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 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 Quarter 3Applications of DifferentiationIntegrationOverviewDuring this quarter, students discover some of the many applications of the derivative. Students will continue to graph relationships between f, f’, and f’’ and solve problems that involve rates of change and motion – functionalize, derivate, test for extrema, and solve. Students are taught how to approach a particular problem in calculus, and use the calculator as a tool in the solution process. The unit on Integration will supply the students with the capability of integrating a variety of function types. It is necessary for them to integrate by hand as well as with a calculator. The relationship between the Riemann Sums and the definite integral is a major point of interest and The Trapezoidal Rule helps to demonstrate the idea of the definite integral representing summation, once again.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 & ResourcesChapter 3: Applications of Differentiation (Allow approximately 3 weeks for instruction, review, and assessment)Domain: Computing and Applying DerivativesCluster: Apply differentiation techniquesD-AD Use first and second derivatives to analyze a function7. Relate the increasing and decreasing behavior of f to the sign of f’ both analytically and graphically. 8. Use the first derivative to find extrema (local and global).9. Analytically locate the intervals on which a function is increasing, decreasing, or neither. 10. Relate the concavity of f to the sign of f” both analytically and graphically. 11. Use the second derivative to find points of inflection as points where concavity changes.12. Analytically locate intervals on which a function is concave up or concave down. Enduring Understandings: The derivative has both theoretical and real life applications. The derivative provides useful information about the behavior of functions and the shapes of graphs. Understanding the rate of change of a function allows you to predict future behavior. Essential Questions: What does the derivative tell us? How can the derivative be used to solve optimization problems? How do rates of change relate in real-life settings? Objectives:Students will:Analyze the graphs of polynomials, rational, radical, piecewise, and transcendental functions using appropriate technology. Discuss which functions behave “nicely” with respect to algebraic properties and which do not. Justify your discussions.Describe asymptotic behavior (analytically and graphically) in terms of infinite limits and limits at infinity.Given a complete set of algebraic information and calculus information, construct a sketch of a function that matches the given data. Display several functions that satisfy one set of data. Describe their differences and similarities.Give a sketch of a graph of a function and completely describe the function in mathematical terms so that the sketch could be replicated from the description and would be close to the original graph3.6: A Summary of Curve SketchingAdditional Resource(s)HYPERLINK ""Larson Calculus Videos – Section 3.6 Visual Calculus TutorialsKhan Academy Calculus VideosCalculus Activities Using the TI-84Precalculus & Calculus TasksChapter 3 Vocabulary (3-6, 3-7 & 3-9): asymptotes, critical numbers, points of inflection, test intervals, primary equation, secondary equation, differential of x, differential of y, propagated error, differential formWriting in Math/DiscussionSuppose f” (t) < 0 for all t in the interval (2, 8). Explain why f (3) > f (5).Domain: Computing and Applying DerivativesCluster: Apply derivatives to solve problemsD-AD 16. Solve optimization problems to find a desired maximum or minimum value.Objectives:Students will:Use optimization to find extreme values (relative and absolute).3.7: Optimization Problems Additional Resource(s)HYPERLINK ""Larson Calculus Videos – Section 3.7Visual Calculus TutorialsKhan Academy Calculus VideosCalculus Activities Using the TI-84Precalculus & Calculus TasksWriting in Math/DiscussionA shampoo bottle is a right circular cylinder. Because the surface area of the bottle does not change when it is squeezed, is it true that the volume remains the same? Explain.Domain: Computing and Applying DerivativesCluster: Apply derivatives to solve problems D-AD 18. Use tangent