Grade 8

?IntroductionIn 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 ready90% of students will graduate on time100% of our students who graduate college or career ready will enroll in a post-secondary opportunityIn 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. 42291023279100-571500-1270The 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:The TN Mathematics StandardsThe Tennessee Mathematics Standards: 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 Practice Mathematical Practice Standards 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 MapsThis 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 judgment 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 9-12 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; 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 MapsOverviewAn 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. Topics Addressed in QuarterExponential & Logarithmic FunctionsAn Introduction to Trigonometry FunctionsTrigonometric Identities, Inverses, and Equations OverviewStudents have worked with exponential models in Algebra I and further in Algebra II. During this quarter students will solve problems involving exponential functions and logarithms and express their answers using logarithm notation. In general, students understand logarithms as functions that undo their corresponding exponential functions. Students will extend their knowledge from geometry of an understanding of trigonometric ratios through the study of right triangles, special triangles, and special angles. Students will relate the concept of trigonometric functions not only by exploring ratios of triangles, but they will relate trigonometric functions to the unit circle. Students also explored the Pythagorean Theorem in geometry and will now explore the Pythagorean identity sin2(θ) + cos2(θ) = 1 and other trigonometric identities. Students will extend their work with trigonometric functions, investigating the reciprocal functions secant, cosecant, and cotangent and their graphs and properties. They will find inverse trigonometric functions by appropriately restricting the domains of the standard trigonometric functions and use them to solve problems that arise in modeling contexts.Fluency The high school standards do not set explicit expectations for fluency, but fluency is important in high school mathematics. Fluency in algebra 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. These fluencies are highlighted to stress the need to provide sufficient supports and opportunities for practice to help students gain fluency. 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: STATE STANDARDSCONTENTINSTRUCTIONAL SUPPORT & RESOURCESExponential and Logarithmic Functions(Allow approximately 3 weeks for instruction, assessment, and review)Domain: Building FunctionsCluster: Build new functions from existing functions.F-BF.5.a-d Find inverse functions (including exponential and logarithmic)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 , and the inverse with respect to composition, f ?1 (x) . Understand the f (x) 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. Produce an invertible function from a non-invertible function by restricting the domain. Recognize a function is invertible if and only if it is one-to-one. 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)Functions need to be understood and interpreted in terms of their context. A function can be represented in different ways; these different representations help with analysis of the function. A function can be used to model the relationship between two quantities. New functions from existing functions can be understood and built. Comparing linear, quadratic, logarithmic and exponential models to solve problems is understood and applied. Essential Question(s):How can you compare the graphs of the sine, cosine, tangent functions and their inverses?Since the trigonometric functions are not one-to-one, how can the domain be restricted to graph the inverse functions? How are inverse trigonometric functions used to find angles in real-world problems? Objective(s):Students will:Identify one-to-one functionsExplore inverse functions using ordered pairsFind inverse functions using an algebraic methodGraph a function and its inverseSolve applications of inverse functions4.1 One-to-One Inverse Functions (Coburn)2.7 Inverse Functions (Blitzer)Additional Resource(s) HYPERLINK "" Algebra & Trigonometry (Coburn), 2nd Edition(Exercise Videos, Lecture Videos, Graphing Calculator Videos, PowerPoint Presentations) Khan Academy: Intro to Inverse Functions HYPERLINK "" Khan Academy: Finding Inverse Functions Khan Academy: Inverse Trigonometric FunctionsEngage NY: Revisiting the