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GENERIC EVALUATION CRITERIA20010-2015MathematicsCalculus

R-E-S-P-O-N-S-ECRITERIA NOTESYes No N/A

I. INTER-ETHNICThe instructional material meets the requirements of inter-ethnic: concepts, content and illustrations, as set by West Virginia Board of Education Policy (Adopted December 1970).

II. EQUAL OPPORTUNITYThe instructional material meets the requirements of equal opportunity: concept, content, illustration, heritage, roles contributions, experiences and achievements of males and females in American and other cultures, as set by West Virginia Board of Education Policy (Adopted May 1975).

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INSTRUCTIONAL MATERIALS ADOPTION: 21st CENTURY LEARNING EVALUATION CRITERIAGENERAL EVALUATION CRITERIA

20010-2015MathematicsCalculus

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(IMR Committee) Responses

I=In-depth A=Adequate M=Minimal N=Nonexistent I A M N

In addition to alignment of Content Standards and Objectives (CSOs), materials must also clearly connect to Learning for the 21st Century which includes opportunities for students to develop

A. Learning Skills

Thinking and Problem-Solving Skills/ Rigor and Depth of ContentContent is presented in a way that deepens student understanding through engagement in meaningful, challenging mathematics that builds on prior knowledge and promotes connections among mathematical concepts.

Thinking and Problem-Solving Skills /Development of Conceptual UnderstandingLearning opportunities require students to develop their own viable mathematical understandings and help them build connections between mathematical ideas.

Information and Communication Skills/Mathematical Language Appropriately introduce and reinforce in multiple ways all necessary terms and symbols.

Personal and Work Place Productivity Skills

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B. 21st Century Tools

Problem-solving tools (such as spreadsheets, decision support, design tools)

Communication, information processing and research tools (such as word processing, e-mail, groupware, presentation, Web development, Internet search tools)

Personal development and productivity tools (such as e-learning, time management/calendar, collaboration tools)

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INSTRUCTIONAL MATERIALS ADOPTION: 21st Century Learning EVALUATION CRITERIA

The general evaluation criteria apply to each grade level and are to be evaluated for each grade level unless otherwise specified. These criteria consist of information critical to the development of all grade levels. In reading the general evaluation criteria and subsequent specific grade level criteria, e.g. means “examples of” and i.e. means that “each of” those items must be addressed. Eighty percent of the combined general and specific criteria must be met with I (In-depth) or A (Adequate) in order to be recommended.

20010-2015MathematicsCalculus

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(IMR Committee) Responses

I=In-depth A=Adequate M=Minimal N=Nonexistent I A M N

For student mastery of content standards and objectives, the instructional materials will provide students with the opportunity to

4. Multimedia

1. offer appropriate multimedia (e.g., software, audio, visual, internet access) materials.

2. provide a website which provides links to relevant sites as well as lesson plans, student activities and parent resources.

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3. Integrate technology seamlessly when appropriate to model mathematical situations, analyze data, calculate results, and solve problems.

B. Scientifically-Based Research Strategies

1. Consistently require students to link prior knowledge to new information to construct their own viable understandings of mathematical ideas.

2. Consistently provide opportunities for students to solve complex problems that have multiple entry points and the possibility of multiple solution processes.

3. Consistently provide opportunities for students to communicate their mathematical thinking processes to others orally, in writing, or pictorially.

4. Routinely require students to develop and defend mathematical conjectures, arguments, reasoning and proof.

5. Provide opportunities for the students to be involved in investigations that enable them to make connections among mathematical ideas.

6. Expect students to develop multiple representations of the mathematics in order to depict reasoning used to explain real world phenomena or solutions to relevant problems and move fluently between those representations.

7. Present varied teaching models with emphasis on differentiated instruction in content, process, and product.

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C. Critical Thinking

1. emphasize questioning models to promote higher order thinking skills based on depth of knowledge.

2. Consistently require students to discuss mathematics with each other and with the teacher, make arguments, conjecture and reason, and justify/clarify their ideas in writing and orally in precise mathematical symbols and language.

3. Present real world application that is current, engaging, integrated throughout the instruction, and promotes and develops critical thinking.

D. Life Skills

1. address life skills (e.g., reading road maps, using reference tools, researching, reading a newspaper, using want ads, completing an application, applying the interview process and goal setting).

