Grade 7 Curriculum Guide s3

ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Introduction

The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction and assessment. It defines the content knowledge, skills, and understandings that are measured by the Standards of Learning assessment. It provides additional guidance to teachers as they develop an instructional program appropriate for their students. It also assists teachers in their lesson planning by identifying essential understandings, defining essential content knowledge, and describing the intellectual skills students need to use. This Guide delineates in greater specificity the content that all teachers should teach and all students should learn.

The format of the Curriculum Guide facilitates teacher planning by identifying the key concepts, knowledge, and skills that should be the focus of instruction for each objective. The Curriculum Guide is divided into sections: Curriculum Information, Essential Knowledge and Skills, Key Vocabulary, Essential Questions and Understandings, Teacher Notes and Elaborations, Resources, and Sample Instructional Strategies and Activities. The purpose of each section is explained below.

Curriculum Information: This section includes the objective and SOL Reporting Category, focus or topic, and in some, not all, foundational objectives that are being built upon.

Essential Knowledge and Skills: Each objective is expanded in this section. What each student should know and be able to do in each objective is outlined. This is not meant to be an exhaustive list nor a list that limits what is taught in the classroom. This section is helpful to teachers when planning classroom assessments as it is a guide to the knowledge and skills that define the objective.

Key Vocabulary: This section includes vocabulary that is key to the objective and many times the first introduction for the student to new concepts and skills.

Essential Questions and Understandings: This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an understanding of the objectives.

Teacher Notes and Elaborations: This section includes background information for the teacher. It contains content that is necessary for teaching this objective and may extend the teachers’ knowledge of the objective beyond the current grade level. It may also contain definitions of key vocabulary to help facilitate student learning.

Resources: This section lists various resources that teachers may use when planning instruction. Teachers are not limited to only these resources.

Sample Instructional Strategies and Activities: This section lists ideas and suggestions that teachers may use when planning instruction. 1 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS

The following chart is the pacing guide for the Prince William County Algebra II Curriculum. The chart outlines the order in which the objectives should be taught; provides the suggested number blocks to teach each unit; and organizes the objectives into Units of Study. The Prince William County cross-content vocabulary terms that are in this course are: analyze, compare and contrast, conclude, evaluate, explain, generalize, question/inquire, sequence, solve, summarize, and synthesize.

ALGEBRA II SOL TEST QUESTION BREAKDOWN (50 QUESTIONS TOTAL) (Based on 2009 SOL Objectives and Reporting Categories)

Expressions and Operations 13 Questions 26% of the Test Equations and Inequalities 13 Questions 26% of the Test Functions and Statistics 24 Questions 48% of the Test

2 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS

Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations SOL Reporting Category The student will use problem solving, Essential Questions Expressions and Operations mathematical communication,  What is a rational expression? mathematical reasoning, connections  How is a rational expression simplified? and representations to:  What is a radical expression? Topic  Add, subtract, multiply, and divide  How are radical expressions simplified? Expressions and Operations rational algebraic expressions.  How do radical expressions apply to real-life situations?  Simplify a rational algebraic expression  How is conversion between radical and rational exponents completed? with common monomial or binomial  When is a polynomial completely factored? Virginia SOL AII/T.1 factors.  What are the patterns to investigate when factoring a polynomial? The student, given rational,  Recognize a complex algebraic radical, or polynomial expressions fraction, and simplify it as a quotient or Essential Understandings will product of simple algebraic fractions.  Computational skills applicable to numerical fractions also apply to rational expressions a. add, subtract, multiply, divide, and  Simplify radical expressions containing involving variables. simplify rational algebraic positive rational numbers and variables. expressions;  Radical expressions can be written and simplified using rational exponents.  Convert from radical notation to  Only radicals with a common radicand and index can be added or subtracted. b. add subtract, multiply, divide and exponential notation and vice versa. simplify radical expressions  A relationship exists among arithmetic complex fractions, algebraic complex fractions,  Add and subtract radical expressions. and rational numbers. containing rational numbers and  Multiply and divide radical expressions variables, and expressions  The complete factorization of polynomials has occurred when each factor is a prime not requiring rationalizing the polynomial. containing rational exponents; denominators. c. write radical expressions as  Pattern recognition can be used to determine complete factorization of a polynomial.  Multiply and divide radical expressions expressions containing rational requiring rationalizing the exponents and vice versa; and Teacher Notes and Elaborations denominators. d. factor polynomials completely. A polynomial is a sum and/or difference of terms. The complete factorization of  Factor polynomials by applying general polynomials has occurred when each factor is a prime polynomial. Pattern recognition can patterns including difference of squares, be used to determine the complete factorization of a polynomial. sum and difference of cubes, and perfect square trinomials. The following steps may be followed when factoring a polynomial:  Factor polynomials completely over the 1. Determine the greatest monomial factor (GCF) as a first step in complete factorization. integers. 2. Check for special patterns  Verify polynomial identities including - Difference of squares [e.g., (4x2 - 25) = (2 x - 5)(2 x + 5) ] the difference of squares, sum and 3 2 difference of cubes, and perfect square - Sum of two cubes [e.g., (x+ 8) = ( x + 2)( x - 2 x + 4) ] trinomials. - Difference of two cubes [e.g., (y3- 125) = ( y - 5)( y 2 + 5 y + 25) ; ( 24x3- 81 = 3(2 x - 3)(4 x 2 + 6 x + 9) ; ( 225x6- 81 y 10 = 9(5 x 3 - 3 y 5 )(5 x 3 + 3 y 5 ) ] - Perfect square trinomials [e.g., (y2- 12 y + 36) = ( y - 6) 2 ] - General trinomials [e.g., (6x2 - 7 x - 5) = (2 x + 1)(3 x - 5) ] 3. If there are four or more terms, grouping should be tried.

3 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS (continued) (continued)

Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations

4 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category (continued) Teacher Notes and Elaborations (continued) Expressions and Operations Key Vocabulary After factoring, multiply the factors to see if the original polynomial results. complex fraction difference of two cubes To divide a polynomial by a binomial either factor or use long division. Factoring and Topic difference of squares simplifying is the preferred method but does not always work. In these cases, long division Expressions and Operations index (indices) is used. perfect square trinomial polynomial Factoring Example: Virginia SOL AII/T.1 radical x2 +8 x + 15 ( x + 5)( x + 3) The student, given rational, radical expression = x+3 ( x + 3) radical, or polynomial expressions radicand will rational expression (x+ 5)( x + 3) = a. add, subtract, multiply, divide, and restricted variable (x + 3) simplify rational algebraic sum of two cubes =(x + 5) expressions; b. add subtract, multiply, divide and simplify radical expressions Long Division Example: containing rational numbers and a2 -4 a - 6 a - 6 6 2 a -6 + variables, and expressions a + 2 a+2 a - 4 a - 6 a + 2 containing rational exponents; -(a2 + 2 a ) c. write radical expressions as expressions containing rational -6a - 6 exponents and vice versa; and -( - 6a - 12) d. factor polynomials completely. +6

A complex fraction is a fraction that has a fraction in its numerator or denominator or in both its numerator and denominator. A rational expression is a polynomial or the quotient of two polynomials. The denominator cannot be 0 (this is called a restricted variable). Rational expressions written as complex fractions can be written as a quotient or product of simple fractions.

Rational expressions must be simplified. A simplified expression meets the following conditions: 1. It has no negative exponents. 2. It has no fractional exponents in the denominator. 3. It is not a complex fraction.

Rational expressions can be added, subtracted, multiplied, and divided.

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Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations 5 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category Teacher Notes and Elaborations (continued) Expressions and Operations When adding or subtracting rational expressions with polynomial factors in denominators, a common denominator must be found. To simplify a rational expression, divide out any common factors of the numerator and denominator.

Topic Example 1: Example 2: Example 3 Expressions and Operations x+1 x - 3 4 3 x2-3 x ( x - 2) 2 + + x3 x x2-16 x 2 + 8 x + 16 x2 -5 x + 6 2x 3(x+ 1) x - 3 4 3 x( x- 3) ( x - 2)( x - 2) Virginia SOL AII/T.1 + + The student, given rational, 3x 3 x (x+ 4)( x - 4) ( x + 4)( x + 4) (x- 2)( x - 3) 2 x radical, or polynomial expressions 3x+ 3 x - 3 4(x+ 4) 3( x - 4) (x - 2) will + 2+ 2 3x 3 x (x+ 4) ( x - 4) ( x + 4) ( x - 4) a. add, subtract, multiply, divide, and 2 simplify rational algebraic 3x+ 3 + x - 3 4x+ 16 3 x - 12 2+ 2 expressions; 3x (x+ 4) ( x - 4) ( x + 4) ( x - 4) b. add subtract, multiply, divide and 4x 7x + 4 simplify radical expressions 2 containing rational numbers and 3x (x+ 4) ( x - 4) variables, and expressions 4

containing rational exponents; 3 c. write radical expressions as

expressions containing rational 1 n exponents and vice versa; and A radical expression is an expression that contains a radical and is in the form a or a n . 5 is called a radical. The number 5 under the d. factor polynomials completely. radical sign is called the radicand. In 3 7 , 3 is called the index.

Radicals with common radicands and common indices are added, subtracted, or multiplied the same way monomials are added, subtracted, or multiplied. The following rules will aid in the multiplication and division of radicals:

Product Property of Radicals – For any real numbers a and b and any integer n, n > 1, 1. if n is even, then nab= n a n b where a and b are both nonnegative, and 2. if n is odd, then nab= n a n b .

an a Quotient Property of Radicals – For any real numbers a and b, where b ≠ 0, and for any integer n, where n > 1, n = , if all roots b n b are defined.

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6 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Expressions and Operations Expressions containing radicals can be operated on and simplified.

A radical expression is simplified when the following conditions are met: Topic - the index, n, has the least value possible; Expressions and Operations - the radicand contains no factors (other than 1) that are nth powers of an integer or polynomial; - the radicand contains no fractions; and - no radicals appear in the denominator. Virginia SOL AII/T.1 The student, given rational, radical, or polynomial expressions m For any nonzero real number b, and any integers m and n, with n > 1, n n mn m except when b < 0 and n is even. will b= b = ( b ) a. add, subtract, multiply, divide, and simplify rational algebraic 17 11 Example: 617 11 6 6 6 5 6 5 2 6 5 5 6 6 expressions; 729q r= 3 q q q r r = 3 q r q r or 3 q r b. add subtract, multiply, divide and simplify radical expressions containing rational numbers and variables, and expressions Rationalizing a denominator is a procedure for transforming a quotient with a radical in the denominator into an expression with no radical containing rational exponents; in the denominator. The following are examples of rationalizing the denominator of radical expressions. c. write radical expressions as x x 骣 3 expressions containing rational Example 1: = 琪 (Multiply by 1.) exponents and vice versa; and 3 3桫 3 d. factor polynomials completely. x 3 = 3 3 x 3 = 3

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Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Expressions and Operations 2 2骣 1- 3 Example 2: = 琪 (Use the conjugate of 1+ 3 to multiply by 1.) Topic 1+ 3 1 + 3桫 1 - 3 Expressions and Operations 2- 2 3 = 1- 3 + 3 - 3 2- 2 3 Virginia SOL AII/T.1 = The student, given rational, -2 radical, or polynomial expressions = -1 + 3 will a. add, subtract, multiply, divide, and 23 2 simplify rational algebraic Example 3: 3 = expressions; 3x 3 3x b. add subtract, multiply, divide and 3 2骣3 32x 2 simplify radical expressions = 琪 3 琪3 2 2 containing rational numbers and 3x 桫 3 x variables, and expressions 3 2 32x 2 containing rational exponents; = c. write radical expressions as 3 3x 32 x 2 expressions containing rational 3 18x2 exponents and vice versa; and = d. factor polynomials completely. 3 33x 3 3 18x2 = 3x

Students need multiple experiences simplifying radical expressions with and without radicals in the denominator.

An equation that is true for all real numbers for which both sides are defined is called an identity. A polynomial identity is two equivalent polynomial expressions. To verify or prove polynomial identities, work with the expressions on each side of the equation until they are the same. The following are examples of proving polynomial identities.

Example 1: Prove that x2- y 2 =( x - y )( x + y ) =x2 + xy - xy - y 2 =x2 - y 2

8 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS (continued)

Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations

9 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category Teacher Notes and Elaborations (continued) Expressions and Operations Example 2: Prove that (a+ b )( a2 - ab + b 2 ) = a 3 + b 3 First multiply each term of a2- ab + b 2 by the term a in the binomial (a + b), and then multiply each term of a2- ab + b 2 by the term Topic Expressions and Operations b in the binomial (a + b). (a+ b )( a2 - ab + b 2 ) = a 3 + b 3 a3- a 2 b + ab 2 + a 2 b - ab 2 + b 3 = Virginia SOL AII/T.1 a3- a 2 b + a 2 b + ab 2 - ab 2 + b 3 = The student, given rational, a3+ b 3 = radical, or polynomial expressions will 2 2 a. add, subtract, multiply, divide, and Example 3: Prove that (a+ b) +( a - b) =2 a2 + 2 b 2 simplify rational algebraic First square each binomial. expressions; 2 2 2 2 b. add subtract, multiply, divide and (a+ b) +( a - b) =2 a + 2 b simplify radical expressions a2+2 ab + b 2 + a 2 - 2 ab + b 2 = containing rational numbers and a2+ a 2 +2 ab - 2 ab + b 2 + b 2 = variables, and expressions 2 2 2a+ 2 b = containing rational exponents; c. write radical expressions as 2 2 2 2 2 2 2 expressions containing rational Example 4: Use polynomial identities to describe numerical relationships such as (x+ y ) = ( x - y ) + (2 xy ) to generate Pythagorean exponents and vice versa; and triples. d. factor polynomials completely. (x2+ y 2 ) 2 = ( x 2 - y 2 ) 2 + (2 xy ) 2 x4+ x 2 y 2 + x 2 y 2 + y 4 = x 4 - x 2 y 2 - x 2 y 2 + y 4 + 4 x 2 y 2 x4+2 x 2 y 2 + y 4 = x 4 - 2 x 2 y 2 + y 4 + 4 x 2 y 2 x4+2 x 2 y 2 + y 4 = x 4 + 4 x 2 y 2 - 2 x 2 y 2 + y 4 x4+2 x 2 y 2 + y 4 = x 4 + 2 x 2 y 2 + y 4

10 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Resources Sample Instructional Strategies and Activities

SOL Reporting Category Text: - Consider the activity "I have: . . . Who has. . . ?" Expressions and Operations Prentice Hall Algebra II, Virginia A sample card might say: Edition,©2012, Charles et al., Pearson Topic Education I have x2 - 4 Expressions and Operations Who has the factors of 6x2 - 7 x - 5 Virginia SOL AII/T.1 PWC Mathematics website http://pwcs.math.schoolfusion.us/ Foundational Objectives The person having a card that says “I have (2x + 1) (3x – 5)” would then read their card. 8.1a Virginia Department of Education website When making a set of cards, be sure that the answer to the question on the last card The student will simplify numerical http://www.doe.virginia.gov/instruction/ma answer will be on the first card. This game can be used for many different types of expressions involving positive thematics/index.shtml problems (i.e., factoring polynomials; and adding, subtracting, multiplying, and dividing exponents, using rational numbers, integers, rational numbers, complex numbers, etc.). Each student must work the order of operations and properties of problems to determine if they have the card with the correct answer. operations with real numbers.  Students will be divided into cooperative groups of four and be given a pair of rational 8.5 expressions to be added, to be subtracted, to be multiplied, and to be divided. They will, The student will in turn, choose an operation, perform it and then explain it to the other group members. a. determine whether a given number As a part of this process, the group members will correct each other's mistakes. is a perfect square; and  Model problems containing roots such as ratios of areas and volumes. b. find the two consecutive whole numbers between which a square  Divide the class into groups of four students. Give each group four problems that root lies. involve radicals. Each of the four operations should be used. Have each group explain A.2 how to do the problems step by step. The groups should make sure that every person in The student will perform operations on the group understands each problem before presenting them. polynomials, including a. applying the laws of exponents to perform operations on expressions; b. adding, subtracting, multiplying, and dividing polynomials; and c. factoring completely first- and second-degree binomials and trinomials in one or two variables. Graphing calculators will be used as a tool for factoring and for confirming algebraic factorizations. A.3 The student will express the square roots and cube roots of whole numbers and the square root of a monomial algebraic expression in simplest radical form.

