Mathematics Content Standards K-12 s3
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GENERIC EVALUATION CRITERIA 20010-2015 Mathematics Algebra III
R-E-S-P-O-N-S-E Yes No N/A CRITERIA NOTES
I. INTER-ETHNIC The 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 OPPORTUNITY The 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).
1 INSTRUCTIONAL MATERIALS ADOPTION: 21st CENTURY LEARNING EVALUATION CRITERIA
GENERAL EVALUATION CRITERIA 20010-2015 Mathematics Algebra III
(IMR Committee) Responses (Vendor/Publisher) SPECIFIC LOCATION OF I=In-depth A=Adequate M=Minimal N=Nonexistent CONTENT WITHIN PRODUCT 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 Content Content 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 Understanding Learning 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
2 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)
3 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.
20010-2015 Mathematics Algebra III
(IMR Committee) Responses (Vendor/Publisher) SPECIFIC LOCATION OF I=In-depth A=Adequate M=Minimal N=Nonexistent CONTENT WITHIN PRODUCT 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.
4 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.
5 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).
6 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.
7 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.
8 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.
9 SPECIFIC EVALUATION CRITERIA Mathematics Algebra III
Algebra III is intended for students who have mastered the concepts of Algebra I, Geometry, and Algebra II. Algebra III objectives develop and extend properties of higher degree polynomial functions, rational functions, exponential functions and logarithmic functions using the common concepts and language of algebraic, graphical, and tabular representations. The use of analytic geometry for sense making, conceptual understanding of abstract ideas and modeling real world applications is stressed, making use of calculators, computers, and interactive activities. 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: Algebra Through 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.
10 (IMR Committee) Responses (Vendor/Publisher) SPECIFIC LOCATION OF I=In-depth A=Adequate M=Minimal N=Nonexistent CONTENT WITHIN PRODUCT 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 a variety of examples and exercises to use properties of analytic geometry that justify and use the distance and midpoint formulas and negative reciprocal criterion for non-vertical perpendicular lines.
2. Provide opportunities to factor higher order polynomials by using techniques that can be applied to the factoring of second degree polynomials.
3. Provide a variety of examples and exercises to relate factored forms of polynomials to graphs, tables, and solutions to problems in context.
11 4. Provide opportunities for students to relate analytical attributes such as: characteristics of zeros, x- and y- intercepts, symmetry, asymptotes, end behavior, maximum and minimum points, and domain and range to graphical and algebraic representations of polynomials and rational functions.
5. Provide exercises that allow students to analyze the discriminant to classify the roots of quadratic equations with real coefficients, and relate the existence of x- intercepts of the graph to information obtained from the discriminant.
6. Provide opportunities for students to solve equations with extraneous roots and explain, using precise mathematical language, why the extraneous roots are excluded from the solution set.
7. Allow students the opportunity to compare and contrast the use of interval notation, set notation, and number line representations to express the domain and range of functions.
12 8. Provide the opportunity to compare and contrast the domain and range of a modeling function with the restricted domain and range used in a real world situation; justify the restricted domain and range choice for a problem in context.
9. Provide a variety of examples and exercises to: differentiate between functions and relations; evaluate, add, subtract, multiply, divide, rationalize, simplify, and compose functions (including rational, radical and those with fractional exponents); express domain and range of functions.
10. Provide the opportunity to formulate strategies to solve real life problems including compound interest and exponential growth and decay.
11. Appropriately introduce and reinforce how to restrict the possible rational zeros of a polynomial function by using the Rational Zeros Theorem and Descartes’ Rule of Signs; confirm the real zeros of a polynomial function by using the Remainder and Factor Theorems.
12.Provide the opportunity to approximate zeros of a polynomial or rational function using a graphing utility and the Intermediate Value Theorem.
13 13. Provides students the opportunity to compare and contrast the cases when 0
14.Provide students examples of Interpreting the number e as a limit and use e to build exponential functions modeling real world applications.
15.Provides students with the opportunity to use common and natural logarithms in the evaluation of logarithmic functions whose base is neither 10 nor e.
16.Provide students the opportunity to incorporate the change of base formula and properties of logarithms to simplify and expand algebraic expressions and to solve logarithmic and exponential equations.
17.Provides a variety of examples and exercise to solve equations involving radical, exponential, and logarithmic expressions
18. Provides students with opportunities to formulate strategies to solve real life problems involving compound interest and exponential growth and decay.
14 19.Builds conceptual understanding through opportunities for students to build on the skills of solving linear equations in two variables using elimination, substitution, or matrix methods to solve systems with three or more unknowns involving real world applications and to categorize systems of equations as zero, one, or infinitely many solutions, by both geometric and algebraic methods.
20.Provide students with materials and resources that support their work in groups to choose a real life situation that could be modeled by a polynomial, rational, exponential, or logarithmic function, and make a hypothesis, design an experiment, gather data, analyze data, refine the hypothesis into an appropriate mathematical model, use the model to make a prediction, test the prediction using the experimental setup, and compare the results. Present the collaboration as a project using words, graphs, tables, equations, and appropriate presentation tools.
15 21.Provide opportunities for students to convert between graphs and equations of circles identifying important features from either representation; translate from general form to standard form by completing the square and describe readily usable characteristics of each form; represent a circle as two functions graphically and algebraically.
22.Provide a variety of examples and exercises to analyze a piecewise defined function in multiple representations, to give its domain, intercepts, range, constituent pieces as elementary functions, and end behavior; apply to real world data.
23.Enable students to: determine the average rate of change of a function between any two points on its graph and use this rate to find the equation of a secant line; interpret the average rate of change to solve real world problems; relate signs of average rate of change to the function increasing or decreasing; and demonstrate a geometrical and conceptual understanding of the difference quotient.
16 24.Provide a variety of examples and exercises to use synthetic division to divide a polynomial, verify a factor, and determine its roots; compare and contrast synthetic division to long division.
25.Provide opportunities to investigate how the multiplicity of zeros of polynomial functions affects the graph; characterize a polynomial given the zeros, the behavior of the graph at the zeros, and the end-behavior.
26.Use of multiple strategies, interactive software, websites, and/or applets to investigate the characteristics of a transformation involving polynomial, radical, absolute value, logarithmic, or exponential functions, unravel the effect of a series of transformations using multiple representations
17 27.Define and discusses one-to-one functions including: the role of the Vertical and Horizontal Line Tests; uses multiple representations in describing the relationship between a function and its inverse, including the domain and range of each; identifies and explains the need for appropriate restrictions necessary to guarantee an inverse function; discusses the symmetrical relationship associated with the line y=x between the function and its inverse and explain the geometric reason the symmetry exists; demonstrates how to algebraically verify that two functions are inverses of each other.
28.Provides students the opportunity to prioritize relevant techniques to graph a given rational function: explaining the relevance of symmetry, end behavior, and domain and range; using zeros of the denominator to differentiate between vertical asymptotes and points of discontinuity; using long division to determine end behavior and explain the role of quotient and remainder in the process; explaining how the factors of the numerator and denominator can be used to analytically and graphically determine where the graph will fall above or below the x-axis.
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