lines to approximate function values and changes in function values when inputs change (linearization). Objectives:Students will:Understand the concept of a tangent line pare the value of the differential, dy, with the actual change in y, Δy.Estimate the propagated error using a differential.Find the differential of a function using differentiation formulas.3.9: Differentials Additional Resource(s)HYPERLINK ""Larson Calculus Videos – Section 3.9Visual Calculus TutorialsKhan Academy Calculus VideosCalculus Activities Using the TI-84Precalculus & Calculus TasksWriting in Math/DiscussionDescribe the change in accuracy of dy as an approximation for Δy when Δx is decreased.When using differentials, what is meant by the terms propagated error, relative error, and percent error?Chapter 4: Integration (Allow approximately 6 weeks for instruction, review, and assessment)Domain: Understanding IntegralsCluster: Understand and apply the Fundamental Theorem of CalculusI-UI 4. Recognize differentiation and antidifferentiation as inverse operations.Enduring Understandings: Derivatives and anti-derivatives have an inverse relationship to each other. The area under the curve is the geometric meaning of anti-derivatives. The anti-derivative has both theoretical and real life applications. Essential Question(s)How are the rules for differentiation used to develop the basic rules of integration? How can we use the measure of area under a curve to discuss net change of a function over time? How is the anti-derivative related to the accumulation function? How are area under the curve and the definite integral related? How are the properties of definite integrals related to the Riemann sum definition? How can one apply numerical techniques to compute an integral without knowing the associated antiderivative? Objectives:Students will:Write the general solution of a differential equation.Use indefinite integral notation for antiderivatives.Use basic integration rules to find antiderivatives.Find a particular solution of a differential equation.4.1: Antiderivatives and Indefinite IntegrationAdditional Resource(s)Visual Calculus TutorialsHYPERLINK ""Larson Calculus Videos – Section 4.1Calculus Tutorial Videos HYPERLINK "" Khan Academy Calculus VideosCalculus Activities Using the TI-84Precalculus & Calculus TasksChapter 4 Vocabulary: Antiderivative, constant of integration, general derivative, general solution, differential equation, antidifferentiation (indefinite integration), indefinite integral, particular solution, initial condition, sigma notation, index of summation, upper and lower bounds of summation, inscribed rectangle, circumscribed rectangle, lower sum, upper sum, Riemann sum, integrable, definite integral, lower limit, upper limit, Fundamental Theorem of Calculus (I and II), net change, displacement, Integration by Substitution, Pattern Recognition, change of variables, integration of odd and even functions, Trapezoidal Rule, Simpson’s Rule Writing in Math/DiscussionWhat is the difference, if any, between finding the antiderivative of f(x) and evaluating the integral ∫f(x) dx?Domain: Understanding IntegralsCluster: Demonstrate understanding of a Definite Integral I-UI 3. Use Riemann sums (left, right, and midpoint) and trapezoidal sums to approximate definite integrals of functions, represented graphically, numerically, and by tables of values.Objectives:Students will:Use Sigma notation to write and evaluate a sum.Approximate the area of a plane region using limits.Find the area of a plane region using limits.4.2: AreaAdditional Resource(s)HYPERLINK ""Larson Calculus Videos – Section 4.2Visual Calculus Tutorials HYPERLINK "" Khan Academy Calculus VideosCalculus Activities Using the TI-84Precalculus & Calculus TasksWriting in Math/DiscussionIn your own words using appropriate figures, describe the methods of upper sums and lower sums in approximating the area of a region.Give the definition of the area of a region in the plane.Domain: Understanding IntegralsCluster: Demonstrate understanding of a Definite IntegralI-UI Define the definite integral as the limit of Riemann sums and as the net accumulation of change. Correctly write a Riemann sum that represents the definition of a definite integral.Objectives:Students will:Understand the definition of Riemann sums.Evaluate a definite integral using limits.Evaluate a definite integral