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Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations SOL Reporting Category The student will use problem solving, Essential Questions Functions and Statistics mathematical communication,  What is the difference between a series and a sequence? mathematical reasoning, connections  What is the difference between arithmetic and geometric sequences and series? and representations to:  What is Sigma notation (Σ)? Topic  Distinguish between a sequence and a  What real-world situations use sequences and series? Functions series.  Generalize patterns in a sequence using Essential Understandings explicit and recursive formulas.  Sequences and series arise from real-world situations. Virginia SOL AII/T.2  Use and interpret the notation: Σ, n, nth  The study of sequences and series is an application of the investigation of patterns. The student will investigate and apply term, and an .  A sequence is a function whose domain is the set of natural numbers. the properties of arithmetic and th  Given the formula, find an (the n  Sequences can be defined explicitly and recursively. geometric sequences and series to solve real-world problems including writing term) for an arithmetic or a geometric Teacher Notes and Elaborations the first n terms, finding the nth term, sequence. A sequence is a mathematical pattern of numbers. A series is an indicated sum of a and evaluating summation formulas.  Given formulas, write the first n terms sequence of terms. Both sequences and series may be finite or infinite. and find the sum, Sn , of the first n Notation will include  and an . terms of an arithmetic or geometric Two types of sequences are arithmetic and geometric. An arithmetic sequence is a sequence series. in which each term, after the first term, is found by repeated addition of a constant, called  Given the formula, find the sum of a the common difference to the previous term. A geometric sequence is a sequence in which convergent infinite series. each term, after the first term, can be found by multiplying the preceding term by a nonzero  Model real-world situations using constant, called the common ratio. The terms between any two nonconsecutive terms of a sequences and series. geometric/arithmetic sequence are called the geometric/arithmetic means.

A series is the expression for the sum of the terms of a sequence. Finite sequences and series Key Vocabulary have terms that can be counted individually from 1 to a final whole number n. Infinite arithmetic sequence sequences and series continue without end. Infinite sequences or series are indicated with common difference ellipsis points (3 dots indicating the missing part of a statement). common ratio Example: Finite sequence 4, 8, 12, 16, 20 convergent Finite series 4 + 8+ 12 + 16 + 20 divergent Infinite sequence 2, 7, 12, 17, … explicit formula Infinite series 2 + 7 + 12 + 17 +… geometric sequence recursive formula sequence series sigma notation (summation notation)

14 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category Teacher Notes and Elaborations (continued)

Functions and Statistics There are formulas that assist in determining elements of the sequence (an ) or sum of a series. The sigma notation or summation notation (Σ) is one way of writing a series. Σ is the Greek letter sigma, the equivalent of the English letter S (for summation). Use limits to indicate how many terms are added. Limits are the least and greatest integral values of n. Topic Example: Functions Use summation notation to write the series 4+ 8 + 12 + ... for 23 terms. 23 (4n ) where 23 is the upper limit (greatest value of n), 1 is the lower limit (least value of n) and the explicit formula for the series is Virginia SOL AII/T.2 n=1 The student will investigate and apply 4n. the properties of arithmetic and geometric sequences and series to solve A recursive formula for a sequence describes how to find the nth term from the terms before it. A recursive formula defines the terms in a real-world problems including writing sequence by relating each term to the ones before it. An explicit formula expresses the nth term in terms of n. th the first n terms, finding the n term, and evaluating summation formulas. Arithmetic Sequence Formulas Recursive Formula Explicit Formula Notation will include  and an . a1 = a given value, an= a n-1 + d an = a1 +( n - 1) d

th In these formulas, an is the n term, a1 is the first term, n is the number of the term, and d is the common difference.

Geometric Sequence Formulas Recursive Formula Explicit Formula n-1 a1= a given value, an= a n-1 r an = a1 r

th In these formulas, an is the n term, a1 is the first term, n is the number of the term, and r is the common ratio.

In some cases, an infinite geometric series can be evaluated. When r < 1, the series is convergent, or gets closer and closer to the sum S. When r 1, the series is divergent, or approaches no limit.

Real-world applications such as fractals, growth, tax, and interest are solved using sequences and series.

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15 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics Sum of a Finite Arithmetic Series

n th The Sn of a finite arithmetic series a1+ a 2 + a 3 +... + an is S=( a + a ) where a1 is the first term, an is the n term, and n is the n2 1 n Topic number of terms. Functions

n th Another formula for the sum of a finite arithmetic series is Sn =[2 a1 + ( n - 1) d ] where a1 is the first term, an is the n term, and n is the Virginia SOL AII/T.2 2 The student will investigate and apply number of terms. the properties of arithmetic and geometric sequences and series to solve real-world problems including writing Sum of a Finite Geometric Series th n the first n terms, finding the n term, a1 (1- r ) The Sn of a finite geometric series a1+ a 2 + a 3 +... + an , r ≠ 1, is S = where a1 is the first term, r is the common ratio, and n is and evaluating summation formulas. n 1- r

Notation will include  and an . the number of terms.

Sum of an Infinite Geometric Series

a1 An infinite geometric series with r < 1 converges to the sum S = where a1 is the first term and r is the common ratio. 1- r

Curriculum Information Resources Sample Instructional Strategies and Activities

SOL Reporting Category Text:  Model real-life situations with arithmetic sequences, such as increasing lengths of rows Functions and Statistics Prentice Hall Algebra II, Virginia in some concert halls. Edition,©2012, Charles et al., Pearson  Model real-life situations with geometric sequences, such as revenues that increase at a Topic Education constant percent. Functions  Model a real-life situation with geometric series, such as total distance traveled by a bouncing ball. Virginia SOL AII/T.2 PWC Mathematics website  Assign groups of students to make up a geometric series and write it in sigma notation. http://pwcs.math.schoolfusion.us/ Groups exchange problems and find the sum of the geometric series. Return the results Foundational Objectives to the original group to check accuracy. Virginia Department of Education website  For a geometric series, students are offered a penny for the first day of the month, to be The student will identify and extend http://www.doe.virginia.gov/instruction/ma doubled everyday there after for the month. Have students determine the amount on the geometric and arithmetic sequences. thematics/index.shtml 30th day and the total for the thirty days. 7.2  Compare and contrast between arithmetic and geometric series: Students are offered two The student will describe and represent, jobs one with a constant annual raise verses a percentage raise. Which job will have the arithmetic and geometric sequences highest salary after twenty years? using variable expressions. 8.14 The student will make connections between any two representations (tables, graphs, words, and rules) of a given relationship.

18 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category The student will use problem solving, Essential Questions Expressions and Operations mathematical communication,  How are real, imaginary, and complex numbers related? mathematical reasoning, connections  What properties extend from the real numbers to the complex numbers? and representations to:  What is the relationship between a complex number and its conjugate? Topic  Recognize that the square root of -1 is  What are the patterns of the powers of i? Expressions and Operations represented as i.  Determine which field properties apply Essential Understandings to the complex number system.  Complex numbers are organized into a hierarchy of subsets. Virginia SOL AII/T.3  Simplify radical expressions containing  A complex number multiplied by its conjugate is a real number. The student will perform operations on negative rational numbers and express  Equations having no real number solutions may have solutions in the set of complex complex numbers, express the results in in a+bi form. numbers. simplest form using patterns of the  Simplify powers of i.  Field properties apply to complex numbers as well as real numbers. powers of i, and identify field  Add, subtract, and multiply complex  All complex numbers can be written in the form a+bi where a and b are real numbers properties that are valid for the complex numbers. and i is -1 . numbers.  Place the following sets of numbers in a hierarchy of subsets; complex, pure Teacher Notes and Elaborations imaginary, real, rational, irrational, Complex numbers are a superset of real numbers and, as a system, contain solutions for integers, whole, and natural. equations that are not solvable over the set of real numbers.  Write a real number in a+bi form.  Write a pure imaginary number in a+bi Complex numbers are organized into a hierarchy of subsets with properties applicable to form. each subset.

Complex Numbers (a + bi) Key Vocabulary complex number conjugate Real Numbers ( b = 0) Imaginary Numbers imaginary number (b ≠ 0) Rational Numbers Irrational Numbers Integers Pure Imaginary Numbers (a = 0) Whole Numbers

Natural Numbers

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19 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Expressions and Operations A complex number (the sum of a real number and an imaginary number) can be written in the form a + bi where a and b are real numbers and i = -1 . An imaginary number (the square root of a negative number) is a complex number (a +bi) where b ≠ 0, and i = -1 . An imaginary number where a = 0 is called a pure imaginary number. Topic Expressions and Operations The set of complex numbers form a field (Properties of closure, associative, commutative, inverse, and identity of addition and multiplication with the distributive property are valid.).

Virginia SOL AII/T.3 Axioms of equality (reflexive, symmetric, transitive, substitution, addition, and multiplication) and axioms of inequality and order The student will perform operations on (trichotomy, transitive, addition, and multiplication) can also be identified using the set of complex numbers. complex numbers, express the results in simplest form using patterns of the Experiences should include identifying the field properties and/or axioms of equality that are valid for complex numbers presented as a powers of i, and identify field single statement or equation (e.g., 2i+ 5 i = (2 + 5) i is an example of the distributive property; if 2i+ 5 i = y , then y=2 i + 5 i is an example properties that are valid for the complex of the symmetric property). Experiences should also include providing justifications of steps when simplifying an expression or solving an numbers. equation.

Operations of addition, subtraction, and multiplication of complex numbers are performed in the same manner as the respective operations for radicals. The conjugate (expressions that differ only in the sign of the second term) of a + bi is a – bi. When dividing complex numbers the conjugate of the denominator must be used to simplify the expression.

Trichotomy is the property of the real line, that given elements x and y, exactly one of the following is true: x< y, y < x , x = y .

Real numbers are not adequate to determine the solutions of an equation such as x2 + 1 = 0 , because there is no real number x that can be squared to produce –1. With the definition of i2 = -1, and i = -1 , the sets of imaginary and complex numbers are created. The standard form of any complex number is a+ bi with a, b 温 .

Operations of addition, subtraction, and multiplication are performed in the same manner as the respective operations for radicals. All answers must be simplified to standard form using the pattern of the powers of i (e.g., i3 = - i , i4 = 1 ).

SOL Reporting Category Text:  Teachers and students, beginning with natural numbers and progressing to complex Expressions and Operations Prentice Hall Algebra II, Virginia numbers, use flow charts and Venn Diagrams, as classroom activities. Edition,©2012, Charles et al., Pearson  Make a set of about 30 cards, one per card, i to a certain power. Pair the students and Topic Education have them write on a white board the simplified result. After completing the cards, Expressions and Operations students should be able to generalize how to simplify i raised to a power.

Virginia SOL AII/T.3 PWC Mathematics website http://pwcs.math.schoolfusion.us/ Foundational Objectives 8.2 Virginia Department of Education website The student will describe orally and in http://www.doe.virginia.gov/instruction/ma writing the relationships between the thematics/index.shtml subsets of the real number system. 8.15c The student will identify properties of operations used to solve an equation. A.4b The student will solve multi-step linear and quadratic equations in two variables, including justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets.

Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations SOL Reporting Category The student will use problem solving, Essential Questions Equations and Inequalities mathematical communication,  What are the characteristics of an absolute value function? mathematical reasoning, connections  Why can an absolute value equation take on more than one solution? Topic and representations to:  What are the methods used to solve quadratic equations? Equations and Inequalities  Solve absolute value equations and  How is the discriminant of a quadratic equation calculated and what is its significance? inequalities algebraically and  How does a graphing calculator confirm algebraic solutions of quadratic equation? graphically.  What is a real-world example for a quadratic, absolute value, radical, or rational Virginia SOL AII/T.4  Write absolute value equations or equation? The student will solve, algebraically inequalities for given graphs.  When working with a real-world problem, how are solution(s) verified? and graphically,  Solve quadratic equations over the set a. absolute value equations and  How is an equation containing rational expressions solved? of complex numbers using an  How is an equation containing radical expressions solved? inequalities; appropriate strategy. b. quadratic equations over the set of  What are the possible number of solutions for each type of equation?  Calculate the discriminant of a  How is an absolute value equation solved? complex numbers; quadratic equation to determine the  How can the solution for an absolute value inequality be described? c. equations containing rational number of real and complex solutions. algebraic expressions; and  Solve equations containing rational d. equations containing radical Essential Understandings algebraic expressions with monomial or expressions.  Equations can be solved in a variety of ways. binomial denominators algebraically  The quadratic formula can be used to solve any quadratic equation. and graphically. Graphing calculators will be used for  A quadratic function whose graph does not intersect the x-axis has roots with imaginary  Solve an equation containing a radical solving and for confirming the components. expression algebraically and algebraic solutions.  The value of the discriminant of a quadratic equation can be used to describe the number graphically. of real and complex solutions.  Verify possible solutions to an equation containing rational or radical  The definition of absolute value (for any real numbers a and b, where b 0 , if a= b , expressions. then a= b or a= - b ) is used in solving absolute value equations and inequalities.  Apply an appropriate equation to solve  Absolute value inequalities can be solved graphically or by using a compound statement. a real-world problem.  Real-world problems can be interpreted, represented, and solved using equations and  Recognize that the quadratic formula inequalities. can be derived by applying the  The process of solving radical or rational equations can lead to extraneous solutions. completion of squares to any quadratic  Set builder notation may be used to represent solution sets of equations and inequalities. equation in standard form. Teacher Notes and Elaborations When solving equations it is important to check possible solutions in the original equation Key Vocabulary as one or more may be an extraneous solution. An extraneous solution is a solution of an absolute value equations and inequalities equation derived from an original equation that is not a solution of the original equation. completing the square Absolute value, radical, and rational equations may have extraneous solutions. compound sentence extraneous solution Absolute value equations and inequalities can be used to model problems dealing with a quadratic equation range of acceptable measurements. radical expression (continued)

23 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS rational expression

24 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category Teacher Notes and Elaborations (continued) Equations and Inequalities The absolute value of a number x is the distance the number is from zero on a number line. When solving absolute value equations (equations that contain absolute values), the expression inside the absolute value symbol can be any real number. An absolute value Topic equation may have two solutions. Equations and Inequalities An absolute value inequality may be solved by transforming the inequality into a compound sentence using the words “and” or “or.” Generally, in an absolute value inequality if it is a “less than” statement it is “and” and if it is a “greater than” statement it is “or”. Virginia SOL AII/T.4 The student will solve, algebraically The sketch of absolute value inequalities and functions presents an opportunity to picture the absolute value relationship. The graphing and graphically, calculator illustrates various transformations and aids in the solution of absolute value equations and inequalities. a. absolute value equations and inequalities; A quadratic equation is a polynomial equation containing a variable to the second degree. The standard form of a quadratic equation is b. quadratic equations over the set of y= ax2 + bx + c , where a, b, and c are real numbers and a 0 . The graph of a quadratic function is called a parabola. Quadratic functions complex numbers; 2 c. equations containing rational can be written in the form y= a( x - h ) + k to facilitate graphing by transformations and finding maxima and minima. algebraic expressions; and d. equations containing radical Quadratic equations can be solved by factoring, graphing, using the quadratic formula, and completing the square. They can have real or expressions. complex solutions. Graphing calculators should be used as a primary tool in solving quadratics and aid in visualizing or confirming solutions. Calculators can also be used to determine a quadratic equation that best fits a data set. Graphing calculators will be used for solving and for confirming the -b� b2 4 ac algebraic solutions. The quadratic formula is x = . 2a

The discriminant of a quadratic equation, b2 - 4 ac , provides information about the nature and number of roots of the equation. The following table summarizes all possibilities.