Graphs of the Trigonometric FunctionsEngage NY: Inverse Trig FunctionsTask(s) Solve exponential equations using logarithms. Students should be able to use this exponent logarithm relationship when finding the inverse of a function. For instance, we have the function f(x) = 3x , which we cant treat as y=3x. Finding the inverse means switching x and y, and then solving for y. So what we have is really x = 3y. We can now use the logarithms to solve for y. Exponential Growth Project: Rabbits, Rabbits, and More Rabbits VocabularyOne-to-one function, horizontal line test, algebraic methodWriting in MathExplain how to determine if two functions are inverses of each other.Describe how to find the inverse of a one-to-one function.What is the horizontal line test and what does it indicate?Describe how to use the graph of a one-to-one function to draw the graph of its inverse.Domain: Interpreting Functions Cluster: Analyze functions using different representations.F-IF.2 Identify or analyze the distinguishing properties of exponential, polynomial, logarithmic, trigonometric, and rational functions from tables, graphs, and equations.Enduring Understanding(s):Functions need to be understood and interpreted in terms of their context. A function can be represented in different ways; these different representations help with analysis of the function. A function can be used to model the relationship between two quantities. New functions from existing functions can be understood and built. Comparing linear, quadratic, logarithmic and exponential models to solve problems is understood and applied. Essential Question(s):How do you evaluate exponential functions for given values?How do you use exponential models so solve real-world problems?Objective(s):Students will:Evaluate an exponential functionGraph general exponential functionsGraph base-e exponential functionsSolve exponential equations and applications4.2 Exponential Functions (Coburn)4.1 Exponential Functions (Blitzer)Additional Resource(s) HYPERLINK "" Algebra & Trigonometry (Coburn), 2nd Edition(Exercise Videos, Lecture Videos, Graphing Calculator Videos, PowerPoint Presentations)Khan Academy: Exponential FunctionsTask(s)Modeling Exponential Growth HYPERLINK "" Achieve: Spread of DiseaseIllustrative Math: Foxes and Rabbits 2 Math Vision Project: Linear and Exponential Functions (Select from the tasks based upon assessment purpose, i.e., “Developing Understanding”, “Solidifying Understanding”, or “Practice Understanding”)VocabularyExponential function, natural exponential functionWriting in MathDescribe the relationship between and equation in logarithmic form and an equation in exponential form.Explain why the logarithm of 1 with a base b is 0.Explain how to find the domain of a logarithmic function.Name some real-world examples of the use of exponential functions.Domain: HYPERLINK ""N-NE – Number ExpressionsCluster: Represent, intercept, compare, and simplify number expressions.N-NE.2 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Enduring Understanding(s):Functions need to be understood and interpreted in terms of their context. A function can be represented in different ways; these different representations help with analysis of the function. A function can be used to model the relationship between two quantities. New functions from existing functions can be understood and built. Comparing linear, quadratic, logarithmic and exponential models to solve problems is understood and applied. Essential Question(s):How do you use transformations to sketch graphs of exponential and logarithmic functions? How do you solve exponential and logarithmic equations? Objective(s):Students will:Write exponential equations in logarithmic formFind common logarithms and natural logarithmsGraph logarithmic functionsFind the domain of a logarithmic functionSolve applications of logarithmic functions.4.3 Logarithms and Logarithmic Functions (Coburn)4.2 Logarithmic Functions (Blitzer)Additional Resource(s) HYPERLINK "" Algebra & Trigonometry (Coburn), 2nd Edition(Exercise Videos, Lecture Videos, Graphing Calculator Videos, PowerPoint Presentations)Examples of Properties of LogarithmsTask(s)GSE Algebra II/ Advanced Math: Unit 5 Exponential and Logarithmic Functions Select from the following tasks:What is a Logarithm? p. 40 Evaluating Logarithms that are not Common or Natural .p. 51 The Logarithmic Function p. 57 How Long Does It Take? p.64 Zombies Revisited – Can You Model Zombie Growth? p.72 Half-Life p.78 How Does Your Money Grow? p. 86 Applications of Logarithmic Functions p. 99 Newton’s Law of Cooling-Coffee, Donuts, and (later) Corpses p. 107 Graphing Logarithmic and Exponential Functions p.133 John and Leonhard at the Café Mathematica