Prentice Hall CME Project, Algebra
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1 Prentice Hall CME Project, Algebra Grades 9-12 C O R R E L A T E D T O Grades 9-12
2 CME Project Algebra Introduction The Correlation of CME Project, Algebra to the Idaho Content Standards, Algebra II demonstrates the alignment of content between CME Project and the Idaho Content Standards. This document contains references from the Teacher Editions (TE) and Student Editions (SE) to the relevant standards in Algebra II. References from two additional resources, Data Analysis and Probability Workbook and High School Math Skills Review and Practice were added to this correlation to further enhance the alignment of CME Project, Algebra 2 to the Idaho Content Standards. The pages referenced will be available on phschool.com. As part of the CME Project program, teachers also have these resources to draw from for professional development, implementing and teaching, assessment, and practice. Additional CME Project Teacher Resources include: Implementing and Teaching Guide, Algebra 2 Teaching Resources, Algebra 2 Practice Workbook, Algebra 2 Assessment Resources, Algebra 2 Solutions Manual, and the TIn-spire Handbook Additional CME Project Technology Resources include: ExamView Test Generator CD-ROM, TeacherEXPRESS CD-ROM, StudentEXPRESS CD-ROM, Success Tracker, and PearsonSchool.com The Center for Mathematics Education Project (CME Project) is a new high school program funded through a National Science Foundation (NSF) Grant. The CME Project is a research-based program developed by a team from Education Development Center (EDC). Led by Dr. Al Cuoco, the program is designed to help students develop a deep understanding of mathematics. The development of this program was also informed by the involvement of well-known and respected mathematicians in the field as well as teachers, educational researchers, teacher educators, and field test teachers. It is organized around the familiar structure of Algebra 1, Geometry, Algebra 2, and Precalculus. The CME Project meets the dual goals of mathematical rigor and accessibility for all students. The CME Project is a coherent, four-year high school program designed around how knowledge is organized and generated within mathematics: the themes of algebra, geometry, and analysis. The CME Project sees these branches of mathematics not only as compartments for certain kinds of results, but also as descriptors for methods and approaches the habits of mind that determine how knowledge is organized and generated within mathematics itself. As such, they deserve to be centerpieces of a curriculum, not its by-products. The CME Project uses the traditional American structure of algebra, geometry, advanced algebra, and precalculus. Its primary goal is to develop in students, robust mathematical proficiency. The program builds on lessons learned from high-performing countries: develop an idea thoroughly and then revisit it only to deepen it; organize ideas in a way that is faithful to how they are organized in mathematics; and reduce clutter and extraneous topics. It also employs the best American models that call for struggling with ideas and problems as preparation for instruction, moving from concrete problems to abstractions and general theories, and situating mathematics in engaging contexts. Each level of the CME Project contains 8 chapters. 3-4 Investigations make up each chapter. Each Investigation is then composed of 3-6 lessons. The basic mathematics of each Investigation is accessible to all, and each Investigation can ultimately challenge the best students. The Investigation introduction, essentially for the teacher, describes what to expect. The students work from a more informal to formal understanding of the mathematical topic explored in that particular chapter. The Investigation wrap-up, called Mathematical Reflections, provides an opportunity to review and summarize at the end of the chapter. 1