Value of Discriminant a Nature of Roots Nature of Graph b2 - 4 ac perfect square? > 0 Yes 2 real, rational Intersects x-axis twice > 0 No 2 real, irrational Intersects x-axis twice < 0 --- 2 imaginary Doesn’t intersect x-axis = 0 --- 1 real Intersects x-axis once

Solutions to quadratic equations may be represented in different formats. 镲禳1-i 2 1 + i 2 1 i 2 1 i 2 Example: Solve and name both roots for 3x2 - 2 x + 1 = 0 . Solution: 睚 , or x= or x = 铪镲 3 3 3 3 3 (continued)

Curriculum Information Essential Questions and Understandings

25 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Teacher Notes and Elaborations

SOL Reporting Category Teacher Notes and Elaborations (continued) Equations and Inequalities A rational expression is written as a quotient of polynomials in simplest form; the divisor is never zero. Equations containing rational expressions can be solved by finding the least common denominator. Topic Equations and Inequalities A radical expression is an expression that contains a square root. Equations with variables in the radicals are called radical equations. To solve this type of equation, isolate the radical and then raise each side of the equation to a power that equals the root of the equation. For example, if the root index is 3, then each side of the equation needs to be raised to the 3rd power. If the root index is 4, then each side is Virginia SOL AII/T.4 raised to the 4th power. The student will solve, algebraically and graphically, A solution of an equation makes the equation true for a given value or set of values. Equations containing rational expressions can be a. absolute value equations and solved in a variety of ways. inequalities; b. quadratic equations over the set of The solution of an equation in one variable can be found by graphing each member of the equation separately and noting the x-coordinate complex numbers; of the point of intersection. c. equations containing rational algebraic expressions; and Set builder notation is used to represent solutions. For example, if the solution is y = 10 then in set notation the answer is written d. equations containing radical {y: y = 10}. expressions. Graphing calculators are powerful tools for solving and confirming algebraic solutions. Practical problems can be interpreted, represented, Graphing calculators will be used for and solved using equations. solving and for confirming the algebraic solutions. Completing the square is the process of finding the last term of a perfect square trinomial. If one side of an equation is not a perfect square trinomial, it can be converted into a perfect square trinomial by rewriting the constant term. 2 2 2 骣b 骣 b Use the relationship x+ bx +琪 = 琪 x + to find the term that will complete the square. 桫2 桫 2

Example: Solve by completing the square. x2 +10 x - 9 = 0 2 2 骣10 骣b 琪 = 25 Find 琪桫2 桫2 x2 +10 x = 9 Rewrite so all terms containing x are on one side. x2 +10 x + 25 = 9 + 25 Complete the square by adding 25 to each side. (x + 5)2 = 34 Factor the perfect square trinomial. x +5 = 34 Find the square root of each side. x = -5 34 Solve for x.

Conjugate is an adjective used to describe two items having features in common but inverses or opposites in some aspect. Number pairs of the form a+ b and a - b are conjugates. Complex solutions occur in pairs (conjugates)

SOL Reporting Category Text:  Working in cooperative groups, students discover, by point plotting, the "and/or Equations and Inequalities Prentice Hall Algebra II, Virginia concept" of absolute value. Edition,©2012, Charles et al., Pearson Topic  Students, working in cooperative groups of four, write a type of equation (absolute Equations and Inequalities Education value, quadratic, rational, and radical). Collect and redistribute to different groups. Next, students solve, discuss, and graph the solution. Choose spokesperson to present findings Virginia SOL AII/T.4 to the class. PWC Mathematics website  Teach the concept of absolute value, beginning with simplest equation and progressing Foundational Objectives http://pwcs.math.schoolfusion.us/ systematically until complex forms are presented. 8.15 The student will  As a cooperative learning activity, students will solve the quadratic equation, a. solve multi-step linear equations in Virginia Department of Education website ax2 + bx + c = 0 , by using the completion of the square method to derive the quadratic http://www.doe.virginia.gov/instruction/ma one variable on one and two sides of formula. Students, working in cooperative groups, solve x2 -5 x + 6 = 0 in three the equation; thematics/index.shtml different ways. Compare and contrast the methods and use the graphing calculator to b. solve two-step linear inequalities verify the roots of the equation. and graph the results on a number  Consider the activity "I have …Who has?" Last card answer will be on the first card. line; and This game can be used for many different types of equations. Each student must work A.2 The student will perform the problems to determine if they have the card with the correct answer. operations on polynomials, including  Working in cooperative groups, ask students to discuss an appropriate method for a. applying the laws of exponents to solving equations involving two radicals. Students will verbalize each step needed and perform operations on expressions; discuss the proper steps needed. b. adding, subtracting, multiplying, and dividing polynomials; and c. factoring completely first- and second-degree binomials and trinomials in one or two variables. A.3 The student will express the square roots and cube roots of whole numbers and the square root of a monomial algebraic expression in simplest radical form. A.4 The student will solve multi-step linear and quadratic equations in two variables, including c. solving quadratic equations algebraically and graphically; d. solving multi-step linear equations Foundational Objectives (continued) algebraically and graphically; and A.5a The student will solve multi-step f. solving real-world problems linear inequalities in two variables, involving equations and systems of including solving multi-step linear equations. inequalities algebraically and graphically.

28 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category The student will use problem solving, Essential Questions Equations and Inequalities mathematical communication,  What is a linear function? In what form(s) can it be written? mathematical reasoning, connections  What is a quadratic function? In what form(s) can it be written? and representations to:  How does a graphing calculator confirm algebraic solutions of quadratic functions? Topic  Predict the number of solutions to a  What is a real-world example of a non-linear system of equations? Equations and Inequalities nonlinear system of two equations.  What are the different ways that the graph of a line and a quadratic can intersect?  Solve a linear-quadratic system of two  What are the different ways that the graphs of two quadratics can intersect? equations algebraically and graphically. Virginia SOL AII/T.5  Solve a quadratic-quadratic system of Essential Understandings The student will solve nonlinear two equations algebraically and  Solutions of a nonlinear system of equations are numerical values that satisfy every systems of equations, including linear- graphically. equation in the system. quadratic and quadratic-quadratic,  The coordinates of points of intersection in any system of equations are solutions to the algebraically and graphically. Graphing system. calculators will be used as a tool to Key Vocabulary  Real-world problems can be interpreted, represented, and solved using systems of visualize graphs and predict the number linear-quadratic equations equations. of solutions. nonlinear systems of equations quadratic-quadratic equations Teacher Notes and Elaborations An equation in which one or more terms have a variable of degree 2 or higher is called a nonlinear equation. A nonlinear system of equations contains at least one nonlinear equation.

A system of equations is a set of two or more equations that use the same variable. If the graph of each equation in a system of two variables is a line, then the system is a linear system. Nonlinear systems of equations can be classified as linear-quadratic or quadratic- quadratic. Both systems can be solved algebraically and graphically.

Solutions of a nonlinear system of equations are values that satisfy every equation in the system.

Points of intersection in nonlinear systems are solutions to the system.

Each point of intersection of the graphs of the equations in a system represents a real solution of the system. A linear-quadratic system of equations has two solutions, one solution or no solution. A quadratic-quadratic system of equations has four solutions, three solutions, two solutions, one solution, or no solution.

Graphing calculators can be used to visualize a nonlinear system of two equations and predict the number of solutions.

(continued)

SOL Reporting Category Text:  Working in cooperative groups, ask students to discuss an appropriate method for Equations and Inequalities Prentice Hall Algebra II, Virginia solving a given system. Students will verbalize each step needed and discuss the proper Edition,©2012, Charles et al., Pearson steps needed. Topic Education  In order to show students the relationship between the different ways in which two Equations and Inequalities second degree equations can intersect and the number of real solutions, have students draw a circle on graph paper and a parabola on patty paper. Tell students such a system Virginia SOL AII/T.5 PWC Mathematics website can have zero, one, two, three, or four real solutions. Instruct them to hold the circle http://pwcs.math.schoolfusion.us/ stationary and move the parabola to illustrate each number of possible solutions. Foundational Objectives A.4e, f The student will solve multi- Virginia Department of Education website step linear and quadratic equations in http://www.doe.virginia.gov/instruction/ma two variables, including thematics/index.shtml e. solving systems of two linear equations in two variables algebraically and graphically; and f. solving real-world problems involving equations and systems of equations. A.5 The student will solve multi-step linear inequalities in two variables, including a. solving multi-step linear inequalities algebraically and graphically; b. justifying steps used in solving inequalities, using axioms of inequality and properties of order that are valid for the set of real numbers and its subsets; c. solving real-world problems involving inequalities; and d. solving systems of inequalities. A.6 The student will graph linear equations and linear inequalities in two Foundational Objectives (continued) variables, including b. writing the equation of a line when a. determining the slope of a line when given the graph of the line, two points given an equation of the line, the on the line, or the slope and a point on graph of the line, or two points on the line. the line. Slope will be described as G.12 The student, given the coordinates of rate of change and will be positive, the center of a circle and a point on the negative, zero, or undefined; and circle, will write the equation of the circle. (continued)

Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations SOL Reporting Category The student will use problem solving, Essential Questions Functions and Statistics mathematical communication,  What are different representations of functions? mathematical reasoning, connections  What is the transformational approach to graphing? Topic and representations to:  What is the connection between the algebraic and graphical representation of a Functions  Recognize graphs of parent functions. transformation?  Given a transformation of a parent  What is the relationship between exponential and logarithmic functions? function, identify the graph of the  How can the calculator be used to investigate these functions (absolute value, square Virginia SOL AII/T.6 transformed function. root, cube root, rational, polynomial, exponential, and logarithmic)? The student will recognize the general  Given the equation and using a shape of function (absolute value, transformational approach, graph a Essential Understandings square root, cube root, rational, function.  The graphs/equations for a family of functions can be determined using a polynomial, exponential, and  Given the graph of a function, identify transformational approach. logarithmic) families and will convert the parent function.  A parent graph is an anchor graph from which other graphs are derived with between graphic and symbolic forms of  Given the graph of a function, identify transformations. functions. A transformational approach the transformations that map the  Transformations of graphs include translations, reflections, and dilations. to graphing will be employed. Graphing preimage to the image in order to calculators will be used as a tool to determine the equation of the image. Teacher Notes and Elaborations investigate the shapes and behaviors of  Using a transformational approach, A function is a correspondence in which values of one variable determine the values of these functions. write the equation of a function given another. It is a rule of correspondence between two sets such that there is a unique element its graph. in one set assigned to each element in the other.

A polynomial function is a function of one variable whose exponents are natural numbers. Key Vocabulary The degree of a polynomial function determines its graphing behavior. A polynomial absolute value function function is linear, quadratic, cubic, quartic, etc., according to its degree, 1, 2, 3, 4, …, cube root function respectively. The degree of the polynomial will help determine the graph of the polynomial cubic function function. dilation exponential function The graphs and/or equations for a family of functions can be determined using a family of functions transformational approach. A family of functions is a group of functions with common logarithmic function characteristics. A parent function is the simplest function with these characteristics. A parabola parent function and one or more transformations make up a family of functions. Shapes and parent function behavior of graphs of polynomials can be determined by analyzing transformations of polynomial function parent functions. quartic function rational function The following is one example of a parent function and family of functions. reflection Parent Function Family of Functions square root function f( x ) = x2 f( x )= 3( x - 4)2 + 5 transformations of graphs 2 translation (vertical and horizontal) f( x )= - 2( x + 1) - 7

32 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics A quadratic equation is a polynomial equation containing a variable to the second degree. The standard form of a quadratic equation is y= ax2 + bx + c , where a, b, and c are real numbers and a 0 . The graph of a quadratic function is called a parabola. Parabolas have an Topic axis of symmetry, a line that divides the parabola into two parts that are mirror images of each other. The vertex of a parabola is either the Functions lowest point on the graph or the highest point on the graph. Quadratic functions can be written in the form y= a( x - h )2 + k to facilitate graphing by transformations and finding maxima and minima. The chart below summarizes the characteristics of the graph of Virginia SOL AII/T.6 y= a( x - h )2 + k . The student will recognize the general shape of function (absolute value, y= a( x - h )2 + k a is positive a is negative square root, cube root, rational, polynomial, exponential, and Vertex (h, k) (h, k) logarithmic) families and will convert Axis of Symmetry x= h x= h between graphic and symbolic forms of Direction of Opening upward (minima) downward (maxima) functions. A transformational approach As the value of the absolute value of a increases, to graphing will be employed. Graphing the graph of y= a( x - h )2 + k narrows. calculators will be used as a tool to investigate the shapes and behaviors of An exponential function is a function of the form y= ax , where a is a positive constant not equal to one. Population growth and viral these functions. growth are among examples of exponential functions. Logarithmic functions are inverses of exponential functions.

A function is continuous if the graph can be drawn without lifting the pencil from the paper. A graph is discontinuous if it has jumps, breaks, or holes in it.

P( x ) A rational function f(x) can be written as f( x ) = , where P(x) and Q(x) are polynomial functions and Q(x) ≠ 0. Q( x )

A radical function is a function that is defined by a radical expression. The square root of a number, x , is a number that when multiplied by itself produces the given number, x. Since the domain of a function is the set of all real number values of x for which a function , f, is defined, the domain of the square root function, f( x ) = x , does not include negative numbers. A function f is called a cube root function if f( x ) = 3 x for all real numbers.

A u-turn in a graph occurs when the function values change from increasing to decreasing or vice versa and indicates either a local maximum or a local minimum. The number of u-turns in a graph is no greater than 1 less than the degree of the function.

(continued)

33 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics A cubic function is of degree three and a quartic function is of degree four.

Topic Absolute value function is a function described by f( x ) = x . The graph of the absolute value function has a characteristic V-shape. Functions Experiences with shapes and transformations of functions should include the following: Virginia SOL AII/T.6 Quadratic Cubic Quartic Square Root The student will recognize the general 2 3 4 3 shape of function (absolute value, f( x )= 2 x - 3 f( x )= 2 x - 3 x - 1 f( x )= x - 2 x + 2 x = 1 y= x square root, cube root, rational, y y y y polynomial, exponential, and 4 3 3 4 logarithmic) families and will convert 3 2 2 between graphic and symbolic forms of 2 1 1 2 functions. A transformational approach 1 to graphing will be employed. Graphing x x x -4 -3 -2 -1 1 2 3 4 -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 x -1 calculators will be used as a tool to -1 -1 2 4 6 investigate the shapes and behaviors of -2 -2 -2 these functions. -3 -2 -3 -3 -4

Rational Exponential Logarithmic Absolute Value 4 f( x )= - 1.5 f( x )= 2x f( x )= log x f( x ) = x x - 3 2 y y y y 8 8 4 8

6 6 3 6

4 4 2 4

2 2 1 2 x x x x -8 -6 -4 -2 2 4 6 8 -8 -6 -4 -2 2 4 6 8 -4 -3 -2 -1 1 2 3 4 -8 -6 -4 -2 2 4 6 8 -2 -2 -1 -2

-4 -4 -2 -4

-6 -6 -3 -6

-8 -8 -4 -8

(continued)

Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics A transformation of a function is an alteration of the function rule that results in an alteration of its graph.

Topic If y= f( x ) , then y= f( x ) + k gives a vertical translation of the graph of f. The translation is k units up for k > 0 and k units down for Functions k < 0. If y= f( x ) , then y= f( x - h ) gives a horizontal translation of the graph of f. The translation is h units to the right for h > 0 and h units to the left for h < 0. Virginia SOL AII/T.6 The student will recognize the general If y= f( x ) , then y= - f( x ) gives a reflection of the graph of f across the x-axis. If y= f( x ) , then y= f( - x ) gives a reflection of the shape of function (absolute value, square root, cube root, rational, graph of f across the y-axis. polynomial, exponential, and logarithmic) families and will convert A curve’s shape is determined by the rule or relation which defines it. Very often multiplication or division is involved. A line’s slope and between graphic and symbolic forms of a parabola’s stretch or compression is attributed to dilation – expansion, growing or shrinking, multiplication. If y= f( x ) , then y= af( x ) functions. A transformational approach gives a vertical stretch or vertical compression of the graph of f. If a > 1, the graph is stretched vertically by a factor of a. If 0 < a < 1, the to graphing will be employed. Graphing graph is compressed vertically by a factor of a. If y= f( x ) , then y= f( bx ) gives a horizontal stretch or horizontal compression of the calculators will be used as a tool to 1 1 investigate the shapes and behaviors of graph of f. If b > 1, the graph is compressed horizontally by a factor of . If 0 < b < 1, the graph is stretched horizontally by a factor of . b b these functions.

36 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category Text:  The students are divided into small groups and given graph representations of the Functions and Statistics Prentice Hall Algebra II, Virginia following functions: absolute value, square root, cube root, rational, polynomial, Edition,©2012, Charles et al., Pearson Topic exponential, and logarithmic. They must identify the graph representations and explain Functions Education the transformations from the basic graph of each function. Each group must verify their answers on the graphing calculator. Virginia SOL AII/T.6  Use compound interest, population growth, and/or rate of decay as examples of PWC Mathematics website exponential functions. Foundational Objectives http://pwcs.math.schoolfusion.us/ A.4c The student will solve multi-step linear Virginia Department of Education website and quadratic equations in two http://www.doe.virginia.gov/instruction/ma variables, including solving quadratic thematics/index.shtml equations algebraically and graphically. A.6 The student will graph linear equations and linear inequalities in two variables, including a. determining the slope of a line when given an equation of the line, the graph of the line, or two points on the line. Slope will be described as rate of change and will be positive, negative, zero, or undefined; and b. writing the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line.

