Algebra 2 Unit and Lesson Overviews...

Algebra II Unit & Lesson Overviews Mathematics

Revised 8/24/2014

Unit: 1.1 Quadratic Functions and Equations Days : 25

Essential Questions

What patterns govern transformations of functions? How can you graph the function f(x)=a(x-h)^2+k? How is the structure of a quadratic equation related to the structure of the parabola it describes? What is the number i? How do you simplify a radical expression? How do you determine where the graph of a quadratic function crosses the x-axis? How do you convert quadratic functions to vertex form? When does a quadratic equation have non-real solutions? How do you find all of the possible roots of a Quadratic Equation?

Content to be Learned Skills

Graphing Quadratic Functions using Transformations

Properties of Quadratic Functions in Standard Form Creating a table of values using technology Students will rewrite functions in intercept form, given in standard form Students will identify x-intercepts of a quadratic function when written in intercept form Students will identify vertex of a quadratic function by averaging the zeros Simplify radical expressions Define i as a number in the set of Complex numbers such that i2=-1 Square root of a negative number is an imaginary root. Rewrite a quadratic function in vertex form Students will use the discriminant to identify the number and types of solutions Students will identify the solutions of a quadratic function via the quadratic formula

Graphing with transformations Identifying key features of parabolas Rewrite an equation into vertex form by completing the square Determine the zeros of a quadratic equation by graphing, factoring, completing the square, or the quadratic formula. Identify imaginary roots of radical expressions or quadratic equations.


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Assessments Standards

Quizzes, Test, Common Task CC.9-12.F.BF.3,.5 CC.9-12.A.CED.2 CC.9-12.F.IF.2,7, 7a, 8, 8a, 9 CC.9-12.A.SSE.3, 3a CC.9-12.N.CN.7 CC.9-12.A.REI.4, 4b CC.9-12.N.CN.1, 2, 8

Sample Instructional Activities Resources

“Explorations in CORE MATH” Algebra 2 chapter 2 “Explorations in CORE MATH” Algebra 2


Revised 8/24/2014

Unit 1.1 Intro to Parent Functions Lesson 1 of 8 Days 2

Lesson Focus 1. Standards Addressed 2. Content to be Learned 3. Mathematical Practices 4. Essential Questions

F.BF.3, F.IF.5 Transformations of graphs Look for and make use of structure: As students graph the functions, they will notice patterns.

What patterns govern transformations of functions?

5. Prerequisite Knowledge 6. Essential Vocabulary 7. Possible Misconceptions 8. Teaching Materials

Linear functions, quadratic functions, geometry

Reflecting, translations, stretch, compression

f(x) -> f(x-h) translates the graph to the right h units.

Text

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Introduce how the value of a number on a number line changes when you (a) add or subtract to/from it and (b) multiply it. A students time to work through the activity in chunks, pausing to check for understanding and debrief. Review questioning strategies, particularly “If you multiply the x-coordinate of a point by a whole number, does the point move closer to or further from the y-axis?” The teacher must be sure that the students know that when the function f(x) is replaced by f(x-h), the resulting graph is a horizontal translation h units to the right. Likewise, f(x+h) is a horizontal translation to the h units to the left.

Working in pairs or small groups, students will plotting points and translating them on the coordinate plane. They will be given graphs of functions that will be translated and reflected across the coordinate plane. Through this process, students are expected to notice patterns and then summarize what they’ve noticed. If time permits, students can make posters of their observations to be posted in the classroom for future reference throughout the year.


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Unit 1.1 Transformations of Quadratic Functions

Lesson 2 of 8 Days 3


A.CED.2, F.IF.2, F.IF.7, F.BF.3,F.IF.9

Graphing Quadratic Functions using Transformations

Model with Mathematics: Students will describe the quadratic relationship between the height of a bridge support and the horizontal distance from the base of the support. They will sketch a graph of the quadratic function and identify and use the zeros and the vertex of the function to determine physical quantities.

How can you graph the function f(x)=a(x-h)^2+k?


Graphing functions of the form f(x)=(x-h)^2+k and of the form f(x)=x^2

Vertex form Zero of a function Maximum / Minimum Vertex

Text, graphing software (if possible, one that includes sliders)

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Introduce: The teacher should use a formative assessment before beginning this activity to see if the students have the prerequisites required. If students lack the prerequisites, the teacher will need to complete a mini-lesson. Introduce by asking the students what effect h and k have on the graph of the function f(x)=(x-h)^2+k If accessible to a projector and internet , access the website desmos.com and create a function with sliders prior to this lesson. (If possible to access a computer lab, this may be another option). As the quadratic function transforms across the coordinate plane, ask students to think of real-world

Students are observing the transformations of the parent function f(x)=x^2. They are identify the coordinates of the vertex and making the connection to how that coordinate relates to the function f(x)=a(x-h)^2. Students will learn what a zero of a function is and then given the graph of a quadratic will be able to identify the zeros. Students will reflect on how the a value in f(x)=a(x-h)^2+k effect the graph of f(x)=x^2. Students will label the vertex as either a maximum or minimum and will be able to state the value.


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images that remind them of the shape they are noticing (Particularly when a is -1<a<0 …If no students mentions a bridge, show the image found here http://bit.ly/1k0Izdl) A students time to work through the activity in chunks, pausing to check for understanding and debrief. Review questioning strategies, paying extra attention to make sure students know what the zeros and vertex of a quadratic function are.


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Unit 1.1 Quadratic Functions in Standard Form Lesson 3 of 8 Days 3


F.IF.7 Properties of Quadratic Functions in Standard Form Creating a table of values using technology.

Look for and make use of structure: Students study the structure of parabolas.

How is the structure of a quadratic equation related to the structure of the parabola it describes?


Using transformations to graph quadratic functions.

Parabola Axis of symmetry

Calculator errors Text

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Start the lesson by discussing symmetry. Lead the discussion into the symmetry in a parabola (define parabola), and how symmetry can be used to find points on the parabola.

Students will share-out objects that are symmetric and describe the line of symmetry. Students will create a table of values for quadratic functions using a scientific calculator’s table feature.


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Unit 1.1 Solving Quadratic Equations by Graphing and

Factoring Lesson 4 of 8 Days 3


CC.9-12.A.SSE.3, 3a CC.9-12.CED.2 CC.9-12.F.IF.2, 7, 7a, 8, 8a

• Students will rewrite functions in intercept form, given in standard form

• Students will identify x-intercepts of a quadratic function when written in intercept form.

• Students will identify vertex of a quadratic function by averaging the zeros

• Standard 4 – Modeling Mathematics by completing Example 3 within the lesson

How do you determine where the graph of a quadratic function crosses the x-axis?


• Factoring quadratics in form

Intercept Form • Given ,

students may identify x-intercepts at x = 3 and x = 2, rather than at x = -3, and x = -2

• Given ,

students may identify an x-intercept at x = 3, rather than at x = 3/2.

• Explorations in Core Math – Algebra 2


Revised 8/24/2014

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing)

• Warm Up – Factor a quadratic function in the form in standard form. How can you check to make sure that you have factored correctly?

• Through questioning, highlight key characteristics that students have found for quadratic functions (vertex, zeros, etc.) and introduce intercept form for a quadratic function.

• Monitor student progress through Section 2-3 pgs. 55-59

• Facilitate class discussions led by student presentations to examples within the section.

• Students should begin working on the warm up upon arriving to class.

• Students should work through Section 2-3 pgs. 55-59

• Students will present their work to examples within the activity


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Unit 1.1 Simplify Radicals & Define Imaginary numbers Lesson 5 of 8 Days 2


CC9-12.N.CN.1 Simplify radical expressions Define i as a number in the set of Complex numbers such that i2=-1 Square root of a negative number is an imaginary root.

Look for and make use of structure.

What is the number i? How do you simplify a radical expression?


Square root of a negative number is not a real root. Perfect squares

Complex Number System Imaginary Numbers

Students may inaccurately state

that , or .

Supplementary materials are needed

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Pre-assess the students’ ability to evaluate and simplify square roots of positive numbers. Teacher would then determine if a further review of simplifying radicals is required or if the class can move on to radicals of negative numbers. The teacher must define the number i.

Students should be simplifying radical expression with real and complex roots.


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Unit 1.1 Completing the Square to Solve Quadratics Lesson 6 of 8 Days 3


CC.9-12.A.SSE.3, 3b CC.9-12.CED.2 CC.9-12.F.IF.2, 7, 7a, 8, 8a

• Students will rewrite a quadratic function in vertex form from standard form.

• Standard 8 – Look for and express regularity in repeated reasoning

• How do you convert quadratic functions to vertex form?


• Graphing quadratic functions in vertex form

• Standard form • Students may not factor out a value from

• Students may forget to keep

the function the same by either adding or subtracting.

• Students may forget if there is an a value, to distribute to the last term to determine what needs to be added/subtracted to keep the function the same

• Explorations in Core Math – Algebra 2 section 2-4


• Warm Up – Rewrite a function in standard form from vertex form

• Through questioning, highlight that different forms for a quadratic function (vertex, standard, intercept) and emphasize that it is possible to covert between the forms, as seen by the warm up problem. Introduce the lesson by asking the essential question, How do you convert a quadratic function in standard from to vertex form?

NOTE TO INSTRUCTOR This lesson may be better presented via direct instruction

• Lesson is presented on pgs. 63-67

• Students should begin working on the warm up upon arriving to class.


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Unit 1.1 Solving Quadratic Equations using the Quadratic

Formula Lesson 7 of 8 Days 1


CC.9-12.N.CN.7 CC.9-12.A.REI.4, 4b CC.9-12.N.CN.1, 2, 8

• Students will use the discriminant to identify the number and types of solutions

• Students will identify the solutions of a quadratic function via the quadratic formula

Standard 6 – Attend to Precision • When does a quadratic equation have non-real solutions?

• How do you find all of the possible roots of a Quadratic Equation?


• Quadratic Formula (NOTE - this may or may not have been covered in Algebra 1

• Multiplying binomials

• Multiplying complex numbers

• Discriminant

• Students may state that there is no solution when the determinant is negative

• Students may forget that the quadratic formula yields two

answers due to the

• Explorations in Core Math – Algebra 2

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) NOTE TO INSTRUCTOR This lesson only asks students to identify solutions which are complex within the lesson. It may be a good idea to incorporate Additional Practice on p. 81 #1-6 which asks students to identify real solutions of quadratic equations to get them comfortable/reacquaint them with the quadratic formula.

• Introduce/Reacquaint students with the Quadratic Formula as a method to finding solutions of quadratic equations

• Monitor student progress through Section 2-6 pgs. 77-79

• Facilitate class discussions led by student presentations to examples within the section.

