Curriculum and Instruction – Mathematics

Quarter 1 Pre-Calculus

Introduction

In 2014, the Shelby County Schools Board of Education adopted a set of ambitious, yet attainable goals for school and student performance. The District is committed to these goals, as further described in our strategic plan, Destination2025. By 2025,

80% of our students will graduate from high school college or career ready 90% of students will graduate on time 100% of our students who graduate college or career ready will enroll in a post-secondary opportunity

In order to achieve these ambitious goals, we must collectively work to provide our students with high quality, college and career ready aligned instruction. The Tennessee State Standards provide a common set of expectations for what students will know and be able to do at the end of a grade. College and career readiness is rooted in the knowledge and skills students need to succeed in post-secondary study or careers. The TN State Standards represent three fundamental shifts in mathematics instruction: focus, coherence and rigor.

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Focus

The Standards call for a greater focus in mathematics. Rather than racing to cover topics in a mile-wide, inch-deep curriculum, the Standards require us to significantly narrow and deepen the way time and energy is spent in the math classroom. We focus deeply on the major work of each grade so that students can gain strong foundations: solid conceptual understanding, a high degree of procedural skill and fluency, and the ability to apply the math they know to solve problems inside and outside the math classroom.

Coherence

Thinking across grades:The Standards are designed around coherent progressions from grade to grade. Learning is carefully connected across grades so that students can build new understanding on to foundations built in previous years. Each standard is not a new event, but an extension of previous learning. Linking to major topics:Instead of allowing additional or supporting topics to detract from the focus of the grade, these concepts serve the grade level focus. For example, instead of data displays as an end in themselves, they are an opportunity to do grade-level word problems.

Rigor

Conceptual understanding: The Standards call for conceptual understanding of key concepts, such as place value and ratios. Students must be able to access concepts from a number of perspectives so that they are able to see math as more than a set of mnemonics or discrete procedures. Procedural skill and fluency: The Standards call for speed and accuracy in calculation. While high school standards for math do not list high school fluencies, there are fluency standards for algebra 1, geometry, and algebra 2..Application: The Standards call for students to use math flexibly for applications in problem-solving contexts. In content areas outside of math, particularly science, students are given the opportunity to use math to make meaning of and access content.

Curriculum and Instruction – Mathematics

Quarter 1 Pre-Calculus

The Standards for Mathematical Practice describe varieties of expertise, habits of minds and productive dispositions that mathematics educators at all levels should seek to develop in their students. These practices rest on important National Council of Teachers of Mathematics (NCTM) “processes and proficiencies” with longstanding

importance in mathematics education. Throughout the year, students should continue to develop proficiency with the eight Standards for Mathematical Practice.

This curriculum map is designed to help teachers make effective decisions about what mathematical content to teach so that, ultimately our students, can reach Destination 2025. To reach our collective student achievement goals, we know that teachers must change their practice so that it is in alignment with the three mathematics instructional shifts.

Throughout this curriculum map, you will see resources as well as links to tasks that will support you in ensuring that students are able to reach the demands of the standards in your classroom. In addition to the resources embedded in the map, there are some high-leverage resources around the content standards and mathematical practice standards that teachers should consistently access:

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The TN Mathematics StandardsThe Tennessee Mathematics Standards:https://www.tn.gov/education/article/mathematics-standards

Teachers can access the Tennessee State standards, which are featured throughout this curriculum map and represent college and career ready learning at reach respective grade level.

Standards for Mathematical PracticeMathematical Practice Standardshttps://drive.google.com/file/d/0B926oAMrdzI4RUpMd1pGdEJTYkE/view

Teachers can access the Mathematical Practice Standards, which are featured throughout this curriculum map. This link contains more a more detailed explanation of each practice along with implications for instructions.

Purpose of the Mathematics Curriculum Maps

This curriculum framework or map is meant to help teachers and their support providers (e.g., coaches, leaders) on their path to effective, college and career ready (CCR) aligned instruction and our pursuit of Destination 2025. It is a resource for organizing instruction around the TN State Standards, which define what to teach and what students need to learn at each grade level. The framework is designed to reinforce the grade/course-specific standards and content—the major work of the grade (scope)—and provides a suggested sequencing and pacing and time frames, aligned resources—including sample questions, tasks and other planning tools. Our hope is that by curating and organizing a variety of standards-aligned resources, teachers will be able to spend less time wondering what to teach and searching for quality materials (though they may both select from and/or supplement those included here) and have more time to plan, teach, assess, and reflect with colleagues to continuously improve practice and best meet the needs of their students.