38 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations SOL Reporting Category The student will use problem solving, Essential Questions Functions and Statistics mathematical communication,  What is a function? mathematical reasoning, connections  What is the relationship between domain and range? and representations to:  What is the relationship between a function and its inverse? Topic  Identify the domain, range, zeros, and  What is a zero of a function? Functions intercepts of a function presented  What operations can be performed on functions? algebraically or graphically.  What is meant by composition of functions?  Describe restricted/discontinuous  What is the relationship between the degree of a function and the graph of a function? Virginia SOL AII/T.7 domains and ranges. The student will investigate and analyze  What is the relationship between exponential and logarithmic functions?  Given the graph of a function, identify functions algebraically and graphically.  How can the calculator be used to investigate the shape and behavior of polynomial, intervals on which the function is Key concepts include exponential, and logarithmic functions? increasing and decreasing. a. domain and range, including limited  How are the x- and y-intercepts determined?  Find the equations of vertical and and discontinuous domains and  What is an asymptote? horizontal asymptotes of functions. ranges;  What role do asymptotes have in graphing functions?  Describe the end behavior of a function. b. zeros;  What is meant by the end behavior of a function?  Find the inverse of a function. c. x-and y-intercepts;  How can a hole in the graph of a function be determined?  Graph the inverse of a function as a d. intervals in which a function is  What is meant by the turning points of a function and how are they found? increasing or decreasing; reflection across the line y = x. e. asymptotes;  Investigate exponential and logarithmic f. end behavior; functions using the graphing calculator. Essential Understandings g. inverse of a function; and  Convert between logarithmic and  Functions may be used to model real-world situations. exponential forms of an equation with h. composition of multiple functions.  The domain and range of a function may be restricted algebraically or by the real-world bases consisting of natural numbers. situation modeled by the function.  Find the composition of two functions. Graphing calculators will be used as a  A function can be described on an interval as increasing, decreasing, or constant.  Use composition of functions to verify tool to assist in investigation of  Asymptotes may describe both local and global behavior of functions. functions. two functions are inverses.  End behavior describes a function as x approaches positive and negative infinity.  A zero of a function is a value of x that makes f(x) equal zero.  If (a, b) is an element of a function, then (b, a) is an element of the inverse of the function. x  Exponential (y = a ) and logarithmic (y = logax) functions are inverses of each other.  Functions can be combined using composition of functions.

Teacher Notes and Elaborations Functions describe the relationship between two variables. A function is continuous if the graph can be drawn without lifting the pencil from the paper. A graph is discontinuous if it has jumps, breaks, or holes in it. Each function, whether continuous or discontinuous, has a distinct domain, range, zero(s), y-intercept, and inverse.

(continued) (continued) (continued)

Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations SOL Reporting Category (continued) Teacher Notes and Elaborations (continued) Functions and Statistics Key Vocabulary The domain is the set of all possible values for the first coordinate of a function. The range asymptotes of a function is the set of all possible values for the second coordinate of a function. composition of functions Topic continuous The x-intercept is the x-coordinate of the point where the graph crosses the x-axis. The Functions decreasing y-intercept is the y-coordinate of the point where the graph crosses the y-axis. discontinuous domain An asymptote is a line that a graph approaches as x or y increases in absolute value. An Virginia SOL AII/T.7 end behavior asymptote of a curve is a line such that the distance between the curve and the line The student will investigate and increasing approaches zero as it tends to infinity. analyze functions algebraically and intercepts (x and y) graphically. Key concepts include inverse of a function The procedure for asymptotes is the following: a. domain and range, including limited polynomial function - set the denominator equal to zero and solve and discontinuous domains and range the zeroes (if any) are the vertical asymptotes, everything else is the domain; ranges; strictly decreasing - compare the degrees of the numerator and the denominator b. zeros; strictly increasing if the degrees are the same, then the horizontal asymptote is at c. x-and y-intercepts; zero(s) numerator 's leading coefficient d. intervals in which a function is y = , denominator 's leading coefficient increasing or decreasing; e. asymptotes; if the denominator's degree is greater (by any margin), then the horizontal f. end behavior; asymptote is at y = 0 (the x-axis). g. inverse of a function; and - Extension: If the numerator's degree is greater (by a margin of 1), then a slant h. composition of multiple functions. asymptote exists.

Graphing calculators will be used as a If x- b is a factor of the numerator and the denominator of a rational function, then there is tool to assist in investigation of a hole in the graph of the function when x= b unless x= b is a vertical asymptote. functions. The domain of every polynomial function is the set of all real numbers. As a result, the graph of a polynomial function extends infinitely. What happens to a polynomial function as its domain values get very small and very large is called the end behavior of a polynomial function.

Examples such as this should be included: x2 +3 x + 1 : Given the function: f( x ) = , determine the domain and range, find the 4x2 - 9 asymptotes, and determine the end behavior.

Experiences determining domain and range, finding asymptotes, and determining end behavior should include logarithmic and exponential functions.

(continued) 40 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS

Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics A polynomial function is a function of one variable whose exponents are natural numbers. The graph of a polynomial function can be generated by a table or a transformation of the parent function. The degree of a polynomial function determines its graphing behavior. A polynomial is linear, quadratic, cubic, quartic, etc., according to its degree, 1, 2, 3, 4, respectively. The degree of the polynomial will help Topic determine the graph of a polynomial function. Functions n n-1 If a polynomial function is written in standard form, f( x )= an x + a n-1 x + ... + a 1 x + a 0 , the leading coefficient is an . The leading

Virginia SOL AII/T.7 coefficient is the coefficient of the term of greatest degree in the polynomial. an and n determine the end behavior of the graph of any The student will investigate and polynomial function. analyze functions algebraically and - When the degree of the function is odd and the leading coefficient is positive, the graph falls on the left and rises on the right. graphically. Key concepts include - When the degree of the function is odd and the leading coefficient is negative, the graph rises on the left and falls on the right. a. domain and range, including limited - When the degree of the function is even and the leading coefficient is positive, the graph rises on the left and on the right. and discontinuous domains and - When the degree of the function is even and the leading coefficient is negative, the graph falls on the left and the right. ranges; b. zeros; The points on the graph of a polynomial function that correspond to local maxima and local minima are called turning points. c. x-and y-intercepts; d. intervals in which a function is Functions can be combined through addition, subtraction, multiplication, division, and composition. increasing or decreasing; e. asymptotes; The inverse of a function consisting of the ordered pairs (x, y) is the set of all ordered pairs (y, x). The domain of the inverse is the range of f. end behavior; the original relation. The range of the inverse is the domain of the original relation. g. inverse of a function; and h. composition of multiple functions. Examples of finding the inverse of a function such as the following should be included: 3 3 -1 x + 6 Graphing calculators will be used as a Example 1: Given f( x )= 8 x - 6 , the inverse is f( x ) = . 2 tool to assist in investigation of -1 2 functions. Example 2: Given f( x )= x + 3 where x � 3 , the inverse is f( x )= x - 3 where x 0 .

Graphs of functions that are inverses of each other are reflections across the line y = x.

Graphing calculators are used to assist in the investigation of functions.

Composition of functions refers to the forming of a new function h (the composite function) from given functions g and f by the rule: h( x )= g ( f ( x )) or h( x )= ( g° f )( x ) for all x in the domain of f for which f(x) is in the domain of g. This function is read as “g of f”. The order in which functions are combined is important.

The composition of a function and its inverse is the identity function.

Exponential and logarithmic functions, inverses of each other, are either strictly increasing or strictly decreasing. Exponential and logarithmic functions have asymptotes.

42 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics A function is increasing on an interval if its graph always rises as it moves from left to right over the interval. It is decreasing on an interval if its graph always falls as it moves from left to right over the interval. If a function is strictly increasing or strictly decreasing no flatness is allowed. Topic Functions For a function y= f( x ) :

- When x1< x 2, then f ( x 1 ) f ( x 2 ) the function is increasing.

Virginia SOL AII/T.7 - When x1< x 2, then f ( x 1 ) < f ( x 2 ) the function is strictly increasing. The student will investigate and analyze - When x1> x 2, then f ( x 1 ) f ( x 2 ) the function is decreasing. functions algebraically and graphically. x> x, then f ( x ) > f ( x ) Key concepts include - When 1 2 1 2 the function is strictly decreasing. a. domain and range, including limited and discontinuous domains and ranges; A function is constant on an interval if its graph is horizontal over the interval. For any x1 and x 2 in the interval, where b. zeros; x1< x 2, then f ( x 1 ) = f ( x 2 ) c. x-and y-intercepts; d. intervals in which a function is increasing or decreasing; e. asymptotes; f. end behavior; Example: g. inverse of a function; and h. composition of multiple functions. y 9 8 Graphing calculators will be used as a tool 7 6 to assist in investigation of functions. 5 4 3 2 1 x

-9 -8 -7 -6 -5 -4 -3 -2 -1-1 1 2 3 4 5 6 7 8 9 -2 -3 -4 -5 -6 -7 -8 -9

This function is constant on the interval [-5, 2], decreasing on the interval (2, 3), and increasing on the interval [3, ∞).

SOL Reporting Category Text:  Students, working in small groups, will be given several selective polynomial functions. Functions and Statistics Prentice Hall Algebra II, Virginia They will use the graphing calculator to categorize these functions. Next, each group Edition,©2012, Charles et al., Pearson will draw conclusions about the general shape, end behavior, zeros, and y-intercept of Topic Education the functions. Functions  Students, working in small groups, will be given several selective logarithmic and exponential functions. They will use the graphing calculator to categorize these Virginia SOL AII/T.7 PWC Mathematics website functions. Next, each group will draw conclusions about the general shape and end http://pwcs.math.schoolfusion.us/ behavior of the functions. Foundational Objectives  On cards write 20 functions beginning with a(x), b(x), c(x) …. Mix the cards and select A.4 Virginia Department of Education website The student will solve multi-step linear two at random. Have students write the composition of the two functions [e.g., a(d(x))]. http://www.doe.virginia.gov/instruction/ma Using the same functions have students reverse the order [e.g., (d(a(x))]. Is composition and quadratic equations in two thematics/index.shtml variables, including of functions commutative? Use the same cards and have students perform the four c. solving quadratic equations operations. algebraically and graphically; d. solving multi-step linear equations algebraically and graphically; A.7 The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including a. determining whether a relation is a function; b. domain and range; c. zeros of a function; d. x- and y-intercepts; e. finding the values of a function for elements in its domain; and f. making connections between and among multiple representations of functions including concrete, verbal, numeric, graphic, and algebraic.

Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations SOL Reporting Category The student will use problem solving, Essential Questions Functions and Statistics mathematical communication,  What is a polynomial function? mathematical reasoning, connections  What theorems assist in investigating the behavior of polynomial functions? and representations to:  What is the relationship between the solution of an equation, zeros of a function, x- Topic  Describe the relationships among intercepts of a graph, and factors of a polynomial expression? Functions solutions of an equation, zeros of a  What is meant by the multiplicity of a zero and how does that affect the graph of the function, x-intercepts of a graph, and function? factors of a polynomial expression. Virginia SOL AII/T.8  Define a polynomial function, given its Essential Understandings The student will investigate and zeros. describe the relationships among  The Fundamental Theorem of Algebra states that, including complex and repeated  Determine a factored form of a solutions, an nth degree polynomial equation has exactly n roots (solutions). solutions of an equation, zeros of a polynomial expression from the function, x-intercepts of a graph, and  The following statements are equivalent: x-intercepts of the graph of its factors of a polynomial expression. - k is a zero of the polynomial function, f ; corresponding function. - (x – k) is a factor of f( x ) ;  For a function, identify zeros of multiplicity greater than 1 and describe - k is a solution of the polynomial equation f( x )= 0 ; the effect of those zeros on the graph of - k is an x-intercept for the graph of y= f( x ) ; and the function. - k is a root of the polynomial function f( x ) .  Given a polynomial equation, determine the number of real solutions Teacher Notes and Elaborations and non-real solutions. A polynomial function is a function of one variable whose exponents are natural numbers. The graph of a polynomial function can be generated by a table of values. The degree of a polynomial function determines its graphing behavior. Key Vocabulary factor A factor is a number or expression that is multiplied by one or more other numbers or multiplicity expressions to yield a product. A factor of a polynomial is a polynomial by which a given root polynomial is divisible. solution x-intercept A root is a solution to an equation. A solution is a value that can replace the variable that zero of a function makes an equation or inequality true.

The x-intercept is the x-coordinate of the point where the graph crosses the x-axis.

The zero of a function is any number x such that f( x )= 0 . The real zeros of a polynomial function correspond to the x-intercepts of the graph of the function.

47 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics The relationship among the zeros of the polynomial function, the factors of the polynomial, and the solutions to the accompanying polynomial equation should be demonstrated graphically and algebraically.

Topic If a linear factor of a polynomial is repeated, then the zero is repeated. A repeated zero is called a multiple zero. A multiple zero has a Functions multiplicity equal to the number of times the zero occurs. Multiplicity is the number of times that a factor is repeated in the factorization of a polynomial expression.

Virginia SOL AII/T.8 Equivalent Statements about Polynomials The student will investigate and - (x + 4) is a factor of x2 +3 x - 4 describe the relationships among 2 solutions of an equation, zeros of a - -4 is a solution of x+3 x - 4 = 0 function, x-intercepts of a graph, and - -4 is an x-intercept of the graph of f( x )= x2 + 3 x - 4 factors of a polynomial expression. - -4 is a zero of f( x )= x2 + 3 x - 4 - -4 is a root of f( x )= x2 + 3 x - 4

Theorems that assist in explaining the relationships are: Factor Theorem, Remainder Theorem, Fundamental Theorem of Algebra, Rational Root Theorem, Complex Conjugates Theorem, and the Zero Product Property. Division may also be used to obtain a rational root.

- Factor Theorem: The expression s – a is a linear factor of a polynomial if and only if the value a is a zero or the related polynomial function.

- Remainder Theorem: If a polynomial P(x) of degree n ≥ 1 is divided by (x – a). where a is a constant, then the remainder is P(a).

p Rational Root Theorem: If is in simplest form and is a rational root of the polynomial equation with integer coefficients, the p - q must be a factor of the constant and q must be a factor of the leading coefficient.

- Complex Conjugates Theorem: If the imaginary number a + bi is a root of a polynomial equation with real coefficients, then the conjugate a – bi also is a root.

- Zero Product Property: If ab = 0, a = 0 and/or b = 0.

Synthetic division, a shorten form of long division with polynomials, may be used to obtain a rational root. In synthetic division variables and exponents are not written.

Synthetic division can be used to divide a polynomial only by a linear binomial of the form x- r . When dividing by nonlinear divisors, long division is used.

48 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category Text:  The graphing calculator should be integrated throughout the study of polynomials for Functions and Statistics Prentice Hall Algebra II, Virginia predicting solutions, determining the reasonableness of solutions, and exploring the Edition,©2012, Charles et al., Pearson behavior of polynomials. Topic Education Functions

Virginia SOL AII/T.8 PWC Mathematics website http://pwcs.math.schoolfusion.us/ Foundational Objectives A.2c Virginia Department of Education website The student will perform operations on http://www.doe.virginia.gov/instruction/ma polynomials, including factoring thematics/index.shtml completely first- and second-degree binomials and trinomials in one or two variables. A.4c The student will solve multi-step linear and quadratic equations in two variables, including solving quadratic equations algebraically and graphically. A.7 The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including c. zeros of a function; d. x- and y-intercepts; and e. finding the values of a function for elements in its domain.