p. 134 Culminating Task: Jason’s Graduation Presents p. 146VocabularyTranscendental function, logarithmic function, common logarithm, natural logarithmWriting in MathCompare and contrast exponential functions and logarithmic functions.Describe the product rule for logarithms and give an example.Describe the quotient rule for logarithms and give an example.Describe the power rule for logarithms and give an example.You overhear a student talking about a property of logarithms in which division becomes subtraction. Explain what the student means by this.Domain: HYPERLINK ""N-NE – Number ExpressionsCluster: Represent, intercept, compare, and simplify number expressions.N-NE.1 Use the laws of exponents and logarithms to expand or collect terms in expressions; simplify expressions or modify them in order to analyze them or compare them.N-NE.3 Classify real numbers and order real numbers that include transcendental expressions, including roots and fractions of pi and e.Enduring Understanding(s):Functions need to be understood and interpreted in terms of their context. A function can be represented in different ways; these different representations help with analysis of the function. A function can be used to model the relationship between two quantities. New functions from existing functions can be understood and built. Comparing linear, quadratic, logarithmic and exponential models to solve problems is understood and applied. Essential Question(s):How do you use transformations to sketch graphs of exponential and logarithmic functions? How do you solve exponential and logarithmic equations? How do you solve real-world problems involving exponential and logarithmic functions?Objective(s):Students will:Solve logarithmic equations using the fundamental properties of logarithmsApply the product, quotient, and power properties of logarithmsSolve general logarithmic and exponential equationsSolve applications involving logistic, exponential, and logarithmic functions4.4 Properties of Logarithms: Solving Exponential and Logarithmic Equations (Coburn)4.3 Properties of Logarithms (Blitzer)4.4 Exponential and Logarithmic Equations (Blitzer)Additional Resource(s) HYPERLINK "" Algebra & Trigonometry (Coburn), 2nd Edition(Exercise Videos, Lecture Videos, Graphing Calculator Videos, PowerPoint Presentations)Examples of Solving Logarithmic EquationsTask(s)GSE Algebra II/ Advanced Math: Unit 5 Exponential and Logarithmic Functions (See above tasks from Sections 4.3/4.2)VocabularyExponential equation, logarithmic equation, uniqueness property of logarithmsWriting in MathExplain how to solve an exponential equation using 3x = 140 in your explanation.Explain how to solve an logarithmic equation using log3 (x – 1 ) = 4 in your explanation.Domain: Model with DataCluster: Model data using regression equations HYPERLINK ""S-MD.1 Create scatter plots, analyze patterns and describe relationships for bivariate data (linear, polynomial, trigonometric or exponential) to model real-world phenomena and to make predictions.S-MD.2 Determine a regression equation to model a set of bivariate data. Justify why this equation best first the data.S-MD.3 Use a regression equation modeling bivariate data to make predictions. Identify possibly considerations regarding the accuracy of predictions when interpolating or extrapolating.Enduring Understanding(s):Exponential functions are typically either growth functions or decay functions. The base of the exponent determines whether or not the function will be a growth function or a decay function. All exponential functions retain the same basic shape and are subject to the same transformations as all other functions.Essential Question(s):How do you differentiate between an exponential and a logarithmic function? How do the multiple representation of functions aid in building more efficient and more accurate models? How can technology be employed to help build mathematical models, and how can it be used to assess the appropriateness of a specific model?Objective(s):Students will:Calculate simple interest and compound interestCalculate interest compounded continuouslySolve applications involving annuities and amortizationSolve applications of exponential growth and decayMWTI – Modeling with Technology I (pp.283-292) (Coburn)MWTII – Modeling with Technology II (pp.491-502) (Coburn)4.5 Exponential Growth and Decay; Modeling Data (Blitzer)4.5 Applications From Business, Finance and Science (Coburn)Additional Resource(s) HYPERLINK "" Algebra & Trigonometry (Coburn), 2nd Edition(Exercise Videos, Lecture Videos, Graphing Calculator Videos, PowerPoint Presentations)Task(s)Math Vision Project: Linear and Exponential Functions (Select from the tasks based upon assessment