3 CME Project Algebra IDAHO CONTENT STANDARDS IDAHO MATHEMATICS ALGEBRA II CONTENT STANDARDS Students are expected to know content and apply skills from Algebra I and prior math courses. Mathematical reasoning and problem solving processes will be incorporated throughout all mathematics standards. Students will demonstrate knowledge and communicate mathematical thinking through words, numbers, symbols, charts, graphs, tables, diagrams, and models. Objectives provide the focus for this course. They will be taught using a variety of methods and applications so that students attain a deep understanding of these concepts and are able to apply them to solve real-world problems. Skill Statements provide clarity and direction to achieve each objective. Students need to demonstrate proficiency in these skills upon completion of this course. The appropriate use of technological tools is encouraged to assist students in solving problems and the formation and testing of conjectures. Standard 1: Number and Operation Goal 1.1: Understand numbers, ways of representing numbers, relationships among numbers, and number system. AII Compare and contrast the properties of numbers and number systems within the complex number system to include rational, irrational, and imaginary numbers. AII Demonstrate meaning of complex numbers as solutions to polynomial equations that do not have real solutions. AII logarithms. Represent powers using AII Recognize matrices as a method of arranging data. AII Know that matrices have some of the properties of the real number system. AII Develop an understanding of the properties of logarithmic expressions and expressions with rational exponents. Goal 1.2: SE/TE: 91, 114, 212, 218, 321, 346, 401, SE/TE: 202, , 206, , , 223, 241, 249, 251, 265, 266, 268, , , 275, 279 SE/TE: 461, 462, , , , , , , , , 493, 496, 497, 584 SE/TE: 291, 293, 296, 299, 301, , , , , , , 353, , , , , , SE/TE: 316, , 327, , , , , , 352, 353 SE/TE: , , 423, , , , Understand meanings of operations and how they relate to one another. AII Develop an understanding of the properties of, and representations for, the addition, subtraction, and multiplication of matrices. SE/TE: , , , , , , 339, , 348, 349, 352, 353, , 359, , , , 392, 393, 394 2
4 Goal 1.3: IDAHO CONTENT STANDARDS CME Project Algebra Compute fluently and make reasonable estimates. AII Use the properties of real numbers to simplify expressions. AII matrices. Perform computations with AII Add, subtract, multiply and divide radical expressions. AII Perform computations with complex numbers. AII Perform computations with logarithmic expressions and expressions with rational exponents. Skill Statements: The student will be able to: 1. Apply the properties of exponents including rational exponents to simplify expressions. 2. Define and explain the meaning of i, represented as i= -1, as a solution to the equation x 2 = Identify real and imaginary roots for polynomial equations. 4. Identify expressions of the form a + bi as complex numbers. SE/TE: 92, 114, 184, , 187, 191, 212, 215, SE/TE: , , , , 339, , 348, 349, 352, 353, , 359, , , , 392, 393, 394 SE/TE: , , , 241, 251, , , 433, 459, 497, 777 SE/TE: , , , , , , , , 233, , , , , 251, 253, , , 265, 277, 279 SE/TE: , , 423, 435, 459, 461, 462, , , , , 488, 491, 493, , 497 SE/TE: 398, , , , , , 417, , , 423, 424, 435, , 447, 449, 451, 458, 459, 469, 495, 497, 583 SE/TE: 203, 204, 208, 212, 223, 260 SE/TE: 168, 181, 202, , , 217, 223, 232, 241, 249, 251, 265, 266, 268, , 273, 275, 277, 279, 394 SE/TE: 197, 198, 209, , , 218, 221, 223, 233, 235, 242, 250, Identify complex conjugates. SE/TE: , , , 232, 234, 242, 250, 251, 254, 261, , 277, 279, Express the square root of a negative number in the form bi, where b is real. 7. Apply the properties (to include commutative, associative, distributive, inverse and identity) to simplify computations in the complex number system. 8. Identify a logarithmic function as the inverse of an exponential function. 9. Represent a number in both logarithmic and exponential forms. Example: 5 3 = 125 can be written as log 5 = 3 SE/TE: , 210, 223, 279, 585, 777 SE/TE: , , , , , , 223, 228, 233, , SE/TE: 462, 463, 466, , 474 SE/TE: 462, , , , 496, 497, 583 3
