CURRICULUM FOR HONORS ALGEBRA II GRADES …...Honors Algebra II September 18, 2012 Grades 10-12 RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW UNIT: One Content Area: Algebra Unit

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This curriculum is part of the Educational Program of Studies of the Rahway Public Schools.

ACKNOWLEDGMENTS

Christine H. Salcito, Director of Curriculum and Instruction Kevin Robinson, Program Supervisor of Stem

The Board acknowledges the following who contributed to the preparation of this curriculum.

Rachael Gulley

Subject/Course Title: Date of Board Adoptions:

Honors Algebra II September 18, 2012 Grades 10-12

RAHWAY PUBLIC SCHOOLS CURRICULUM

UNIT OVERVIEW

UNIT: One

Content Area: Algebra

Unit Title: Expressions, Equations, and Inequalities

Target Course/Grade Level: Honors Algebra II/10th-12th Grade

Unit Summary:

Unit 1 expands on students’ understandings and skills related to expressions, equations, and inequalities.

Students will develop the answers to the following essential questions: • How do variables help you model real-world situations? • How can you use the properties of real numbers to simplify algebraic equations? • How do you solve an equation or an inequality?

Approximate Length of Unit:

• 7 days

Primary interdisciplinary connections:

• Science, Geometry, Statistics, Business, Family/Consumer Science, Industrial Arts, Physical Education, Social Studies, Language Arts

LEARNING TARGETS

21st Century Workplace Readiness Standards

Standard/Strand Cumulative Progress Indicator CPI#

Personal Financial Literacy/Money Management Compare strategies for saving and investing and the factors that influence how much should be saved or invested to meet financial goals.

9.2.12.B.2

Personal Financial Literacy/Credit and Debt Management

Compare and compute interest and compound interest. 9.2.12.C.2

21st Century Career and Technical Education/Science, Technology, Engineering & Mathematics Career Cluster

Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, engineering, and mathematics problems.

9.4.12.O.(1).1


Develop an understanding of how science and mathematics function to provide results, answers, and algorithms for engineering activities to solve problems and issues in the real world.

9.4.12.O.(2).1

Content Area Domain Content Area Cluster Standard Seeing Structure in Expressions Interpret the structure of expressions A.SSE.1.b, A.SSE.2

Seeing Structure in Expressions Write expressions in equivalent forms to solve problems A.SSE.3, A.SSE.4

Creating Equations Create equations that describe numbers or relationships A.CED.1, A.CED.4

English Language Arts Standards: Science and Technical Subjects

Key Ideas and Details RST.11-12.2


Craft and Structure RST.11-12.4, RST.11-12.6

Unit Understandings

Students will understand that… • properties of numbers and equality and inverse operations can be used to solve an equation by finding increasingly simpler equations that have

the same solution as the original equation • important properties of equality include reflexive, symmetric, transitive, substitution, addition, subtraction, multiplication and division • just as properties of equality can be used to solve equations, properties of inequality can be used to solve inequalities • properties of numbers and inequality can be used to solve an inequality by finding increasingly simpler inequalities that have the same solution

as the original inequality • an absolute value quantity is nonnegative • since opposites have the same absolute value, an absolute value equation can have two solutions • absolute value inequalities can be written as compound inequalities without absolute value signs

Unit Essential Questions

• How do variables help you model real-world situations? • How can you use the properties of real numbers to simplify algebraic equations? • How do you solve an equation or an inequality?

Knowledge and Skills

Students will know… • Vocabulary –

o absolute value o algebraic expression o compound inequality o like terms o literal equations o term o variable o Properties of Equality o Properties of Inequality

Students will be able to…

• use an expression to model the nth term of a pattern • use variables to represent unknown quantities in real-world situations • apply properties of real numbers to simplify algebraic expressions • apply the Properties of Equality to solve an equation • apply the Properties of Inequality to solve an inequality • find all of the values of a variable that make an equation or inequality true

EVIDENCE OF LEARNING

Assessment

What evidence will be collected and deemed acceptable to show that students truly “understand”? • Unit tests, quizzes, quarterly formative assessments • Open-ended problems that involve written responses • Daily student work • Student/group presentations • Daily Homework • Long term projects

Learning Activities

What differentiated learning experiences and instruction will enable all students to achieve the desired results? • Individualized practice and instruction using Khanacademy.org videos and exercises • Study Island (if available) • Cooperative learning opportunities

RESOURCES

Teacher Resources:

• Algebra II Math Textbook: Teachers’ Edition & accompanying resources, e.g. Lesson presentations, practice worksheets, assessments, writing assignments

• Teacher developed worksheets and activities Equipment Needed:

• Graphing calculators • Internet access for all students


UNIT OVERVIEW

UNIT: Two

Content Area: Algebra, Functions

Unit Title: Functions, Equations, and Graphs

Target Course/Grade Level: : Honors Algebra II/10th-12th Grade

Unit Summary:

Unit 2 expands on students’ understandings and skills related to functions, equations, and graphs.

Students will develop the answers to the following essential questions: • Does it matter which form of a linear equation you use? • How do you use transformations to help graph absolute value functions? • How can you model data with a linear function?