purposes, i.e., “Developing Understanding”, “Solidifying Understanding”, or “Practice Understanding”)VocabularySimple interest, principal, interest rate, compound interest, accumulated value, present value equation, interest compounded continuously, annuity, exponential growth, exponential decay, growth rate, decay rateWriting in MathNigeria has a growth rate of 0.031 or 3.1%. Describe what this means.How can you tell if an exponential model describes exponential growth or exponential decay?Describe the shape of a scatter plot that suggests modeling the data with an exponential function.An Introduction to Trigonometry Functions (Allow approximately 3 weeks for instruction, assessment, and review)Domain: Applied Trigonometry Cluster: Use trigonometry to solve problems.G-AT.3 Derive and apply the formulas for the area of a sector of a circle (Coburn).HYPERLINK ""G-AT.4 Calculate the arc length of a circle subtended by a central angle.Domain: Trigonometric FunctionsCluster: Extend the domain of trigonometric functions using the unit circle.F-TF.1 Convert from radians to degrees and from degrees to radians.Enduring Understanding(s):The relationship between the sides and angles of right triangles leads to the exploration of trigonometric functions.Essential Question(s):What are the six trig functions for the acute angles in a right triangle?Why are the trig ratios in similar triangles equal?How is trig used to solve right triangles, including real-world applications?Objective(s):Students will:Use the vocabulary associated with a study of angles and triangles.Find fixed ratios of the sides of special trianglesUse radians for angle measure and compute circular arc length and area using radians.Convert between degrees and radians for nonstandard angles.Solve applications involving angular velocity and linear velocity using radians.5.1 Angles Measures, Special Triangles, Special Angles (Coburn)5.1 Angles and Radian Measure (Blitzer)Task(s):Illustrative Math: Defining Trig Ratios Edutoolbox: Making Right TrianglesEdutoolbox: Relating Trigonometric FunctionsAdditional Resources: HYPERLINK "" Algebra & Trigonometry (Coburn), 2nd Edition(Exercise Videos, Lecture Videos, Graphing Calculator Videos, PowerPoint Presentations)Learnzillion: Right Angles and TrigonometryMathshell: Trigonometry FunctionsBetter Lesson: Problem Solving with Isosceles Triangles and CirclesVocabularyAngle, vertex, protractor, straight angle, right angle, complementary, supplementary, acute, obtuse, minutes, seconds, decimal degrees, 45-45-90 triangle, 30-60-90 triangle, initial side, terminal side, positive angles, negative angles, coterminal angles, standard position, quandrantal angles, central angle, area of a circular sectorWriting in MathList the special triangles given. Why are they important to memorize? Compare and contrast degree measure and radian measure.Explain what is meant by one radian.Describe how to convert an angle in degrees to radians.Describe how to convert an angle in radians to degrees.Domain: Applied Trigonometry Cluster: Use trigonometry to solve problems.G-AT.1 Use the definitions of the basic trigonometric ratios of sides in a right triangle to solve problems about lengths of sides and measures of angles.Enduring Understanding(s)There are many instances of periodic data in the world around us. Trigonometric functions can be used to model real world data that is periodic in nature. There is a direct relationship between right triangle trigonometry and trigonometric functions. Essential Question(s)How do you evaluate trigonometric functions for given values, periods, and intervals? How trigonometric functions relate to the unit circle? How do we model real-world scenarios to trigonometric functions?Objective(s):Students will:Find values of the six trigonometric functions from their ratio definitions.Solve a right triangle given one angle and one side.Solve a right triangle given two sides.Use cofunctions and complements to write equivalent expressions.Solve applications involving angles of elevation and depression.Solve general applications of right triangles.5.2 The Trigonometry of Right Triangles (Coburn)5.2 Right Triangle Trigonometry (Blitzer) Additional Resource(s) HYPERLINK "" Algebra & Trigonometry (Coburn), 2nd Edition(Exercise Videos, Lecture Videos, Graphing Calculator Videos, PowerPoint Presentations)Khan Academy: Right Triangle Trigonometry HYPERLINK "" Review of Special Right TrianglesTask(s)Math Shell: Pythagorean TriplesVocabularyOpposite, adjacent, hypotenuse, sine, cosine, tangent, functions of an acute angle, cofunctions, cotangent, secant, cosecantWriting in MathGive a mnemonic device for the trigonometric functions. If you are given the lengths of the sides of a