• Students will work through Section 2-6 pgs. 77-79

• Students will present their work to examples within the activity


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Unit 1.1 Modeling with Quadratic Functions Lesson 8 of 8 Days 2


CC.9-12.N.Q.1 CC.9-12.N.Q.2 CC.9-12.A.SSE.1 CC.9-12.A.APR.1 CC.9-12.A.CED.1 CC.9-12.A.REI.3 CC.9-12.F.IF.2 CC.9-12.F.BF.1

How to model real-world phenomena using a quadratic function. (The major topics of this unit are embedded and applied throughout this activity.)

Modeling with Math; Construct Viable Arguments and critique the arguments of others.

How can you model the changes in revenue from season ticket sales using a quadratic function?


Quadratic functions in intercept form; quadratic functions in vertex form; real solutions of quadratic equations; n-th term

Revenue, profit, discrete, upper-bound, lower-bound

Students may have difficulty writing an expression for the number of tickets holders after n price increases. Have the students create a table of values.

Explorations in Core-Math

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Have students discuss what happens when a business raises the price of an item (Why? Is it justified?)

Students should be working through section 2-8 pgs. 87-90 and presenting their work throughout the lesson.


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Unit: 1.2 Polynomial Expressions Days : 7

Essential Questions

How do you add, subtract, and multiply complex numbers? How do you add and subtract polynomials? How do you multiply polynomials?


Operations with Complex Numbers, Operations with Polynomial Expressions

• Adding and Subtracting Complex Numbers

• Multiplying Complex Numbers

• Dividing Complex Numbers

• Conjugates of Complex Numbers

• Adding Polynomials

• Subtracting Polynomials

• Multiplying Polynomials

Assessments Assessments

TBD CC.9.12.N.CN.1, 2, 3(+) CC.9.12.A.APR.1 CC.9.12.F.BF.1, 1b CC.9-12.A.SSE.1 CC.9.12.A.APR.1, 4, 5(+)


“Explorations in CORE MATH” Algebra 2 chapter 2.9, 3.1, and

3.2

“Explorations in CORE MATH” Algebra 2


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Unit 1.2 Operations on Complex Numbers Lesson 1 of 3 Days 2


CC.9.12.N.CN.1 CC.9.12.N.CN.2 CC.9-12.CN.3(+)

• Adding and Subtracting Complex Numbers

• Multiplying Complex Numbers

• Dividing Complex Numbers

• Conjugates of Complex Numbers

• Understanding Absolute Values of Complex Numbers

Mathematical Practice Standard 7 – Look for and make use of structure.

How do you add, subtract, and multiply complex numbers?


• Solving quadratic equations

• Adding and multiplying binomials

• Pythagorean Theorem

• Imaginary unit

• Imaginary number

• Conjugate

• Complex number

• Pure imaginary number

• Absolute value

Students sometimes confuse i = -1 instead of i2 = -1. As well as they believe the square root of -1 = i. Students also tend to leave i2 when the foil complex numbers and do not further simplify

Explorations in Core Math – Algebra 2 book Calculator (if preferred)


Revised 8/24/2014

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Introduce the lesson by reviewing the discriminant b2 – 4ac of the quadratic formula. Remind the students that if the discriminant is negative, a quadratic function has no real solutions because it would require taking the square root of a negative number. Students should also have a quick review or reminder of multiplying binomials using the FOIL method. Be sure to remind them that i2 can replace -1 in order to simplify. Students can then be instructed to work on the activity in their books on pg.95 – 99. The teacher should be facilitating the students as they are working on the activity. They may use the questioning strategies in the book to further help the students understand. This section may need to be separated into a two day lesson. It may be appropriate to stop the students at Example 4 on day one. Homework can be found on pg. 100 with additional practice problems on pg. 101 and a

Students should be following the teacher’s introduction to the lesson. Working on any review problems/questions the teacher poses. Students will then work on the activity on pg. 95 – 99 in the student edition. Students should be following up with assigned homework problems.


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Unit 1.2 Addition and Subtraction of Polynomial Expressions Lesson 2 of 3 Days 1


CC.9.12.A.APR.1 CC.9.12.F.BF.1 CC.9-12.F.BF.1b Also:CC.9-12.A.SSE.1

• Adding Polynomials

• Subtracting Polynomials

Mathematical Practice Standard 4- Model with mathematics

How do you add and subtract polynomials?


• Closure of the set of integers

• Defining of polynomials

• No new vocabulary introduced in this unit.

Students sometimes do not distribute the subtraction operation to all terms of the second polynomial, only the first term is subtracted and they then add the rest. Students should be reminded of this and to check their work.

Explorations in Core Math – Algebra 2 book

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Introduce the lesson by reviewing operations on integers, emphasizing that the integer system is closed under the operations of addition, subtraction, and multiplication. The teacher may also want to review what it means to be “like” terms. Students can then be instructed to complete the activity on page 111-113 as the teacher walks around the room and facilitates the students learning. Homework or reinforcement problems can be found on pg.114-115 with a problem solving problem on p. 116.

Students should be following the teacher’s introduction to the lesson. Working on any review problems/questions the teacher poses. Students will then work on the activity on pg. 111-113 in the student edition. Students should be following up with assigned homework problems.


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Unit 1.2 Multiplication of Polynomial Expression Lesson 3 of 3 Days 3


CC.9.12.A.APR.1 CC.9.12.A.APR.4 CC.9-12.A.APR.5(+)

• Multiplying Polynomials

•

Mathematical Practice Standard 3-Construct viable arguments and critique the reasoning of others.

How do you multiply polynomials?


• Closure of the set of integers

• Multiplying binomials

• No new vocabulary introduced in this unit.

When using the rules for special products, students often forget to apply the rules of exponents to the coefficients of terms in the binomial. .

Explorations in Core Math – Algebra 2 book

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Introduce the lesson by reviewing polynomial addition/subtraction emphasizing that like integers, polynomials are closed under addition and subtraction. Students may also benefit from reviewing the FOIL method, specifying the FOIL method only works when multiplying binomials but the process of multiplying each term of the first polynomial must be multiplied by each term of the second polynomial. Teacher will then facilitate the students as they proceed with the activity on pgs. 117-123* Teachers of Algebra 2B may skip the Explore 4 & 5 regarding Pascale’s Triangle/Binomial Thereom. Homework/Practice problem can be found on pgs. 124-126*

Students should be following the teacher’s introduction to the lesson. Working on any review problems/questions the teacher poses. Students will then work on the activity on pg. 117-123* in the student edition. Students should be following up with assigned homework problems.


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Unit: 2.1 Title: Polynomial Equations and Functions Days : 18

Essential Questions

How can you graph polynomial functions using characteristics and zeros? How can you determine all real zeros of a polynomial function? Can you determine and verify under which operations the set of polynomial expressions is closed? (+)How can you find the zeros over the set of Complex numbers of a polynomial with real coefficients?


Factor a polynomial expression to find its zeros Divide 2 polynomial expressions Use the remainder theorem to determine if a number is a root/zero Use the factor theorem to determine if a number is a root/zero Graph a polynomial function using its roots and end behavior Write an equation of a polynomial using its known zeros Show polynomials are closed under same operations as integers Use the Fundamental Theorem of Algebra to show polynomials are solvable over the set of Complex numbers

Operations with polynomials, specifically division and factoring; using rational root and remainder theorems to determine zeros; transformations of parent graphs, using end behavior and zeroes to graph polynomials; modeling volume using polynomial expressions and equations


Quizzes, test, common task A.APR.1 Understand polynomials are a system similar to the integers, namely, they are closed under the

operations of addition, subtraction, and multiplication.

A.APR.2 Know and apply the Remainder Theorem. For a polynomial p(x) and a number a, the remainder on

division by x-a is p(a), so p(a) = 0 if and only if (x-a) is a factor of p(x).

A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to

construct a rough graph of the function defined by the polynomial.

A.CED.2 Create equations in two variables to represent relationships between quantities; graph equations.

F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases

and using technology for more complicated cases.

F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and

showing end behavior.

N.CN.9(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic equations.

F.BF.3 Identify the effects on the graph of f(x) by replacing(x) by f(kx), f(x) + k, kf(x), and f(x+k) for both

positive and negative values of k.


“Explorations in CORE MATH” Unit 3, sections 3-9. “Explorations in CORE MATH” Algebra 2


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Unit 2.1 Dividing Polynomials Lesson 1 of 7 Days 3

Lesson Focus 1. Standards Addressed 2. Content to be Learned 3. Mathematical Practices 4. Essential Questions A.APR.1 Understand polynomials

are a system similar to the integers,

namely, they are closed under the

operations of addition, subtraction,

and multiplication.

A.APR.2 Know and apply the

Remainder Theorem. For a

polynomial p(x) and a number a, the

remainder on division by x-a is p(a),

so p(a) = 0 if and only if (x-a) is a

factor of p(x).

Divide 2 polynomial expressions Use the remainder theorem to determine if a number is a root/zero Use the factor theorem to determine if a number is a root/zero


What is the relationship between polynomial division and the remainder theorem?


Addition, subtraction, and multiplication of polynomials The closure property as it relates to the set of polynomials under addition, subtraction, and multiplication Long division of integers

Polynomial division Synthetic division Synthetic substitution

Synthetic division is a process of division that is seemingly outside of the context of division. Students should be warned about common errors, such as copying incorrect signs of coefficients, or using the incorrect sign of the root being used as the divisor, and the importance of a zero place holder for “missing” powers of x in the dividend. Students also should recognize that synthetic division only applies when the divisor is a first degree binomial.

Explorations in CORE MATH Algebra 2, section 3.3 pg. 129

Graphing calculators/technology


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Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Review the closure property of integers under the operations of addition, subtraction, and multiplication. Ask the students to justify if the set of integers is closed under division and show a counterexample. Have the students show that polynomials are also not closed under division and show a counter example using long division. Teachers should have students work on examples of division using long division and look for patterns in the number of terms in the quotient as well as the resulting degree of the quotient. Students should be given examples of polynomials in descending order which have “missing” terms and recognize that these terms have a coefficient of zero so that they will understand why a placeholder of zero may be used in both long and synthetic division. Teachers may then show the students the process of synthetic division as a shortcut for division by a binomial of degree one. Teachers should show the students the connection between the remainder theorem and division of polynomials. Students at the upper level may benefit from proving this theorem.

Students should attempt to find a counter example to show that the set of polynomials is not closed under the operation of division. Students may benefit from working together. Students should be working on examples of long division for at least one day and then examples division using the synthetic division process. Students should be showing that the remainder theorem yields the same value as the remainder that resulted from division of a polynomial if it is divided by a first degree binomial. Students should then be using the remainder theorem to determine the value of the remainder if the two polynomials were to be divided.


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Unit 2.1 Factoring Polynomials Lesson 2 of 7 Days 1


A.APR.3 Identify zeros of

polynomials when suitable

factorizations are available, and use

the zeros to construct a rough graph

of the function defined by the

polynomial.

Factor a polynomial expression to find its zeros Divide 2 polynomial expressions Use the factor theorem to determine if a number is a root/zero

Attend to Precision. What is the relationship between polynomial division and the Factor Theorem?