50 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations SOL Reporting Category The student will use problem solving, Essential Questions Functions and Statistics mathematical communication,  How do various algebraic equations fit real world data? mathematical reasoning, connections  How can the curve-of-best-fit help predict trends of data? and representations to:  How are the equations-of-best-fit determined on a graphing calculator? Topic  Collect and analyze data. Statistics  Investigate scatterplots to determine if Essential Understandings patterns exist and then identify the  Data and scatterplots may indicate patterns that can be modeled with an algebraic patterns. equation. Virginia SOL AII/T.9  Find an equation for the curve of best The student will collect and analyze  Graphing calculators can be used to collect, organize, picture, and create an algebraic fit for data using a graphing calculator. model of the data. data, determine the equation of the Models will include polynomial, n n-1 curve of best fit, make predictions, and  Data that fit polynomial ( f( x )= an x + a n-1 x + ... + a 1 x + a 0 , where n is a nonnegative exponential, and logarithmic functions. solve real-world problems using x  Make predictions using data, integer, and the coefficients are real numbers), exponential ( y= b ), and logarithmic ( mathematical models. Mathematical scatterplots, or the equation of the curve y= log x ) models arise from real-world situations. models will include polynomial, b of best fit. exponential, and logarithmic functions.  Given a set of data, determine the Teacher Notes and Elaborations model that would best describe the data. In Algebra I objective 11, students collected and analyzed data and determined the equation of the curve of best fit in order to make predictions, and solve real-world problems using models which included linear and quadratic functions. Key Vocabulary curve of best fit A mathematical model usually describes a system by a set of variables and a set of mathematical model equations that establish relationships between the variables. scatterplot Data that fit polynomial, exponential, and logarithmic models arise from practical situations. The analyzing of data to determine a curve of best fit has numerous real-world applications such as oceanography, business, economics, and agriculture.

A scatterplot visually shows the nature of a relationship and both its shape and dispersion. A curve of best fit may be drawn to show the approximate relationship formed by the plotted points.

The use of scatterplots on a graphing calculator will determine if the relationship is polynomial, exponential, or logarithmic.

Graphing calculators can be used to collect, organize, picture and create an algebraic model of the data.

SOL Reporting Category Text:  Collect information from students on how many miles they drove that week and how Functions and Statistics Prentice Hall Algebra II, Virginia many gallons of gasoline they used. Draw a scatterplot and make a prediction about how Edition,©2012, Charles et al., Pearson much gasoline they will use each year. Next, students will estimate the amount of money Topic Education they will spend on gas this year, if gasoline cost $ 3.89 per gallon. Statistics  Students try to determine how tall a person is whose femur is 17 inches long. They measure their own femurs and their heights and enter the data into a graphing calculator Virginia SOL AII/T.9 PWC Mathematics website or computer to get a scatterplot. Find the equation for the curve of best fit and use it to http://pwcs.math.schoolfusion.us/ make predictions. Foundational Objectives A.11 Virginia Department of Education website The student will collect and analyze http://www.doe.virginia.gov/instruction/ma data, determine the equation of the thematics/index.shtml curve of best fit in order to make predictions, and solve real-world problems, using mathematical models. Mathematical models will include linear and quadratic functions.

53 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category The student will use problem solving, Essential Questions Functions and Statistics mathematical communication,  What is the difference between direct and inverse variation? mathematical reasoning, connections  What is joint variation? and representations to:  What is combined variation? Topic  Translate “y varies jointly as x and z” as Statistics y= kxz . Essential Understandings  Translate “y is directly proportional to  Real-world problems can be modeled and solved by using inverse variation, joint x” as y= kx . variation, and a combination of direct and inverse variations.  Joint variation is a combination of direct variations. The student will identify, create, and  Translate “y is inversely proportional to solve real-world problems involving k inverse variation, joint variation, and a x” as y = . Teacher Notes and Elaborations x combination of direct and inverse Two variable quantities are proportional if one of them is always the product of the other variations.  Given a situation, determine the value and a constant quantity. of the constant of proportionality.  Set up and solve problems, including Rational equations, to include direct and inverse variation, can be solved algebraically. real-world problems, involving inverse variation, joint variation, and a Direct variation: combination of direct and inverse When two variables are related so that their ratio remains constant, one of the variables is variations. said to vary directly to the other variables or the variables are said to vary proportionately. A linear function defined by an equation of the form y= kx, where k 0 , represents direct Key Vocabulary variation. The constant of variation (constant of proportionality) is k. For a linear function combined variation to be a direct variation the graph must be a non-horizontal line through the origin. constant of proportionality direct variation Inverse variation: inverse variation When the ratio of one variable is constant to the reciprocal of the other variable, one of the joint variation variables is said to vary inversely to the other variable. A function of the form proportional k y= or xy = k , where k 0 , is an inverse variation. The constant of variation (constant of x proportionality) is k.

Joint variation: If y= kxz then y varies jointly as x and z, and the constant of variation is k. In joint variation one quantity varies directly as two quantities. This is a combination of direct variations. For example: In a rectangular prism the equation for volume is V= lwh . If h = 8 then the volume varies jointly as the length (l) and width (w). The constant of variation is 8.

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54 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics Combined variation: If a variable varies directly as one variable and inversely as another, the resulting relationship is called a combined variation For example, kx yz Topic if z varies directly as x and inversely as y, the relationships can be expressed in the following equations: zy= kx or z = or k = . y x Statistics Practical problems can be modeled and solved by using direct and/or inverse variations.

The student will identify, create, and solve real-world problems involving inverse variation, joint variation, and a combination of direct and inverse variations. .

SOL Reporting Category Text:  In groups have students develop problems that involve the different variations. Students Functions and Statistics Prentice Hall Algebra II, Virginia explain how they know what type of variation this problem represents. Edition,©2012, Charles et al., Pearson Topic Education Statistics

PWC Mathematics website http://pwcs.math.schoolfusion.us/ Foundational Objectives Virginia Department of Education website The student, given a situation in a real- http://www.doe.virginia.gov/instruction/ma world context, will analyze a relation to thematics/index.shtml determine whether a direct or inverse variation exists, and represent a direct variation algebraically and graphically and an inverse variation algebraically.

57 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category The student will use problem solving, Essential Questions Functions and Statistics mathematical communication,  What is a normal distribution curve and how is the graph constructed? mathematical reasoning, connections  How can the amount of data that lies within 1, 2, 3, or k standard deviations of the mean and representations to: be found? Topic  Identify the properties of a normal  How does the standard normal distribution curve correspond to probability? Statistics probability distribution.  How can the area under the standard normal curve be found?  Describe how the standard deviation  How is a standard normal probability table used and applied in problem solving? and the mean affect the graph of the Virginia SOL AII/T.11 normal distribution. The student will identify properties of a Essential Understandings  Compare two sets of normally normal distribution and apply those  A normal distribution curve is a symmetrical, bell-shaped curve defined by the mean distributed data using a standard normal properties to determine probabilities and the standard deviation of a data set. The mean is located on the line of symmetry of distribution and z-scores. associated with areas under the standard the curve.  Represent probability as area under the normal curve.  Areas under the curve represent probabilities associated with continuous distributions. curve of a standard normal probability  The normal curve is a probability distribution and the total area under the curve is 1. distribution.  For a normal distribution, approximately 68 percent of the data fall within one standard  Use the graphing calculator or a deviation of the mean, approximately 95 percent of the data fall within two standard standard normal probability table to deviations of the mean, and approximately 99.7 percent of the data fall within three determine probabilities or percentiles standard deviations of the mean. based on z-scores.  The mean of the data in a standard normal distribution is 0, and the standard deviation is 1.  The standard normal curve allows for the comparison of data from different normal Key Vocabulary distributions. area under a curve  A z-score is a measure of position derived from the mean and standard deviation of data. mean  A z-score expresses, in standard deviation units, how far an element falls from the mean normal distribution curve of the data set. normal probability distribution  A z-score is a derived score from a given normal distribution. percentile population  A standard normal distribution is the set of all z-scores. standard deviation standard normal curve Teacher Notes and Elaborations z-score Statistics is the science of collecting, analyzing, and drawing conclusions from data. Methods for organizing and summarizing data make up the branch of statistics called descriptive statistics.

Measures of dispersion indicate the extent to which values are spread around a central value such as the mean when doing standard deviation or the median when doing box-and- whisker plots.

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Curriculum Information Essential Questions and Understandings

58 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Teacher Notes and Elaborations

SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics Differences from the mean, x - µ, are called deviations. The deviation of an entry x in a population data set is the difference between the entry and the mean µ of the data set. The mean is the balance point of the distribution, so the set of deviations from the mean will always add to zero. Topic Statistics The following technical assistance is provided by the VDOE for use with Algebra I, objective 9. This objective is intended to extend the study of descriptive statistics beyond the measures of center studied during the middle grades. Although calculation is included in this objective, instruction and assessment emphasis should be on understanding and interpreting Virginia SOL AII/T.11 statistical values associated with a data set including standard deviation, mean absolute deviation, and z-score. While not explicitly The student will identify properties of a included in this objective, the arithmetic mean will be integral to the study of descriptive statistics. normal distribution and apply those properties to determine probabilities The study of statistics includes gathering, displaying, analyzing, interpreting, and making predictions about a larger group of data associated with areas under the standard (population) from a sample of those data. Data can be gathered through scientific experimentation, surveys, and/or observation of groups normal curve. or phenomena. Numerical data gathered can be displayed numerically or graphically (examples would include line plots, histograms, and stem-and-leaf plots). Methods for organizing and summarizing data make up the branch of statistics called descriptive statistics.

Sample vs. Population Data Sample data can be collected from a defined statistical population. Examples of a statistical population might include SOL scores of all Algebra I students in Virginia, the heights of every U.S. president, or the ages of every mathematics teacher in Virginia. Sample data can be analyzed to make inferences about the population. A data set, whether a sample or population, is comprised of individual data points referred to as elements of the data set.

th An element of a data set will be represented as xi , where i represents the i term of the data set.

When beginning to teach this standard, start with small, defined population data sets of approximately 30 items or less to assist in focusing on development of understanding and interpretation of statistical values and how they are related to and affected by the elements of the data set.

Related to the discussion of samples versus populations of data are discussions about notation and variable use. In formal statistics, the arithmetic mean (average) of a population is represented by the Greek letter μ (mu), while the calculated arithmetic mean of a sample is represented by x , read “x bar.”

The arithmetic mean of a data set will be represented by μ.

Only population data sets (not sample data sets) will be used in the statistics units of study in Algebra I and Algebra II. The formulas provided by the state in the Technical Assistance Manuals for Algebra I and Algebra II and SOL formula sheets use n to represent population size. Typically this is represented by N.

On both brands of approved graphing calculators in Virginia, the calculated arithmetic mean of a data set is represented by x .

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Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics Mean Absolute Deviation vs. Variance and Standard Deviation Statisticians like to measure and analyze the dispersion (spread) of the data set about the mean in order to assist in making inferences about the population. One measure of spread would be to find the sum of the deviations between each element and the mean; however, this sum Topic is always zero. There are two methods to overcome this mathematical dilemma: 1) take the absolute value of the deviations before finding Statistics the average or 2) square the deviations before finding the average. The mean absolute deviation uses the first method and the variance and standard deviation uses the second. If either of these measures is to be computed by hand, do not require students to use data sets of more than about 10 elements. Virginia SOL AII/T.11 The student will identify properties of a NOTE: Students have not been introduced to summation notation prior to Algebra I. An introductory lesson on how to interpret the normal distribution and apply those notation will be necessary. properties to determine probabilities associated with areas under the standard 5 4 normal curve. Examples of summation notation: i =1 + 2 + 3 + 4 + 5 xi = x1 + x 2 + x 3 + x 4 i=1 i=1

Mean Absolute Deviation Mean absolute deviation is one measure of spread about the mean of a data set, as it is a way to address the dilemma of the sum of the deviations of elements from the mean being equal to zero. The mean absolute deviation is the arithmetic mean of the absolute values of the deviations of elements from the mean of a data set.

n x - m Mean absolute deviation i , where µ represents the mean of the data set, n represents the number of elements in = i=1 n th the data set, and xi represents the i element of the data set.

The mean absolute deviation is less affected by outlier data than the variance and standard deviation. Outliers are elements that fall at least

1.5 times the interquartile range (IQR) below the first quartile (Q1) or above the third quartile (Q3). Graphing calculators identify Q1 and Q3 in the list of computed 1-varible statistics. Mean absolute deviation cannot be directly computed on the graphing calculator as can the standard deviation. The mean absolute deviation must be computed by hand or by a series of keystrokes using computation with lists of data (See examples of calculator keystrokes in the Algebra I Curriculum Guide.).

Variance The second way to address the dilemma of the sum of the deviations of elements from the mean being equal to zero is to square the deviations prior to finding the arithmetic mean. The average of the squared deviations from the mean is known as the variance, and is another measure of the spread of the elements in a data set.

61 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics n (x - m )2 Variance i , where µ represents the mean of the data set, n represents the number of elements in the (s 2 ) = i=1 Topic n th Statistics data set, and xi represents the i element of the data set.

The differences between the elements and the arithmetic mean are squared so that the differences do not cancel each other out when Virginia SOL AII/T.11 finding the sum. When squaring the differences, the units of measure are squared and larger differences are “weighted” more heavily than The student will identify properties of a smaller differences. In order to provide a measure of variation in terms of the original units of the data, the square root of the variance is normal distribution and apply those taken, yielding the standard deviation. properties to determine probabilities associated with areas under the standard Standard Deviation normal curve. The standard deviation is the positive square root of the variance of the data set. The greater the value of the standard deviation, the more spread out the data are about the mean. The lesser (closer to 0) the value of the standard deviation, the closer the data are clustered about the mean.

(x - m )2 Standard deviation i , where µ represents the mean of the data set, n represents the number of elements in (s ) = i=1 n th the data set, and xi represents the i element of the data set.

To find standard deviation: - Find the mean of the data set m . - Find the difference between each value and the mean: x - m - Square each difference: (x - m )2 n (x - m )2 - Find the mean of these squares (also referred to as the variance): s 2 = i=1 n n (x - m )2 - Take the square root to find the standard deviation: s = i=1 n

If the difference between mean absolute deviation and standard deviation is large, then the data has a great amount of variability.

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Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations 62 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics Often, textbooks will use two distinct formulas for standard deviation. In these formulas, the Greek letter “σ ”, written and read “sigma”, represents the standard deviation of a population, and “s” represents the sample standard deviation. The population standard deviation can be estimated by calculating the sample standard deviation. The formulas for sample and population standard deviation look very similar Topic except that in the sample standard deviation formula, n – 1 is used instead of n in the denominator. The reason for this is to account for the Statistics possibility of greater variability of data in the population than what is seen in the sample. When n – 1 is used in the denominator, the result is a larger number. So, the calculated value of the sample standard deviation will be larger than the population standard deviation. As sample sizes get larger (n gets larger), the difference between the sample standard deviation and the population standard deviation gets Virginia SOL AII/T.11 smaller. The use of n – 1 to calculate the sample standard deviation is known as Bessel’s correction. Use the formula for standard deviation The student will identify properties of a with n in the denominator as noted previously. normal distribution and apply those properties to determine probabilities When using Casio or Texas Instruments (TI) graphing calculators to compute the standard deviation for a data set, two computations for associated with areas under the standard the standard deviation are given, one for a population (using n in the denominator) and one for a sample (using n − 1 in the denominator). normal curve. Students should be asked to use the computation of standard deviation for population data in instruction and assessments. On a Casio calculator, it is indicated with “xσ n” and on a TI graphing calculator as “σ x”. More information (keystrokes and screenshots) on using graphing calculators to compute this can be found in the Algebra I Curriculum Guide in objective nine.

z-Scores A z-score, also called a standard score, is a measure of position derived from the mean and standard deviation of the data set. In Algebra I, the z-score will be used to determine how many standard deviations an element is above or below the mean of the data set. It can also be used to determine the value of the element, given the z-score of an unknown element and the mean and standard deviation of a data set. The z-score has a positive value if the element lies above the mean and a negative value if the element lies below the mean. A z-score associated with an element of a data set is calculated by subtracting the mean of the data set from the element and dividing the result by the standard deviation of the data set.

x - m z-score (z ) = , where x represents an element of the data set, µ represents the mean of the data set, and σ represents the s standard deviation of the data set.

A z-score can be computed for any element of a data set; however, they are most useful in the analysis of data sets that are normally distributed. In Algebra II, z-scores will be used to determine the relative position of elements within a normally distributed data set, to compare two or more distinct data sets that are distributed normally, and to determine percentiles and probabilities associated with occurrence of data values within a normally distributed data set.