5 CME Project Algebra IDAHO CONTENT STANDARDS 10. Convert between expressions containing radical form and those containing rational exponents. 11. Use properties of logarithms to evaluate and simplify logarithmic expressions. 12. Simplify expressions containing rational and irrational numbers to include rational exponents. 13. Perform operations on radical expressions. 14. Perform operations on rational expressions. SE/TE: 414, 416, , , 423, 459, 473, 475, 495 SE/TE: , , , , , , 488, 491, 493, 496, 497 SE/TE:180, 202, 206, , 217, 223, 232, 251, 269, 273, 277, 279, 399, 400, 401, , , , , , , , 433, 459, 497 SE/TE: , , , 241, 251, , , 433, 459, 497, 777 SE/TE: , , 187, 191, 192, 195, Identify the dimensions of a matrix. SE/TE: 292, 307, , 313, 324, , 336, 338, 348, , 392, Perform operations with matrices to include scalar multiplication, addition, subtraction, and matrix multiplication (2 by 2). SE/TE: , , , , 339, , 348, 349, 352, 353, , 359, , , , 392, 393, Identify the degree of a polynomial. SE/TE: 90, 95, 134, 136, 140, , 147, , , , 161, Factor polynomials. SE/TE: 155, 163, 164, , , , , , , , , 187, 191, 192, 195, Divide a polynomial by a lower degree polynomial. 20. Rewrite complex fractions composed of simple rational expressions as a simple fraction in lowest terms. 21. Simplify and estimate radicals having various indices. SE/TE: , , 160, 183, , 191, 192, 195, 395, 585 can be found on the following pages: SE/TE: , Practice: 107, 251 SE/TE: 180, 202, 203, , , 212, 223, 228, 232, 241, 251, , , , , 433, 459, 497, 777 Suggested vocabulary and symbols base, complex number (a+bi), complex conjugate, conjugate, exponent, index, logarithm, power, radicand, radical, natural logarithm (ln), common logarithm, change of base for logs, principal square root, imaginary number (i), rationalize, dimensions of a matrix, scalar multiplication, simplest form of a radical, degree of a polynomial. 4
6 IDAHO CONTENT STANDARDS CME Project Algebra Standard 2: Concepts and Principles of Measurement Goal 2.1: Understand measurable attributes of objects and the units, systems, and processes of measurement. AII Recognize the relationship between radian and degree measures. can be found on the following pages: SE/TE: , Practice: , 283 Goal 2.2: Apply appropriate techniques, tools, and formulas to determine measurements. Skill Statement: The student will be able to: 1. Convert between degree and radian measures. Suggested vocabulary and symbols Radian measure. Standard 3: Goal 3.1: Concepts and Language of Algebra and Functions Understand patterns, relations, and functions. AII Represent patterns and functional relationships in a table and as a graph. AII Describe the graph of a quadratic equation and discuss its attributes in terms of the basic concepts of maximum, minimum, intercepts, and roots. AII Graph and analyze the graph of an absolute value equation and its characteristics. can be found on the following pages: SE/TE: , Practice: 276 SE/TE: 5, 6-7, 8, 10-11, 12-15, 16-18, 19-21, 22-23, 24-28, 29-34, 34-38, 39, 41-42, 43-45, 46-48, 49-52, 53-54, 56-59, 63-65, , , , , , 139, 425, , , , , , 584, 716, , , SE/TE: 123, 130, 147, 160, 168, 502, 504, 510, , , 530, 531, , 580, 582, 585 SE/TE: 501, 503, 504, 518, 528, 531, 555, 570, 580, 582 AII Understand and represent transformations by using sketches, coordinates, and function notation. SE/TE: 501, , , , , , , 531, 555, , , 577, 580, 582, , 777 5
7 IDAHO CONTENT STANDARDS CME Project Algebra Goal 3.2: Represent and analyze mathematical situations and structures using algebraic symbols. AII Write equations and inequalities in multiple forms. AII Solve equations and inequalities and systems of equations and inequalities. AII Perform operations on simple rational expressions. SE/TE: , 105, , , 155, 159, , 173, 399, 400, , 408, , 417, 461, 462, , 468, , 474, 476 SE/TE: 168, 181, 202, , , 217, 223, 232, 241, 249, 251, 265, 266, 268, , 273, 275, 277, 279, 282, , , , , , 301, , , 352, 353, , 393, 394, 399, 400, 403, 409, 414, 426, , 439, 445, , 458, 459, 461, 464, , 474, 476, , 493, 496, 497, 584 SE/TE: , , 187, 191, 192, 195, 222 Goal 3.3: Use mathematical models to represent and understand quantitative relationships. Goal 3.4: Analyze change in various contexts. AII Interpret how changes to an equation affect the parent graph of the equation. Skill Statements: The student will be able to: 1. Solve systems of equations and inequalities. 