right triangle, describe how to find the sine of either acute angle.Describe one similarity and one difference between the definitions of sin Θ and cos Θ, where Θ is an acute angle of a right triangle.Domain: Interpreting Functions Cluster: Analyze functions using different representations.F-IF.2 Identify or analyze the distinguishing properties of exponential, polynomial, logarithmic, trigonometric, and rational functions from tables, graphs, and equations.Enduring Understanding(s)There are many instances of periodic data in the world around us. Trigonometric functions can be used to model real world data that is periodic in nature. There is a direct relationship between right triangle trigonometry and trigonometric functions. Essential Question(s)How do you evaluate trigonometric functions for given values, periods, and intervals? How trigonometric functions relate to the unit circle? How do we model real-world scenarios to trigonometric functions?Objective(s):Students will:Define the trigonometric functions using the coordinates of a point in QI Use reference angles to evaluate the trig functions for any angleSolve applications using the trig functions of any angle5.3 Trigonometric and The Coordinate Plane (Coburn)5.3 Trigonometric Functions of Any Angle (Blitzer)Additional Resource(s) HYPERLINK "" Algebra & Trigonometry (Coburn), 2nd Edition(Exercise Videos, Lecture Videos, Graphing Calculator Videos, PowerPoint Presentations)Trig-reference Task(s) HYPERLINK "" CCGPS Advanced Algebra: Unit 5 Trigonometric Functions Select from the following tasks:Clock Problem p.11 Figuring Out All the Angles p. 20 Real Numbers and the Unit Circle p. 38 Trigonometric Functions on the Unit Circle p. 66 Un-Wrapping the Unit Circle p. 74 Exploring Sine and Cosine Graphs p. 84 A Better Mouse Trap p. 97 Transforming Sinusoidal Graphs p. 134 Modeling with Sinusoidal Functions p. 145 Discovering a Pythagorean Identity p. 160 Culminating Task: Graphing Other Trigonometric Functions p. 168VocabularyReference angleWriting in MathIf you are given a point on the terminal side of an angle Θ, explain how to find sin Θ.Explain why tan 90? is undefined.If cos Θ > 0 and tan Θ < 0, explain how to find the quadrant in which Θ lies.Explain how reference angles are used to evaluate trigonometric functions. Give an example with your description.Domain: Trigonometric FunctionsCluster: Extend the domain of trigonometric functions using the unit circle.HYPERLINK ""F-TF.A.2 Use special triangles to determine geometrically the values of sine, cosine, tangent, for π/3, π/4, and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π-x, π+x, and 2π-x in terms of their values for x, where x is any real number. F-TF.A.3 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.Enduring Understanding(s)There are many instances of periodic data in the world around us. Trigonometric functions can be used to model real world data that is periodic in nature. There is a direct relationship between right triangle trigonometry and trigonometric functions. Essential Question(s)How do you evaluate trigonometric functions for given values, periods, and intervals? How trigonometric functions relate to the unit circle? How do we model real-world scenarios to trigonometric functions?Objective(s):Students will:Locate points on a unit circle and use symmetry to locate other pointsUse special triangles to find points on a unit circle and locate other points using symmetry Define the six trig functions in terms of a point on the unit circleDefine the six trig functions in terms of a real number tFind the real number t corresponding to given values of sin t, cos t, and tan t5.4 Unit Circles and the Trigonometry of Real Numbers (Coburn)5.4 Trigonometric Functions of Real Numbers; Periodic Functions (Blitzer)Additional Resource(s) HYPERLINK "" Algebra & Trigonometry (Coburn), 2nd Edition(Exercise Videos, Lecture Videos, Graphing Calculator Videos, PowerPoint Presentations)Circle Trig Reference:Task(s)CCGPS Pre-Calculus: Unit 2 Trigonometric Functions Right Triangles and the Unit Circle (p.12) HYPERLINK "" CCGPS Advanced Algebra: Unit 5 Trigonometric Functions Select from the tasks in Section 5.3VocabularyRadius, center, central circle, unit circle, quadrantal points, circular functions, reference arc, Writing in MathWhy are the trigonometric functions sometimes called the circular functions?What is the range of the sine function? Use the unit circle to explain where this range comes from.What do we mean by even trigonometric functions? Which of the six functions fall into this category?What is a periodic function? Why are the sine and the cosine functions periodic?Domain: Graphing Trigonometric FunctionsCluster: Model periodic phenomena