Division of Polynomials Multiplication of Polynomials

Factors, Factoring It is unnecessary to use the nested form as described in the CORE MATH text.

Explorations in CORE MATH Algebra 2, section 3.4 pg. 135


Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Teachers could ask students, what is the remainder of a number when it is divided by one of its factors? They can then connect this idea to the division of polynomials. Teachers may have students use the remainder theorem, multiplication, or their graphing calculators to check the results obtained by division and justify if in fact a divisor is a factor of the polynomial.

Students should be connecting the remainder theorem to the factor theorem using division of polynomials. Students should be able to identify which binomials are factors of a given polynomial and which are not based on the remainder when division or the remainder theorem is used.


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Unit 2.1 Finding Real Roots of Polynomial Equations Lesson 3 of 7 Days: 4


A.APR.3 Identify zeros of

polynomials when suitable

factorizations are available, and use

the zeros to construct a rough graph

of the function defined by the

polynomial.

A.CED.2 Create equations in two

variables to represent relationships

between quantities; graph

equations.

F.IF.7 Graph functions expressed

symbolically and show key features

of the graph, by hand in simple

cases and using technology for more

complicated cases.

F.IF.7c Graph polynomial functions,

identifying zeros when suitable

factorizations are available, and


N.CN.9(+) Know the Fundamental

Theorem of Algebra; show that it is

true for quadratic equations.

Determine the zeros of a polynomial equation by factoring, using the Factor Theorem or the Remainder Theorem and the Rational Root Theorem. Graph a polynomial using roots/zeros and end behavior.

Construct viable arguments and critique the reasoning of others. Look for and express regularity in repeated reasoning.

How can you determine the real roots of a polynomial? How can you use the real roots and end behavior to graph a polynomial function written in factored or standard form?


Understanding end behavior Determine end behavior Dividing Polynomials

Polynomial, degree of a monomial or polynomial, polynomial function

The degree of a polynomial is the highest-degree term; but, in factored form the students must multiply all of the first terms in the parenthesis to determine the highest-degree. If a parenthesis is to a power, then the students must repeat this term that number

Explorations in CORE MATH

Algebra 2, section 3.5 pg. 139



Revised 8/24/2014

of times. Students often have a difficult time understanding how the multiplicity of a zero affects the graph; for example, they may have a difficult time with the graph being tangent to the x-axis if there is a double root but will pass through the x-axis if there is a triple root.

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Review the degree of polynomials with single-variable monomials. Draw a general polynomial on the board. Discuss the key features of the graph, such as end behavior, roots, and critical points. Ask students how they could determine the degree of the polynomial represented by the graph. Guide students through activities/questions that connect the end behaviors of parent graphs to more complicated polynomials of the same degree for both positive and negative lead coefficients, identifying zeros from graphs, using the number of zeros to determine the degree, using the number of maxima or minima to determine the degree. Have students graph polynomials in factored form using the zeros and the end behavior, and the multiplicity of each zero of the functions based on the number of times the zero is repeated in the equation or from the graph(i.e. does the graph bounce off the axis or pass through the axis?) Teachers could have students graph the polynomials on graphing calculators and zoom in at the zero to see that if the zero occurs once, it looks like y = x, if the zero occurs twice, it looks like y = x2, if the zero occurs tree times, then it looks like y = x3. Have students compare the zeros of polynomials found from the factored form to the expanded standard form of the equation. Can the students find a pattern that leads to the Rational Root Theorem? Have students use the Rational Root Theorem to determine all possible rational roots.

Students should be connecting the end behavior of polynomials in standard from to the end behavior of x2, x3, or x4. Students should be comparing the graphs of polynomials with positive lead coefficients versus negative lead coefficients. Students should be creating rough graphs of polynomials in factored form using the zeros and the end behavior. Students should be using the rational root theorem to determine possible rational roots of the polynomial. Students should use the Remainder or Factor Theorems to determine which of the possible rational roots are real zeros. Once the rational roots have been determined, students may use division to reduce the polynomial to linear or quadratic factors which may be solved to determine irrational or complex roots. Students may use graphing calculators to determine real zeros as well as to check their solutions for real zeros. Students are writing the equations of polynomials given its rational zeros.


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Unit 2.1 Fundamental Theorem of Algebra Lesson 4 of 7 Days 2

Lesson Focus 1. Standards Addressed 2. Content to be Learned 3. Mathematical Practices 4. Essential Questions A.APR.2 Know and apply the

Remainder Theorem. For a

polynomial p(x) and a number a, the

remainder on division by x-a is p(a),

so p(a) = 0 if and only if (x-a) is a

factor of p(x).

N.CN.9(+) Know the Fundamental

Theorem of Algebra; show that it is

true for quadratic equations.

Use the Fundamental Theorem of Algebra to show polynomials are solvable over the set of Complex numbers.


How can you find the zeros of polynomial functions?


Recognize special products Expand powers of binomials Finding rational zeros

Polynomial, degree of a polynomial, rational zeros

Students will sometimes misrepresent the exponents when factoring special patterns such as x4-25 as (x+5)(x-5) instead of (x2+5)(x2-5). Students will sometimes incorrectly state that the zero of x+4=0 is x=4 instead of x=-4. Students may not realize that the rational zero theorem does not work when the polynomial has no rational zeros.

Explorations in CORE MATH

Algebra 2, section 3.6 pg. 152


Revised 8/24/2014

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Present students with an activity to recognize special patterns of factoring. Show students how to use the rational zeros they have found to factor a polynomial by synthetic or long division. Have students recognize that if a polynomial of degree n has at least 1 complex zero and exactly n zeros is repeated zeros are counted multiple times. Have students identify different methods that may be used to determine the zeros of a polynomial in the set of complex numbers. Have students use technology to determine irrational zeros of a polynomial that may or may not have a rational zero.

Students should be using synthetic division to identify rational zeros of a polynomial and then any other factors of the polynomial. Students should reflect on the zeros of a polynomial. They should be performing activities and answering questions such as: How does the number of zeros compare to the degree of the polynomial? Did the list of possible rational zeros contain all of the zeros of the polynomial, if not which were excluded and how can these zeros be obtained? How does the list of possible zeros assist in determining the zeros of the polynomial? How can you determine the zeros of a polynomial that does not have any rational zeros? Etc.


Revised 8/24/2014

Unit 2.1 Investigating Graphs of Polynomials Lesson 5 of 7 Days 1






complicated cases.





F.BF.3 Identify the effect on the

graph of replacing f(x) by …f(kx)

for specific values of k…

Students should be able to determine if a function is odd/even in the form f(x) = xn. Students should be able to determine and describe the end behavior of a function in the form of f(x) = xn.

Use appropriate tools strategically. Look for and make use of structure. Look for and express regularity in repeated reasoning.

How does the value of n affect the behavior of the function f(x)=xn, where n is any whole number?


Graphs of the functions f(x) = x and f(x) = x2 as well as their end behavior and symmetry.

End Behavior Even Function Odd Function

Students will sometimes confuse f(x) = -x2 with f(x) = (-x)2, thus will end up with a positive y value when substituting a negative x value into f(x) instead of the negative or vice versa.

Graph paper Graphing calculator Explorations in CORE MATH Algebra 2, section 3.7 pg. 159.

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Teacher introduces the lesson by reviewing the parent graph of f(x) = x and f(x) = x2, asking the students to point out common characteristics. Teacher should introduce the vocabulary of end behavior and even versus odd functions. Teacher helps students while they are working on their graphs of polynomials of various degrees. Teacher brings the class together and discusses the findings and eliminates any common misconceptions.

Students are using a table of values or a graphing tool to graph functions in the form f(x) = xn, for various positive integer values of n. Students should then reflect about what is happening as x increases without bound or as x decreases without bound. Students should make a conjecture relating the end behavior of f(x) to the value of n when n is odd verses even. Students should then practice using this relationship to sketch graphs.


Revised 8/24/2014

Unit 2.1 Transforming Polynomial Functions Lesson 6 of 7 Days 1






complicated cases.





F.BF.3 Identify the effect on the

graph of replacing f(x) by f(x)+k,

kf(x) and f(x+k) for specific

values of k both positive and

negative…

Students will be able to determine the effects of the constants a, h and k on the graph of the function f(x) in the form of f(x) = af(x-h) +k.

Use appropriate tools strategically. Look for and make use of structure. Look for and express regularity in repeated reasoning.

What are the effects of the constants a, h and k on the graph of f(x) = a(x-h)n + k?


Graph f(x) = xn Translations of graphs

No new vocabulary Students may misinterpret the sign of h in (x-h)n and obtain the opposite horizontal translation, saying the graph should shift left when in fact it is shift right.

Graph paper Graphing calculator Explorations in CORE MATH Algebra 2, section 3.8, pg. 165


Revised 8/24/2014

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Teacher will give a quick warm-up on the parent graphs of f(x) = xn for n as a positive integer (such as x2, x3, and x4). Teacher will either use the book or create their own activity to explore the effects of a, h, and k on the graphs of common polynomials. The activity should explore both positive and negative values of a, h, and k. To save time, you may wish to break the students into groups and have each group work on a different transformation and then share out to the entire class. The teacher should facilitate the sharing out of the students’ work and ask questions such as “When graphing horizontal and vertical translations in the same graph, does it matter if you graph the horizontal and then the vertical or vertical and then horizontal?” “Does the effect of a, h, or k change depending on the degree of f(x)?” Teacher should present graphs which have already been translated and have the students determine the algebraic representation for the function.

Students are working on creating graphs to explore the rules of transformations. Students may use technology to create the graphs since the focus is on the effect of the constants in the equation and the resulting transformation. Students may benefit from working in groups on this activity. Students will be making conjectures and summarizing the results of this activity.


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Unit 2.1 3-9 Curve Fitting with Polynomial Models Lesson 7 of 7 Days 2


A.SSE.1 Interpret expressions

that represent a quantity in terms

of its context.

A.CED.2 Create equation in two

variables to represent

relationships between quantities.

F.IF. 5 Relate the domain of a

function to the quantitative

relationship it describes. F.BF. 1a Determine an explicit

expression from a context. F.IF.7 Graph functions expressed




complicated cases.





• Write a function that represents volume of a box

• Determine the domain and range of volume function

• Write and solve a polynomial equation

• Graph polynomial function

• Determine maximum values

Standard 4 – Model with Mathematics

How can you use polynomial functions to model and solve real-world problems?


• Graph polynomial functions

• Operation on polynomials

• Solving polynomial equations

• Rational Zero Theorem

• Volume

• Students may graph the volume function without restricting the domain

• Students may associate the height with “zeros” and not dimensions of the box

• Explorations in Core Math

• Graph paper

• Graphing calculator

• Paper to model situation (i.e. actually cutting out corners of a sheet of paper and folding it into an open rectangular box

Suggested Learning Practices


Revised 8/24/2014

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Teacher can begin by modeling the situation (cutting out corners of a

box and folding it into an open rectangular box) and explain that they will be asked to find dimensions (length, width, height) that will create a box with a given volume.