• 12 days



LEARNING TARGETS




9.2.12.B.2





9.4.12.O.(1).1



9.4.12.O.(2).1

Content Area Domain Content Area Cluster Standard Creating Equations Create equations that describe numbers or relationships A.CED.2

Interpreting Functions Understand the concept of a function and use function notation F.IF.1,

Interpreting Functions Interpret functions that arise in applications in terms of the context F.IF.4, F.IF.6

Interpreting Functions Analyze functions using different representations F.IF.7, F.IF.7.b, F.IF.8, F.IF.9

Building Functions Build a function that models a relationship between two quantities F.BF.1

Building Functions Build new functions from existing functions F.BF.3





Unit Understandings

Students will understand that… • some quantities are in a relationship where the ratio of corresponding values is constant • the formula y/x = k says that the ratio of all output-input pairs equals the constant k, the constant of variation • when moving from any point on a non-vertical line in the coordinate plane to any other point on the line, the ratio of the vertical change to the

horizontal change is constant. That constant ratio describes the slope of the line • slope can be calculated by finding the ratio of the difference in the y-coordinates to the difference in the x-coordinates for any two points on the

line • the slopes of two lines in the same plane indicate how the lines are related • linear functions can be represented by either the slope-intercept, point-slope, or standard form. One version can be transformed to another as

needed • sometimes it is possible to model data from a real-world situation with a linear equation. The equation can then be used to draw conclusions

about the situation • the equation of a trend line, or line of best fit, can be used to model data that cluster in a linear pattern • a scatter plot can be used to determine the strength of the relationship, or correlation between data sets • there are sets of functions called families, in which each function is a transformation of a special function called the parent • a parent function is the simplest form of a set of functions that form a family. Each function in the family is a transformation of the parent

function • just as the absolute value of x is its distance from 0, the absolute value of f(x), or f(x), gives the distance from the line y = 0 for each value f(x) • the simplest example of an absolute value function is f(x) = x • the values of a, b, and k, in the form y = ax – h + k determine how the parent function y = x can be transformed • the graph of the linear inequality contains all points on one side of a line and may or may not include the points on the line


• Does it matter which form of a linear equation you use? • How do you use transformations to help graph absolute value functions? • How can you model data with a linear function?



o correlation o direct variation o domain o function o linear equation o range o relation o slope o slope-intercept form of a linear equation o point-slope form of a linear equation o standard form of a linear equation


• identify different forms of linear equations • determine which from of a linear equation is most easily found with the given information • convert between various forms of linear equations • identify the different kinds of transformations • determine whether a transformation changes the location or shape of the graph or both • make a scatter plot of linear data • determine the correlation of linear data • use linear regression to find the line of best fit of linear data with a graphing calculator • use the correlation coefficient to analyze linear data with a graphing calculator


Assessment


Learning Activities


RESOURCES Teacher Resources:


• Teacher developed worksheets and activities

Equipment Needed:


RAHWAY PUBLIC SCHOOLS CURRICULUM for HONORS ALGEBRA II

UNIT OVERVIEW

UNIT: Three

Content Area: Algebra

Unit Title: Linear Systems


Unit Summary:


Students will develop the answers to the following essential questions: • How does representing functions graphically help you solve a system of equations? • How does writing equivalent equations help you solve a system of equations? • How are the properties of equality used in the matrix solution of a system of equations?


• 12 days



LEARNING TARGETS




9.2.12.B.2





9.4.12.O.(1).1



9.4.12.O.(2).1

Content Area Domain Content Area Cluster Standard Creating Equations Create equations that describe numbers or relationships A.CED.2, A.CED.3

Reasoning with Equations and Inequalities Solve systems of equations A.REI.5, A.REI.6, A.REI.8(+)

Reasoning with Equations and Inequalities Represent and solve equations and inequalities graphically A.REI.11, A.REI.12





Unit Understandings

Students will understand that… • a system of equations is solved by finding a set of values that replace the variables in the equations and make each equation true • a point of intersection (x,y) of the graphs of the functions f and g is a solution of the system y = f(x), y = g(x) • a system of equations can be solved by writing equivalent systems until the value of one variable is clear, then substituting to find the value(s) of

the other variable • if the equations of two systems are equivalent, then a solution of the system that is easier to solve is also a solution of the more difficult system • a system of inequalities can be solved in more than one way. Graphing is usually the most appropriate method to solve a system of inequalities • the solution set is the set of all points that are the solutions of each inequality in the system • some real-world problems involve multiple linear relationships. Linear programming accounts for all these linear relationships and gives the

solution to the problem • the feasible region contains all the points that satisfy all the constraints • systems of three equations in three variables can be solved using some of the same algebraic methods used to solve systems of two equations

and two variables • matrices can be used to represent and solve a system of equations without writing the variables • the matrix row operations of adding rows and multiplying by a constant are equivalent to the addition and multiplication properties of equality


• How does representing functions graphically help you solve a system of equations? • How does writing equivalent equations help you solve a system of equations? • How are the properties of equality used in the matrix solution of a system of equations?



o dependent system o equivalent system o independent system o linear system o matrix o matrix element o row operation o system of equations


• solve a system of linear equations by graphing the equations to find points of intersection • use substitution and elimination methods to write equivalent equations until they get an equation with only one variable • use the Addition Property of Equality to add rows of matrices and the Multiplication Property of Equality to multiply rows by a constant • use row operations to get reduced row echelon form, which gives the solution


Assessment


Learning Activities


RESOURCES

Teacher Resources:





UNIT OVERVIEW

UNIT: Four

Content Area: Number and Quantity, Algebra, Functions

Unit Title: Quadratic Functions and Equations


Unit Summary:

Unit 4 expands on students’ understandings and skills related to functions, equations, and graphs. Students will develop the answers to the following essential questions:

• What are the advantages of a quadratic equation in vertex form? In standard form? • How is any quadratic function related to the parent quadratic function y = x2 ? • How are the real solutions of a quadratic equation related to the graph of the related quadratic function?