The map is meant to support effective planning and instruction to rigorous standards; it is not meant to replace teacher planning or prescribe pacing or instructional practice. In fact, our goal is not to merely “cover the curriculum,” but rather to “uncover” it by developing students’ deep understanding of the content and mastery of the standards. Teachers who are knowledgeable about and intentionally align the learning target (standards and objectives), topic, task, and needs (and assessment) of the learners are best-positioned to make decisions about how to support student learning toward such mastery. Teachers are therefore expected--with the support of their colleagues, coaches, leaders, and other support providers--to exercise their professional judgement aligned to our shared vision of effective instruction, the Teacher Effectiveness Measure (TEM) and related best practices. However, while the framework allows for flexibility and encourages each teacher/teacher team to make it their own, our expectations for student learning are non-negotiable. We must ensure all of our children have access to rigor—high-quality teaching and learning to grade-level specific standards, including purposeful support of literacy and language learning across the content areas.

Additional Instructional SupportShelby County Schools adopted our current math textbooks for grades 6-8 in 2010-2011. The textbook adoption process at that time followed the requirements set forth by the Tennessee Department of Education and took into consideration all texts approved by the TDOE as appropriate. We now have new standards;

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Curriculum and Instruction – Mathematics

Quarter 1 Pre-Calculus

therefore, the textbook(s) have been vetted using the Instructional Materials Evaluation Tool (IMET). This tool was developed in partnership with Achieve, the Council of Chief State Officers (CCSSO) and the Council of Great City Schools. The review revealed some gaps in the content, scope, sequencing, and rigor (including the balance of conceptual knowledge development and application of these concepts), of our current materials. The additional materials purposefully address the identified gaps in alignment to meet the expectations of the CCR standards and related instructional shifts while still incorporating the current materials to which schools have access. Materials selected for inclusion in the Curriculum Maps, both those from the textbooks and external/supplemental resources (e.g., EngageNY), have been evaluated by district staff to ensure that they meet the IMET criteria.

How to Use the Mathematics Curriculum Maps

OverviewAn overview is provided for each quarter. The information given is intended to aid teachers, coaches and administrators develop an understanding of the content the students will learn in the quarter, how the content addresses prior knowledge and future learning, and may provide some non-summative assessment items.

Tennessee State StandardsThe TN State Standards are located in the left column. Each content standard is identified as the following: Major Work, Supporting Content or Additional Content.; a key can be found at the bottom of the map. The major work of the grade should comprise 65-85% of your instructional time. Supporting Content are standards that supports student’s learning of the major work. Therefore, you will see supporting and additional standards taught in conjunction with major work. It is the teacher’s responsibility to examine the standards and skills needed in order to ensure student mastery of the indicated standard.

ContentTeachers are expected to carefully craft weekly and daily learning objectives/ based on their knowledge of TEM Teach 1. In addition, teachers should include related best practices based upon the TN State Standards, related shifts, and knowledge of students from a variety of sources (e.g., student work samples, MAP, etc.). Support for the development of these lesson objectives can be found under the column titled ‘Content’. The enduring understandings will help clarify the “big picture” of the standard. The essential questions break that picture down into smaller questions and the objectives provide specific outcomes for that standard(s). Best practices tell us that clearly communicating and making objectives measureable leads to greater student mastery.

Instructional Support and ResourcesDistrict and web-based resources have been provided in the Instructional Resources column. Throughout the map you will find instructional/performance tasks, i-Ready lessons and additional resources that align with the standards in that module. The additional resources provided are supplementary and should be used as needed for content support and differentiation.