• Teacher monitors students as they complete the activity.

• Students will work through Activity 3-9 (p. 177-180)


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Unit: 2.2 Exponential and Logarithmic Functions Days : 15

Essential Questions

What are the characteristics of an exponential function? How do you find the inverse of a function, and how is the original function related to its inverse? What are the characteristics of logarithmic functions? What are the properties of logarithms and how are they proved? How can logarithms be used to solve exponential equations? What are some of the general processes to solving equations that contain logarithms?

How does the graph of compare to graphs of exponential functions with other bases?

How does changing the values of a, h, and k affect the graph of ? How can you model the time it takes a radioactive substance to decay as a function of the percent of the substance remaining?


Graphing exponential functions. The inverse relation of a function can be used to input the original functions range value(or output value) to obtain the original functions domain value (or input value). The inverse of an exponential function is a logarithmic function. The properties used to simplify logarithmic expressions. Use logarithms to solve exponential equations.

Graphing exponential functions, graphing logarithmic functions, identifying asymptotes, identify the domain and range of exponential and logarithmic functions, simplifying and evaluating logarithmic expressions, and solving equations that contain exponential or logarithmic terms.


TBA F.BF.5 F.BF.4 and 4a F.BF.3

A.CED.1 A.CED.2

A.SSE.3c

A.REI.11

F.LE.3 F.LE.5 F.IF2 F.IF.7 and 7e


“Explorations in CORE MATH” Algebra 2 activities in unit 4.1-

4.8



Revised 8/24/2014

Unit 2.2 Exponential Functions, Growth and Decay Lesson 1 of 8 Days 2


F.IF.7e F.LE.3

Graphing exponentials by hand Look for and express regularity in repeated reasoning (The outputs keep doubling so f(x) can be determined without actually evaluating the function.)

What are the characteristics of an exponential function?


Evaluating integer exponents; Properties of exponents

Exponential function; exponential growth function; exponential decay function; Asymptote

f(x)=x2 verses f(x)=2x

2-3 = -6 (negative exponents) Exponent rules (product verses power properties)

Calculators with graphing and/or table of values capabilities; Explorations in CORE MATH

Algebra 2 pages 191-194; Chart paper, whiteboards

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) I. Providing warm-ups that refresh integer exponents and properties of exponents (NECAP 2012 #10; NECAP 2010 #17) II Monitor student activity as they work in pairs or small groups through the activity. The teachers should familiarize themselves with the questioning strategies. III The teacher poses the following scenario: “For the next 365 days, would you rather be given $100 each day or instead be given 1 penny on day 1, two pennies on day 2, four pennies on day 3, and so on. (Each sequential day you would get double the number of pennies you got on the previous Assuming you want to have the most money possible, of course. Explain and include details.

I. Using whiteboards in pairs or small groups responding to the warm-ups. II. Following the directions throughout the activity. III Work in groups of 3 or 4 and coming to a conclusion to the “Money” problem. If able, allow students to create posters of their thoughts and post them around the room. At the end of the unit, students can have an opportunity to respond to their initial thoughts – either defend or change their initial opinions.


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Unit 2.2 Inverses of Relations and Functions Lesson 2 of 8 Days 1


CC.9-12.F.BF.4 CC.9-12.F.BF.4a

A function takes an input, applies a rule, and gives an output. The inverse of that function will use that output as its input and apply a rule to give the input of the original function as its output.

Model with mathematics (After students find the inverse of a function, they must be able to interpret it in the context of the situation. Discuss how units are switched in the domain and range because the input and the output of the function are reversed.)

How do you find the inverse of a function, and how is the original function related to its inverse?


Functions Relations mapping Graphing linear functions Solving equations for a variable

Inverse functions After an activity addressing that a negative exponent is the reciprocal of its base, we introduce the inverse of a function’s notation as f-1(x). However, this is strictly notation, not the reciprocal of f(x). Students having difficulty working backwards can benefit from a flow chart approach.

Calculators with graphing and/or table of values capabilities; Explorations in CORE MATH

Algebra 2 pages 191-194


Revised 8/24/2014

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) I. I. Providing warm-ups that refresh solving an equation for a variable (2x-3y=9…solve for y), and graphing linear functions (must include the line y=x). II. Play the function game: Give a (mathematically strong) student an index with the following rules: [i] multiply input by 2 [ii] add 3 to the result [iii] give the output. The class will shout out various inputs and the student with the index card will give the corresponding output. Have another student create a mapping of the inputs/outputs in front of the class. All other students are to figure out what the rules are that are being applied. Once the rule is determined, ask the students to figure out what input gives an output of 39. Make sure they are clear and concise in their response. III Guide students through the activity 4-2 on pages 199 through 202.

I In pairs or small groups, students are working on the warm-ups in their daily journals or on whiteboards. They will present their work with other groups or in a whole class discussion. II. The students will play the function game as described in the Instruction practices. They will figure out how to determine the input when 39 is the output on whiteboards and will present their whiteboards with another group for critique. III Students can work through the activity


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Unit 2.2 Logarithmic Functions Lesson 3 of 8 Days 2


F.IF.7e F.BF.5

The inverse of an exponential function is a logarithmic function. Therefore, the range of an exponential function becomes the domain of a logarithmic function and the domain of an exponential function becomes the range of the logarithmic function. Students will evaluate simple logarithmic expressions and graph logarithmic functions by using a table of values.

Reason abstractly and quantitatively. (Have students find the two consecutive integers that f(40) lies between for [f(x)=log4 x] and have them explain why it lies there.)

What are the characteristics of logarithmic functions?


Basic exponential functions Inverses of functions Mentally calculating bm for 0<b<5 and -5<m<5.

One-to-one functions Logarithm Logarithmic function Common logarithm Natural logarithm

Point out that the base is written as a subscript in a logarithm. Have students write “10” as the subscript until they are comfortable using common logarithms

Explorations in CORE Math (page208-210) *Calculators are NOT recommended for this lesson* Multicolor Post-It notes


Revised 8/24/2014

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) I. Warm-up: Follow the “Introduce” instructions on page 207. (Students are presented with an exponential equation, but the exponent is missing.) II. On the board, write f(x) = 3x, but replace x with a sticky note (1 color) and replace f(x) with a different color sticky note. Have students give a value of a x, then the corresponding f(x). Create a mapping under the function and move the x-sticky note to the domain and the f(x)-sticky note to the range. Repeat this until you have 5 or 6 values (try to include zero, 1, and some negative x-values) in the domain. Next create a new mapping and remove the values in the range and place them in the new mapping’s domain, and remove the values in the domain and place them in the new mapping’s range. Depending on the level of the class, you may challenge them to ask if they know a rule/function that models the mapping. Common answers are Sqrt(x) or xth root of x. Address these misconceptions or show how the function doesn’t model the mapping. Write “the power3 that gives x is f(x)” and then one at a time, remove a value from the mapping’s domain and range and place them over the x and f(x) in the statement. Continue to replace the values from the domain and range into the statement and have students practice answering them. (relate this to the warm-up). Finally, explain that mathematicians use the word logarithm in place of the words “the power”. You may want to have students research the history as to why (John Napier in the 1500s). III. Monitor students as they complete the activities on pages 207 through 210. Allow time to debrief and to clarify any misconceptions. Monitor the students conversations, and familiarize yourself with the questioning strategies in the teacher’s edition.

I. Students will work in pairs or small groups determining the value of the missing exponent that makes the equation true. II. The students will follow along to the teacher’s guided instructions / lesson . III. Students will complete the activities in pairs or small groups.


Revised 8/24/2014

Unit 2.2 Properties of Logarithms Lesson 4 of 8 Days 2


F.BF.5 Properties of Logarithms Construct viable arguments and critique the reasoning of others (Students write formal proofs for the properties, verify the properties by showing they work in certain cases, and confirm the properties by graphing equivalent equations on a graphing calculator.)

What are the properties of logarithms and how are they proved?


Properties of exponents Logarithmic functions as inverses of exponential functions

N/A log

log loglog

aa b

b− =

A graphing calculator; Explorations in CORE Math (pages 215-216)


Revised 8/24/2014

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) I. Warm-up: Follow the “Introduce” instructions on page 215. (Students are presented with an exponential expressions and applying the properties of exponents to simplify expressions.) II. Give students examples of each property of logarithms using integers. Give students some of the common misconceptions (see above) and explain/ask why they are wrong. If possible, (Highly recommended), go to http://www.desmos.com and have students verify that f(x)=log(ax) and g(x)=log(a) + log (x) are equivalent functions by comparing their graphs. III Criteria of organizer: Comparison of logarithm properties with corresponding exponent properties; graphs of 1 or 2 examples of incorrect properties of logarithms commonly seen.

I. Students will work in pairs or small groups determining the value of the missing exponent that makes the equation true. II. Students will copy the examples the teacher gives on the board into their books next to the properties of logarithms (page 215). Have students verify other properties of logarithms using their graphing calculators. Also students will verify that log (a+b) is not equal to log (a) + log (b). III Students will create a graphic organizer / foldable that compares the properties of logarithms with the corresponding properties of exponents.


Revised 8/24/2014

Unit 2.2 Solving Exponential and Logarithmic Equations Lesson 5 of 8 Days 2


A.CED.1 F.IF.7 A.SSE.3c A.REI.11

Day1: Students will learn how to solve exponential equations algebraically with a table or graph using their graphing calculators. The equations are all in a form such that each sides is a power with the same base.(Read teacher notes page 221) Day2: Students will solve exponential equations by taking the logarithm of both sides. Students will solve some logarithmic equations graphically.

Use appropriate tools strategically. (For all equations solved algebraically, insist that students show their work on paper so that you can see they understand the property of equality for exponential equations. They should be allowed and encouraged to use a calculator to check those answers.)

What is the general process for solving exponential and logarithmic equations?


Exponential growth and decay functions; Properties of exponents; Interest compounded continuously; Solving exponential equations; Properties of logarithms

N/a

( )

1 2 1

2 11

9 3

9 3

x x

xx

+ +

++

≠

= ---------------------------------- log85

log 2 On a calculator, be sure to close the parentheses for log 85 before dividing by log 2.

Explorations in CORE Math (page208-210) Graphing calculator


Revised 8/24/2014

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) I. Warm-up: Follow the “Introduce” instructions on page 221. (Students are presented with an exponential equations to solve.) II. The activities in this activity are designed to be student centered. Allow students plenty of time to make it “their own” and to follow the instructions in the book. Avoid answering questions too quickly – instead redirect students to where they can find the answer to their questions. Monitor students working in groups. Encourage the use of whiteboards and share-out exemplary work with the class. Familiarize yourself with the questioning strategies presented throughout the section in the teacher’s edition.