• 16 days



LEARNING TARGETS




9.2.12.B.2



Content Area Domain Content Area Cluster Standard The Complex Number System Perform arithmetic operations with complex numbers N.CN.1, N.CN.2

The Complex Number System Use complex numbers in polynomial identities and equations N.CN.7, N.CN.8

Seeing Structure in Expressions Interpret the structure of expressions A.SSE.1.a, A.SSE.2

Arithmetic with Polynomials and Rational Expressions Understand the relationship between zeros and factors of polynomials A.APR.3

Creating Equations Create equations that describe numbers or relationships A.CED.1,A.CED.2,A.CED.3

Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning A.REI.7

Interpreting Functions Interpret functions that arise in applications in terms of the context F.IF.4, F.IF.5, F.IF.6

Interpreting Functions Analyze functions using different representations F.IF.7, F.IF.8, F.IF.9

Building functions Build a function that models a relationship between two quantities F.BF.1

Building functions Build new functions from existing functions F.BF.3







9.4.12.O.(1).1



9.4.12.O.(2).1

Unit Understandings

Students will understand that… • the graph of any quadratic function is a transformation of the parent function, y = x2 • the vertex form of a quadratic function is f(x) = a( x – h )2 + k, where a ≠ 0 • the axis of symmetry is a line that divides the parabola into two mirror images • the vertex of the parabola is (h,k), the intersection of the parabola and its axis of symmetry • any quadratic function is a stretch, compression, reflection, and/or a translation of y = x2 • the standard form of a quadratic function is is f(x) = ax2 + bx + c, where a ≠ 0 • for any quadratic function f(x) = ax2 + bx + c, the values of a, b, and c provide key information about ifs graph • standard form is “calculator ready” and a transformation of y = x2 • three noncollinear points, no two of which are in line vertically, are on the graph of exactly one quadratic function • many quadratic trinomials (ax2 + bx + c) can be factored into products of two binomials • the Distributive Property or FOIL method can be used to multiply two binomials. FOIL can be used in reverse to factor • the zeros of a quadratic function y = ax2 + bx + c can be found by solving the related quadratic equation • some quadratic equations in standard form can be solved by factoring the quadratic expression and using the Zero-Product Property • the real solutions of a quadratic equation show the zeros of the related quadratic function and the x-intercepts of its graph • every quadratic equation has complex number solutions (that sometimes are real numbers) • the imaginary unit is the complex number whose square is -1. So i2 = -1 and i = √-1 • systems involving quadratic equations can be solved using methods similar to the ones used to solve systems of linear equations • the points where the graphs of the equations intersect represent the solutions of a system


• What are the advantages of a quadratic equation in vertex form? In standard form? • How is any quadratic function related to the parent quadratic function y = x2 ? • How are the real solutions of a quadratic equation related to the graph of the related quadratic function?



o axis of symmetry o complex number o discriminant o greatest common factor o imaginary number o parabola o quadratic function o standard form o vertex form o zero of a function o Quadratic formula


• identify the vertex, line of symmetry, maximum or minimum, domain, range, and translations of a quadratic function • graph quadratic functions with and without graphing calculators • use quadratic functions as models • graph transformations of the parent quadratic function y = x2 • compare translations to the parent quadratic function • identify the x-intercepts of the graphs of related quadratic functions


Assessment


Learning Activities







UNIT OVERVIEW

UNIT: Five

Content Area: Number and Quantity, Algebra, and Functions

Unit Title: Polynomials and Polynomial Functions


Unit Summary:


Students will develop the answers to the following essential questions: • What does the degree of a polynomial tell you about its related polynomial function? • For a polynomial function, how are the factors, zeros, and x-intercepts related? • For a polynomial equation, how are factors and roots related?


• 14 days



LEARNING TARGETS




9.2.12.B.2



Content Area Domain Content Area Cluster Standard The Complex Number System Use complex numbers in polynomial identities and equations N.CN.7, N.CN.8, N.CN.9

Seeing Structure in Expressions Interpret the structure of expressions A.SSE.1.a, A.SSE.1.b, A.SSE.2

Arithmetic with Polynomials and Rational Expressions Understand the relationship between zeros and factors of polynomials A.APR.2, A.APR.3

Arithmetic with Polynomials and Rational Expressions Understand the relationship between zeros and factors of polynomials A.APR.4, A.APR.5, A.APR.6

Reasoning with Equations and Inequalities Represent and solve equations and inequalities graphically A.REI.11

Interpreting Functions Interpret functions that arise in applications in terms of the context F.IF.4, F.IF.5, F.IF.6

Interpreting Functions Analyze functions using different representations F.IF.7, F.IF.8, F.IF.9








9.4.12.O.(1).1



9.4.12.O.(2).1

Unit Understandings

Students will understand that… • the algebraic form of a polynomial function gives information about its graph. Its graph gives information about its algebraic form • the shape and end behavior of the graph of a polynomial is determined by the degree of the polynomial and by the sign of the leading coefficient • it’s possible to determine the degree of a polynomial by analyzing the first, second, etc., differences of its y-values • knowing the zeros of a polynomial function gives information about its graph • a polynomial of degree n has n linear factors. The graph of the related function crosses the x-axis an even or odd number of times depending on

whether n ix even or odd • ( x – a ) is a linear factor if and only if a is a zero • if ( x – a ) is a factor of a polynomial, then the polynomial has value 0 when x = a. If a is a real number, then the graph of the polynomial has