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63 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics The following technical assistance is provided by the VDOE for use with Algebra II, Objective 11. In Algebra I SOL A.9, students study descriptive statistics through exploration of standard deviation, mean absolute deviation, and z- Topic scores of data sets. Students compute these values for small defined populations, and the focus of instruction is on interpretation of these Statistics descriptive statistics. Algebra II, Objective 11 continues the study of descriptive statistics as students analyze properties of normal distributions and apply those properties to determine probabilities associated with areas under the standard normal curve. Students will be provided with the mean, standard deviation, and/or elements of samples or populations of normal distributions and asked to apply the Virginia SOL AII/T.11 properties of normal distributions to calculate probabilities associated with given elements of the data set. The student will identify properties of a normal distribution and apply those Normal Distributions properties to determine probabilities Students examine many sets of data in their study of statistics. Some data have been provided to them and some data they have collected associated with areas under the standard through surveys, experiments, or observations. Representation of data can take on many forms (line plots, stem-and-leaf plots, box-and- normal curve. whisker plots, bar graphs, circle graphs, and histograms). Often teachers try to describe the physical shape of these representations in words or by using measures such as the arithmetic mean, the balance point of the data, or the median, the point at which the data is split into two equal amounts of data points. Another useful type of graphical representation is called a density curve, which models the pattern of a distribution. The distribution of certain types of data take on the appearance of a bell-shaped density curve called a normal curve. These collections of data are often naturally-occurring data or data produced by repetition in a mechanical process.

Examples of data that can be modeled by normal distributions include:  heights of corn stalks in similar growing environments;  heights of 16-year-old girls;  blood pressures of 18-year-old males;  weights of pennies in a given production year;  lifespan of a specific electric motor; and  standardized test scores like the ACT® or SAT®.

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Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics Standard Deviation The standard deviation (σ) of a data set is a measure of the spread of data about the mean. The greater the value of the standard deviation, the more spread out the data are about the mean. The lesser (closer to 0) the value of the standard Topic deviation, the closer the data are clustered about the mean. Statistics Properties of normal distributions (normal probability distribution for continuous data) Normal distributions are represented by a family of symmetric, bell-shaped curves called “normal” curves. Normal curves are defined by Virginia SOL AII/T.11 the mean and standard deviation of the data set. The arithmetic mean is located on the line of symmetry of the curve. In a normal The student will identify properties of a distribution, the arithmetic mean is equivalent to the median and mode of the data set. Approximately 68 percent of the data values fall normal distribution and apply those within one standard deviation (σ) of the mean (μ), approximately 95 percent of the data values fall within two standard deviations of the properties to determine probabilities mean, and approximately 99.7 percent of the data values fall within three standard deviations of the mean. This is often referred to as the associated with areas under the standard 68-95-99.7 rule. normal curve. Figure 1

Figure 1 shows the approximate percentage of observations that fall within different partitions of the normal distribution.

In normal distributions, the total area under the curve is always equal to 1. The mean and standard deviation of a normal distribution affect the location and shape of the curve. The vertical line of symmetry of the normal distribution falls at the mean and the width or spread of the curve is determined by the standard deviation. The greater the standard deviation, the wider (“flatter” or “less peaked”) the distribution of the data.

66 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics Summary of the Properties of the Normal Distribution - The curve is bell shaped. Topic - The mean, median, and mode are equal and located at the center. Statistics - The curve is unimodal, it has only one mode. - The curve is symmetric about the mean. - The curve is continuous. Virginia SOL AII/T.11 - The curve never touches the x-axis, which is the asymptote to the curve. The student will identify properties of a - The total area under the curve is equal to 1.00 or 100%. normal distribution and apply those - The area which lies within one standard deviation is 68%, two standard deviations is 95%, and three standard deviations is 99.7%. properties to determine probabilities associated with areas under the standard normal curve. Figure 2

Figure 2 shows how mean (μ) and standard deviation (σ) affect the graph of the normal distribution.

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67 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics A normal distribution with μ = 5 and σ = 1.5 can be graphed on a graphing calculator.

Texas Instruments (TI-83/84) Topic Calculator function parameters - normalpdf(x, µ, σ) Statistics

Virginia SOL AII/T.11 1. Press Y = 2nd VARS (to get to the distribution menu) The student will identify properties of a normal distribution and apply those properties to determine probabilities associated with areas under the standard normal curve.

2. Select "1:normalpdf(" or press ENTER

3. Press X, T, θ, n , 5 , 1 . 5 ) ENTER

4. Set the window as follows:

1 Hint: When graphing normal distributions, set Ymax = . 2s

5. Press GRAPH and TRACE (to see the function and values)

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69 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics Casio (9750/9850/9860) NOTE: Casio calculators will not graph a normal distribution given only the mean and standard deviation. An element (x) of the data set must also be entered in order to enable the graphing feature. Topic Statistics 1. From the menu screen select and press EXE 2. Pres F5 (to choose "DIST") Virginia SOL AII/T.11 The student will identify properties of a 3. Press F1 (to choose "NORM") normal distribution and apply those 4. Press F1 (to choose "Npd") properties to determine probabilities associated with areas under the standard 5. Enter the values for x, µ, and σ normal curve. 6. Arrow down and highlight "Execute"

7. Press F6 (to choose "DRAW")

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70 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics Determining probabilities associated with normal distributions The cumulative probability of a specified range of values can be represented as the area under a normal distribution curve between the lower and upper bounds. When the range of values is an interval with lower and upper bounds equal to the mean or mean plus or minus Topic one, two, or three standard deviations, the 68-95-99.7 rule can be used to determine probabilities. Graphing calculators can also be used to Statistics compute and graph the areas under normal curves.

Example 1 Virginia SOL AII/T.11 Given a normally distributed data set of 500 observations measuring tree heights in a forest, what is the approximate number of The student will identify properties of a observations that fall within two standard deviations from the mean? normal distribution and apply those properties to determine probabilities Solution associated with areas under the standard A quick sketch of the normal distribution will assist in solving this problem (refer to Figure 1). We know from the 68-95-99.7 rule that normal curve. 95 percent of the data falls within two standard deviations from the mean. Therefore, approximately 500 · 0.95 = 475 of the trees’ height observations fall within two standard deviations from the mean.

Example 2 A normally distributed data set containing the number of ball bearings produced during a specified interval of time has a mean of 150 and a standard deviation of 10. What percentage of the observed values fall between 140 and 160?

Solution From the sketch of a normally distributed data set, 140 is one standard deviation below the mean while 160 is one standard deviation above the mean. Therefore, approximately 34.1% + 34.1% = 68.2% of the data in this distribution falls between 140 and 160.

Example 3 Donna’s boss asked her to purchase a large number of 20-watt florescent light bulbs for their company. She has narrowed her search to two companies offering 20-watt bulbs for the same price. The Bulb Emporium and Lights-R-Us each claim that the mean lifespan for their 20- watt bulbs is 10,000 hours. The lifespan of light bulbs has a distribution that is approximately normal. The Bulb Emporium’s distribution of the lifespan for 20-watt bulbs has a standard deviation of 1,000 hours and Lights-R-Us’ distribution of the lifespan of 20-watt bulbs has a standard deviation of 750 hours. Donna’s boss asked her to use probabilities associated with these normal distributions to make a purchasing decision.

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71 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics Donna decided that she would compare the proportion of light bulbs from each company that would be expected to last for different intervals of time. She started with calculating the probability that a light bulb would be expected to last less than or equal to 9,000 hours. Letting x represent the lifespan of a light bulb, P(x ≤ 9,000 hours) represents the probability that the lifespan of a light bulb would fall less Topic than or equal to 9,000 hours in its normal distribution. Donna continued by finding P(9,000 ≤ x ≤ 11,000 hours) and P(x ≥ 11,000 hours) Statistics for each company.

There are two ways to find the cumulative probability within a range of values on TI graphing calculators. The probability can be Virginia SOL AII/T.11 computed by finding the area under the curve bounded by a range of values using the ShadeNorm function, or it can be directly computed The student will identify properties of a using the normalcdf function. normal distribution and apply those properties to determine probabilities Texas Instruments (TI-83/84) associated with areas under the standard Computing probabilities using the ShadeNorm function: normal curve. Calculator function parameters - normalpdf(x, μ, σ) ShadeNorm(lowerbound,upperbound, μ, σ) 1. Graph the normal distribution of the lightbulbs using the function Y1= normalpdf(x,10000,1000) and the window shown.

NOTE: The calculator automatically converts very small numbers to scientific notation.

To shade and compute the area under the curve for P(x ≤ 9,000 hours) 2. 2nd VARS ► to select "Draw" 3. Choose "ShadeNorm("or press ENTER 4. Press 0 , 9 0 0 0 , 1 0 0 0 0 , 1 0 0 0 ) ENTER

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72 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics

Topic Statistics The area under the curve is equal to 0.1587.

Virginia SOL AII/T.11 The student will identify properties of a normal distribution and apply those properties to determine probabilities associated with areas under the standard To shade and compute the area under the curve for P(9,000 ≤ x ≤ 11,000 hours), use ShadeNorm(9000,11000,10000,1000) normal curve. to get 0.6827. NOTE: Use the ClrDraw function (2nd PRGM 1 ENTER ) to clear graph shading between each shading. To shade and compute the area under the curve for P(x ≥ 11,000 hours), use ShadeNorm(11000,50000,10000,1000) to get 0.1587

Computing probabilities using the normalcdf function Calculator function parameters - normalcdf(lowerbound,upperbound, µ, σ)

To compute P(x ≤ 9,000 hours) 1. Press 2nd VARS 2 (to bring "normalcdf(" to the home screen) 2. Press 0 , 9 0 0 0 , 1 0 0 0 0 , 1 0 0 0 ) ENTER

P(x ≤ 9,000 hours) = 0.1587

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73 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics To compute P((9,000 ≤ x ≤ 11,000 hours) = 0.6827, use the syntax below.

Topic Statistics

Virginia SOL AII/T.11 The student will identify properties of a normal distribution and apply those properties to determine probabilities associated with areas under the standard To compute P(x ≥ 11,000 hours) = 0.1587, use the syntax below. normal curve.

Note: When computing cumulative probabilities greater than a value using the normalcdf function, the upper bound should be at least µ+ 4σ.

On Casio graphing calculators, a feature similar to the TI’s ShadeNorm is only available in terms of the standard normal distribution using z-scores (see Example 4, Solution B on page 60). The cumulative probabilities within a range of values can be computed on Casio graphing calculators.

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74 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics Casio (9750/9850/9860) Computing probabilities using the Ncd function To compute P(x ≤ 9,000 hours) Topic Statistics 1. From the menu screen select and press EXE 2. Press F5 (to choose "DIST")

Virginia SOL AII/T.11 3. Press F1 (to choose "NORM") The student will identify properties of a 4. Press F2 (to choose "Ncd") normal distribution and apply those properties to determine probabilities 5. Enter values associated with areas under the standard Lower bound = 9000 normal curve. Upper bound = 11000 σ = 1000 µ = 10,000 6. Arrow down to highlight "Execute" 7. Press F1 (to choose "CALC")

NOTE: For information on what "z:" means, see section on standard normal distribution and z-scores.

After recording the probabilities for the Bulb Emporium as shown in Figure 3, Donna calculated the same interval probabilities for Lights- R-Us and recorded them. Figure 3 Bulb Emporium (σ = 1000) Lights-R-Us (σ = 750) P(x ≤ 9,000 hours) 0.1587 0.0912 P(9,000 ≤ x ≤ 11,000 hours) 0.6827 0.8176 P(x ≥ 11,000 hours) 0.1587 0.0912

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75 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics In analyzing her data, Donna noticed that a lightbulb at Bulb Emporium had a higher probability of lasting less than 9,000 hours than a bulb at Lights-R-Us. However, a lightbulb at Bulb Emporium also had a higher probability of lasting longer than 11,000 hours than a bulb at Lights-R-Us. She determined that these two statistical probabilities offset one another. The middle interval of data showed that a bulb at Topic Bulb Emporium had a lower probability of lasting between 9,000 and 11,000 hours, inclusively, than a bulb at Lights-R-Us. She therefore Statistics chose to purchase lightbulbs from Lights-R-Us.

z-Scores Virginia SOL AII/T.11 A z-score, also called a standard score, is a measure of position derived from the mean and standard deviation of the data set. The student will identify properties of a The z-score is a measure of how many standard deviations an element falls above or below the mean of the data set. The z- normal distribution and apply those score has a positive value if the element lies above the mean and a negative value if the element lies below the mean. properties to determine probabilities A z-score associated with an element of a data set is calculated by subtracting the mean of the data set from the element and associated with areas under the standard dividing the result by the standard deviation of the data set. normal curve. x - m z-score (z ) = , where x represents an element of the data set, μ represents the mean of the data set, and σ represents s the standard deviation of the data set.

The standard normal curve is a normal distribution that has a mean of 0 and a standard deviation of 1. It is used to model z-scores obtained from normally distributed data. Prior to calculator technologies that can determine probabilities associated with normal distributions, the table of Standard Normal Probabilities, commonly referred to as a “z-table” was used to determine normal distribution probabilities. See pages 63 and 64 for a copy of the table of Standard Normal Probabilities provided for the Algebra II End-of-Course (EOC) SOL test.

Given a z-score (z), the table of Standard Normal Probabilities shows the area under the curve to the left of z. This area represents the proportion of observations with a z-score less than the one specified. Table rows show the z-score’s whole number and tenths place. Table columns show the hundredths place. In the table of Standard Normal Probabilities provided for the state EOC SOL test in Algebra II, the cumulative probability from negative infinity to the z-score appears in table cells. Other tables of Standard Normal Probabilities show probabilities from the mean to the z-score.

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76 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics Interpreting values from the table of Standard Normal Probabilities A z-score associated with an element of a normal distribution is computed to be 1.23. The probability from the table of Standard Normal Probabilities associated with a z-score of 1.23 can be determined as indicated in Figure 4. The probability can be used differently based Topic upon the context of the question. Statistics - The probability that a data value will fall below the data value associated with a z-score of 1.23 is 0.8907 (89.07%). Virginia SOL AII/T.11 The student will identify properties of a - The data value associated with a z-score of 1.23 falls in the 89th percentile. normal distribution and apply those This means that 89 percent of the data in the distribution fall below the properties to determine probabilities value associated with a z-score of 1.23. associated with areas under the standard normal curve. - The probability that a value from the data set will fall above this value is 1 - 0.8907 = 0.1093 (10.93%).

Figure 4

Figure 4 shows the cumulative probability associated with a z-score of 1.23 using the table of standard normal probabilities. (continued)

77 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics From Example 3, Donna could have determined P(x ≤ 9,000 hours) for Lights-R-Us using the table of Standard Normal Probabilities.

Given μ = 10,000 and σ = 750, the z-score for 9,000 hours can be computed using the formula for z-score. Topic 9000- 10000 Statistics z= = - 1.33 750 In the table of Standard Normal Probabilities, z = 1.33 is associated with a cumulative probability of 0.0918, a value very close (difference Virginia SOL AII/T.11 of 0.0006) to the one found for Lights-R-Us in Figure 3, using a graphing calculator. The student will identify properties of a normal distribution and apply those Example 4 properties to determine probabilities In statistics, a percentile is the value of a variable below which a certain percent of observations fall. For example, the 20th percentile is associated with areas under the standard the value (or score) below which 20 percent of the observations may be found. The term percentile and the related term percentile rank are normal curve. often used in the reporting of scores from norm-referenced tests. The 25th percentile is also known as the first quartile (Q1), the 50th percentile as the median or second quartile (Q2), and the 75th percentile as the third quartile (Q3).

The ACT® is an achievement test given nationally with normally distributed scores. Amy scored a 31 on the mathematics portion of her 2009 ACT®. The mean for the mathematics portion of the ACT® in 2009 was 21.0 and the standard deviation was 5.3. What percent of the population scored higher than Amy on the mathematics portion of the ACT®?

There are two “typical” approaches to find the percentage of the population that scored higher than Amy.

Solution A – Find the z-score and associated cumulative probability using the table of standard normal probabilities and subtract the cumulative probability from 1.

31- 21.0 Calculate the z-score: z = = 1.89 . Look up the cumulative probability associated with a z-score of 1.89 on the table of 5.3 standard normal probabilities. The probability of a test taker scoring a 31 (z-score = 1.89) is 0.9706 or 97 percent. This means that Amy scored in the 97th percentile and only 2.94 percent (1 – 0.9706 = 0.0294) scored higher than Amy.

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Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics Example 4 extension Solution B – Find the probability of scoring higher than 31 using the graphing calculator.