I Students will work in pairs or small groups solving basic exponential equations.

II. Students are working in groups of 3 or 4 throughout the activity. They should be presenting their work on whiteboards and then once the group agrees to an answer, the should copy the work into their books.


Revised 8/24/2014

Unit 2.2 4-6 The Natural Base, Lesson 6 of 8 Days 1


A.CED.2 Create equation in two

variables to represent relationships

between quantities; graph equations

on coordinate axes with labels and

scales.

F.IF2 Use function notation,

evaluate functions for inputs in their

domains.





complicated cases.

F.IF.7e Graph exponential…

functions, showing intercepts and

end behavior…

F.BF.3 Identify the effects on the

graph of f(x) by replacing(x) by

f(kx), f(x) + k, kf(x), and f(x+k) for

both positive and negative values of

k. F.LE. 5 Interpret the parameters in

a … exponential function in terms of

a context.

• Students will investigate the meaning of both graphically and numerically

• Students will compare the graph of to exponential functions with other bases

• Translations of

Standard 7 – Look for and Make Use of Structure

How does the graph of

compare to graphs of exponential functions with other bases?


• Graph basic exponential functions

• Transforming exponential growth functions

• Definition of irrational numbers

• as an irrational number • Students may treat e as a variable rather than a constant

• Explorations in Core Math section 4-6 pg. 233



Revised 8/24/2014

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Teacher demonstrates with a graphing calculator how to input

and create a table of values

• Teacher monitors students as they work through the activity and debrief after each set of examples and reflection questions

• Students should be working through the activity on pages 233-236


Revised 8/24/2014

Unit 2.2 Transforming Exponential and Logarithmic

Functions Lesson 7 of 8 Days 1






complicated cases.

F.IF.7e Graph exponential…

functions, showing intercepts and

end behavior… F.BF.3 Identify the effects on the

graph of f(x) by replacing(x) by

f(kx), f(x) + k, kf(x), and f(x+k) for

both positive and negative values of

k.

• Determining how a, h, and k affect the graph of

Standard 7 – Look for and Make Use of Structure

• How does changing the values of a, h, and k affect the graph of

?


• Graphing basic exponential functions and logarithmic functions

• Graphing inverses

• Students may write a horizontal asymptote in the

form of and vertical asymptotes in the form of

• Students may interchange the definitions of stretch vs. shrink

• Students may interpret an a value as a factor of determining if an exponential function is growth or decay

• Students may think that h is negative when seeing functions in form

• Explorations in Core Math section 4-7 pg. 239



Revised 8/24/2014

and h is positive when given

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Teacher can provide examples of a quadratic or polynomial function

and ask students to identify a, h, and k values as well as describe all transformations on the parent function.

• Teacher will need to supplement this section with transformations of exponential function as the book addresses solely logarithmic functions.

• Students should work through the activity on pages 239-241.

• Teacher can assign additional practice after the activity.


Revised 8/24/2014

Unit 2.2 Curve Fitting with Exponential and Logarithmic

Models Lesson 8 of 8 Days 2


A.CED.2, 4 F.IF.2, 4 F.BF.1, 1a

• Model radioactive decay with an exponential function

• Converting between exponential and logarithmic functions

• Converting to the common logarithm

Standard 4 – Model with Mathematics

How can you model the time it takes a radioactive substance to decay as a function of the percent of the substance remaining?


• Exponential decay functions

• Definition of logarithms

• Properties of logarithms

• Half-life • Students may misinterpret the initial amount of percent remaining as 0, rather than 100%



Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) • Teacher can introduce how scientists use half-life to determine the age

of a fossil using the levels of decaying carbon-14.

• Teacher will recall general form for exponential decay .

• Students will work through Activity 4-8 (p. 247-250)


Revised 8/24/2014

Unit: 3.1 Rational Functions and Equations Days : 14

Essential Questions

What is the effect of changing the value of a on the graph of ? How do you find products and quotients of rational expressions? How do you find sums and differences of rational expressions?

What effects do the values of h and k have on the graph of ?

How can you rewrite f(x) in the form so that you may use the rules of transformations to graph? (note: p(x) and q(x) are of first degree)


For rational functions in the form f(x) = a/x

• Identify horizontal/vertical asymptotes, domain/range, two reference points, and end behavior

• Use the above information to graph rational functions

• Describe all transformations of the parent function Multiply and divide rational expressions Add and subtract rational expressions Identify excluded values

Sketch a graph of a function in the form of y = a/x. Describe transformations of graphs in the family of (x) = a/x. Sketch a graph using the transformations of the parent graph a/x. Identify restrictions or excluded values of rational expressions. Add, subtract, multiply, and divide rational expressions. Solve equations containing rational expressions. Identify extraneous solutions.


TBA CC.9-12.A.CED.1, 2 CC.9-12.F.IF.2, 4, 7, 7d CC.9-12.F.BF.1, 1a, 1b,3 CC.9-12.A.SSE.1, 1b CC.9-12.A.APR.6, 7 CC.9-12.A.REI.2




Revised 8/24/2014

Unit 3.1 Graphing y = a/x Lesson 1 of 5 Days 1


CC.9-12.A.CED.2 CC.9-12.F.IF.2, 4, 7, 7d CC.9-12.F.BF.1, 1a, 3

For rational functions in the form

• Identify horizontal/vertical

asymptotes, domain/range, two reference points, and end behavior

• Use the above information to graph rational functions

• Describe all transformations of the parent function

• Look for and express regularity in repeated reasoning

• What is the effect of changing the value of a on the graph of

5. Prerequisite Knowledge 6. Essential Vocabulary 7. Possible Misconceptions 8. Necessary Materials

• Transformations on parent functions

• Parent function

• Transformations

• Reflection

• Vertical stretch

• Vertical shrink

• End behavior

• Asymptotes (horizontal/vertical)

• Reference points

• Branches

• Students may not understand why horizontal and vertical asymptotes exist for rational functions

• Students may sketch branches of their graphs intersecting asymptotes

• Interchanging the meaning of vertical stretch/shrink

• Explorations in Core Math Section 5.1

• Projector/Elmo



Revised 8/24/2014

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing)

• Warm up questions could include o Discuss the following transformations on… focusing

on how the values of a, h, and k change any parent function

o Identify the end behavior of…

• The parent function is graphed for students on page 261. It may help for the instructor to lead the class through a discussion about the graph including

o Why the function is graphed in pieces o Why the table does not allow x to be equal to zero o Why asymptotes are included in the graph and how

to determine if a function has asymptotes. o How end behavior is different for rational functions o How to write end behavior for rational functions

• Monitor as students work through p. 261-264 in Core Math

• Warm up questions

• Working through p. 261-264

• Presenting samples of work o Results of each example should be discussed using

samples of student work


Revised 8/24/2014

Unit 3.1 Graphing Translations of y = a/x Lesson 2 of 5 Days 2


CC.9-12.A.APR.6 CC.9-12.F.IF.4 CC.9-12F.IF.7 CC.9-12.F.IF.7d(+) CC.9-12.F.BF.3

Identify the effects of h and k on the graph of f(x) = a/x. Use polynomial division to rewrite f(x) = p(x) /q(x) into the

form and then graph the functions. Identify horizontal asymptotes as y = k and vertical asymptotes as x = h from an equation in graphing form.

Look for and make use of structure. Use appropriate tools strategically.

What effects do the values of h

and k have on the graph of f(x) = a/x when in the form of

. ? How can you rewrite f(x) in the form

so that you may use the rules of transformations to graph?


Rules of transformations of polynomial functions in the form of f(x) = a(x –h)n +k. Polynomial division. Use of reference points to sketch a graph by transforming a parent graph. Graph of f(x) = a/x.

Rational Function Asymptotes

Vertical asymptote is x = -h, for example if f(x) = a/x+2, students may place the asymptote at x = 2 instead of x = -2. Students may confuse horizontal and vertical asymptotes. Students may confuse the notation when stating a restriction in the domain with the domain of the function and the equation of the asymptote. For example, a student may write x = 3 for the domain when they mean that the asymptote is x = 3 and the domain is all reals except x = 3 or x ≠3.

“Explorations in CORE MATH” Algebra 2section 5-4 Graphing Calculators



Revised 8/24/2014

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) It may be helpful to complete this lesson after 5.1 (graphing y = a/x) in the On Core Math book.

Teacher can relate the process of graphing to the process of graphing polynomials in the form of f(x) = a(x –h)n +k. Teacher may ask the students to identify the transformations from the equation and then discuss how these transformations will affect the placement of the asymptotes as well as the points on the branches of the rational functions. Teacher may direct the students to use division to rewrite a quotient in the form of ax+b/(cx + d) into the graphing form so that they may use transformations to sketch a graph of the function. Teacher should have the students investigate the relationship between the end behavior of the graph and the equations of the asymptotes.

Students should be working through examples, questions, and activities that connect the rules of transformations to the graph of rational functions. Students should be able to state the restrictions of the domain and range using the asymptotes and an investigation of the end behaviors of the branches of a rational function. The students could make tables to investigate the end behavior of the branches both as x increases and decreases without bound to which value does y approach, as well as y increases or decreases without bound to which value does x approach. Students may also use graphing calculators with the trace function to explore these relationships. Students need to be able to explain how the domain and range of a rational function are related to the graph of the function.


Revised 8/24/2014

Unit 3.1 Multiplying and Dividing Rational Expressions Lesson 3 of 5 Days 2


CC.9-12.A.SSE.1, 1b CC.9-12.A.APR.7 CC.9-12.F.BF.1, 1b

• Multiply and divide rational expressions

• Identify excluded values


• How do you find products and quotients of rational expressions?

5. Prerequisite Knowledge 6. Essential Vocabulary 7. Possible Misconceptions 8. Necessary Materials

• Multiplying polynomials

• Factoring polynomials

• Identifying excluded values

• Excluded values • Students may try to get a common denominator, remembering incorrect operations on fractions


• Projector/Elmo


• Warm up questions could include o Multiplication/division of fractions to help review the

process so that it can be applied in the lesson

• Monitor as students work through p. 269-270

• Provide additional examples

• Formative assessment


• Working through p. 269-270 and additional examples

• Presenting samples of work o Results of each examples should be discussed using




Revised 8/24/2014

Unit 3.1 Addition and Subtraction of Rational Expressions Lesson 4 of 5 Days 3


CC.9-12.A.SSE.1, 1b CC.9-12.A.APR.7 CC.9-12.F.BF.1, 1b

• Add and subtract rational expressions

Identify excluded values


How do you find sums and differences of rational expressions?