(a,0) as an intercept • polynomials can be divided using steps that are similar to the long division steps that are used to divide whole numbers • the factors of the numbers an and a0 in P(x) = an xn + an-1 xn-1 + … + a1 x + a0 can be used to factor P(x) and solve the equation P(x) = 0 • ( x – a ) is a linear factor if and only if a is a root of the related polynomial equation • the degree of a polynomial equation tells how many roots the equation has • a pattern of coefficients and the pattern an, an-1b, an-2b2,…, a1bn-1, bn can be used to write the expansion of (a + b)n • to expand the power of a binomial in general, first multiply as needed. Then write the polynomial in standard form • the behavior of the graphs of polynomial functions of different degrees can suggest which will best fit a particular real-world data set • the graph of the function y = af( x – h ) + k is a vertical stretch or compression by the factor a, a horizontal shift of h units, and a vertical

shift of k units of the graph y = f(x) Unit Essential Questions

• What does the degree of a polynomial tell you about its related polynomial function? • For a polynomial function, how are the factors, zeros, and x-intercepts related? • For a polynomial equation, how are factors and roots related?



o end behavior o monomial o multiplicity o Pascal’s Triangle o polynomial function o relative maximum o relative minimum o standard form of a polynomial function o synthetic division o turning point


• write a polynomial function given a polynomial equation • identify the degree of a polynomial equation • identify the highest power of a polynomial function • write a polynomial given its factors or zeros • identify the zeros of a polynomial function by finding the x-intercepts of its graph • factor a polynomial equation • apply the Zero-Product Property


Assessment

What evidence will be collected and deemed acceptable to show that students truly “understand”? • Unit tests, quizzes, quarterly formative assessments • Open-ended problems that involve written responses • Daily student work

• Student/group presentations • Daily Homework • Long term projects

Learning Activities







UNIT OVERVIEW

UNIT: Six


Unit Title: Radical Functions and Rational Exponents


Unit Summary:

Unit 6 expands on students’ understandings and skills related to radical functions and rational exponents.

Students will develop the answers to the following essential questions: • To simplify the nth root of an expression, what must be true about the expression? • When you square each side of an equation, is the resulting equation equivalent to the original? • How are a function and its inverse function related?


• 13 days



LEARNING TARGETS




9.2.12.B.2





9.4.12.O.(1).1



9.4.12.O.(2).1

Content Area Domain Content Area Cluster Standard Seeing Structure in Expressions Interpret the structure of expressions A.SSE.2

Creating Equations Create equations that describe numbers or relationships A.CED.4

Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning A.REI.2

Interpreting Functions Analyze functions using different representations F.IF.7, F.IF.8

Building Functions Build a function that models a relationship between two quantities F.BF.1.b

Building Functions Build new functions from existing functions F.BF.4.a, F.BF.4.c





Unit Understandings

Students will understand that… • corresponding to every power, there is a root. For example, just as there are squares (second powers), there are square roots. Just as there are

cubes (third powers), there are cube roots, and so on • the nth root of an expression that contains an nth power as a factor can be simplified n√xn = xn/n = x, if n is odd or x if n is even • if n√a and n√b are real numbers, then n√a • n√b = n√(ab) • a radical expression can be simplified when the exponent of one factor of the radicand is a multiple of the radical’s index • the product of powers that have the same exponent can be simplified • the product of radicals that have the same index can be simplified • like radicals can be combined using properties of real numbers • n√x = 1/xn for any positive integer n. Like the radical form, the exponent form indicates the principal root • solving a square root equation may require squaring each side of the equation. This can introduce extraneous solutions • a radical equation can be solved by isolating the radical on one side of the equation, then raising each side to the power suggested by the index • functions can be added, subtracted, multiplied and divided based on how these operations are performed for real numbers. One difference

however is that the domain of each function must be considered • the inverse of a function may or may not be a function • when you square each side of an equation, the resulting equation may have more solutions than the original equation • if f and f-1 are functions and if either maps a to b, then the other maps b to a, i.e. (f° f-1)(a) = . (f-1° f)(a) =a • the range of the relation is the domain of the inverse. The domain of the relation is the range of the inverse • a square root function is the inverse of a quadratic function that has a restricted domain


• To simplify the nth root of an expression, what must be true about the expression? • When you square each side of an equation, is the resulting equation equivalent to the original? • How are a function and its inverse function related?



o composite function o inverse function o nth root o principal root o radical equation o radicand o rational exponent o rationalize the denominator o square root equation o square root function


• simplify radical expressions • solve radical equations • determine the domain of radical functions • check for extraneous solutions • find inverse functions • graph functions and their inverses


Assessment


Learning Activities What differentiated learning experiences and instruction will enable all students to achieve the desired results?

• Individualized practice and instruction using Khanacademy.org videos and exercises • Study Island (if available) • Cooperative learning opportunities






UNIT OVERVIEW

UNIT: Seven


Unit Title: Exponential and Logarithmic Functions


Unit Summary:

Unit 7 expands on students’ understandings and skills related to radical functions and rational exponents.

Students will develop the answers to the following essential questions: • How do you model a quantity that changes regularly over time by the same percentage? • How are exponents and logarithms related? • How are exponential functions and logarithmic functions related?


