Note: This curriculum was updated July, 2002 to reflect the 2001 Mathematics SOL
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Course |
Discrete Math |
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Adopted Text |
Excursions in Modern Mathematics, Prentice Hall |
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Grade Level(s) |
11-12 |
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Prerequisite |
Algebra I (or Algebra I, Parts 1 and 2), Geometry, and Algebra II |
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This semester course is intended for the college bound student who may otherwise elect not to take a math course during his junior or senior year. Discrete Mathematics is a study of contemporary mathematices including many varied topics such as networks (graphs), counting methods, scheduling, voting theory, logic, fair apportionment, recursion, game theory, and matrices. The mathematics included in this course is very practical and would involve answering questions like these: How many ways can a newspaper delivery route be scheduled? Which route is the most efficient? How are the House of Representatives apportioned among the fifty states? How does an annuity work? Students will use technology such as spreadsheets and/or graphics calculators during the course. The course may be taken before, after, or concurrently with Analysis. Note that students who are considering any math related field should not take Discrete Math in lieu of Analysis/Calculus but may wish to take it in addition to these courses. Discrete Math will be offered in alternating semesters with Statistics and Probability.
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THE STUDENT WILL BE ABLE TO: |
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DM.1 Discuss general facts about preference ballots and schedules and write a preference schedule from a given set of preference ballots. DM.2 Determine a winner and rank of candidates from a preference schedule using any of the four voting methods. (Plurality, Borda count, plurality with elimination, and pairwise comparision.) DM.3 Discuss how to break ties and how the four voting methods relate to fairness criteria, such as majority criterion, Condorcet criterion, monotonicity criterion, and independence of irrelevant alternatives criterion. DM.4 Describe a weighted voted system and identify its associated notation and terminology. DM.5 Determine the power distribution of a given weighted voting system using the Banzhaf power index and the Shapley-Shubik power index. DM.6 Discuss fair division schemes and apply an assortment of methods for solving both continous and discrete fair division problems. DM.7 Identify and be able to discuss the advantages and disadvantages of various apportionment methods. DM.8 Apply various apportionment methods in solving given problems. DM.9 Identify and define various terms and concepts of graph theory. DM.10 Eulerize a given graph and apply Fleury's algorithm to find an Euler's circuit for the given graph. DM.11 Apply Fleury's algorithm to analyze and solve real world routing problems. DM.12 Identify and discuss weighted graphs, optimal routes, and Hamilton circuits. DM.13 Apply various algorithms to analyze and solve real world routing problems (like the traveling salesman problem). DM.14 Identify and define various terms and concepts relating to trees, spanning trees, and networks. DM.15 Apply Kruskal's algorithm to find a minimum spanning tree for a given graph. DM.16 Determine the Steiner point(s) needed for a given graph and then construct a Steiner tree that produces a shortest network. DM.17 Identify and define various terms and concepts relating to directed graphs. DM.18 Apply various algorithms to analyze and solve scheduling problems. DM.19 Explore optional topics at the discretion of the instructor. (These may include logic, fractals, sequences, recursion formulas, and exponential growth.) |
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Objective: |
DM.1 The student will be able to discuss general facts about preference ballots and schedules and write a preference schedule from a given set of preference ballots. |
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Text Resources: |
Excursions in Modern Mathematics (Prentice-Hall) Ch. 1 |
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Related Web Sites: |
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Suggested Manipulatives: |
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Technology Resources: |
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Other Resources: |
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Assessment Suggestions: |
Instructors's Manual Excursions in Modern Mathematics |
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Suggestions for Integration: |
. |
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Objective: |
DM.2 The student will be able to determine a winner and rank of candidates from a preference schedule using any of the four voting methods. (Plurality, Borda count, plurality with elimination, and pairwise comparision.) |
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Text Resources: |
Excursions in Modern Mathematics (Prentice-Hall) Ch. 1 |
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Related Web Sites: |
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Suggested Manipulatives: |
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Technology Resources: |
Graphing Calculators |
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Other Resources: |
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Assessment Suggestions: |
Instructors's Manual Excursions in Modern Mathematics |
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Suggestions for Integration: |
Use matrices with the Borda count method. |
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Objective: |
DM.3 The student will be able to discuss how to break ties and how the four voting methods relate to fairness criteria, such as majority criterion, Condorcet criterion, monotonicity criterion, and independence of irrelevant alternatives criterion. |
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Text Resources: |
Excursions in Modern Mathematics (Prentice-Hall) Ch. 1 |
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Related Web Sites: |
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Suggested Manipulatives: |
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Technology Resources: |
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Other Resources: |
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Assessment Suggestions: |
Instructors's Manual Excursions in Modern Mathematics |
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Suggestions for Integration: |
. |
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Objective: |
DM.4 The student will be able to describe a weighted voted system and identify its associated notation and terminology. |