2. Solve radical equations and inequalities. SE/TE: 181 SE/TE: 501, , , , , , , 531, 555, , , 577, 580, 582, , 777 SE/TE: 282, , , , , , 301, , , 352, 353, , 393, Solve polynomial functions. SE/TE: 168, 181, 202, , , 217, 223, 232, 241, 249, 251, 265, 266, 268, , 273, 275, 277, 279, Solve rational equations. can be found on pages SE/TE: , Practice: 110, Solve logarithmic equations. SE/TE: 472, 474, 476, , 493, 496, 497, Solve equations containing a variable in the exponent. SE/TE: 399, 400, 403, 409, 414, 426, , 439, 445, , 458, 459, 461, 464, , 474, 496, 497, 483 6
8 CME Project Algebra IDAHO CONTENT STANDARDS 7. Compare and contrast the graphs of ƒ(x)=x 2 to ƒ(x)=a(x-h) 2 + k. SE/TE: 501, , 511, , 517, , , 527, 530, 531, 555, 569, 580, 582, Graph absolute value functions. SE/TE: 501, 503, 504, 518, 528, 531, 555, 570, 580, Graph quadratic equations and inequalities. SE/TE: 123, 130, 133, 160, 502, 504, , 519, , , , , , 580, 582, Graph polynomial functions. SE/TE: 117, , 125, 127, 129, 135, 141, 501, 504, , , , 528, 530, Graph exponential functions. SE/TE: , , 436, 444, 449, 451, 453, , 459, 463, 492, 495, 497, 505, 521, , Graph circles. SE/TE: , 512, 516, 518, 526, , 531, , 570, 580, 582, Rewrite equations of parabolas and circles in standard form. 14. Recognize contexts in which quadratic models are appropriate; determine and interpret quadratic models that describe quadratic behavior. 15. Use the quadratic formula and completing the square to solve any quadratic equation in one variable. 16. Determine the nature of the roots of an equation by using the discriminant. can be found on the following pages: SE/TE: 501, 502, , , 519, 529 Practice: 260 SE/TE: 29-34, 34-38, 39, , , , 136, 144, 145, 147 SE/TE: 202, 206, , 217, 223, 232, 251, 269, 273, 277, 279 can be found on page: SE/TE: 202 Practice: Find the all the roots of polynomials. SE/TE: 168, 181, 202, , , 217, 223, 232, 241, 249, 251, 265, 266, 268, , 273, 275, 277, 279, Recognize graphs of the following: y=x, y=x 2, y=x 3, y=1/x, y= x, and y= x. SE/TE: 501, , , Suggested vocabulary and symbols Complex fraction, rational expression, degree, extraneous roots, inverse, constraints, feasible region, independent variables, dependent variables, factor (verb and noun), zeros of a function, root, domain, range, coincident, consistent systems, inconsistent systems, maximum, minimum, bounded regions, unbounded regions, f(x), discriminant, linear programming, vertex form of a quadratic, synthetic division, synthetic substitution, standard form, parabola, focus of parabola, joint variation, direct variation, inverse variation, exponential growth and decay, cubic, quartic, quadratic, vertex, vertices, focus, directrix, axis of symmetry. 7
9 IDAHO CONTENT STANDARDS Standard 4: CME Project Algebra Concepts and Principles of Geometry Goal 4.1 Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. AII Use trigonometric relationships to determine lengths and angle measures. SE/TE: 683, , 737, 739, 741, , , , , 770, 774, 775 Goal 4.2 Specify locations and describe spatial relationships using coordinate geometry and other representational systems. AII Analyze the graphs of circles and SE/TE: 31-32, 34, 290, 455, 501, , parabolas , , , , 526, , 531, 546, , , 580, 582 Goal 4.3: Apply transformations and use symmetry to analyze mathematical situations. Goal 4.4: Use visualization, spatial reasoning, and geometric models to solve problems. Skill Statement: The student will be able to: 1. Demonstrate the proper use of the Law of Sines and the Law of Cosines to solve triangles. 2. Graph circles and parabolas and their transformations. Suggested vocabulary and symbols Sine, cosine, tangent, secant, cosecant, cotangent. Standard 5: Data Analysis, Probability, and Statistics SE/TE: , , , , 770, 774, 775 SE/TE: 501, 502, , 512, , 519, , 526, , , , 580, 582, 691 8
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