• 12 days



LEARNING TARGETS




9.2.12.B.2



Content Area Domain Content Area Cluster Standard Seeing Structure in Expressions Interpret the structure of expressions A.SSE.1.a, A.SSE.1.b

Creating Equations Create equations that describe numbers or relationships A.CED.1, A.CED.2, A.CED.3


Interpreting Functions Analyze functions using different representations F.IF.7.e, F.IF.8, F.IF.9

Building Functions Build a function that models a relationship between two quantities F.BF.1.b

Building Functions Build new functions from existing functions F.BF.4.a

Linear and Exponential Models Construct and compare linear and exponential models and solve

problems

F.LE.4







9.4.12.O.(1).1



9.4.12.O.(2).1

Unit Understandings

Students will understand that… • repeated multiplication can be represented with a function in the form y = abx where b is a positive number other than 1 • an exponential function is a function with the general form y = abx , a ≠ 0, with b > 0, and b ≠ 1. In an exponential function, the base b is a

constant. the exponent x is the independent variable with domain the set of real numbers • the factor a in y = abx can stretch or compress, and possibly reflect the graph of the parent function y = bx • the function y = abx , a > 0, b > 1, models exponential decay if 0 < b < 1 • the exponential function y = bx is one-to-one, so its inverse x = by is a function. To express “y as a function of x” for the inverse, write y =

logbx • logarithms are exponents. In fact, logba = c if and only if bc = a • logarithms and exponents have corresponding properties • logarithms can be used to solve exponential equations. Exponents can be used to solve logarithmic equations • the function y = ex and y = ln x are inverse functions. Just as before, this means that if a = eb and b = ln a and vice versa


• How do you model a quantity that changes regularly over time by the same percentage? • How are exponents and logarithms related? • How are exponential functions and logarithmic functions related?



o asymptote o Change of Base Formula o common logarithm o exponential equation o exponential function o exponential growth o logarithm o logarithmic equation o logarithmic function o natural logarithmic function

Students will be able to… • model situations with exponential functions • use exponents to solve logarithmic equations and logarithms to solve exponential equations • show that exponents and logarithms are inverse functions • graph exponential and logarithmic functions


Assessment


Learning Activities


RESOURCES

Teacher Resources:





UNIT OVERVIEW

UNIT: Eight

Content Area: Number and Quantity, Algebra, and Functions

Unit Title: Rational Functions


Unit Summary:


Students will develop the answers to the following essential questions: • Are two quantities inversely proportional if an increase in one corresponds to a decrease in the other? • What kinds of asymptotes are possible for a rational function? • Are a rational expression and its simplified form equivalent?


• 11 days



LEARNING TARGETS

Content Area Domain Content Area Cluster Standard Seeing Structure in Expressions Interpret the structure of expressions A.SSE.1.a, A.SSE.1.b, A.SSE.2

Creating Equations Create equations that describe numbers or relationships A.CED.1, A.CED.2

Arithmetic with Polynomials and Rational

Expressions

Understand the relationship between zeros and factors of polynomials A.APR.3

Arithmetic with Polynomials and Rational

Expressions

Use polynomial identities to solve problems A.APR.6, A.APR.7

Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the

reasoning

A.REI.2


Building Functions Build a function that models a relationship between two quantities F.BF.1, F.BF.1.b









9.2.12.B.2





9.4.12.O.(1).1



9.4.12.O.(2).1

Unit Understandings

Students will understand that… • if a product is constant, a decrease in the value of one factor must accompany an increase in the value of the other factor • in a direct variation, two positive quantities either increase together or decrease together. In an inverse variation, as one quantity increases the

other decreases • quantities x and y are inversely proportional if and only if increasing x by the factor k (k ≠ 0) means shrinking y by the factor 1/k • transformations of the parent reciprocal function include stretches, compressions (or shrinks), reflections, and horizontal and vertical translations • a rational function may have zero or one horizontal or oblique asymptote and zero or more vertical asymptotes • a rational function is a ratio of polynomial functions • if a function has polynomials in its denominator, its graph has a gap at each zero of the polynomial. The gap could be a one-point hole in the

graph, or it could be the location of a vertical asymptote for the graph • a reasonable graph for a rational function can be sketched by finding all intercepts and asymptotes. Sometimes a few extra points should be

plotted to get a good sense of the shape of the graph • much of what is true about multiplying and dividing fractions can be used to multiply and divide rational expressions • a rational expression is in its simplest form when its numerator and denominator are polynomials that have no common divisors • functions such as f(x) = (x+a)/(x2-a2) and g(x) = 1/(x-a) , x≠ ±a , are equivalent • much of what is true about operating with fractions can be used to operate with rational expressions. Rational expressions can be added of

subtracted by first finding a common denominator – preferably the least common multiple (LCM) of the denominators • the LCM of denominators is the product of their prime factors, each raised to the greatest power that occurs in any of the expressions • solving an equation containing rational expressions begins by multiplying each side by the least common denominator of the rational

expressions. Doing this, however, can introduce extraneous solutions Unit Essential Questions

• Are two quantities inversely proportional if an increase in one corresponds to a decrease in the other? • What kinds of asymptotes are possible for a rational function? • Are a rational expression and its simplified form equivalent?



o combined variation o complex fraction o continuous graph o discontinuous graph o inverse variation o joint variation o point of discontinuity o rational equation o rational expression o rational function o reciprocal function


• identify and describe inverse and direct variation functions • graph asymptotes of rational functions • identify whether a rational function has an asymptote • differentiate between vertical, horizontal, and oblique asymptotes • define the domains of simplified rational expressions to make them equivalent to the originals


Assessment


Learning Activities


RESOURCES

Teacher Resources:





UNIT OVERVIEW

UNIT: Nine


Unit Title: Sequences and Series


Unit Summary:

Unit 9 expands on students’ understandings and skills related to sequences and series.

Students will develop the answers to the following essential questions: • How can you represent the terms of a sequence explicitly? How can you represent them recursively? • What are equivalent explicit and recursive definitions for an arithmetic sequence? • How can you model a geometric sequence? How can you model its sum?