• Multiplying polynomials

• Factoring polynomials

• Identifying excluded values

• Excluded values • Students may try to use the process of multiplying/dividing rational expressions when adding/subtracting


• Projector/Elmo


• Warm up questions could include o Addition/subtraction of fractions to help review the

process so that it can be applied in the lesson o Factoring polynomials

• Monitor as students work through p. 275-277

• Provide additional examples



• Working through p. 275-277 and additional examples

• Presenting samples of work o Results of each examples should be discussed using




Revised 8/24/2014

Unit 3.1 Solving Rational Equations and Inequalities Lesson 5 of 5 Days 4


CC.9-12.A.CED.1 CC.9-12.A.REI.2 CC.9-12.A.REI.11

Multiplying both sides of an equation by the LCM to turn the rational equation into a polynomial equation. Graph both sides of an equation as two functions f(x) and g(x) and show that the point of intersection is the solution to the equation f(x) = g(x).

Look for and express regularity in repeated reasoning.

What methods are there for solving rational equations?


Multiplying rational expressions. Adding rational expressions. Finding the zeros of polynomial functions. Determining excluded values of a rational expression.

Extraneous Solutions Students may forget to reject all extraneous solutions.

“Explorations in CORE MATH” Algebra 2 section 5-5


Revised 8/24/2014

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Teachers can begin by reviewing factoring expressions and determining the LCM from factored form. Teachers could warm up using the LCM to rewrite a rational equation as a polynomial equation with equations that contain numerical denominators (these are actually polynomial equations with rational coefficients). Then show students an equation that contains rational expressions and have the students describe a strategy to simplify and then solve the equation. The teacher should have the students check all solutions to be certain there are no extraneous solutions. It is also a good practice to have the students write the excluded values of all expressions in the equation before solving and have the students check to see if their solutions need to be rejected. Teachers could have students solve more complicated equations, such as

by graphing both sides of the equation using a graphing calculator and looking for the point of intersection.

Students should be algebraically solving equations containing rational expressions both with and without extraneous solutions. Students should be able to determine why it is important to list excluded values and factor all denominators before attempting to solve algebraically. Students should be able to describe strategies to solve rational equations, such as multiplying both sides of an equation by the LCM to turn the rational equation into a polynomial equation. Students should be able to use a graphing calculator to solve more complex rational equations by looking for the point of intersection of two functions created by graphing both sides of the equal sign.


Revised 8/24/2014

Unit: 3.2 Radical Functions and Equations Days : 10

Essential Questions

How are radical and rational exponents related? How can you graph transformations of the parent square root and cube root function? How can you solve equations involving square roots and cube roots?


Understanding the relationship between radicals and rational exponents Transformations on the parent graph of square and cube root functions. Writing a cube/square root equation from a graph. Solving equations involving square roots and cube roots. Determine an if an extraneous solution exists.

Rewrite expression containing rational exponents in radical form, or an expression in radical form as an expression with a rational exponent. Simplify radical expressions. Multiply, add, and subtract rational expressions. Graphing radical functions as transformations of the square root and cube root parent graphs. Solve equations containing square root and cube root expressions. Check solutions to test for extraneous solutions. Relate possible solutions to the domain of radical functions.


TBA CC.9-12.N.RN.1 CC.9-12.N.RN.2 CC.9-12.F.IF.7 CC.9-12.F.IF.7b CC.9-12.F.BF.3 CC.9-12.F.BF.9 CC.9-12.A.REI.2 CC.9-12.A.REI.11 CC.9-12.A.CED.1*




Revised 8/24/2014

Unit 3.2 Radical Expressions and Rational Exponents Lesson 1 of 3 Days 4


CC.9-12.N.RN.1 CC.9-12.N.RN.2

Understanding the relationship between radicals and rational exponents

Standard 7: Look for and make use of structure

How are radical and rational exponents related?


Square/Cube Roots Properties of Integer Exponents

Radical Expression Students may forget that x is the same as x1

Students may give x2 + x2 as x4 and not add to be 2x2

“Explorations in CORE

MATH” Algebra 2 5.6

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) The teacher may introduce the lesson by recapping the rules for integer exponents to avoid some of the misconceptions. Then proceed to monitor and facilitate the students with the activity on pg. 299-301. Suggestion: After the students to example 3 on pg. 301, it may be a good idea to ask them what would be different if the problem was written as 27(x9)2/3 Also: Students will need to recall that the square root of a negative number is not defined in the real number system Example 19, it is relevant to the following section, which includes imaginary numbers

Students should work on the activity on pages 299 – 301.Followed by practice problems on pages 302


Revised 8/24/2014

Unit 3.2 Graphing Radical Functions Lesson 2 of 3 Days 1


CC.9-12.F.IF.7 CC.9-12.F.IF.7b CC.9-12.F.BF.3 CC.9-12.F.BF.9

Transformations on the parent graph of square and cube root functions. Writing a cube/square root equation from a graph

Mathematical Practice Standard 7: Look for and make use of structure

How can you graph transformations of the parent square root and cube root function


Inverses of quadratic functions Inverses of cubic functions

No new vocabulary in this section



Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) The teacher may introduce the lesson by reviewing how the letters a, h, and k have effected other parent graphs such as f(x)= x2. Next, write the equation of the parent square root functions and ask students to write af(x), f(x) + k and f(x – h). Show students that in the general form for a square root function a= 1, h=0 and k = 0. Finally, ask students to write the general form for a cubic function. Then have students work on the activity in the workbook as the teacher walks around the room and facilitates. As time permits this section may be broken up into a two day lesson possibly stopping on day one on pg. 308 Example 3.

Students should be listening to the introduction of the lesson given by the teacher and be able to then write the general form of both the square root and cubic functions. Students will then work on the activity in their work books on pgs. 305-308. (see teacher note for timing schedule) Followed by the practice problems on page 309-310.


Revised 8/24/2014

Unit 3.2 Solving Radical Equations and Inequalities Lesson 3 of 3 Days 3


CC.9-12.A.REI.2 CC.9-12.A.REI.11 CC.9-12.A.CED.1*

• Solving equations involving square roots and cube roots.

• How to determine an extraneous solutions

Mathematical Practice Standard 5: Use appropriate tools

How can you solve equations involving square roots and cube roots?


• Graphing square root and cube root functions

• Inverse Functions

Extraneous solutions • Students often enter numbers such as negative five squared in their calculators without the parenthesis resulting in an answer of -25. Be sure that they remember to use the parenthesis.

• Students may need to be reminded that if you cannot solve the quadratic function by factoring they can use the quadratic formula.



Calculator *teacher discretion

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) The teacher could introduce the lesson by asking a chain of questions such as: What is 5 cubed? What is the cube root of 125? What is six squared? What is the square root of 25? How is cubing and finding cube roots different from squaring and finding square roots? Follow by assigning the activity 5-8 on pgs 313-316. Walk around the room and facilitate as the students work on the activity.

Students should work on the activity on 5-8 pgs. 313-316. It should be followed up with the practice problems on p. 317-318


Revised 8/24/2014

Unit: 4.1 Trigonometric Functions Days : 12

Essential Questions

What is radian measure and how are radians related to degrees? What is an angle rotation and what does it measure? How can the sine, cosine and tangent functions be defined using the unit circle? How can you construct the inverse of a trigonometric function? How can you prove the Law of Sines and use it to find side lengths and angle measures in a triangle?

How can you prove the Law of Cosines and use it to find side lengths and angle measures in a triangle?


• Ratio of arc length to radius

• Converting radians to degrees and degrees to radian

• Understand angles of rotations

• Sketch angles of rotation greater than 360° and less than 0°

• Find infinite number of co-terminal angles

• Identify reference angles

• Evaluate trigonometric functions

• Evaluate inverse trigonometric functions

• Identify domain of inverse trigonometric functions

• Derive and use the Law of Sines in completing triangle applications

• Explore the ambiguous case

• Derive and use the Law of Cosines to find different angle measures within a triangle

In this unit students will deepen their understanding of trigonometric functions and how these functions are related to the trigonometric ratios of angles in right triangles. Students will define these trigonometric functions on the unit circle and they will learn and use the Law of Sines and the Law of Cosines to evaluate parts of a triangle.


TBA CC.9-12.N.Q.1 CC.9-12.A.CED.1, 4 CC.9-12.G.C.5 CC.9-12.F.TF.1, 2, 3(+), 6(+), 7(+) CC.9-12.F.IF.1 CC.9-12.G.SRT.10(+), SRT.11(+)


“Explorations in Core Math” – Unit 10, Sections 1-6 “Explorations in CORE MATH” Algebra II

Unit 4.1 Right Angle Trigonometry Lesson 1 of 6 Days 2


Revised 8/24/2014


CC.9-12.N.Q.1 CC.9-12.CED.4 CC.9-12.G.C.5

• Ratio of arc length to radius

• Converting radians to degrees and degrees to radian

Standard 2 - Reason abstractly and quantitatively

What is radian measure and how are radians related to degrees?


• Circles

• Arcs

• Angles

• Intercepted arcs

• Radian Measure

In this lesson, students will attempt/seek to add a unit symbol or add an abbreviation to radian measures. Although they may sometimes see the abbreviation “rad” used to refer to radians, radian measures do not require a unit symbol or abbreviation. Degrees must always have a degree symbol. Those without a symbol are assumed to be radian measures. Also, students can confuse the circumference of a circle with the degree measure of a circle.

“Explorations in CORE MATH” Algebra II, section 10-1


• Review basic facts about circles. Students should understand the difference between the radius, diameter, and the circumference of a circle.

• Instructor should monitor student progress of the completion of the lesson as well as facilitate discussion after the completion of each objective.

• Students should work throughout the lesson and participate in group/class discussions after the completion of each objective.

Unit 4.1 Angles of Rotation Lesson 2 of 6 Days 2

Lesson Focus


Revised 8/24/2014

1. Standards Addressed 2. Content to be Learned 3. Mathematical Practices 4. Essential Questions

CC.9-12.F.TF.1

• Understand angles of rotations

• Sketch angles of rotation greater than 360° and less than 0°

• Find infinite number of co-terminal angles

Standard 8 - Look for and express regularity in repeated reasoning

What is an angle rotation and what does it measure?


• Circles

• Arcs

• Angles

• Radian Measure

• Angle rotation

• Standard position

• Initial side

• Terminal side

• Unit circle

• Co terminal

• In this lesson, students will investigate angles of rotation. These angles are not restricted to measures less than 180 degrees, as they often are in Geometry courses. They may measure more than 180 and also have negative measures

• Students will often want to associate positive angles going in the clockwise direction.



• Remind students that one revolution has 360 degrees.

• Teacher will assign and facilitate students in Activity 10.2 on pgs. 563-565.

• Practice problems are located on p. 560

• Additional problems are on p. 561-562

Students should be working in their On Core book on activity 10.2 found on pages 563-565

Unit 4.1 The Unit Circle Lesson 3 of 6 Days 2



Revised 8/24/2014

CC.9-12.F.IF.1 CC.9-12.F.TF.2 CC.9-12.F.TF.3(+)

• Connect past experience with triangle trigonometry to the study of circular trigonometry.

Look for and express regularity in repeated reasoning.