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Text Resources: |
Excursions in Modern Mathematics (Prentice-Hall) Ch. 2 |
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Related Web Sites: |
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Suggested Manipulatives: |
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Technology Resources: |
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Other Resources: |
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Assessment Suggestions: |
Instructors's Manual Excursions in Modern Mathematics |
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Suggestions for Integration: |
. |
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Objective: |
DM.5 The student will be able to determine the power distribution of a given weighted voting system using the Banzhaf power index and the Shapley-Shubik power index. |
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Text Resources: |
Excursions in Modern Mathematics (Prentice-Hall) Ch. 2 |
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Related Web Sites: |
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Suggested Manipulatives: |
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Technology Resources: |
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Other Resources: |
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Assessment Suggestions: |
Instructors's Manual Excursions in Modern Mathematics |
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Suggestions for Integration: |
. |
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Objective: |
DM.6 The student will be able to discuss fair division schemes and apply an assortment of methods for solving both continous and discrete fair division problems. |
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Text Resources: |
Excursions in Modern Mathematics (Prentice-Hall) Ch. 3 |
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Related Web Sites: |
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Suggested Manipulatives: |
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Technology Resources: |
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Other Resources: |
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Assessment Suggestions: |
Instructors's Manual Excursions in Modern Mathematics |
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Suggestions for Integration: |
. |
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Objective: |
DM.7 The student will be able to identify and be able to discuss the advantages and disadvantages of various apportionment methods. |
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Text Resources: |
Excursions in Modern Mathematics (Prentice-Hall) Ch. 4 |
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Related Web Sites: |
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Suggested Manipulatives: |
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Technology Resources: |
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Other Resources: |
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Assessment Suggestions: |
Instructors's Manual Excursions in Modern Mathematics |
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Suggestions for Integration: |
. |
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Objective: |
DM.8 The student will be able to apply various apportionment methods in solving given problems. |
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Text Resources: |
Excursions in Modern Mathematics (Prentice-Hall) Ch. 4 |
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Related Web Sites: |
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Suggested Manipulatives: |
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Technology Resources: |
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Other Resources: |
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Assessment Suggestions: |
Instructors's Manual Excursions in Modern Mathematics |
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Suggestions for Integration: |
. |
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Objective: |
DM.9 The student will be able to identify and define various terms and concepts of graph theory. |
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Text Resources: |
Excursions in Modern Mathematics (Prentice-Hall) Ch. 5 |
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Related Web Sites: |
http://www.c3.lanl.gov/mega-math/ |
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Suggested Manipulatives: |
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Technology Resources: |
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Other Resources: |
Discrete Mathematics and Its Applications |
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Assessment Suggestions: |
Instructors's Manual Excursions in Modern Mathematics |
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Suggestions for Integration: |
. |
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Objective: |
DM.10 The student will be able to eulerize a given graph and apply Fleury's algorithm to find an Euler's circuit for the given graph. |
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Text Resources: |
Excursions in Modern Mathematics (Prentice-Hall) Ch. 5 |
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Related Web Sites: |
http://www.c3.lanl.gov/mega-math/ |
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Suggested Manipulatives: |
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Technology Resources: |
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Other Resources: |
Discrete Mathematics and Its Applications |
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Assessment Suggestions: |
Instructors's Manual Excursions in Modern Mathematics |
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Suggestions for Integration: |
. |
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Objective: |
DM.11 The student will be able to apply Fleury's algorithm to analyze and solve real world routing problems. |
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Text Resources: |
Excursions in Modern Mathematics (Prentice-Hall) Ch. 5 |
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Related Web Sites: |
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Suggested Manipulatives: |
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Technology Resources: |
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Other Resources: |
Discrete Mathematics and Its Applications |
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Assessment Suggestions: |
Instructors's Manual Excursions in Modern Mathematics |
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Suggestions for Integration: |
. |
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Objective: |