• 5 days



LEARNING TARGETS




9.2.12.B.2





9.4.12.O.(1).1



9.4.12.O.(2).1

Content Area Domain Content Area Cluster Standard Seeing Structure in Expressions Write expressions in equivalent forms to solve problems A.SSE.4

Interpreting Functions Understand the concept of a function and use function notation F.IF.3





Unit Understandings

Students will understand that… • a sequence can be defined explicitly by describing its nth term with a formula using n or recursively by stating its first term and a formula for

its nth term using the (n-1) term • a geometric sequence can be modeled explicitly or recursively • when two terms and the number of terms in a finite arithmetic sequence are known, the sum of the terms can be found • just as with a finite arithmetic series, the sum of a finite geometric series can be found using a formula. It is necessary to know the first term,

the number of terms, and the common ratio • the sum of the first n terms of a geometric sequence is a1(1 – rn)/(1 – r)


• How can you represent the terms of a sequence explicitly? How can you represent them recursively? • What are equivalent explicit and recursive definitions for an arithmetic sequence? • How can you model a geometric sequence? How can you model its sum?



o arithmetic sequence o arithmetic series o common difference o common ratio o converge o diverge o explicit formula o geometric sequence o geometric series o limits o recursive formula


• identify mathematical patterns found in a sequence • find a rule to describe a pattern • find the common difference of an arithmetic sequence • find the common ratio of a geometric sequence • determine whether a geometric series converges


Assessment


Learning Activities







UNIT OVERVIEW

UNIT: Ten

Content Area: Functions, Geometry

Unit Title: Quadratic Relations and Conic Sections


Unit Summary:

Students will develop the answers to the following essential questions: • What is the intersection of a cone and a plane parallel to a line along the side of a cone? • What is the graph of x2/9 + y2/9 = 1? • What is the difference between the algebraic representations of ellipses and hyperbolas?


• 12 days


• Science, Geometry, Industrial Arts, Physical Education, Social Studies, Language Arts

LEARNING TARGETS




9.2.12.B.2





9.4.12.O.(1).1



9.4.12.O.(2).1

Content Area Domain Content Area Cluster Standard Interpreting Functions Analyze functions using different representations F.IF.8

Expressing Geometric Properties with

Equations

Translate between the geometric description and the equation for a conic

section

G.GPE.1, G.GPE.2, G.GPE.3





Unit Understandings

Students will understand that… • there are four types of curves known as conic sections: parabolas, circles, ellipses, and hyperbolas. Each curve has its own distinct shape and

properties • a conic section is a curve obtained by intersecting a plane and a double cone • each point of a parabola is equidistant from a point called the focus and a line called the directrix • the intersection or a cone and a plane parallel to a line along its side is a parabola • an equation of a circle with center (0,0) and radius r in the coordinate plane is x2 + y2 = r2 • not every circle has its center at the origin. Suppose a circle with radius r has center (h,k). Then r is the distance from (h,k) to any point (x,y) on

the circle. The equation is r2 = (x-h)2 + (y-k)2 • x2/ r2 + y2/ r2 = 1 is an equation of a circle centered at the origin with radius r. Multiply each side by r2 to get x2 + y2 = r2 • a circle is the set of points a fixed distance from one point. An ellipse “stretches” a circle in one direction and is the set of points that have a total

fixed distance from two points • the x2 and y2 terms of the algebraic form of an ellipse are both positive. For hyperbola, one term is negative • the shape of a hyperbola is guided by asymptotes • in an x-y relationship, replacing x by x – h and y by y – k (with h > 0 and k > 0 ) translates the graph of the relation h units to the right and k units

up Unit Essential Questions

• What is the intersection of a cone and a plane parallel to a line along the side of a cone? • What is the graph of x2/9 + y2/9 = 1? • What is the difference between the algebraic representations of ellipses and hyperbolas?



o center of a circle o circle o conic section o directrix o ellipse o hyperbola o radius o standard form of an equation of a circle


• identify the possible conic sections formed depending on the angle of intersection of the cone and plane • graph functions of circles • identify conic sections based on their equations • differentiate between ellipses and hyperbolas algebraically and graphically


Assessment


Learning Activities







UNIT OVERVIEW

UNIT: Eleven

Content Area: Statistics and Probability

Unit Title: Probability and Statistics


Unit Summary:

Unit 11 expands on students’ understandings and skills related to probability and statistics.

Students will develop the answers to the following essential questions: • What is the difference between a permutation and a combination? • What is the difference between experimental and theoretical probability? • How are measures of central tendency different from standard deviation?


• 16 days



LEARNING TARGETS

Content Area Domain Content Area Cluster Standard Making Inferences and Justifying Conclusions Understand and evaluate random processes underlying statistical experiments S.IC.1, S.IC.2

Making Inferences and Justifying Conclusions Make inferences and justify conclusions from sample surveys, experiments, and

observational studies

S.IC.3, S.IC.4, S.IC.15 S.IC.6

Interpreting Categorical and Quantitative Data Summarize, represent, and interpret data on a single count or measurement variable S.ID.4

Use Probability to Make Decisions Use probability to evaluate outcomes of decisions S.MD.6, S.MD.7

Conditional Probability and the Rues of

Probability

Understand independence and conditional probability and use them to interpret data S.CP.2, S.CP.3, S.CP.4, S.CP.5

Conditional Probability and the Rues of

Probability

Use the rules of probability to compute probabilities of compound events in a

uniform probability model

S.CP.6, S.CP.7, S.CP.8, S.CP.9








9.2.12.B.2





9.4.12.O.(1).1



9.4.12.O.(2).1

Unit Understandings

Students will understand that… • multiplication can be used to count the number of ways that certain things can happen • the probability, p, of an event is a number such that 0 ≤ p ≤ 1 • the probability of an impossible event is 0 • the probability of a certain event is 1 • to find the probability of two events occurring together, it is necessary to determine whether the occurrence of one event affects the probability

that the other event will occur • a conditional probability is the probability that one event occurs, given that another event has occurred • probability models can be used to analyze situations and make fair decisions • data sets can be described using various statistical measures, depending on what characteristics are being studied • standard deviation is a measure of how far the numbers in a data set deviate from the mean • you can get good statistical information about a population by studying a sample of the population • binomial properties can be used to model situations in which there are two possible outcomes • normal distributions model many common natural phenomena, such as human height, weight, and blood pressure