2. address habits of mind activities (e.g., literacy skills, interpersonal communications, problem solving and self-directional skills).

E. Classroom Management

1. include opportunities for large group, small group, and independent learning.

2. Consistently require students to explore mathematical ideas, individually and collaboratively, while integrating the process standards (see Section I of this rubric).

3. provide suggestions for differentiated instruction (e.g., practice activities, learning stations, assessment, lesson plans).

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F. Instructional Materials

1. Are organized according to WV content standards or other increments that allow students to investigate and explore major mathematical ideas; provide a variety of lessons, activities, and projects from which to choose; and emphasize connections between mathematical ideas.

2. Consistently integrate tasks that engage students and invite them to speculate and hypothesize, are open-ended, and require them to determine appropriate strategies.

3. Provide teachers with guiding questions to aid students’ development of mathematical discourse to further mathematical understanding.

4. Provide additional resources that are organized in a way that is easy to access and use.

5. Include various instructional models to address varied learning styles of students.

6. Provide extensive and varied opportunities to differentiate individual needs for skill-building.

7. Provide supplemental materials for intervention and enrichment.

8. Provide teachers with support to properly integrate the process standards using the available resources.

9. Include a teacher resource that builds content knowledge for the teacher.

10. Spiral previously taught skills and strategies with new content.

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G. Assessment

1. provide assessment formats commensurate with WV assessment programs (e.g., WESTEST, NAEP, State Writing Assessment, informal assessments, PLAN, EXPLORE, ACT and SAT).

2. provide opportunities for assessment based on performance-based measures, open-ended questioning, portfolio evaluation, rubrics and multimedia simulations.

3. provide benchmark and ongoing progress monitoring.

4. provide rubric-based differentiated assessment.

5. provide an electronic system for managing assessment data to facilitate the implementation of tiered instruction

6. integrate student self-assessment for and of learning by providing tools and organizers that are linked to clearly identified learning goals.

7. Integrate formal and informal means of assessment in the materials for diagnostic, formative, and summative purposes.

8. include various types of assessments: performance tasks, multiple choice, short answer, and free response.

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H. Process Standards

1. Problem Solving: Provide frequent opportunities for students to formulate, grapple with, and solve complex problems that require a significant amount of effort and have multiple viable solution paths.

2. Communication: Routinely challenge students to communicate their thinking to others orally, in writing, and/or pictorially, using precise mathematical language.

3. Reasoning and Proof: Provide frequent opportunities for students to complete mathematical investigations with and without technology; develop conjectures, mathematical arguments and proofs to confirm those conjectures.

4. Connections with Mathematics: Consistently establish connections, and provide opportunities for students to establish connections, among mathematical concepts and their real-world applications.

5. Representations: Provide frequent opportunities for students to develop multiple representations of the mathematics in order to depict reasoning used to explain real world phenomena or solutions to relevant problems and move fluently between those representations.

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SPECIFIC EVALUATION CRITERIAMathematicsCalculus

Calculus objectives are designed for students who have completed Algebra I, Geometry, Algebra II, Trigonometry, and Pre-Calculus. Study includes functions and continuity, limits, differentiation and applications of derivatives, integration and its application to area, volume, and displacement. The Rule of Four (Numerical, Analytical, Graphical and Verbal) will be applied throughout the course. Available technology will be used by students and teachers to enhance learning. Graphing utilities will be used to investigate concepts and to evaluate derivatives and integrals. The West Virginia Standards for 21st Century Learning include the following components: 21st Century Content Standards and Objectives and 21st Century Learning Skills and Technology Tools. All West Virginia teachers are responsible for classroom instruction that integrates learning skills, technology tools, and content standards and objectives.

Standard 2: AlgebraThrough communication, representation, reasoning and proof, problem solving, and making connections within and beyond the field of mathematics, students will

demonstrate understanding of patterns, relations and functions, represent and analyze mathematical situations and structures using algebraic symbols, use mathematical models to represent and understand quantitative relationships, and analyze change in various contexts.

Standard 3: GeometryThrough communication, representation, reasoning and proof, problem solving, and making connections within and beyond the field of mathematics, students will

analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships,

specify locations and describe spatial relationships using coordinate geometry and other representational systems, apply transformations and use symmetry to analyze mathematical situations, and solve problems using visualization, spatial reasoning, and geometric modeling.