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Quarter 1 Pre-Calculus

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Topics Addressed in Quarter Various Functions & Their Graphs Polynomials & Polynomial Functions Inverse Functions

OverviewStudents develop conceptual knowledge of functions that set the stage for the learning of other standards in Pre-calculus. Students apply the standards in Interpreting Functions and Building Functions in the cases of polynomial functions of degree greater than two, more complicated rational functions, the reciprocal trigonometric functions, and inverse trigonometric functions. Students will examine end behavior of functions and learn how to find asymptotes. Students further their understanding of inverse functions and construct inverse functions by appropriately restricting domains.

Fluency The high school standards do not set explicit expectations for fluency, but fluency is important in high school mathematics. Fluency in algebraic concepts can help students get past the need to manage computational and algebraic manipulation details so that they can observe structure and patterns in problems. Such fluency can also allow for smooth progress toward readiness for further study/careers in science, technology, engineering, and mathematics (STEM) fields. Fluency is not meant to come at the expense of conceptual understanding. Rather, it should be an outcome resulting from a progression of learning and thoughtful practice. It is important to provide the conceptual building blocks that develop understanding along with skill toward developing fluency.

References: http://www.tn.gov/education/article/mathematics-standards http://www.corestandards.org/ http://www.nctm.org/ http://achievethecore.org/

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Quarter 1 Pre-Calculus

TN STATE STANDARDS CONTENT INSTRUCTIONAL SUPPORT & RESOURCESGLENCOE - Chapter 1: Functions from a Calculus Perspective

SULLIVAN – Chapter 2: Functions & Their Graphs Chapter 5: Exponential & Logarithmic Functions (Allow approximately 6 weeks for instruction, review, and assessment)

Domain: Interpreting FunctionsCluster: Analyze functions using different representations.F-IF -1 : Determine whether a function is even, odd, or neither.F-IF -3 Identify or analyze the distinguishing properties of exponential, polynomial, logarithmic, trigonometric, and rational functions from tables, graphs, and equations.F-IF -4 Identify the real zeros of a function and explain the relationship between the real zeros and the x-intercepts of the graph of a function (exponential, polynomial, logarithmic, trigonometric, and rational).

F-IF -7 :Graph rational functions, identifying zeros, asymptotes (including slant), and holes (when suitable factorizations are available) and showing end-behavior.

Enduring Understanding(s): Functions are a mathematical way to

describe relationships between two quantities that vary.

When the equation of a function is changed the graph of the function also changes.

Zeroes of a function can be used to construct a graph of the function.

Essential Question(s): How can functions describe real-world

situations, model predictions and solve problems?

How can you use the number of sign changes in a function to determine the number and type of real zeros of a function?

Objective(s):Students will: Identify and evaluate functions and

state their domains in symbolic and verbal forms.

Use graphs of functions to estimate function values and find domains, ranges, y-intercepts, and zeros of functions.

Determine symmetry of graphs and identify even and odd functions.

Determine intervals on which a function is continuous.

Describe end behavior of functions.

Glencoe1-1: Functions1-2: Analyzing Graphs of Functions and Relations1-3: Continuity, End Behavior, and Limits

Sullivan2.1: Functions2.2: The Graph of a Function2.3: Properties of Functions

Tasks:Illustrative Math: Identifying Even and Odd FunctionsIdentifying Even and Odd Functions

Additional Resources:Function Notation: Even and OddKhan Academy: Recognizing Even and Odd FunctionsLearnzillion: Identify Zeros of Polynomials and Construct Rough Graphs of Polynomial Functions

Vocabulary: set-builder notation, interval notation, implied domain, piecewise-defined function, relevant domain, continuous, limit discontinuous, infinite, jump, point, removable and non-removable discontinuities, end behavior

Writing in Math: Write two things you already know about

functions.

Give an example of a real-life situation that is a function, and explain the type of function (linear, quadratic) it is. Explain whether and how the domain and range of the function are restricted, given the situation. Explain what the x-intercept andthe y-intercept represent in the situation.

Domain: Interpreting FunctionsCluster: Analyze functions using different representations.F-IF -6 : Visually locate critical points on the

Enduring Understanding(s):Functions can have varied rates of change over specified intervals.