• What is the difference between a permutation and a combination? • What is the difference between experimental and theoretical probability? • How are measures of central tendency different from standard deviation?



o combination o conditional probability o experimental probability o measure of central tendency o mutually exclusive events o normal distribution o permutation o simulation o theoretical probability


• find permutations and combinations of data sets using formulas • use simulation to model experimental probability • find the theoretical probability of events using a formula • find and analyze the measures of central tendency of given data sets • find the standard deviation of given data sets


Assessment


Learning Activities







UNIT OVERVIEW

UNIT: Twelve

Content Area: Functions

Unit Title: Periodic Functions and Trigonometry


Unit Summary:

Unit 12 expands on students’ understandings and skills related to periodic functions and trigonometry.

Students will develop the answers to the following essential questions: • How can you model periodic behavior? • What function has as its graph a sine curve with amplitude, period π, and a minimum at the origin? • If you know the value of sin θ, how can you find cos θ, tan θ, csc θ, sec θ, and cot θ?


• 11 days


• Science, Geometry, Statistics, Business, Family/Consumer Science, Industrial Arts, Music, Physical Education, Social Studies, Language Arts

LEARNING TARGETS




9.2.12.B.2





9.4.12.O.(1).1



9.4.12.O.(2).1

Content Area Domain Content Area Cluster Standard Interpreting Functions Interpret functions that arise in applications in terms of the context F.IF.4

Interpreting Functions Analyze functions using different representations F.IF.7.e

Trigonometric Functions Extend the domain of trigonometric functions using the unit circle F.TF.1, F.TF.2

Trigonometric Functions Model periodic phenomena with trigonometric functions F.TF.5





Unit Understandings

Students will understand that… • periodic behavior repeats over intervals of constant length • the amplitude of a periodic function is half the difference of the maximum and minimum values of the function • an angle with a full circle rotation measures 2π radians. An angle with a semicircle rotation measures π radians • to convert degrees to radians, multiply by (π radians)/180° • to convert radians to degrees, multiply by 180° /(π radians) • for a circle of radius r and a central angle θ (in radians), the length of the intercepted arc is s = rθ • as the terminal side of an angle rotates about the origin (beginning at 0°), its sine value on the unit circle increases from 0 to 1, decreases from 1

to -1, and then increases back to 0 • a nontranslated sine function can be completely described in terms of its amplitude and period • as the terminal side of an angle rotates about the origin (beginning at 0°), its cosine value on the unit circle decreases from 1 to -1, and then

increases back to 1 • a nontranslated cosine function can be completely described in terms of its amplitude and period • the tangent function has infinitely many points of discontinuity with a vertical asymptote at each point. Its range is all real numbers. Its period is

π • suppose y = a tan bθ, with b > 0 and θ measured in radians. Then π/b is the period of the function, one cycle occurs in the interval -π/(2b) to

π/(2b), and there are vertical asymptotes at each end of the cycle • periodic functions can be translated the same way as other functions • all functions in the sine and cosine family of functions can be completely described by their amplitude, period, vertical shift, and horizontal shift

(or phase shift) • cosine, sine and tangent have reciprocals. Cosine and secant are reciprocals, as are sine and cosecant. Tangent and cotangent are also reciprocals • reciprocal trigonometric functions can be evaluated using what you know about sine, cosine, and tangent • the graphs of reciprocal trigonometric functions have asymptotes when their denominators equal 0


• How can you model periodic behavior? • What function has as its graph a sine curve with amplitude, period π, and a minimum at the origin? • If you know the value of sin θ, how can you find cos θ, tan θ, csc θ, sec θ, and cot θ?



o amplitude o central angle o cosine o cycle o midline o period o periodic function o phase shift o radian o sine o tangent o unit circle

Students will be able to… • identify and explore periodic behavior • graph periodic functions • write the formulas of trigonometric functions • find amplitude, period, minimums, and maximums of trigonometric functions • find the value of the reciprocal trigonometric functions based on the corresponding trigonometric functions


Assessment


Learning Activities




• Teacher developed worksheets and activities

Equipment Needed:



UNIT OVERVIEW

UNIT: Thirteen

Content Area: Functions, Geometry

Unit Title: Trigonometric Identities and Equations


Unit Summary:

Unit 13 expands on students’ understandings and skills related to periodic functions and trigonometry.

Students will develop the answers to the following essential questions: • How do you verify that an equation involving the variable x is an identity? • A trigonometric function corresponds one number to many, so how can its inverse be a function? • How do the trigonometric functions relate to the trigonometric ratios for a right triangle


• 11 days


• Science, Geometry, Industrial Arts, Physical Education, Social Studies, Language Arts

LEARNING TARGETS




9.2.12.B.2





9.4.12.O.(1).1



9.4.12.O.(2).1

Content Area Domain Content Area Cluster Standard Trigonometric Functions Model periodic phenomena with trigonometric functions F.TF.6, F.TF.7

Trigonometric Functions Prove and apply trigonometric identities F.TF.8, F.TF.9

Similarity, Right Triangles, and Trigonometry Define trigonometric ratios and solve problems involving right

triangles

G.SRT.6, G.SRT.8

Similarity, Right Triangles, and Trigonometry Apply trigonometry to general triangles G.SRT.9, G.SRT.10, G.SRT.11





