroes when suitable factorizations are available and showing end behavior.)Concepts and Skills to Master:• Graph higher degree polynomial functions from their zeroes and end behavior.• Identify relative extremes.• Descartes Rule of Signs (HONORS)• x = y2 Parabolas (HONORS)SUPPORTS FOR TEACHERSCritical Background Knowledge:• Zeroes as x-intercepts on a graphAcademic Vocabulary:Relative extremes, zeroesSuggested Instructional Strategies:• Relate everything we’ve done this unit with higher-order polynomials (solving by graphing/factoring/synthetic division, end behaviors, key points, etc.) to graphing higher-order polynomials.• Show how graphs of some polynomials can have minimum and maximum values for different intervals (use graphing technology to set left bounds and right bounds).

NCDPI Unpacking:

Resources:Algebra 2 Textbook Correlation: 5-2, 5-5, 5-9

Cubic Graphs MARS Task: http://map.mathshell.org/materials/tasks.php? taskid=265&subpage=apprentice

Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.PAGE 13

Course Name: Algebra 2/Math III Unit #1Unit Title:Polynomials

Part a., b., and c. are learned by students sequentially in Courses I – III. Part e. is carried through Courses I – III with a focus on exponential in Course I and moving towards logarithms in Courses II and III. Part d. is introduced in course III with further development in a fourth course option.This standard should be seen as related to F-IF.4 with the key difference being students can create graphs, by hand and using technology, from the symbolic function in this standard.Sample Assessment TasksSkill-based task:1) Find the relative maximum, relative minimum, and zeroes of: y = 2x3 – 23x2 + 78x – 72

Problem Task:1) Textbook pg. 294 #46

2) Find the zeroes and relative minimum and maximum values for y = (x + 1)4, y = (x + 3)4, and y = (x + 1)4 + 2. What are the similarities and differences between the values? Explain why this occurs based on what you have learned about transformations.

NCPDI Examples:Ex. Graph f (x) = 2x+1, identifying its intercepts and asymptotes, and describe the end behavior of the function.

Ex. Graph f(x) = x2 + 6x + 5 and f(x) = x2 + 6x – 16, identifying and comparing the key characteristics in these two graphs.

Ex.A roller coaster’s track design can be modeled by the polynomial f(x)=x4- 8x3+16x2. Analyze the graph of this function and describe the ride of the roller coaster. Is there a possible error to using this function to model the roller coaster? Why or why not?

CORE CONTENTCluster Title: Represent and Solve Equations and Inequalities GraphicallyStandard: A.REI-10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).Concepts and Skills to Master:• Write polynomial functions from their points in real-world situations (regression)• Use polynomial functions to predict future values and solve for needed values

SUPPORTS FOR TEACHERSCritical Background Knowledge:• Finding regression equations in the calculator• Evaluating Functions• Solving Polynomial Equations Using Technology and Algebraically

Academic Vocabulary:Regression, Polynomial, Function

Suggested Instructional Strategies:- Begin by relating polynomial regression to linear and quadratic, which students covered in Math 1 and 2- Expand to cubic and quartic regression to evaluate and predict future values- Use technology to solve cubics and quartics for independent variable when given the dependent variable in a problem- HONORS: Can expand to using common differences to determine appropriate regression

Resources:Algebra 2 Textbook Correlation: 5-8

Pearson Activity 5-8 (uses polynomials to show monetary growth, good intro to when we will use exponentials later): Accessible from Pearson Success Net or wiki

Pearson Enrichment 5-8 (building a cube from cardboard): Accessible from Pearson Success Net or wiki

Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.PAGE 14

Course Name: Algebra 2/Math III Unit #1Unit Title:Polynomials

NCDPI Unpacking:The solutions to equations in two variables can be shown in a coordinate plane where every ordered pair that appears on the graph of the equation is a solution. Understand that all points on the graph of a two-variable equation are solutions because when substituted into the equation, they make the equation true.Sample Assessment TasksSkill-based task:(Give table): What is the value of the cubic function at x = 8?

Problem Task:(Give table of values for quartic): The amount of liquid y that flows through a tube of radius x each hour is given by the table. If _____ gallons of water needs to pass through the tube in one hour, what radius does it need to be?

Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.PAGE 15