A Survey of Mathematics with Applications 8 th Edition, 2009
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1 A Correlation of A Survey of Mathematics with Applications 8 th Edition, 2009 South Carolina Discrete Mathematics Sample Course Outline including Alternate Topics and Related Objectives
2 INTRODUCTION This document demonstrates how A Survey of Mathematics with Applications, 8th Edition 2009, meets the indicators of the South Carolina Sample Course Outline including. Correlation page references are to the Student Edition and Instructor s Edition Manual and are cited at the page level. This best-selling text balances solid mathematical coverage with a comprehensive overview of mathematical concepts as they relate to varied disciplines. The text provides an appreciation of mathematics, highlighting mathematical history, and applications of math arts and sciences. It is an ideal book for students who require a general overview of mathematics, especially those majoring in liberal arts, the social sciences, business, nursing and allied health fields. Features: A readable text showing students how mathematics affects the world around them. Unique, real-life applied examples and exercises motivate and engage students. Students are introduced to problem solving and critical thinking, which is then continued throughout the text. Special problem-solving exercises are presented in the exercise sets and sections on Inductive Reasoning, and the important skills of Estimation and Dimensional Analysis are also featured. Colorful, engaging, and lively boxed features called Did You Know? highlight the connection of mathematics to history, arts and sciences, to technology, and to a broad variety of disciplines. Online Teacher Resources include Active Learning Questions and PowerPoints. This document demonstrates the success students will achieve by using A Survey of Mathematics with Applications. 2
3 I. Social decision making. A. Use group ranking methods to obtain a single result. B. Determine power indexes for weighted voting systems. A Survey of Mathematics with Applications 2009 SE/TE: , Students work with a variety of voting systems, including the Borda count method, in which data are weighted according to rank. They calculate weighted averages and construct weighted graphs to solve optimization problems. SE/TE: , 944, , , C. Solve fair division problems. 1. Continuous examples. Students solve apportionment problems with discrete data. SE/TE: Discrete examples. Students encounter standard divisors and standard quotas in apportionment problems. SE/TE: , D. Solve apportionment problems using a variety of methods. 1. Hamilton method. SE/TE: , , Hill method. In addition Hamilton, Jefferson, and Webster methods for solving apportionment problems, students use the Adams method. SE/TE: , Jefferson method. SE/TE: , , Webster method. SE/TE: , , II. Graph theory. A. Define basic terms and concepts. 1. Graphs. SE/TE: , , , , 843, 850, , , , , , , , , Vertices. SE/TE: , , , , AB-1-AB-6 3. Edges. SE/TE: , , , , AB-1-AB-6 3
4 A Survey of Mathematics with Applications Connected graph. SE/TE: , , , , AB-1-AB-6 5. Complete graph. SE/TE: , , AB-1-AB-6 6. Digraphs. Students examine directed paths along graph edges. SE/TE: 923, 925, AB-1, AB-3 7. Adjacency matrix. Students use matrices to represent information and solve word problems. SE/TE: , , Bipartite graph. Some tree diagrams are bipartite graphs in which each edge is a bridge. SE/TE: 391, Planarity. Planarity is a property of connected and complete graphs. This concept could be incorporated in the lessons on the following pages: SE/TE: , , , , AB-1-AB-6 B. Represent real-world situations using graphs. SE/TE: , , , , , , , , , , , , , , C. Find the critical path. SE/TE: , , D. Define Euler circuits and paths. SE/TE: E. Define Hamiltonian circuits and paths. SE/TE: F. Color graphs. SE/TE: 585, , , 593 G. Define and identify trees. SE/TE: H. Find minimum spanning trees. SE/TE: III. Matrix theory. A. Write an adjacency matrix for a graph. Students use matrices to represent information and solve word problems. SE/TE: , , B. Perform basic matrix operations. SE/TE: ,