Unit Understandings

Students will understand that… • the interrelationships among the six basic trigonometric functions make it possible to write trigonometric expressions in various equivalent

forms, some of which can be significantly easier to work with than others, in mathematical applications • known identities can be used to verify other identities • trigonometric identities can be used to simplify trigonometric expressions • some trigonometric equations can be solved using an inverse trigonometric function to find one solution. Then periodicity can be used to find all

solutions • the values of inverse trigonometric functions are measures of angles. The unit circle can be used to find the values in either radians or degrees • if the domains of the trigonometric functions are restricted to angle measures between 0° and 90°, the function values are the trigonometric

ratios associated with the acute angles of a right triangle • in right triangle trigonometry, the value of one trigonometric ratio determines the values of the others • if two angles and a side of a triangle are known, trigonometry can be used to solve the triangle. If two sides and the included angle are known,

the area of the triangle can be found • the Law of Sines relates the sines of the angles of a triangle to the side lengths • if the measures of enough parts of a triangle to completely determine the triangle are known, the triangle can be solved • the Law of Cosines relates the length of a side of any triangle to the measure of the opposite angle and the other tow side lengths • several trigonometric identities involve a single angle. Other trigonometric identities involve two angles. No important trigonometric identity is

additive; for example, sin(A + B) ≠ sin A + sin B • the cosine curve is the sine curve translated π/2 radians to the left. The cotangent curve is the tangent curve reflected over the x-axis and

translated π/2 radians horizontally • the double-angle identities are special cases of the angle-sum identities. The half-angle identities result from substituting θ/2 for θ in certain

double-angle identities • double-angle and half-angle identities can be used to find exact trigonometric values for certain angle measures


• How do you verify that an equation involving the variable x is an identity? • A trigonometric function corresponds one number to many, so how can its inverse be a function? • How do the trigonometric functions relate to the trigonometric ratios for a right triangle



o Law of Cosines o Law of Sines o trigonometric identities o trigonometric ratios for a right triangle

Students will be able to… • show that trigonometric identities are true for all values of the variable with the domain of validity • restrict the domains of trigonometric functions to make their inverses functions • restrict the domain of the trigonometric function so that the function values are equivalent to the trigonometric ratios for a right triangle


Assessment What evidence will be collected and deemed acceptable to show that students truly “understand”?

• Unit tests, quizzes, quarterly formative assessments • Open-ended problems that involve written responses • Daily student work • Student/group presentations • Daily Homework • Long term projects

Learning Activities







UNIT OVERVIEW

UNIT: Fourteen

Content Area: Numbers and Quantity, Geometry

Unit Title: Matrices


Unit Summary:

Unit 14 expands on students’ understandings and skills related to matrices and systems of equations.

Students will develop the answers to the following essential questions: • How can you use a matrix to organize data? • How can you use a matrix equation to model a real-world situation? • How can a matrix represent a transformation of a geometric figure in the plane?


• 14 days



LEARNING TARGETS




9.2.12.B.2





9.4.12.O.(1).1



9.4.12.O.(2).1

Content Area Domain Content Area Cluster Standard Vector and Matrix Quantities Represent and model with vector quantities N.VM.1, N.VM.2, N.VM.3

Vector and Matrix Quantities Perform operations on vectors N.VM.4.a, N.VM.4.b, N.VM.4.c, N.VM.5.a, N.VM.5.b

Vector and Matrix Quantities Perform operations on matrices and use matrices in applications N.VM.6, N.VM.7, N.VM.8, N.VM.9, N.VM.10, N.VM.11, N.VM.12

Congruence Experiment with transformations in the plane G.CO.2, G.CO.5





Unit Understandings

Students will understand that… • the addition and subtraction of numbers can be extended to matrices • matrices can be added or subtracted by adding or subtracting corresponding elements • to multiply a matrix by a scalar, multiply each element of the matrix by the scalar • if A is an m x n matrix, and B is an n x p matrix, then AB is an m x p matrix • the product of a matrix and its inverse is the multiplicative identity matrix. Not all matrices have inverse matrices • a matrix has an inverse if and only if its determinant does not equal 0 • systems of linear equations can be modeled by matrix equations • some matrix equations of the form AX = B can be solved by multiplying each side of the equation by A-1, the inverse of matrix A • transformations that can be performed using matrices include translations, dilations, rotations, and reflections • a 2 x 1 matrix representing a point can be multiplied by a 2 x 2 matrix to rotate the point about the origin or reflect the point across the line • a vector is a mathematical object that has both magnitude and direction • vectors can be added or subtracted by adding or subtracting corresponding elements • the dot product of two vectors is the sum of the products of corresponding elements


• How can you use a matrix to organize data? • How can you use a matrix equation to model a real-world situation? • How can a matrix represent a transformation of a geometric figure in the plane?



o determinant o dilation o equal matrices o image o matrix equation o preimage o scalar multiplication o variable matrix o zero matrix


• use matrices to compare data • solve systems of equations with matrix equations • use matrix operations to transform geometric figures


Assessment


Learning Activities What differentiated learning experiences and instruction will enable all students to achieve the desired results?

• Individualized practice and instruction using Khanacademy.org videos and exercises • Study Island (if available) • Cooperative learning opportunities





Documents

CURRICULUM FOR HONORS ALGEBRA II GRADES …...Honors Algebra II September 18, 2012 Grades 10-12 RAHWAY PUBLIC SCHOOLS CURRICULUM UNIT OVERVIEW UNIT: One Content Area: Algebra Unit