How can the sine, cosine, and tangent functions be defined using the unit circle?


• Special right triangles

• Right triangle ratios that define Sine, Cosine, and Tangent

• Reference Angle • Students assume their calculators are set in degrees when most are defaulted to radians. Students are confused as to when the calculator must be in which mode.

“Explorations in CORE MATH” Algebra II, section 10-3 Calculators

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Teachers should have students recall from Geometry the ratios of the sides of the special right triangles 30-60-90 and 45-45-90. They should also have students recall the definitions of the ratios for sine, cosine, and tangent using a right triangle. They may then engage the students in an activity to connect these definitions to angles of rotation using the unit circle and constructing right triangles using the reference angles. This will allow the students to derive the definitions of the ratios of the three trigonometric functions using coordinates on the terminal side of the angle of rotation. Teachers should have students consider examples of angles whose terminal side falls in each of the four quadrants so that the students are able to connect the sign of the functions with the quadrants as well as the repetition of the ratios formed by angles with the same reference angles. Teachers may have the students work with exact solutions or ratios for the special angles and have the students evaluate the three trigonometric functions using a calculator.

Students are practicing setting up ratios from the special triangles with respect to a given angle. Students are drawing triangles using an angle of rotation whose reference angle is a special angle and connecting the lengths of the sides of the triangle to the coordinates of a point on the terminal side and the radius of a circle. Students should develop new definitions for sine, cosine, and tangent using circles. Students should then be able to use these definitions to determine the values of the trigonometric functions for any angle of rotation.

Unit 4.1 Inverses of Trigonometric Functions Lesson 4 of 6 Days 2


CC.9-12.A.CED.1 • Use the inverse of a Standard 2 - Reason abstractly How can you construct the


Revised 8/24/2014

CC.9-12.F.TF.6+ CC.9-12.F.TF.7+

trigonometric function to determine the measure of an angle.

• Use the inverse of a trigonometric function to solve an equation.

and quantitatively inverse of a trigonometric function?


• Right-angle trigonometry

• The unit circle

• Arcsine

• Arccosine

• Arctangent

Students may try to use the same domain for sine, cosine and tangent. (see instructional practices for a better description) Students may confuse the reciprocal of the function with its inverse. You should stress the sin-1 notation is the same as f-1 notation and that it represents the inverse of the function and is not an exponent of -1.

“Explorations in CORE MATH” Algebra II, section 10-4 Calculators


Revised 8/24/2014

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Teachers should begin by discussing what it means to be a one-to-one function. Give examples of functions which share this property and some that do not. Have the students discuss if sine, cosine, or tangent are one-to-one. Lead them to the conclusion that the domains must be restricted to create a one-to one-function that still generates all of the possible range values for each function. You may wish to connect it to the graphs of x2 and the radical function as an example from polynomials. Teachers should point out the difference in notation of Sin-1 vs. sin-1, the On-Core book uses the correct notations but never points out the capital letter vs. the lower case letter in the notation and could be easily missed by the students.

Students should understand that their calculators are in function mode and will only yield the value of the angle of the Arcsine, Arccosine, and Arctangent functions versus the arcsine, arccosine, arctangent relations which have multiple possible solutions. This should help them to understand why sometimes they receive an answer of -60 degrees for something they see as 300degrees. Students should be using the inverse functions to determine the values of the angles and solving trigonometric equations.


Revised 8/24/2014

Unit 4.1 The Law of Sines Lesson 5 of 6 Days 2


CC.9-12.G.SRT.10(+) CC.9-12.G.SRT.11(+)

• Write the area of a triangle using sines

• Derive the Law of Sines

• Explore the ambiguous case

• Use the Law of Sines in completing triangle applications

Standard 2 – Reason abstractly and quantitatively Standard 1 – Make sense of problems and persevere in solving them

How can you proves the Law of Sines and use it to find side lengths and angle measures in a triangle?


• Right angle trigonometry

• Angles of rotation

• Unit circle

• Law of Sines Students may forget to check for other possible triangles. Quadrant I and II have angles that have a given positive sine and two triangles may be possible when using the Law of Sines



• Lesson introduction may include review of how to find the area of a triangle, stressing the importance of any side can be considered the base and how the sine function is defined within a right triangle, and how right triangles are created when an altitude is drawn in an acute triangle.




Revised 8/24/2014

Unit 4.1 The Law of Cosines Lesson 6 of 6 Days 2


CC.9-12.G.SRT.10(+) CC.9-12.G.SRT.11(+)

• Derive and use the Law of Cosines to find different angle measures within a triangle

Standard 3 – Constrict viable arguments and critique the reasoning of others

How can you prove and use the Law of Cosines to find side lengths and angle measures in a triangle?


• The Law of Sines • Law of Sines

• Law of Cosines

• Students may use the incorrect law in solving a triangle

• Students may incorrectly substitute an angle for a side of vise versa



• Lesson introduction may include reviewing that triangles consist of three angles and three sides and that a triangle can be solved if three measures are known (with the exception of only knowing three angle measures)




Revised 8/24/2014

Unit: 4.2 Trigonometric Graphs and Identities Days: 11

Essential Questions

What are the key features of the graphs of the sine, cosine, and tangent functions? How do the constants a, b, h, and k in the function, g(Ɵ)=asinb(Ɵ-h)+k and g(Ɵ)=acosb(Ɵ-h)+k affect their graphs? How can you use a given value of one of the trigonometric functions to calculate the values of the other functions? How can you prove addition and subtraction identities for trigonometric functions? How can you prove trigonometric identities involving double angles and half angles? How can you solve trigonometric equations?


• Students will graph the Cosine, Sine, and Tangent functions.

• Students will identify key features (theta-intercepts, y-intercepts, extreme values, increasing & decreasing intervals) of the cosine, sine, and tangent functions

• Students will graph functions of the form f(x)=Acos(Bx-h)+k and f(x)=Asin(Bx-h)+k.

• Students will derive sum and difference identities for the sine, cosine and tangent functions

• Students will derive the double-angle and half-angle identities

• Solving trigonometric equations

• Writing and revising model functions

• Analyzing graphs of a height function

Students will apply their previous knowledge of transformations to identify key attributes of trigonometric functions and their graphs, including the midline and amplitude. Students will manipulate trigonometric expressions using trigonometric identities to simplify and solve problems.


TBA CC.9-12.F.IF.4, 7 CC.9-12.F.BF.3 CC.9-12.F.TF.2, 5, 9 CC.9-12.G.SRT.5


“Explorations in Core Math” – Unit 11, Sections 1-6 “Explorations in CORE MATH” Algebra 2


Revised 8/24/2014

Unit 4.2 Graphs of Sine and Cosine Lesson 1 of 6 Days 2


CC.9-12.F.IF.7 CC.9-12.F.IF4

• Students will graph the Cosine, Sine, and Tangent functions.

• Students will identify key features (theta-intercepts, y-intercepts, extreme values, increasing & decreasing intervals) of the cosine, sine, and tangent functions

Standard 7 – Look for and make use of structure - Student will recognize the periodicity of the sine and cosine functions. They will compare points on the graphs to points on the unit circle and understand how the periodicity is related to revolutions about the unit circle.

What are the key features of the graphs of the sine, cosine, and tangent functions?


• Key features of graphs

• Unit Circle Trigonometry

• Periodic

• Period

• Amplitude

• Midline

Students might be uncomfortable with the graph of the tangent function due to the multiple vertical asymptotes. However, they are familiar with functions that have asymptotes. Ask students to make a table for and then a graph of f(x) = x-1. Ask them to describe what happens to the value of the function as x gets closer and closer to zero from both the positive and negative sides.



Revised 8/24/2014


• Lesson introduction may include using graphing software to display the cosine function and asking students to list what they notice. (Max/Min values, intercepts, increasing decreasing intervals). If not brought up in discussion, be sure to have students notice that the graph repeats itself (periodic). Repeat the activity for the sine function.

• Have students create the sine and cosine function by hand. Then have them compare and contrast their key features.

• Guide the activity presented in the Core Explorations text




Revised 8/24/2014

Unit 4.2 Graphs of Other Trigonometric Functions Lesson 2 of 6 Days 1


CC.9-12.F.IF.7 CC.9-12.F.BF.3

• Graph functions of the form f(x)=Acos(Bx-h)+k and f(x)=Asin(Bx-h)+k.

Standard 5 – Use appropriate tools strategically: Students should be able to quickly sketch the graphs of the parent functions for the sine and cosine and tangent by hand.

How do the constants a, b, h, and k in the function,

g(Ɵ)=asinb(Ɵ-h)+k and

g(Ɵ)=acosb(Ɵ-h)+k affect their

graphs?


• Graphing the sine and cosine functions

• Graphing the tangent function

• stretching, shrinking, and reflecting trigonometric graphs

None Students may wonder how to determine the key points when graphing a function of the form

g(Ɵ)=asinb(Ɵ-h)+k. Remind

them that for sine (and cosine) the key points are the turning points and the zeros, as they were in previous lessons.



Revised 8/24/2014


• Lesson introduction: Have students recall the effects a, h, and k have on the function f(x)=a(x-h)2+k. If necessary, have students create a chart and keep it hanging in the class for the next few days.

• Guide the activity presented in the Core Explorations text




Revised 8/24/2014

Unit 4.2 Fundamental Trigonometric Identities Lesson 3 of 6 Days 2





Revised 8/24/2014

Unit 4.2 Sum and Difference Identities Lesson 4 of 6 Days 2


CC.9-12.F.TF.9(+) • Derive sum and difference identities for the sine, cosine and tangent functions

Standard 3 – Construct viable arguments and critique the reasoning of others.

How can you prove addition and subtraction identities of trigonometric functions?


• Fundamental Trigonometric Identities

None • Students may be challenged when deriving the difference identity for cosine as it requires precise reasoning

• For the sum and difference identities for the tangent function students may use the incorrect sign within the denominator (sum identity requires a difference in the denominator and the difference identity requires a sum in the denominator)



• Lesson introduction may include evaluating trigonometric functions of angles on the unit circle as this lesson extends to evaluating trigonometric functions to angles not located on the unit circle.



Unit 4.2 Double-Angle and Half-Angle Identities Lesson 5 of 6 Days 2


Revised 8/24/2014


CC.9-12.F.TF.9 • Derive the double-angle and half-angle identities

Standard 3 – Construct viable arguments and critique the reasoning of others.

How can you prove trigonometric identities involving double angles and half angles?


• Right Angle Trigonometry

• Unit Circle

None • It is important that students have a solid foundations of the double-angle identity as it is used to derive the half-angle identity



• Lesson introduction may include a review of the Pythagorean Identity as it will be used throughout the lesson in the derivation of the double-angle identity. Motivate students by explaining that they will learn identities that will help them evaluate the exact values of sine, cosine, and tangent of angles that are not familiar to them.