5 C. Use the Leslie model to find population distributions. D. Use the Leontief input-output model to analyze the flow of goods and services among sectors in an economy. A Survey of Mathematics with Applications 2009 Students use matrices to represent information and solve word problems. They explore population sampling, population growth, and the population paradox. SE/TE: 399, , , , 833, Students use matrices to model economic data and also use inverse matrices to solve systems of linear equations. SE/TE: , E. Identify Markov chains. Markov chains describe events that whose probability is not affected by past or future events. While the term is not specifically used in the textbook, such events are described in a unit on probability. SE/TE: F. Investigate game theory 1. With a single best strategy. SE/TE: Without a single best strategy. SE/TE: 777, 792 IV. Counting techniques. A. Define combinations and permutations. SE/TE: , B. Evaluate expressions indicating permutations or combinations. SE/TE: , , C. Construct and examine Pascal s triangle. SE/TE: 304, 808 D. Use the addition and multiplication principles to solve problems. SE/TE: , E. Define and calculate conditional probabilities. SE/TE: F. Calculate expected value. SE/TE: V. Iteration and recursive relations. A. Determine the terms of a sequence. 1. Explicit. SE/TE: 290, Recursive. SE/TE: ,
6 B. Evaluate sums of geometric series. A Survey of Mathematics with Applications Finite. SE/TE: Infinite. SE/TE: 297 C. Solve problems using recursion and mixed recursion. SE/TE: D. Solve exponential growth problems. SE/TE: , E. Solve finance and population problems. Sample References: SE/TE: 16, , , 330, 334, 404, , , , , , , 843, 847, 849 F. Use the method of finite differences for finding closed-form solutions. Students use explicit formulas for arithmetic sequences and series. SE/TE: G. Construct cobweb diagrams. Data are displayed in circle graphs, histograms, frequency polygons, stem-and-leaf displays, Venn and Euler diagrams, SE/TE: 60-71, 72-82, 164, H. Define fractals. SE/TE: I. Investigate chaos theory. SE/TE: VI. Linear programming. A. Solve real-world problems using linear programming. B. Use technology to solve linear programming problems. VII. Mathematical induction. SE/TE: The lesson on linear programming does not specifically refer use of technology to solve the problems of the lesson, but any of the problems can be at least partially solved using a graphing calculator or computer. SE/TE: The textbook does describe the differences between inductive and deductive reasoning, but the process of proof by mathematical induction is not specifically covered. SE/TE: 3-4 6
7 VIII. Logic. A. Identify a statement as existential, universal, or neither. B. Write and use counterexamples and properties. A Survey of Mathematics with Applications 2009 The textbook describes statements with quantifiers all, none (or no), and some. SE/TE: , SE/TE: 4-5, 75, , 153, , 613 C. Negate statements. SE/TE: , 107, , , 118, , D. Determine the truth of a conditional SE/TE: , , , 144 statement. E. Write a logical expression to describe a simple network. F. Find the contrapositive, converse, or inverse of a conditional statement. SE/TE: , 180 SE/TE: G. Use logic to prove or disprove a statement. SE/TE: , , 144, 173 H. Determine the validity or invalidity of arguments. SE/TE: ,
8 I. Vectors. A Survey of Mathematics with Applications 2009 A. Solve problems involving two-dimensional and three-dimensional vectors. 1. Geometric representation of vectors. SE/TE: , , Geometric representation of complex numbers. Students explore real numbers and their properties. SE/TE: Algebraic representation of vectors. Students use geometric representations of vectors to perform translations of geometric figures. SE/TE: , , Vectors and parametric equations. Students use geometric representations of vectors to perform translations of geometric figures. SE/TE: , , Dot product. Students use geometric representations of vectors to perform translations of geometric figures. SE/TE: , , Cross product. Students use geometric representations of vectors to perform translations of geometric figures. SE/TE: , , Vectors and planes. Students use geometric representations of vectors to perform translations of geometric figures. SE/TE: , , II. Sequences and series. A. Solve problems involving series and sequences. 1. Arithmetic sequences. SE/TE: , Geometric sequences. SE/TE: The nth term of an arithmetic or geometric sequence. SE/TE: Recursive definitions of sequences. SE/TE: ,