Essential Question(s):

Glencoe1-4: Extrema and Average Rates of Change

Sullivan2.3: Properties of Functions

Vocabulary: increasing, decreasing, constant, maximum, minimum, extrema, average rate of change, secant line

Writing in Math:

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TN STATE STANDARDS CONTENT INSTRUCTIONAL SUPPORT & RESOURCES

graphs of functions and determine if each critical point is a minimum, a maximum, or point of inflection. Describe intervals where the function is increasing or decreasing and where different types of concavity occur.

How can we use the definite integral in real-world applications?

Objective(s):Students will: Determine intervals on which

functions are increasing, constant, or decreasing, concave up or concave down, maxima, minima, and points of inflection.

Task(s):Inside Math: Quadratics Performance Task

Additional Resources:Engage NY: End Behavior of Rational FunctionsCurve Sketching Tutorial

Describe how the average rate of change of a function relates to a function when it is increasing, decreasing, and constant on an interval.

Domain: Building FunctionsCluster: Build new functions from existing functions.F-BF -1 : Understand how the algebraic properties of an equation transform the geometric properties of its graph. For example, given a function, describe the transformation of the graph resulting from the manipulation of the algebraic properties of the equation (i.e., translations, stretches, reflections and changes in periodicity and amplitude).

Enduring Understanding(s): When the equation of a function is changed

the graph of the function also changes.

Essential Question(s): How can changing the values of a function

affect the shape of the graph of the function?

Objective(s):Students will: Identify, describe, and graph transformations

of parent functions.

Glencoe1-5: Parent Functions and Transformations

Sullivan2.4: Library of Functions2.5: Graphing Techniques: Transformations

Additional Resource(s):Transformations of Graphs of FunctionsTexas Instruments Activity

Vocabulary: parent function, constant function, zero function, identity function, quadratic function, cubic function, square root function, reciprocal function, absolute value function, step function, greatest integer function, transformation, translation, reflections, dilation

Writing in Math:Use words, graphs, tables and equations to relate parent functions and transformations. Show this relationship through a specific example.

Domain: Building FunctionsCluster: Build new functions from existing functions.F-BF -2 : Develop an understanding of functions as elements that can be operated upon to get new functions: addition, subtraction, multiplication, division, and composition of functions.F-BF -3 :Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the

Enduring Understanding(s): Operations and transformations apply to all

types of functions and can be used to build new functions from existing functions.

Essential Question(s): What relationships exist between quantities

that can be modeled by functions?

Objective(s):Students will: Perform arithmetic operations with

functions. Compose compositions of functions.

Glencoe1-6: Function Operations and Composition of Functions

Sullivan5-1: Composite Functions

Task(s):TNCore Task Arc: Building Polynomial Functions Illustrative Math: Compose Functions

Additional Resources:Engage NY Lesson 16: Function Composition

Vocabulary: compose, composite functionWriting in Math:Explain how to determine the output of a composition of functions, given the input.

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TN STATE STANDARDS CONTENT INSTRUCTIONAL SUPPORT & RESOURCES

temperature at the location of the weather balloon as a function of time.

Engage NY Lesson 17: Solving Problems by Function CompositionComposite Functions ApplicationsTI Nspire Composite Functions ActivityHow to Add, Subtract, Multiply and Divide FunctionsLearnzillion: Arithmetic Operations on Functions

Domain: Building FunctionsCluster: Build new functions from existing functions.F-BF -5 Find inverse functions (including exponential, logarithmic and trigonometric).a. Calculate the inverse of a function, f (x) , with respect to each of the functional operations; in other words, the additive inverse, − f (x) , the multiplicative inverse, 1/f(x), and the inverse with respect to composition, f −1(x) . Understand the algebraic and graphical implications of each type.b. Verify by composition that one function is the inverse of another.c. Read values of an inverse function from a graph or a table, given that the function has an inverse.d. Recognize a function is invertible if and only if it is one-to-one. Produce an invertible function from a non-invertible function by restricting the domain.F-BF -6 :Explain why the graph of a function and its inverse are reflections of one another over the line y=x.

Enduring Understanding(s): Operations and transformations apply

to all types of functions and can be used to build new functions from existing functions.

The inverse functions interchange the domain and the range.

The domain of a non-invertible function needs to be restricted in order to construct its inverse function.

Essential Question(s): What are inverse functions and what

are they being used for? How do we restrict the domain of a

non-invertible function to produce an invertible function?