Revised 8/24/2014

Unit 4.2 Solving Trigonometric Equations Lesson 6 of 6 Days 2


CC.9-12.G.SRT.5 CC.9-12.F.TF.5 CC.9-12.F.IF.2 CC.9-12.F.BF.3

• Solve trigonometric equations

• Write and revise model functions

• Analyze graphs of a height function

Standard 4 – Modeling with Mathematics

How can you solve trigonometric equations? How can you model the height of a gondola on a rotating Ferris wheel?


• Angles of Rotation

• Radian Measure

• Graphing the Sine and Cosine functions

• Transformations of trigonometric graphs

None “Explorations in CORE MATH” Algebra II, section 11-6


• Lesson introduction may include a review of the graph of the sine function in the form h(t) = asinb(t – h) + k.

• Students may benefit from direct instruction of solving

trigonometric equations prior to using the activity to model




Revised 8/24/2014

Unit: 4.3 Data Analysis and Statistics Days: 16

Essential Questions



TBA




Revised 8/24/2014

Unit 4.3 Measures of Central Tendency Lesson 1of 8 Days 2


CC.9-12.S.ID.1 CC.9-12.S.ID.3

• Shape, central, and spread to characterize data distribution

Standard 2 - Reason abstractly and quantitatively

How can you use shape, center, and spread to characterize data distribution?


• Creating line plots

• Histograms

• Box plots

• Data distribution

• Uniform distribution

• Normal distribution

• Skewed distribution

• Mean

• Median

• Mode

• Standard deviation

• Interquartile range

• Skewed left

• Skewed right

When using multiple lists in a calculator, students can confuse them. It may be helpful to have students write down what set of data is in each list to help avoid any confusion.

• “Explorations in CORE MATH” Algebra II, section 8-1

• Calculator

• Graph paper


• Teachers may want introduce the lesson by asking the students what would be most useful to the when researching colleges: a list of all college entrance exams of freshman, the average college entrance exam score, or a graph of the scores and the number of students who received each score

• Teacher will assign and facilitate students on Activity 8.1 on pgs. 563-565.

• Practice problems are located on p. 427-432

• Students should be working in their On Core book on activity 8.1 found on pages 427-432


Revised 8/24/2014

Unit 4.3 Data Gathering Lesson 2 of 8 Days 2


CC.9-12.S.IC.1 • Understand techniques for gathering data

• Find statistics using various sampling methods

• Make predictions from a sample

Standard 6 – Attend to precision What are the different methods for gathering data about a population?


• Calculating the mean of a data set

• Numerical data

• Categorical data

• Population

• Census

• Parameter

• Sampling

• Statistic

• Proportion

• Sampling

• Individuals


• Calculator

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) It may be helpful to ask students to give examples of using statistical surveys. Answers should vary. Ask students how this information may have been gathered. Prompt them is necessary, Have students then engage in the activity 8-2 starting on pg. 435. Teacher should facilitate the students by using question strategies after each example to help further their understanding especially since this unit contains a lot of new vocabulary. Practice problems can be found on pg 439-440 Additional problems on pgs. 441-442

Students should be working on Activity 8.2 in their books starting on pg. 435. Students should be sharing on their answers ad questions in their small groups and sharing out in whole group discussion when prompted by the teacher.


Revised 8/24/2014

Unit 4.3 Surveys, Experiments, and Observational Studies Lesson 3 of 8 Days 2


CC.9-12.S.IC.3 CC.9-12.S.IC.6

• Detect errors in surveys

• Identify observational studies and experiments

• Identify control groups and treatment groups

• Evaluate a media report

Standard 3 – Construct viable arguments and critique the reasoning of others

What kinds of statistical research are there, and which ones can establish cause and effect relationships between variables?


• Data-gathering techniques • Survey

• Observational study

• Factor

• Experiment

• Treatment

• Randomized comparative experiment

• Treatment group

• Control group

Results of observational studies are often incorrectly interpreted as establishing a cause and effect relationship. Students may need to be reminded that observational studies can establish only an association between a factor and a characteristic of interest.



Revised 8/24/2014


• Lesson introduction may include discussing how surveys to identify possible ways that results of surveys could have been influenced by how the survey questions were worded, how accuracy could have be affected based on who asks the question in an interview.


• Students may benefit from acting out interview surveys, designing survey questions, to include questions that are biased. Students should identify potential sources of error for each survey and suggest ways that survey accuracy can be improved.



Revised 8/24/2014

Unit 4.3 Significance of Experimental Results Lesson 4 of 8 Days 2


CC.9-12.S.IC.5 • Formulate the null hypothesis

• Use a permutation test

Standard 5 – Use appropriate tools strategically

In an experiment, when is the difference between the control group and treatment group likely to be caused by the treatment?


• Randomized comparative experiments

• Null hypothesis

• Significant result

• Resampling

• Permutation test

When students attempt to find the probability of a difference of means is at least as great as the difference they recorded for the control and treatment groups in the experiment, they may include only the frequencies for the interval in which the difference occurs. Remind students they must find the sum of frequencies for all the intervals including and above the difference.



• Lesson introduction may include asking students to think about whether the differences between control and treatment groups in past experiments were significant (likely to be caused by the treatment and not due to chance).


• Students may benefit from having students state hypotheses about their everyday lives and have the class restate the hypothesis as a null hypothesis to become more comfortable with this concept.



Revised 8/24/2014

Unit 4.3 Sampling Distributions Lesson 5 of 8 Days 2


CC.9-12.S.IC.4

• Use data from sample surveys to estimate a population mean or proportion.

• Use sample statistics to make predictions about population parameters.

Standard 6 - Attend to Precision How can you calculate a confidence interval to estimate the population mean or proportion?


• Mean and population proportions

• Data distributions

• Central Limit Theorem

• z-scores

• Sampling distributions

• Standard error of the mean

• Standard error of the proportion

• Confidence interval

• Margin of error

• Students may confuse the sample mean with the population mean or similarly with population proportions.

• Students may have difficulty understanding that the standard error of the mean is the standard deviation of the values of the sample means of various samples within a population as opposed to the standard deviation of individual data points about the mean of the population.



Revised 8/24/2014

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Teacher should introduce the idea of sampling distributions by breaking the class into groups and conducting a quick survey with a numerical result. Each group represents a sample from the class population. Each group can calculate the mean of their results and then combine the data from the groups to get the mean of the class. Before calculating the class mean you may wish to have the students create a graphical representations of the sample means and form a conjecture as to what they think the class mean should be. You may also extend this idea of sampling to the entire school and have the students collect data from another class. In this model each class is the sample and the school is the population. Teacher can then lead the students through calculating the standard deviation of the sample means; this is also referred to as the standard error of the mean. If you have done both of the sampling distributions discussed above you may investigate what happens to the standard deviation of the means as the sample sizes increases. Teacher may then wish to apply this sampling distribution to population proportions as well as the mean. Teacher may then lead the students through activities to create confidence intervals used to predict the interval with a given confidence level into which the population mean should fall. Teacher should have the students notice what happens to the range of the interval as the confidence level is increased. The use of the common z-scores of 90%, 95%, and 99%, which are found in all statistics texts, may be used to avoid having to recalculate the corresponding z-score for various confidence levels.

Students should be collecting data, identifying sample vs population, and calculating the sample means and population means. Students should be analyzing the means of the samples as compared to the entire population. Students should be able to make a conjecture about the population mean based on a sample distribution. Students should be calculating the standard error of the means and using this to create confidence intervals to aid in conjecting at the population means or proportions.


Revised 8/24/2014

Unit 4.3 Probability Distributions Lesson 6 of 8 Days 2


CC.9-12.S.IC.1 CC.9-12.S.IC.2 CC.9-12.S.MD.3+ CC.0-12.S.MD.5+

• Graph probability distributions (area under the curve is equal to 1).

Standard 8 - Look for and Make use of structure.

What is the probability distribution and how is it displayed?


• Probability

• Data gathering techniques

• Random variable

• Probability distribution

• Cumulative probability

Students are using histograms to represent probability distributions graphically. They may confuse the x-axis values from histograms used in previous sections. A histogram for data distribution has groups of values for the random variable and the probability distributions have all of the possible outcomes for the random variable.

“Explorations in CORE MATH” Algebra II, sec 8-6


Revised 8/24/2014

Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Teachers should point out to the students that when creating probability distributions the events are mutually exclusive and represent all possible outcomes, thus the sum of the areas of the histograms, or the area under the curve, will equal to 1. This can be connecting into a previous concept that P(A) + P(not A) = 1. Teacher should have students compare the x and y axes from sampling distribution from the last section to the probability distributions. Teacher may chose to have the students conduct a probability experiment in groups or to use a simulation on a computer or graphing calculator to create the data.

Students may work through activity 8-6 in the ONCORE MATH series. It contains a introduction to probability distributions from both a theoretical probability and an experimental probability.


Revised 8/24/2014

Unit 4.3 Fitting to a Normal Curve Lesson 7 of 8 Days 2


CC.9-12.S.ID.4 • Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages.

Standard 1 - Make sense of problems and persevere in solving them: Students will need to look for entry points into normal distribution problems. First, they will need to confirm the distribution is, indeed, normal. Then they will plan a solution pathway by determining whether the questions are based on integer multiples of the standard deviation, a special case that can be solved quickly.

How do you find the percents of data and probabilities of events associated with normal distributions?


• Probability Distributions

• Standard normal distribution

• z-score

To help students understand the z-score formula, have them consider a normal distribution with mean u and standard deviation. Let x be a data value from this distribution. Ask: If x=u, what is the corresponding z-score? If x=u +

standard deviation, what is the corresponding z-score? These questions should help students see that the z-score formula associates the mean of normal distribution with 0, the mean of the standard normal distribution. Likewise, the formula associates one standard deviation more than the mean of a normal distribution with 1, the standard deviation of the standard normal distribution.




Revised 8/24/2014

9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing)

• Ask students to recall how to find the mean and standard deviation of a set of data. Have them state in words what those statistics describe. Then have them sketch curves Guide the activity presented in the Core Explorations text




Revised 8/24/2014

Unit 4.3 Analyzing Decisions Lesson 8 of 8 Days 2


CC.9-12.CP.4 • Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified

Standard 4 - Model with mathematics: Students need to interpret mathematical results in the context of the situation. This lesson offers many opportunities for students to practice this skill. When students use Bayes’ Theorem to find a conditional probability, the specific value they find for the probability is less important than understanding what this values means. In the problems presented in this lesson, interpreting the mathematical result is the essential step in determining whether a good decision was made.

How can you use probability to help you analyze decisions?


• Probability and set theory

• Mutually exclusive events

• Overlapping events

• Conditional probability

• Independent events

• Dependent events



Revised 8/24/2014


• Explain to students that they will learn how to use probability to analyze a decision. Point out that many ideas from this unit will come together in this lesson, including two-way tables, conditional probability, and complementary events. Guide the activity presented in the Core Explorations text



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