9 5. Arithmetic series. SE/TE: 291, Geometric series. SE/TE: A Survey of Mathematics with Applications The sum of n terms of an arithmetic or geometric series. SE/TE: 291, Infinite geometric series. SE/TE: Sums of special geometric series. SE/TE: Fibonacci series. SE/TE: B. Write proofs of the validity of summation formulas. 1. Summation formula by Gauss. SE/TE: Mathematical induction. The textbook does describe the differences between inductive and deductive reasoning, but the process of proof by mathematical induction is not specifically covered. SE/TE: 3-4 III. Matrices and determinants. A. Solve problems using matrices. 1. Matrix addition. SE/TE: , Scalar multiplication. SE/TE: , Matrix multiplication. SE/TE: Inverse matrices. Students use augmented matrices and row transformations to solve systems of linear equations. SE/TE: B. Solve problems using matrices and determinants. 1. Systems of equations. SE/TE: Transformations using matrices. SE/TE: Systems with augmented matrices. SE/TE:
10 IV. Graphs and their applications. A Survey of Mathematics with Applications 2009 A. Special functions. 1. Piecewise functions. Students explore relations, functions, and their graphs, including linear, quadratic, and exponential functions. SE/TE: Greatest integer functions. Students explore relations, functions, and their graphs, including linear, quadratic, and exponential functions. SE/TE: B. Venn diagrams. SE/TE: 60-71, 72-82, 82-89, 163 C. Linear programming. SE/TE: D. Paths and circuits. 1. Multigraphs. Although the vocabulary is not specifically introduced in the textbook, some of the graphs studied in the chapter on graph theory are multigraphs. SE/TE: 922, 929, , Loops and parallel edges. SE/TE: 922, 929, , Paths, simple paths, distinct cycles, and their lengths. SE/TE: , , Euler paths or circuits. SE/TE: E. Coloring a graph. 1. Chromatic number for a graph. This concept could be introduced with topology and color graphs. SE/TE: 585, , , Welsh-Powell algorithm to color graph. The use of this algorithm is implied as students explore topology and color maps. SE/TE: 585, , , 593 F. Directed graphs. Students examine directed paths along graph edges. SE/TE: 923, 925, AB-1, AB-3 10
11 V. Recurrence. A Survey of Mathematics with Applications 2009 A. Values and initial conditions. Although the vocabulary is not specifically introduced, students explore recurrence in Fibonacci sequences. SE/TE: B. First-order linear difference equations. Any arithmetic sequence is a first-order linear difference relation, but these sequences are only defined explicitly in the textbook. SE/TE: , C. Second-order homogeneous linear difference equations. Although the vocabulary is not specifically introduced, Fibonacci sequences are secondorder homogeneous linear difference relations. SE/TE: D. Searching and sorting algorithms. Searching and sorting algorithms are not explored in this textbook. VI. Arithmetic. A. Bases (decimal, binary, octal, hexadecimal). SE/TE: , B. Conversions between bases. SE/TE: , C. Arithmetic in non-decimal bases. SE/TE: , , , D. Computer representation of numbers. SE/TE: 198, , 213, E. Computer arithmetic. SE/TE: , 213 VII. Propositional logic. A. Propositions. Propositions are implied throughout the Logic chapter in the textbook. SE/TE: , , , , , , , B. Binary connectives. SE/TE: , , , , C. Truth tables. SE/TE: , , , 143, 146, 152, , 158, , 174 D. Laws of logic. SE/TE: , , , 157 E. Validity of arguments. SE/TE: ,
12 A Survey of Mathematics with Applications 2009 F. Predicate logic. Although the vocabulary is not specifically introduced, predicate logic is implied throughout the Logic chapter in the textbook. SE/TE: , , , , , , , G. Quantifiers. SE/TE: , VIII. Boolean algebra. A. Axiomatic definition. Although the vocabulary is not specifically introduced, axiomatic definitions, or statements which are universally true, are presented as laws of logic, e.g., De Morgan s Laws. SE/TE: , , , 157 B. Simplifying Boolean expressions. Boolean expressions are specifically referenced on the following pages: SE/TE: 101, 106, 173 C. Digital circuits. SE/TE: D. Disjunctive normal form. Although the vocabulary is not specifically introduced, disjunctions of conjunctions are explored in the Logic chapter of the textbook. Sample references: SE/TE: , , E. Karnaugh maps. Karnaugh maps are not introduced in the textbook. Logic statements are represented using symbols, truth tables, Euler diagrams, and switching circuits. Sample references: SE/TE: , , , F. Logic gates. SE/TE: 173,
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