Objective(s):Students will: Use the horizontal line test to determine

whether a function has an inverse function.

Find inverse functions algebraically and graphically.

Verify inverse functions algebraically using a composition of functions and graphically using y=x symmetry.

Glencoe1-7: Inverse Relations and Functions

Sullivan5.2: One-to-One Functions; Inverse Functions

Task(s):Illustrative Math: Invertible or NotIntroduction to Functions (Task is on p. 35)Graphs of Compositions

Additional Resources:Engage NY Lessons: Inverse Functions (Lessons 18-21)Inverse Functions Concept Development Activity

Vocabulary: inverse function, one-to-one, invertible

Writing in Math:Explain what it means for a function to be“one-to-one,” and describe two methods for determining whether or not a function is one-to-one.Justify the identity function, y= x, being the line of reflection for a function and its inverse.

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TN STATE STANDARDS CONTENT INSTRUCTIONAL SUPPORT & RESOURCESSULLIVAN – Chapter 4: Polynomial & Rational Functions

(Allow approximately 3 weeks for instruction, review, and assessment)Domain: Interpreting FunctionsCluster: Analyze functions using different representations.

F-IF -2 Analyze qualities of exponential, polynomial, logarithmic, trigonometric, and rational functions and solve real world problems that can be modeled with these functions (by hand and with appropriate technology).★

F-IF -3 Identify or analyze the distinguishing properties of exponential, polynomial, logarithmic, trigonometric, and rational functions from tables, graphs, and equations.

Enduring Understanding(s):All functions have algebraic, numerical, graphical and verbal representations.

Graphs of functions can explain the observed local and global behavior of a function.

Essential Question(s): What relationships exist between

quantities that can be modeled by functions?

What are some of the characteristics of the graph of an exponential function?

What are some of the characteristics of the graph of a logarithmic function?

Objective(s):Students will: Graph and analyze power functions and

radical functions. Graph polynomial functions. Model real-world data with polynomial

functions.

Glencoe2-1: Power and Radical Functions2-2: Polynomial Functions

Sullivan4.1: Polynomial Functions and Models

Task(s):Ted’s Quest for a Tablet (p.14)

Writing an Exponential Function from a Description

Illustrative Math: Model with Exponential Functions

Exponential and Logarithmic Functions (tasks start on p. 12)

Additional Resources:Learnzillion: Graph of Exponential Growth FunctionAnalyzing-the-Graph-of-a-Rational-Function-Asymptotes-Domain-and-Range

Vocabulary: power function, monomial function, extraneous solutions, polynomial function, polynomial function of degree n, leading coefficient, leading term test, quartic function, turning points, quadratic form, repeated zero, multiplicity

Writing in Math:Describe some similarities and differences between the graphs of exponential, polynomial, logarithmic, trigonometric, and rational functions.

Domain: Interpreting FunctionsCluster: Analyze functions using different representations.F-IF -4: Identify the real zeros of a function and explain the relationship between the real zeros and the x-intercepts of the graph of a function (exponential, polynomial, logarithmic, trigonometric, and rational).

Enduring Understanding(s): The characteristics of polynomial functions

and their representations are useful in solving real-world problems.

The domain and range of polynomial functions can be extended to include the set of complex numbers.

Glencoe2-3: The Remainder and Factor Theorems2-4: Zeros of Polynomial FunctionsSullivan4.5: The Real Zeros of a Polynomial Function4.6: Complex Zeros; Fundamental Theorem of Algebra

Vocabulary: synthetic division, depressed polynomial, synthetic substitution, rational zero theorem, lower bound, upper bound, Decartes’ Rule of Signs, Fundamental Theorem of Algebra, Linear Factorization Theorem, Conjugate Root Theorem, complex conjugate, irreducible over the reals

Writing in Math:

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TN STATE STANDARDS CONTENT INSTRUCTIONAL SUPPORT & RESOURCES

Domain : Complex numbers Cluster: Use complex numbers in polynomial identities and equations.N-CN-7: Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

Essential Question(s): How do polynomial functions model real-

world problems and their solutions? Why are complex numbers necessary?