Topic The Texas Instruments (TI-83/84) graphing calculators can compute the percentage of the population scoring higher than 31 using Statistics the syntax normalcdf(31,43,21.0, 5.3), given lower bound = 31, upper bound = 43, μ = 21.0, and σ = 5.3. The resulting value is 0.0296 or 2.96 percent (beginning on page 54). Note: When choosing an upper bound (43 in this case), choose a number that will encompass all data values above the mean. The number 43 (rounded to 43 using μ + 4 σ = 21+21.2 = 42.2) is far enough above the Virginia SOL AII/T.11 mean to be a “safe” upper bound. The student will identify properties of a normal distribution and apply those The Casio (9750/9850/9860) graphing calculators can compute the percentage properties to determine probabilities of the population that score above a 31 by using the operation “Ncd” or associated with areas under the standard “Normal C.D.” with a lower bound = 31, upper bound = 43, σ = 5.3, normal curve. and μ = 21.0 (beginning on page 55). The calculated percentage of the population scoring above a 31 = 0.0296 or 2.96 percent. The Casio 9750/9860 can “DRAW” the representative area under the curve in terms of z-scores by pressing F6.

Amy took the ACT® and scored 31 on the mathematics portion of the test. Her friend Stephanie scored a 720 on the mathematics portion of her 2009 SAT®. Both the SAT® and the ACT® are achievement tests given nationally with scores that are normally distributed. The mean for the mathematics portion of the SAT® in 2009 was 515 and the standard deviation was 116. For the ACT®, the mean was 21 and the standard deviation was 5.3. Whose achievement was higher on the mathematics portion of their national achievement test?

Solution Since these two national achievement tests have different scoring scales, they cannot be compared directly. One way to compare them would be to find the cumulative probability (percentile) of each score using the associated z-score. Amy’s z-score is 1.89 (97th percentile) and Stephanie’s z-score is 1.77 (96th percentile). Therefore, Amy scored slightly higher than Stephanie on the mathematics portions of their respective national achievement tests.

80 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category Text: Note: All exploration questions should be in the real-world context of normally distributed Functions and Statistics Prentice Hall Algebra II, Virginia data sets. Edition,©2012, Charles et al., Pearson Topic Education 1. Given a normally distributed data set with a specified mean and standard deviation, Statistics explain how the number of values expected to be above or below a certain value can be determined. Virginia SOL AII/T.11 PWC Mathematics website http://pwcs.math.schoolfusion.us/ 2. Given normally distributed data, explain how you can determine how many values or Foundational Objectives what percentage of values are expected to fall within one, two or three standard The student, given a set of data, will Virginia Department of Education website deviations of the mean. interpret variation in real-world http://www.doe.virginia.gov/instruction/ma contexts and calculate and interpret thematics/index.shtml 3. Compare and contrast graphs of normal distributions that have the same mean but mean absolute deviation, standard different standard deviations or different means and the same standard deviation. deviation, and z-scores. 4. Given a normally distributed data set with a specified mean and standard deviation, explain how to determine the probability and/or area under the curve for  an element that has a value greater than a given value;  an element that has a value less than a given value; or  an element that has a value between two given values.

5. Given the mean and standard deviation of two different normally distributed data sets, and a value from each data set, compare the values using their corresponding z-scores and percentiles.

6. Given normally distributed data with specified mean and standard deviation, determine the probability that a randomly selected value will have a z-score within a certain range of values.

Standard Normal Probabilities

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359 0.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753 0.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141 0.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517 Table entry for z is the area under the standard 0.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879 normal curve to the left of z. 0.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224 0.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549 0.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852 0.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133 0.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389 1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621 1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830 1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015 1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177 1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319 1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441 1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545 1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633 1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706 1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767 2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817 2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857 2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890 2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916 2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936 2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952 (continued) (continued) 2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964 Standard Normal Probabilities 2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974 2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981 2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986 3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990 3.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .9993 83 3.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .9995 3.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9997 3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS

Table entry for z is the area under the standard normal curve to the left of z.

84 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS SOL Reporting Category The student will use problem solving, Essential Questions Functions and Statistics mathematical communication,  What is a permutation and how is it determined? mathematical reasoning, connections  What is a combination and how is it determined? Virginia SOL AII/T.12 and representations to:  What is the difference between a permutation and a combination of the same items? The student will compute and  Compare and contrast permutations and  When is a permutation or a combination used? distinguish between permutations and combinations. combinations and use technology for  Calculate the number of permutations Essential Understandings applications. of n objects taken r at a time.  The Fundamental Counting Principle states that if one decision can be made n ways and  Calculate the number of combinations another can be made m ways, then the two decisions can be made nm ways. of n objects taken r at a time.  Permutations are used to calculate the number of possible arrangements of objects.  Use permutations and combinations as  Combinations are used to calculate the number of possible selections of objects without counting techniques to solve real-world regard to the order selected. problems. Teacher Notes and Elaborations A factorial is the product of all the positive integers through the given integer Key Vocabulary (e.g., 5! = 1 ∙ 2 ∙ 3 ∙ 4 ∙ 5). combination factorial A permutation is an arrangement of items in a particular order. It can often be found by permutation using the Fundamental Counting Principle or factorial notation. The number of permutation of n objects is given as n!. n! The permutations of n objects taken r at a time is P(n, r) or P = for 0 ≤ r ≤ n. n r (n- r )!

A combination is a selection in which order does not matter. The number of combinations of n! n items taken r at a time is C(n, r) or C = for 0 ≤ r ≤ n. n r r!( n- r )! The formula for combinations is like the formula for permutations except that it contains the factor r! to compensate for duplicate combinations.

Combination rule: n! The number of combinations of r items selected from n different items is: C = n r r!( n- r )! The following conditions must apply: a total of n different items available, select r of the n items without replacement (consider rearrangements of the same items to be the same – the combination of ABC is the same as CBA).

(continued)

85 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations SOL Reporting Category Teacher Notes and Elaborations (continued) Functions and Statistics Example: A Board of Trustees at the XYZ Company has 9 members. Each year, they elect a 3-person committee to oversee buildings and grounds. Each year, they also elect a chairperson, vice chairperson, and secretary. When the board elects the buildings and grounds committee, how many different 3-person committees are possible?

Solution: Because order is irrelevant when electing the buildings and grounds committee, the number of combinations of r = 3 n! 9! Virginia SOL AII/T.12 people selected from the n = 9 available people; is nCr = = = 84 . Because order does matter with the slates of (n- r )! r ! (9 - 3)!3! The student will compute and distinguish between permutations and candidates, the number of sequences (or permutations) of r = 3 people selected from the n = 9 available people is nPr = combinations and use technology for n! 9! = = 504 applications. (n- r )! (9 - 3)! There are 84 different possible committees of 3 board members, but there are 504 different possible slates of candidates.

Permutation Rule (When Some Items Are Identical to Others) n! If there are n items with n1 alike, n 2 alike,... nk alike, the number of permutations of all n items is n items is . n1! n 2 !... n k !

Example: Consider the letters BBBBBAAAA, which represent a sequence of recent years in which the Dow Jones Industrial Average was below (B) the mean or above (A) the mean. How many ways can the letters BBBBBAAAA be arranged? Does it appear that the sequence is random? Is there a pattern suggesting that it would be wise to invest in stocks?

Solution: In the sequence BBBBBAAAA, n = 9 items, with n1 = 5 alike and n2 = 4 others that are alike. The number of 9! permutations is computed as follows: = 126 . There are 126 different ways that the letters can be arranged. Because there are 5!4! 126 different possible arrangements and only two of them (BBBBBAAAA and AAAABBBBB) result in the letters all grouped together, it appears that the sequence is not random. Because all of the below values occur at the beginning and all of the above values occur at the end, it appears that there is a pattern of increasing stock values. This suggests that it would be wise to invest in stocks.

A permutation problem occurs when different orderings of the same items are counted separately. A combination problem occurs when different orderings of the same items are not counted separately.

SOL Reporting Category Text: Skittles Lab: Functions and Statistics Prentice Hall Algebra II, Virginia 1. How many skittles are in the bag? Edition,©2012, Charles et al., Pearson 2. List the total for each color of skittles in a bag. Education Answer the following questions. (Students need to be thorough in their explanations.) a. How many ways are there to select 5 skittles from your bag, disregarding the order Virginia SOL AII/T.12 of selection? PWC Mathematics website b. In a bag of x skittles how many ways can a red, green, and yellow be selected? Foundational Objectives http://pwcs.math.schoolfusion.us/ c. How many permutations can be made for red skittles in your sample? d. How many different arrangements can be made for red, yellow, green, purple and Virginia Department of Education website orange? http://www.doe.virginia.gov/instruction/ma e. How many different way are there to pick a yellow from your bag? thematics/index.shtml

89 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Topic The student will use problem solving, Essential Questions Triangular and Circular Trigonometric mathematical communication,  What is the standard position of an angle? Functions mathematical reasoning, connections  Given a point on the terminal side of an angle, how are the values of the six and representations to: trigonometric functions determined? Virginia SOL AII/T.13  Define the six triangular trigonometric  What is the relationship between trigonometric and circular functions? The student, given a point other than th functions of an angle in a right triangle. e origin on the terminal side of the angl  Define the six circular trigonometric Essential Understandings e, will use the definitions of the six trig functions of an angle in standard  Triangular trigonometric function definitions are related to circular trigonometric onometric functions to find the sine, cos position. function definitions. ine, tangent, cotangent, secant, and cose  Make the connection between the  Both degrees and radians are units for measuring angles. cant of the angle in standard position. T triangular and circular trigonometric  Drawing an angle in standard position will force the terminal side to lie in a specific rigonometric functions defined on the u functions. quadrant. nit circle will be related to trigonometri  Recognize and draw an angle in  A point on the terminal side of an angle determines a reference triangle from which the c functions defined in right triangles. standard position. values of the six trigonometric functions may be derived.  Show how a point on the terminal side of an angle determines a reference Teacher Notes and Elaborations triangle. As derived from the Greek language, the word trigonometry means “measurement of triangles”.

Key Vocabulary An angle is determined by rotating a ray (half-line) about its endpoint. The starting position circular trigonometric function of the ray is the initial side of the angle, and the position after rotation is the terminal side. degrees initial side The six trigonometric functions of an angle θ are called sine, cosine, tangent, cotangent, radians secant and cosecant. The functions are defined with the angle θ (the Greek letter theta) in reference triangle standard position. terminal side triangular trigonometric function In the rectangular coordinate system an angle with its vertex at the origin and with its initial unit circle side along the positive x-axis is in standard position. For any point P(x, y) on the terminal side of an angle θ in standard position, r is defined as the distance from the vertex to P (r= x2 + y 2 ) . A point on the terminal side of an angle determines a reference triangle from which the values of the six trigonometric functions may be derived.

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90 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations Topic Teacher Notes and Elaborations (continued) Triangular and Circular Trigonometric The six triangular trigonometric functions of θ are: Functions y x sinq = cosq = r r Virginia SOL AII/T.13 y x tanq = cotq = The student, given a point other than x y the origin on the terminal side of the r r angle, will use the definitions of the six cscq = secq = trigonometric functions to find the sine, y x cosine, tangent, cotangent, secant, and cosecant of the angle in standard The properties of the trigonometric functions are connected with the circular function definitions by using a unit circle (a circle with the position. Trigonometric functions radius of one). defined on the unit circle will be related to trigonometric functions defined in If the terminal side of an angle θ in standard position intersects the unit circle at P(x, y), then the six circular trigonometric functions are right triangles. defined as:

sinq = y cosq = x x y tanq = cotq = y x 1 1 cscq = secq = y x

The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. Degrees and radians are equivalent units for angle measurement. One radian is the measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle.

PWC Mathematics website http://pwcs.math.schoolfusion.us/

Virginia Department of Education website http://www.doe.virginia.gov/instruction/ma thematics/index.shtml

94 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Topic The student will use problem solving, Essential Questions Triangular and Circular Trigonometric mathematical communication,  What are the Pythagorean, ratio, and reciprocal identities? Functions mathematical reasoning, connections  Given the value of one trigonometric function, how are the remaining functions and representations to: determined?  Given one trigonometric function value, Virginia SOL AII/T.14 find the other five trigonometric Essential Understandings The student, given the value of one function values.  If one trigonometric function value is known, then a triangle can be formed to use in trigonometric function, will find the  Develop the unit circle, using both finding the other five trigonometric function values. values of the other trigonometric degrees and radians.  Knowledge of the unit circle is a useful tool for finding all six trigonometric values for functions, using the definitions and  Solve problems, using the circular special angles. properties of the trigonometric function definitions and the properties functions. of the unit circle. Teacher Notes and Elaborations  Recognize the connections between the Given the value of one trigonometric function, a triangle can be formed to use in finding the coordinates of points on a unit circle other five trigonometric function values or the remaining functions may also be found using and one of the following methods: - coordinate geometry; Definitions of the trigonometric functions: - cosine and sine values; and y r - lengths of sides of special right sinq = and the cscq = y triangles (30° - 60° - 90° and r x r 45° - 45° - 90°). cosq = and the cscq = r x y x tanq = and the cotq = Key Vocabulary x y degrees Pythagorean identities Relationships between trigonometric functions are identities. radians Reciprocal Identities: ratio (quotient) identities y r 1 1 reciprocal identities Since sinq = and the cscq = , then sinq = and cscq = . unit circle r y cscq sinq Also, cos θ and sec θ are reciprocals as are tan θ and cot θ. The reciprocal identities hold for any angle θ that does not lead to a zero denominator.

Pythagorean Identities: sin2q+ cos 2 q = 1 tan2q+ 1 = sec 2 q 1+ cot2q = cos 2 q

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95 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations Topic Teacher Notes and Elaborations (continued) Triangular and Circular Trigonometric Ratio or Quotient Identities: Functions sinq tanq = cosq cosq Virginia SOL AII/T.14 cotq = The student, given the value of one sinq trigonometric function, will find the values of the other trigonometric Degrees and radians are equivalent units for angle measurement. A central angle with sides and intercepted arcs all the same length functions, using the definitions and measures 1 radian. properties of the trigonometric functions. A unit circle is one that lies on the x-axis, has origin (0, 0), and a radius of 1. Knowledge of the unit circle is a useful tool for finding all six trigonometric values for special angles.

99 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Topic The student will use problem solving, Essential Questions Triangular and Circular Trigonometric mathematical communication,  What is the relationship between radians and degrees? Functions mathematical reasoning, connections  What is the relationship between families of coterminal angles? and representations to:  What is meant by the special angles?  Find trigonometric function values of Virginia SOL AII/T.15 specials angles and their related angles Essential Understandings The student will find, without the aid of in both degrees and radians.  Special angles are widely used in mathematics. a calculator, the values of the  Apply the properties of the unit circle  Unit circle properties will allow special-angle and related-angle trigonometric values to trigonometric functions of the special without using a calculator. be found without the aid of a calculator. angles and their related angles as found  Use a conversion factor to convert from  Degrees and radians are units of angle measure. in the unit circle. This will include radians to degrees and vice versa  A radian is the measure of the central angle that is determined by an arc whose length is converting angle measures from radians without using a calculator. the same as the radius of the circle. to degrees and vice versa. Teacher Notes and Elaborations Key Vocabulary The two most common units used to measure angles are radians and degrees. The radian coterminal angles measure of an angle in standard position is defined as the length of the corresponding arc degrees s quadrantal angles divided by the radius of the circle (q = ). One degree, 1°, is the result from a rotation of radian r 1 revolution of a complete revolution about the vertex in the positive direction. A full revolution unit circle 360 (counterclockwise) corresponds to 360º.

To convert radians to degrees and vice versa, multiply by the appropriate conversion factor 骣 p 180° 琪e.g., 1° = ° rad and 1 rad = . 桫 180 p

Multiples, between 0 and 2π, of first quadrant special angles are found without the aid of a calculator.

Angles that measure greater than 2π can be formed by adding or subtracting a multiple of 2π to its coterminal angle measuring between 0 and 2π.

Two angles in standard position with the same initial and terminal sides are called coterminal angles.

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100 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations Topic Teacher Notes and Elaborations (continued) Triangular and Circular Trigonometric p p Special angles are widely used in mathematics. The first quadrant special angles of a unit circle (a circle with a radius of one) are , , Functions 6 4 p p 3p . The quadrantal angles (any angle with the terminal side on the x-axis or y-axis) of a unit circle are 0, , π, , 2π. 3 2 2 Virginia SOL AII/T.15 The student will find, without the aid of a calculator, the values of the trigonometric functions of the special angles and their related angles as found in the unit circle. This will include converting angle measures from radians to degrees and vice versa.