DM.12 The student will be able to identify and discuss weighted graphs, optimal routes, and Hamilton circuits. |
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Text Resources: |
Excursions in Modern Mathematics (Prentice-Hall) Ch. 6 |
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Related Web Sites: |
http://www.c3.lanl.gov/mega-math/ |
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Suggested Manipulatives: |
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Technology Resources: |
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Other Resources: |
Discrete Mathematics and Its Applications |
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Assessment Suggestions: |
Instructors's Manual Excursions in Modern Mathematics |
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Suggestions for Integration: |
. |
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Objective: |
DM.13 The student will be able to apply various algorithms to analyze and solve real world routing problems (like the traveling salesman problem). |
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Text Resources: |
Excursions in Modern Mathematics (Prentice-Hall) Ch. 6 |
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Related Web Sites: |
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Suggested Manipulatives: |
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Technology Resources: |
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Other Resources: |
Discrete Mathematics and Its Applications |
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Assessment Suggestions: |
Instructors's Manual Excursions in Modern Mathematics |
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Suggestions for Integration: |
. |
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Objective: |
DM.14 The student will be able to identify and define various terms and concepts relating to trees, spanning trees, and networks. |
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Text Resources: |
Excursions in Modern Mathematics (Prentice-Hall) Ch. 7 |
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Related Web Sites: |
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Suggested Manipulatives: |
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Technology Resources: |
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Other Resources: |
Discrete Mathematics and Its Applications |
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Assessment Suggestions: |
Instructors's Manual Excursions in Modern Mathematics |
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Suggestions for Integration: |
. |
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Objective: |
DM.15 The student will be able to apply Kruskal's algorithm to find a minimum spanning tree for a given graph. |
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Text Resources: |
Excursions in Modern Mathematics (Prentice-Hall) Ch. 7 |
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Related Web Sites: |
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Suggested Manipulatives: |
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Technology Resources: |
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Other Resources: |
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Assessment Suggestions: |
Instructors's Manual Excursions in Modern Mathematics |
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Suggestions for Integration: |
. |
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Objective: |
DM.16 The student will be able to determine the Steiner point(s) needed for a given graph and then construct a Steiner tree that produces a shortest network. |
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Text Resources: |
Excursions in Modern Mathematics (Prentice-Hall) Ch. 7 |
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Related Web Sites: |
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Suggested Manipulatives: |
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Technology Resources: |
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Other Resources: |
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Assessment Suggestions: |
Instructors's Manual Excursions in Modern Mathematics |
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Suggestions for Integration: |
. |
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Objective: |
DM.17 The student will be able to determine the Steiner point(s) needed for a given graph and then construct a Steiner tree that produces a shortest network. |
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Text Resources: |
Excursions in Modern Mathematics (Prentice-Hall) Ch. 8 |
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Related Web Sites: |
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Suggested Manipulatives: |
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Technology Resources: |
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Other Resources: |
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Assessment Suggestions: |
Instructors's Manual Excursions in Modern Mathematics |
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Suggestions for Integration: |
. |
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Objective: |
DM.18 The student will be able to apply various algorithms to analyze and solve scheduling problems. |
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Text Resources: |
Excursions in Modern Mathematics (Prentice-Hall) Ch. 8 |
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Related Web Sites: |
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Suggested Manipulatives: |
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Technology Resources: |
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Other Resources: |
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Assessment Suggestions: |
Instructors's Manual Excursions in Modern Mathematics |
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Suggestions for Integration: |
. |
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Objective: |
DM.19 The student will be able to explore optional topics at the discretion of the instructor. (These may include logic, fractals, sequences, recursion formulas, and exponential growth.) |
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Text Resources: |
Excursions in Modern Mathematics (Prentice-Hall) Ch. 9 - 12 |
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Related Web Sites: |
http://math.bu.edu/DYSYS/chaos-game/ |
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Suggested Manipulatives: |
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Technology Resources: |
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Other Resources: |
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Assessment Suggestions: |
Instructors's Manual Excursions in Modern Mathematics |
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Suggestions for Integration: |
. |