using properties of definite integrals.4.3: Riemann Sums and Definite IntegralsAdditional Resource(s)Visual Calculus TutorialsHYPERLINK ""Larson Calculus Videos – Section 4.3Calculus Tutorial Videos HYPERLINK "" Khan Academy Calculus VideosCalculus Activities Using the TI-84Precalculus & Calculus TasksWriting in Math/DiscussionGive an example of a function that is integrable on the interval [-1, 1], but not continuous on [-1, 1]Domain: Understanding IntegralsCluster: Understand and apply the Fundamental Theorem of CalculusI-UI Evaluate definite integrals using the Fundamental Theorem of Calculus. Use the Fundamental Theorem of Calculus to represent a particular antiderivative of a function and to understand when the antiderivative so represented is continuous and differentiable.Apply basic properties of definite integrals.Objectives:Students will:Evaluate a definite integral using the Fundamental Theorem of Calculus.Understand and use the Mean Value Theorem for integrals.Find the average value of a function over a closed interval.Understand and use the Second Fundamental Theorem of Calculus.4.4: The Fundamental Theorem of CalculusAdditional Resource(s)Visual Calculus TutorialsHYPERLINK ""Larson Calculus Videos – Section 4.4Calculus Tutorial VideosHYPERLINK ""Khan Academy Fundamental Theorem of Calculus VideosCalculus Activities Using the TI-84Precalculus & Calculus Tasks∫Domain: Calculate and Apply IntegralsCluster: Apply techniques of antidifferentiation I-AI Develop facility with finding antiderivatives that follow directly from derivatives of basic functions (power, exponential, logarithmic, and trigonometric). Use substitution of variables to calculate antiderivatives (including changing limits for definite integrals).Find specific antiderivatives using initial conditions.Objectives:Students will:Use pattern recognition to find an indefinite integral.Use change of variables to find an indefinite integral.Use the general power rule for integration to find an indefinite integral.Use a change of variables to evaluate a definite integral.Evaluate a definite integral involving an even or odd function.4.5: Integration by SubstitutionAdditional Resource(s)Visual Calculus TutorialsHYPERLINK ""Larson Calculus Videos – Section 4.5Calculus Tutorial Videos HYPERLINK "" Khan Academy Calculus VideosCalculus Activities Using the TI-84Precalculus & Calculus TasksDomain: Understanding IntegralsCluster: Demonstrate understanding of a Definite Integral I-UI 3. Use Riemann sums (left, right, and midpoint) and trapezoidal sums to approximate definite integrals of functions, represented graphically, numerically, and by tables of values.Objectives:Students will:Approximate a definite integral using the Trapezoidal Rule.4.6: Numerical Integration Additional Resource(s)Visual Calculus TutorialsHYPERLINK ""Larson Calculus Videos – Section 4.6Calculus Tutorial Videos HYPERLINK "" Khan Academy Calculus VideosCalculus Activities Using the TI-84Precalculus & Calculus TasksRESOURCE TOOLBOXThe Resource Toolbox provides additional support for comprehension and mastery of subject-level skills and concepts. While some of these resources are embedded in the map, the use of these categorized materials can assist educators with maximizing their instructional practices to meet the needs of all students.?Textbook ResourcesLarson/Edwards Calculus of a Single Variable ? 2010Larson CalculusStandardsCommon Core Standards - MathematicsCommon Core Standards - Mathematics Appendix (formerly TN Core)The Mathematics Common Core ToolboxTennessee’s State Mathematics StandardsState Academic Standards (Calculus) VideosLarson Calculus Videos HYPERLINK "" KhanAcademyHippocampusBrightstormPre-Calculus Review University of Houston VideosCalculatorCalculus Activities Using the TI-84 HYPERLINK "" Texas Instruments EducationCasio EducationTI EmulatorInteractive Manipulatives HYPERLINK "" Interactive ExamplesAdditional Sites Calculus Tutorials HYPERLINK "" Lamar University Tutorial Precalculus & Calculus Tasks Algebra Cheat SheetTrigonometry Cheat SheetOnline Algebra and Trigonometry TutorialStudy Tips for Math CoursesLiteracyGlencoe Reading & Writing in the Mathematics ClassroomLiteracy Skills and Strategies for Content Area Teachers(Math, p. 22)Graphic Organizers (dgelman)Graphic Organizers (9-12) ................
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