104 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Topic The student will use problem solving, Essential Questions Inverse Trigonometric Functions mathematical communication,  What are inverse trigonometric functions? mathematical reasoning, connections and representations to: Essential Understandings Virginia SOL AII/T.16  Use a calculator to find the  The trigonometric function values of any angle can be found by using a calculator. The student will find, with the aid of a trigonometric function values of any  The inverse trigonometric functions can be used to find angle measures whose calculator, the value of any angle in either degrees or radians. trigonometric function values are known. trigonometric function and inverse  Define inverse trigonometric functions.  Calculations of inverse trigonometric function values can be related to the triangular trigonometric function.  Find angle measures by using the definitions of the trigonometric functions. inverse trigonometric functions when the trigonometric function values are Teacher Notes and Elaborations given. The values of the trigonometric functions of any angle can be approximated using a calculator. Most values are approximated to four decimal places. Depending upon the problem, calculators must be in the appropriate mode, whether radian or degree. Key Vocabulary inverse trigonometric functions The inverse trigonometric functions can be used to find angle measures whose trigonometric function values are known. Given the value of any trigonometric function, the angle may be determined by using the appropriate inverse function key on the calculator. Values of inverse trigonometric functions are always in radians.

Definitions of the Inverse Trigonometric Functions:

Function Domain Range p p y= arcsin x if and only if sin y = x -1＃x 1 - ＃y 2 2 y= arccos x if and only if cos y = x -1＃x 1 0 ＃y p p p y= arctan x if and only if tan y = x -� < x -

Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations Topic The student will use problem solving, Essential Questions Trigonometric Identities mathematical communication,  What is an identity? mathematical reasoning, connections  What is the difference between solving equations and verifying identities? and representations to: Virginia SOL AII/T.17  Use trigonometric identities to make Essential Understandings The student will verify basic algebraic substitutions to simplify and  Trigonometric identities can be used to simplify trigonometric expressions, equations, or trigonometric identities and make verify trigonometric identities. The identities. substitutions, using the basic identities. basic trigonometric identities include  Trigonometric identity substitutions can help solve trigonometric equations, verify - reciprocal identities; another identity, or simplify trigonometric expressions. - Pythagorean identities; - sum and difference identities; Teacher Notes and Elaborations - double-angle identities; and An identity is an equation that is true for all possible replacements of the variables. An - half-angle identities. identity involving trigonometric expressions is a trigonometric identity. Trigonometric identities can be used to simplify trigonometric expressions, equations, or identities. The fundamental trigonometric identities are the following: Key Vocabulary - reciprocal identities, identity - Pythagorean identities, double-angle identities - sum and difference identities, half-angle identities - half angle identities, and Pythagorean identities - double angle identities. reciprocal identities sum and difference identities Reciprocal Identities: trigonometric identities y r 1 1 verify Since sinq = and the cscq = , then sinq = and cscq = . r y cscq sinq Also, cos θ and sec θ are reciprocals as are tan θ and cot θ. The reciprocal identities hold for any angle θ that does not lead to a zero denominator.

Pythagorean Identities: sin2q+ cos 2 q = 1 tan2q+ 1 = sec 2 q 1+ cot2q = cos 2 q

sinq Ratio or Quotient Identities: tanq = cosq cosq cotq = sinq

108 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Topic Teacher Notes and Elaborations (continued) Trigonometric Identities 2 tan u Double-Angle Identities: sin 2u= 2sin u cos u tan 2u = 1- tan2 u cos2u= cos2 u - sin 2 u Virginia SOL AII/T.17 2 The student will verify basic = 2cosu - 1 2 trigonometric identities and make = 1- 2sin u substitutions, using the basic identities. Sum and Difference Identities: sin(u+ v ) = sin u cos v + cos u sin v sin(u- v ) = sin u cos v - cos u sin v cos(u+ v ) = cos u cos v - sin u sin v cos(u- v ) = cos u cos v + sin u sin v tanu+ tan v tanu- tan v tan(u+ v ) = tan(u- v ) = 1- tanu tan v 1+ tanu tan v

u1- cos u Half-Angle Identities: sin = 2 2 u1+ cos u cos = 2 2 u1- cos u sin u tan = = 2 sinu 1+ cos u u u u The signs of sin and cos depend on the quadrant in which lies. 2 2 2

To verify a trigonometric identity, either the left or the right side of the equation may be used to deduce the other side. Verifying identities is not the same as solving equations. Techniques used in solving equations, such as adding the same terms to both sides, are not valid when working with identities since the statement to be verified may not be true. To verify an identity, show that one side of the identity can be simplified so that it is identical to the other side.

Guidelines for Verifying Trigonometric Identities 1. Work with one side of the equation at a time. It is often better to work with the more complicated side first. 2. Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator. 3. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents. 4. If the preceding guidelines do not help, try converting all terms to sines and cosines. 5. Always try something. Even making an attempt that leads to a dead end provides insight. 6. Try working backwards from the solution, as it can provide great insight.

Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations Topic The student will use problem solving, Essential Questions Trigonometric Equations, Graphs, and mathematical communication,  What effect does the change in the values A, B, C, and D in the equation Practical Problems mathematical reasoning, connections y = A sin(Bx - C) + D, have on the graph of the function? and representations to:  Why are the terms: phase shift, period, amplitude, vertical shift and asymptote  Determine the amplitude, period, phase important to curve sketching? Virginia SOL AII/T.18 shift, and vertical shift of a The student, given one of the six trigonometric function from the Essential Understandings trigonometric functions in standard equation of the function and from the  The domain and range of a trigonometric function determine the scales of the axes for form, will graph of the function. the graph of the trigonometric function. a. state the domain and the range of  Describe the effect of changing A, B, C,  The amplitude, period, phase shift, and vertical shift are important characteristics of the the function; or D in the standard form of a graph of a trigonometric function, and each has a specific purpose in applications using b. determine the amplitude, period, trigonometric equation trigonometric equations. phase shift, vertical shift; and {e.g., y = A sin(Bx + C) + D, or  The graph of a trigonometric function can be used to display information about the asymptotes; y = A cos[B(x + C)] + D}. periodic behavior of a real-world situation, such as wave motion or the motion of a c. sketch the graph of the function by  State the domain and the range of a Ferris wheel. using transformations for at least a function written in standard form two-period interval; and {e.g., y = A sin(Bx + C) + D or Teacher Notes and Elaborations d. investigate the effect of changing y = A cos[B(x + C)] + D}. Each of the six trigonometric functions is a periodic function whose graph is based on the parameters in a trigonometric  Sketch the graph of a function written repetition. A periodic function is a function f such that f( x )= f ( x + p ) for every real function on the graph of the in standard form number x in the domain of f and for some positive real number p . The smallest possible function. {e.g., y = A sin(Bx + C) + D or positive value of p is the period of the function. The period of the sine, cosine, secant, and y = A cos[B(x + C)] + D } by using cosecant function is 2π. The period of the tangent and cotangent function is π. transformations for at least a two period interval. The amplitude of a function can be interpreted as half the difference between its maximum and minimum values. The amplitude is half the range (difference between maximum and minimum values). Key Vocabulary amplitude Suggested five steps to sketch the parent graph of y = A sin Bx or y = A cos Bx, with B > 0 asymptote are: horizontal phase shift period of the function 骣p 1. Determine the period of repeat, 2琪 . Start at 0 on the x-axis and mark off that periodic function 桫B range distance. vertical phase shift 2. Divide the interval into four equivalent parts. 3. Evaluate the function for each of the five x values resulting from Step 2. The points will be maximum points, minimum points, and x intercepts. 4. Plot those points found in Step 3 and join them with a curve. 5. Draw additional cycles to the left and right of the curve.

(continued) 112 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS

Curriculum Information Essential Questions and Understandings Teacher Notes and Elaborations Topic Teacher Notes and Elaborations (continued) Trigonometric Equations, Graphs, and Transformations to the original graph can be done through phase shifts. The vertical phase shift moves the horizontal axis of the graph Practical Problems along the y-axis. The horizontal phase shift moves the graph along the x-axis.

Steps to sketch the graph of y = A sin(Bx – C) + D or y = A cos(Bx – C) + D, with B > 0 are: Virginia SOL AII/T.18 1. Determine D the vertical phase shift. This will be the new horizontal axis at y = D. The student, given one of the six 2. Determine C the horizontal phase shift. This will lie on the x-axis. trigonometric functions in standard 3. Follow steps 1 - 5 above. form, will a. state the domain and the range of The asymptote is a straight line whose perpendicular distance from a curve decreases to zero as the distance from the origin increases the function; without limit. b. determine the amplitude, period, phase shift, vertical shift; and Reciprocal identities are used to obtain the graphs of the secant and cosecant functions. The cosecant and secant functions will have asymptotes; vertical asymptotes. The asymptotes will have equations of the form x= k , where k is the x-intercept of the sine or cosine function. c. sketch the graph of the function by using transformations for at least a Sketching the graphs of the variations of the tangent and cotangent is similar to sketching the graphs of the transformations of sine and two-period interval; and cosine functions. Key differences are the period of repeat, asymptotes, and the shape of the graph. Tangent and cotangent graphs do not d. investigate the effect of changing have amplitude. the parameters in a trigonometric function on the graph of the The graphing calculator can provide a visual look at how the constants A, B, C, and D affect the graph of a function. Be sure the calculator function. is set for radians. Most calculators have a trig window with domain [-2π, 2π], range [-4, 4], xscl = π, and yscl = 1 . Other settings may be preferable for different equations.

Graphs of trigonometric functions model periodic behavior of real world situations such as wave motion, biorhythms, seasonal temperatures, or the motion of a Ferris wheel.

114 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Topic Text: Trigonometric Equations, Graphs, and Prentice Hall Algebra II, Virginia Practical Problems Edition,©2012, Charles et al., Pearson Virginia SOL AII/T.18 Education

116 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations Topic The student will use problem solving, Essential Questions and Understandings Inverse Trigonometric Functions mathematical communication,  What are the domains and ranges of the inverse trigonometric functions? mathematical reasoning, connections  What are the restrictions on the domain of the inverse trigonometric functions? Virginia SOL AII/T.19 and representations to: The student will identify the domain  Find the domain and range of the Essential Understandings and range of the inverse trigonometric inverse trigonometric functions.  Restrictions on the domains of some inverse trigonometric functions exist. functions and recognize the graphs of  Use the restrictions on the domains of these functions. Restrictions on the the inverse trigonometric functions in Teacher Notes and Elaborations domains of the inverse trigonometric finding the values of the inverse The trigonometric functions are not one-to-one, so it is necessary to determine the functions will be included. trigonometric functions. restrictions on domains to regions that pass the horizontal line test. The inverse  Identify the graphs of the inverse trigonometric functions can be denoted in two ways. For example, the inverse of y= sin x trigonometric functions. may be written as y= arcsin x or y= sin-1 x .

Key Vocabulary Function Domain Range 轾 p p inverse trigonometric function y = arcsin x [-1,1] - , restrictions on domains 臌犏 2 2 y = arccos x [-1,1] [0, π] 轾 p p y = arctan x [-∞,∞] - , 臌犏 2 2 y = arccot x [-∞,∞] [0, π]

Function Domain y = arcsec x [-∞, -1] [1, ∞] y = arccsc x [-∞, -1] [1, ∞]

Function Range p y = arcsec x [0,π], y 2 轾 p p y = arccsc x - , , y 0 臌犏 2 2

The graphs of the inverse trigonometric functions are obtained by interchanging the x- and y- coordinates of the key points of the basic graphs.

119 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Topic The student will use problem solving, Essential Questions and Understandings Trigonometric Equations, Graphs, and mathematical communication,  Do trigonometric equations have unique solutions? Why or why not? Practical Problems mathematical reasoning, connections  What is the relationship of the domain and range to the solution of trigonometric and representations to: equations?  Solve trigonometric equations with Virginia SOL AII/T.20 restricted domains algebraically and by Essential Understandings The student will solve trigonometric using a graphing utility.  Solutions for trigonometric equations will depend on the domains. equations that include both infinite  Solve trigonometric equations with  A calculator can be used to find the solution of a trigonometric equation as the points of

solutions and restricted domain infinite solutions algebraically and by intersection of the graphs when one side of the equation is entered in the calculator as Y1 solutions and solve basic trigonometric using a graphing utility. and the other side is entered as Y2. inequalities.  Check for reasonableness of results, and verify algebraic solutions, using a Teacher Notes and Elaborations graphing utility. Trigonometric equations, like most algebraic equations, are true for some, but not for all values of the variable. Trigonometric equations do not have unique solutions. Solutions for trigonometric equations will depend on the domains. They have infinitely many solutions, Key Vocabulary differing by the period of the function. If the domain of the equations is restricted to one trigonometric equation revolution then only those solutions between 0 and 2π will be determined. trigonometric identities To solve a trigonometric equation, use standard algebraic techniques and fundamental trigonometric identities.

The fundamental trigonometric identities are the following: - reciprocal identities, - Pythagorean identities, - sum and difference identities, - half angle identities, and - double angle identities.

Standard algebraic techniques are used to solve trigonometric inequalities.

Curriculum Information Essential Knowledge and Skills Essential Questions and Understandings Key Vocabulary Teacher Notes and Elaborations Topic The student will use problem solving, Essential Questions and Understandings Trigonometric Equations, Graphs, and mathematical communication,  How are practical problems involving triangles and vectors solved? Practical Problems mathematical reasoning, connections  What is the relationship of a vector to right triangles and trigonometric functions? and representations to:  What is meant by an ambiguous case when determining parts of a triangle? Virginia SOL AII/T.21  Write a real-world problem involving The student will identify, create, and triangles. Essential Understandings solve real-world problems involving  Solve real-world problems involving  A real-world problem may be solved by using one of a variety of techniques associated triangles. Techniques will include using triangles. with triangles. the trigonometric functions, the  Use the trigonometric functions, Pythagorean Theorem, the Law of Pythagorean Theorem, Law of Sines, Teacher Notes and Elaborations Sines, and the Law of Cosines. and Law of Cosines to solve real-world Practical problems involving right triangles can be solved by applying the right triangle problems. definitions of trigonometric functions and the Pythagorean Theorem. Problems involving  Use the trigonometric functions to oblique (non-right) triangles are solved using the Law of Sines or the Law of Cosines model real-world situations. depending upon the given information.  Identify a solution technique that could be used with a given problem. The Law of Sines states that for any triangle with angles of measures A, B, and C, and sides  Prove the addition and subtraction of lengths a, b, and c (a opposite A , b opposite B , and c opposite C ). formulas for sine, cosine, and tangent sinA sin B sin C and use them to solve problems. = = a b c

The Law of Cosines states that for any triangle with sides of lengths a, b, and c then 2 2 2 Key Vocabulary c= a + b - 2 ab cos C . directed line segment Law of Cosines To solve an oblique triangle, the measure of at least one side and any two other parts of the Law of Sines triangle need to be known. This breaks down into the following cases. magnitudes oblique Given Pythagorean Theorem AAS Law of Sines scalar ASA Law of Sines sum and difference formulas SSA Law of Sines (ambiguous case) vector SAS Law of Cosines vector quantity SSS Law of Cosines

Heron’s area formula is used if the lengths of the sides of the triangle are known. If two sides of a triangle and the angle between the two sides are known then the area formula below is used: 1 K = bcsin A 2

123 ALGEBRA II/TRIGONOMETRY CURRICULUM GUIDE (Revised 2013) PRINCE WILLIAM COUNTY SCHOOLS Topic Teacher Notes and Elaborations (continued) Trigonometric Equations, Graphs, and Many quantities in mathematics involve magnitudes. These quantities are called scalar. Other quantities called vector quantities, involve Practical Problems both magnitude and direction. A vector quantity is often represented with a directed line segment, which is called a vector. The length of the vector represents the magnitude of the vector quantity. Each vector has a horizontal and vertical component. Vectors may be added and Virginia SOL AII/T.21 subtracted. The student will identify, create, and solve real-world problems involving Sum and Difference Formulas: triangles. Techniques will include using sin(u+ v ) = sin u cos v + cos u sin v the trigonometric functions, the sin(u- v ) = sin u cos v - cos u sin v Pythagorean Theorem, the Law of cos(u+ v ) = cos u cos v - sin u sin v Sines, and the Law of Cosines. cos(u- v ) = cos u cos v + sin u sin v tanu+ tan v tan(u+ v ) = 1- tanu tan v tanu- tan v tan(u- v ) = 1+ tanu tan v