with trigonometric functions. G-GT .1 Interpret transformations of trigonometric functions.G-GT .2 Match a trigonometric equation with its graph.G-GT .3 Determine the difference made by choice of units for angle measurement when graphing a trigonometric function.G-GT .4 Graph the sine, cosine, and tangent functions and identify characteristics such as period, amplitude, phase shift, and asymptotes. Enduring Understanding(s)There are many instances of periodic data in the world around us. Trigonometric functions can be used to model real world data that is periodic in nature. There is a direct relationship between right triangle trigonometry and trigonometric functions. Essential Question(s)How do you evaluate trigonometric functions for given values, periods, and intervals? How trigonometric functions relate to the unit circle? How do we model real-world scenarios to trigonometric functions?Objective(s):Students will:Graph?f(t)?= sin?t?using special values and symmetryGraph?f(t)?= cos?t?using special values and symmetryGraph sine and cosine functions with various amplitudes and periodsInvestigate graphs of the reciprocal functions?f(t)?= csc (Bt) and?f(t)= sec (Bt)Write the equation for a given graphGraph?y?= tan?t?using asymptotes, zeroes, and the ratio sin?t/cos?tGraph?y?= cot?t?using asymptotes, zeroes, and the ratio cos?t/sin?t Identify and discuss important characteristics of?y?= tan?t?and?y?= cot?tGraph?y?=?A?tan(Bt) and?y?=?A?cot(Bt) with various values of?A?and?BSolve applications of?y?= tan?t?and?y?= cot?tApply vertical translations in contextApply horizontal translations in contextSolve applications involving harmonic motion5.5 Graphs of Sine and Cosine; Cosecant and Secant (Coburn)5.6 Graphs of Tangent and Cotangent (Coburn)5.7 Transformations and Applications of Trigonometry Graphs (Coburn)5.5 Graphs of Sine and Cosine Functions (Blitzer)5.6 Graphs of Other Trigonometric Functions (Blitzer)5.8 Applications of Trigonometric Functions (Blitzer)Additional Resource(s) HYPERLINK "" Algebra & Trigonometry (Coburn), 2nd Edition(Exercise Videos, Lecture Videos, Graphing Calculator Videos, PowerPoint Presentations)Exploration of Transformations of Trig GraphsSummary of Properties of Trig GraphsTask(s)Investigating Trigonometry Graphs HYPERLINK "" CCGPS Advanced Algebra: Unit 5 Trigonometric Functions Select from the tasks in Section 5.3VocabularyPeriodic functions, rule of fourths, even functions, average value, amplitude, period formula, sinusoidal pattern, vertical shift, shifted form, standard form, phase angle, harmonic motion, equilibrium, pressure wave, sound energyWriting in MathWithout drawing a graph, describe the behavior of the basic sine curve.What is the amplitude of the sine function? What does this tell you about the graph?If you are given the equation of a sine function, how do you determine the period?What does a phase shift indicate about the graph of a sine function? How do you determine the phase shift from the function’s equation?Describe a relationship between the graphs of y = sin x and y = cos x.Without drawing a graph, describe the behavior of the basic tangent curve.If you were given the equation of a tangent function, how do you find consecutive asymptotes?Explain how to determine the range of y = csc x from the graph. What is the range?What does it mean to solve a right triangle?Explain how to find one of the acute angles of a right triangle if two sides are known?Trigonometric Identities, Inverses, and Equations (Allow approximately 3 weeks for instruction, assessment, and review)Domain: Trigonometric IdentitiesCluster: Apply trigonometric identities to rewrite expressions and solve equations.G-TI.1 Apply trigonometric identities to verify identities and solve equations. Identities include: Pythagorean, quotient, sum/difference, double-angle, and half-angle.Enduring Understanding(s)Understand the relationship between right triangle trigonometry and unit circle trigonometry Use the unit circle to define trigonometric functions.There are many instances of periodic data in the world around us and trigonometric functions can be used to model real world data that is periodic in nature. The inverses of sine, cosine and tangent functions are not functions unless the domains are limited.Essential Question(s)How do you evaluate trigonometric functions for given values, periods, and intervals? How trigonometric functions relate to the unit circle? How do we model real-world scenarios to trigonometric functions?Objective(s):Students will:Use fundamental identities to help understand and recognize identity “families”Verify other identities using the fundamental identities and basic algebra skillsUse fundamental identities to express a given