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Standard 5: Data Analysis and ProbabilityThrough communication, representation, reasoning and proof, problem solving, and making connections within and beyond the field of mathematics, students will

formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them, select and use appropriate statistical methods to analyze data, develop and evaluate inferences and predictions that are based on models, and apply and demonstrate an understanding of basic concepts of probability.

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CONTENT WITHIN PRODUCT

(IMR Committee) Responses

I=In-depth A=Adequate M=Minimal N=Nonexistent I A M N

For student mastery of content standards and objectives, the instructional materials will provide students with the opportunity to

A. Algebra

1. Provide various opportunities to use abstract notation to apply properties of algebraic, trigonometric, exponential, logarithmic and composite functions, as well as their inverses, represented graphically, numerically, analytically, and verbally; and demonstrate an understanding of the connections among these representations.

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2. Provide various opportunities to demonstrate a conceptual understanding of the definition of a limit via the analysis of continuous and discontinuous functions represented using multiple representations (e.g. graphs and tables).

3. Provide examples and exercises to use the properties of limits including addition, product, quotient, composition, and squeeze/sandwich theorem to calculate the various forms of limits: one-sided limits, limits at infinity, infinite limits, limits that do not exist, and special trigonometric

limits such as , .

4. Provide opportunities to investigate and apply the definition of continuity to determine where a function is continuous or discontinuous including continuity at a point, continuity over an interval, application of the Intermediate Value Theorem, and graphical interpretation of continuity and discontinuity.

5. Provide various opportunities to investigate and apply the definition of the derivative graphically, numerically, and analytically at a point, conceptually interpreting the derivative as an instantaneous rate of change and the slope of the tangent line.

6. Provide opportunities to investigate and discriminate between the average rate of change and the instantaneous rate of change using real-world problems.

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7. Provide various opportunities to justify why differentiability implies continuity and classify functional cases when continuity does not imply differentiability.

8. Provide various opportunities to recognize when the Extreme Value Theorem indicates that function extreme exist.

9. Provide examples and exercises to quickly recall and apply rules of differentiation including the constant multiple rule, sum rule, the difference rule, the product rule, the quotient rule, the power rule, and the chain rule as applied to algebraic, trigonometric, exponential, logarithmic, and inverse trigonometric functions using techniques of both explicit and implicit differentiation.

10.Provide various opportunities to apply Rolle’s Theorem and the Mean Value Theorem to real-world problems.

11. Provide various opportunities to construct and use mathematical models to solve optimization, related-rates, velocity, and acceleration problems.

12.Provide examples and exercises to determine antiderivatives that follow from derivatives of basic functions and apply substitution of variables.

13.Provide opportunities to investigate and apply the definite integral using Riemann sums by evaluating an infinite limit of a sum using summation notation and rules for summation.

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14.Provide examples and exercises to evaluate definite integrals using basic integration properties such as addition, subtraction, constant multipliers, the power rule, substitution, and change of limits.

15.Provide various opportunities to characterize the definite integral as the total change of a function over an interval and use this to solve real-world problems.

16.Provide opportunities to investigate and apply the Fundamental Theorem of Calculus to evaluate definite integrals and to formulate a cumulative area function and interpret the function as it relates to the integrand.

17.Provide various opportunities to use integration to solve problems that involve linear displacement, total distance, position, velocity, acceleration and area between curves by looking at both functions of x and functions of y; utilize units to interpret the physical nature of the calculus process.

B. Geometry

1. Provide opportunities to investigate and apply limits to deduce asymptotic behavior of the graph of a function.

2. Provide various opportunities to compare and contrast the limit definition of continuity and the graphical interpretation of the continuity of a function at a point; recognize different types of discontinuities.

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3. Provide opportunities to investigate and apply tangent lines as best linear approximations to functions near specific points; explain this conceptually; and construct these tangent lines; and apply this concept to Newton’s Method.

4. Provide various opportunities to investigate and explain the relationships among the graphs of a function, its derivative and its second derivative; construct the graph of a function using the first and second derivatives including extrema, points of inflection, and asymptotic behavior.

5. Provide opportunities to investigate and apply approximate areas under a curve using Riemann sums by applying and comparing left, right, and midpoint methods for a finite number of subintervals.

E. Data Analysis and Probability

1. Provide resources that identify a real life situation that involves quantities that change over time that students may use to investigate, analyze, and draw conclusions.

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