Objective(s):Students will: Find the real zeros of polynomial

functions. Find the complex zeros of polynomial

functions and use the Fundamental Theorem of Algebra.

Task(s):Illustrative Math: Zeroes and Factorization of General PolynomialsFactors, Zeroes, and Roots: Oh My! (Teacher notes p. 27 and student pages start on p. 39)

Additional Resources:Khan Academy: Fundamental Theorem of AlgebraKhan Academy: Zeroes or Polynomials and Their Graphs

Explain how you can use a graphing calculator, synthetic division and factoring to completely factor a fifth-degree polynomial with rational coefficients, three integral zeroes and two non-integral, rational zeroes.

Domain: Interpreting FunctionsCluster: Analyze functions using different representations.F-IF -2: Analyze qualities of exponential, polynomial, logarithmic, trigonometric, and rational functions and solve real world problems that can be modeled with these functions (by hand and with appropriate technology).★F-IF -3 Identify or analyze the distinguishing properties of exponential, polynomial, logarithmic, trigonometric, and rational functions from tables, graphs, and equations.F-IF -4 Identify the real zeros of a function and explain the relationship between the real zeros and the x-intercepts of the graph of a function (exponential, polynomial, logarithmic, trigonometric, and rational).F-IF -7 :

Enduring Understanding(s):All functions have algebraic, numerical, graphical and verbal representations.

Graphs of functions can explain the observed local and global behavior of a function.

Essential Questions(s): What relationships exist between

quantities that can be modeled by functions?

Objective(s):Students will: Analyze and graph rational functions,

including horizontal, vertical, and oblique asymptotes, holes, and intercepts.

Solve rational equalities

Glencoe2-5: Rational Functions

Sullivan4.2: Properties of Rational Functions4.3: The Graph of a Rational Function

Task(s):Illustrative Math: Graphing Rational Functions Rational Functions Tasks pgs. 6, 13 & 15

Additional Resources:Learnzillion: Relate the domain of a function to its graph, accounting for asymptotes and restricted domains

Graphing Stories: Graphic Representations of the Real Life Situations

Vocabulary: rational function, asymptote, horizontal asymptote, vertical asymptote, oblique asymptote, holes

Writing in Math:Explain why all of the test intervals must be in order to get an accurate graph of a rational function.

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TN STATE STANDARDS CONTENT INSTRUCTIONAL SUPPORT & RESOURCES

Graph rational functions, identifying zeros, asymptotes (including slant), and holes (when suitable factorizations are available) and showing end-behavior.

RESOURCE TOOLBOXTextbook ResourcesGlencoe Precalculus © 2011http://connected.mcgraw-hill.com/connected/login.doSullivan Precalculus: Enhanced with Graphing Utilities, 5e © 2009.

StandardsCommon Core Standards - MathematicsCommon Core Standards - Mathematics Appendix AEdutoolbox.org (formerly TN Core)The Mathematics Common Core ToolboxCommon Core LessonsTennessee’s State Mathematics StandardsTennessee’s Pre-calculus Standards

VideosKhan AcademyLamar University TutorialThe Futures Channel

CalculatorTexas Instruments Education

Interactive Manipulativeshttp://www.ct4me.net/math_manipulatives_2.htm

Additional Siteshttp://functions.wolfram.com

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RESOURCE TOOLBOXTexas Instruments - Precalculus Activities Casio EducationTI EmulatorMath Nspired

Kuta SoftwareIlluminations (NCTM)Stem ResourcesNational Math Resources

http://www.opensourcemath.orghttp://www.analyzemath.com/Graphing/piecewise_functions.htmlhttp://www.purplemath.com/http://www.onlinemathlearning.com/math-word-problems.htmlhttp://education.ti.com/calculators/downloads/US/Activities/Detail?id=9530UT Dana CenterMars TasksInside Math TasksMath Vision Project Tasks Better LessonAlgebra Cheat SheetTrigonometry Cheat SheetOnline Algebra and Trigonometry TutorialStudy Tips for Math Courses

LiteracyGlencoe Reading & Writing in the Mathematics ClassroomGraphic Organizers (9-12)

ACTTN ACT Information & ResourcesACT College & Career Readiness Mathematics Standards

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