trig function in terms of the other fiveUse counterexamples and contradictions to show an equation is not an identity6.1 Fundamental Identities and Families of Identities (Coburn)6.2 Constructing and Verifying Identities (Coburn)6.1 Verifying Trigonometric Identities (Blitzer)Additional Resource(s) HYPERLINK "" Algebra & Trigonometry (Coburn), 2nd Edition(Exercise Videos, Lecture Videos, Graphing Calculator Videos, PowerPoint Presentations)Khan Academy: Trig IdentitiesTask(s) Identities and EquationsHopewell GeometryCircular ReasoningHYPERLINK ""Accelerated Math III: Unit 6 Trigonometric Identities, Equations, and Applications Select from the following tasksUse Discovering the Pythagorean Identities p. 7 The Sum and Difference Identities p. 11 Riding the Ferris Wheel p. 14 Where’s the Identity? p. 16 Establishing Identities p. 24VocabularyIdentityWriting in MathCreate a toolbox for all of the different trigonometric identities.Explain how to verify an identity.Describe two strategies that can be used to verify identities.Domain: Trigonometric IdentitiesCluster: Apply trigonometric identities to rewrite expressions and solve equations.G-TI.1 Apply trigonometric identities to verify identities and solve equations. Identities include: Pythagorean, quotient, sum/difference, double-angle, and half-angle. Enduring Understanding(s)Understand the relationship between right triangle trigonometry and unit circle trigonometry Use the unit circle to define trigonometric functions.There are many instances of periodic data in the world around us and trigonometric functions can be used to model real world data that is periodic in nature. The inverses of sine, cosine and tangent functions are not functions unless the domains are limited.Essential Question(s)How do you evaluate trigonometric functions for given values, periods, and intervals? How trigonometric functions relate to the unit circle? How do we model real-world scenarios to trigonometric functions?Objective(s):Students will:Develop and use sum and difference identities for cosineUse the cofunction identities to develop the sum and difference identities for sine and tangentUse the sum and difference identities to verify other identities 6.3 The Sum and Difference Identities (Coburn) 6.2 Sum and Difference Identities (Blitzer)Additional Resource(s) HYPERLINK "" Algebra & Trigonometry (Coburn), 2nd Edition(Exercise Videos, Lecture Videos, Graphing Calculator Videos, PowerPoint Presentations)Examples of Sum and Difference FormulasTask(s) HYPERLINK "" Accelerated Math III: Unit 6 Trigonometric Identities, Equations, and Applications,The Sum and Difference Identities p. 11 VocabularyDifference identity for cosine, sum identity for cosineWriting in MathUse words to describe the formula for:The cosine of the difference of two anglesThe cosine of the sum of two anglesThe sine of the sum of two anglesThe sine of the difference of two anglesThe tangent of the sum of two anglesDomain: Trigonometric IdentitiesCluster: Apply trigonometric identities to rewrite expressions and solve equations.G-TI.1 Apply trigonometric identities to verify identities and solve equations. Identities include: Pythagorean, quotient, sum/difference, double-angle, and half-angle.Essential Question(s)How do you evaluate trigonometric functions for given values, periods, and intervals? How trigonometric functions relate to the unit circle? How do we model real-world scenarios to trigonometric functions?Objective(s):Students will:Derive and use the double-angle identities for cosine, tangent, and sineDevelop and use the power reduction and half-angle identitiesDerive and use the product-to-sum and sum-to-product identitiesSolve applications using these identities6.4 The Double-Angle, Half-Angle Formulas, and Product-to-Sum Identities (Coburn)6.3 Double-Angle, Power-Reducing, and Half-Angle Formulas (Blitzer)Additional Resource(s) HYPERLINK "" Algebra & Trigonometry (Coburn), 2nd Edition(Exercise Videos, Lecture Videos, Graphing Calculator Videos, PowerPoint Presentations)Examples of Double and Half Angle FormulasExamples of Product to Sum and Sum to ProductTask(s) HYPERLINK "" GSE Pre-Calculus: Unit 4 Trigonometric IdentitiesDouble-Angle Identities for Sine, Cosine, and Tangent p. 38 The Cosine Double-Angle: A Man with Many Identities p. 43VocabularyDouble-angle formulas, half-angle formulas, power reduction identitiesWriting in MathUse words to describe the formula for:The sine of double an angleThe cosine of double an angle (Describe one of the three formulas.)The tangent of double an angleThe sine of half an angleThe cosine of half an angleThe power-reducing formula for the sine squared of an angleThe power-reducing formula for the cosine squared of an angleDomain: Building FunctionsCluster: Build new functions from existing functions.F-BF.5. Find inverse functions (including exponential and logarithmic)Enduring Understanding(s);Exponential and logarithmic functions are inverse functions.Mathematical functions can be used to solve real world applications.Essential Question(s)How can graphs and equations of functions and their inverses help us to interpret real world problems?How can analytic and graphical methods be used to support each other in the solution of a problem?Objective(s):Students will:Find and graph the inverse sine function and evaluate related expressionsFind and graph the inverse cosine and tangent functions and evaluate related expressionsApply the definition and notation of inverse trig functions to simplify compositionsFind and graph inverse functions for sec?x, csc?x, and cot?xSolve applications involving inverse function6.5 Inverse Trigonometric Functions and their Applications (Coburn)5.7 Inverse Trigonometric Functions (Blitzer)Additional Resource(s) HYPERLINK "" Algebra & Trigonometry (Coburn), 2nd Edition(Exercise Videos, Lecture Videos, Graphing Calculator Videos, PowerPoint Presentations)Khan Academy: Inverse Trigonometric FunctionsTask(s)CCGPS Pre-Calculus: Unit 2 Trigonometric Functions Inverse Trigonometric Functions (p.31)VocabularyInverse functionsWriting in MathExplain why, without restrictions, no trigonometric function has an inverse.How can the graph od y = sin-1x be obtained from the graph of the restricted sine function?Describe the restriction on the sine function so that it has an inverse function.Describe the restriction on the cosine function so that it has an inverse function.Domain: Trigonometric IdentitiesCluster: Apply trigonometric identities to rewrite expressions and solve equations.G-TI.1 Apply trigonometric identities to verify identities and solve equations. Identities include: Pythagorean, quotient, sum/difference, double-angle, and half-angle.Enduring Understanding(s):An identity is a statement that is valid for all values of the variable for which the expressions in the equation are defined. Trigonometric identities are valuable in a wide variety of contexts because they allow for expressions to be represented in more convenient forms.Essential Question(s):How do you solve trig equations by combining like terms?How do you solve trig equations with the square root method?How do you solve trig equations by factoring?Objective(s):Students will:Use a graph to gain information about principal roots, roots in [0, 2π), and roots in?Use inverse functions to solve trig equations for the principal rootSolve trig equations for roots in [0, 2π) or [0, 360°)Solve trig equations for roots in?Solve trig equations using fundamental identitiesSolve trig equations using graphing technologyUse additional algebraic techniques to solve trig equationsSolve trig equations using multiple angle, sum and difference, and sum-to-product identitiesSolve trig equations of the form?A?sin(Bx?+?C) +D?=?kUse a combination of skills to model and solve a variety of applications6.6 Solving Basic Trigonometric Equations (Coburn)6.7 General Trigonometric Equations and Applications (Coburn)6.5 Trigonometric Equations (Blitzer)Additional Resource(s) HYPERLINK "" Algebra & Trigonometry (Coburn), 2nd Edition(Exercise Videos, Lecture Videos, Graphing Calculator Videos, PowerPoint Presentations)Examples of Solving Trig EquationsTrigonometry Practice Test:Task(s) HYPERLINK "" Accelerated Math III: Unit 6 Trigonometric Identities, Equations, and ApplicationsSolving Equations p.27VocabularyTrig equations, principal root, transcendental functionsWriting in MathWhat are the solutions of a trigonometric equation?Describe the difference between verifying a trigonometric identity and solving a trigonometric equation.RESOURCE TOOLBOXTextbook Resources Core Standards - MathematicsCommon Core Standards - Mathematics Appendix A (formerly ) Core LessonsTennessee's State Mathematics Standards HYPERLINK "" TN Advanced Algebra & Trigonometry StandardsCCSS Flip Book (with Examples of each standard)VideosBrightstormTeacher TubeThe Futures ChannelKhan AcademyMath TVLamar University TutorialCalculator Interactive Manipulatives HYPERLINK "" EdugoodiesAdditional Sites Trigonometry Cheat SheetOnline Algebra and Trigonometry TutorialLiteracyGlencoe Reading & Writing in the Mathematics ClassroomGraphic Organizers (9-12)Literacy Skills and Strategies for Content Area Teachers(Math, p. 22)ACTTN ACT Information & ResourcesACT College & Career Readiness Mathematics StandardsTasks/LessonsUT Dana CenterMars TasksInside Math TasksMath Vision Project TasksBetter Lesson ................
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