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DISCRETE MATHEMATICS CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS

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DISCRETE MATHEMATICS CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSIntroduction

The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction and assessment. It defines the content knowledge, skills, and understandings that are measured by the Standards of Learning assessment. It provides additional guidance to teachers as they develop an instructional program appropriate for their students. It also assists teachers in their lesson planning by identifying essential understandings, defining essential content knowledge, and describing the intellectual skills students need to use. This Guide delineates in greater specificity the content that all teachers should teach and all students should learn.

The format of the Curriculum Guide facilitates teacher planning by identifying the key concepts, knowledge, and skills that should be the focus of instruction for each objective. The Curriculum Guide is divided into sections: Curriculum Information, Essential Knowledge and Skills, Key Vocabulary, Essential Questions and Understandings, Teacher Notes and Elaborations, Resources, and Sample Instructional Strategies and Activities. The purpose of each section is explained below.

Curriculum Information:This section includes the objective, focus or topic, and in some, not all, foundational objectives that are being built upon.

Essential Knowledge and Skills:Each objective is expanded in this section. What each student should know and be able to do in each objective is outlined. This is not meant to be an exhaustive list nor a list that limits what is taught in the classroom. This section is helpful to teachers when planning classroom assessments as it is a guide to the knowledge and skills that define the objective.

Key Vocabulary:This section includes vocabulary that is key to the objective and many times the first introduction for the student to new concepts and skills.

Essential Questions and Understandings:This section delineates the key concepts, ideas, and mathematical relationships that all students should grasp to demonstrate an understanding of the objectives.

Teacher Notes and Elaborations:This section includes background information for the teacher. It contains content that is necessary for teaching this objective and may extend the teachers’ knowledge of the objective beyond the current grade level. It may also contain definitions of key vocabulary to help facilitate student learning.

Resources:This section lists various resources that teachers may use when planning instruction. Teachers are not limited to only these resources.

Sample Instructional Strategies and Activities:This section lists ideas and suggestions that teachers may use when planning instruction.

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Discrete Mathematics in Prince William County is a semester course. In addition to the Virginia Department of Education Discrete Mathematics semester course requirements, Prince William County has included additional Standards of Learning from the state curriculum. The following chart lists the objectives for the Prince William County Discrete Mathematics Curriculum organized by units of study. The Prince William County cross-content vocabulary terms that are in this course are: analyze, compare and contrast, conclude, evaluate, explain, generalize, question/inquire, sequence, solve, summarize, and synthesize.

Units of Study ObjectivesGraph Theory DM 1, DM 3, DM 5Circuits, Trees, and Paths DM 2, DM 4Linear Programming DM 6Election Theory DM 8, DM 9Fair Division DM 7

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Curriculum Information Essential Knowledge and SkillsKey Vocabulary

Essential Questions and UnderstandingsTeacher Notes and Elaborations

TopicGraphs

Virginia SOL DM.1The student will model problems, using vertex-edge graphs. The concepts of valence, connectedness, paths, planarity, and directed graphs will be investigated. Adjacency matrices and matrix operations will be used to solve problems (e.g., food chains, number of paths).

Unit of StudyGraph Theory

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Find the valence of each vertex in a

graph. Use graphs to model situations in which

the vertices represent objects, and edges (drawn between vertices) represent a particular relationship between objects.

Represent the vertices and edges of a graph as an adjacency matrix, and use the matrix to solve problems.

Investigate and describe valence and connectedness.

Determine whether a graph is planar or nonplanar.

Use directed graphs (digraphs) to represent situations with restrictions in traversal possibilities.

Key Vocabularyadjacency matrix complete graphconnected graph digraph (directed graph)nonplanarpathplanarplanarityvalence

Essential Questions What is meant by a path and a graph? How is the valence of a vertex determined? How can a matrix represent the vertices of a path/circuit? How can graphs be used to model real-world situations? What is the difference between a planar and nonplanar graph?

Essential Understandings A tournament is a digraph that results from giving directions to the edges of a complete

graph. Adjacent vertices are connected by an edge. In a connected graph, every pair of vertices is adjacent.

Teacher Notes and ElaborationsA path is a connected sequence of edges showing a route on the graph that starts at a vertex and ends at a vertex; a path is usually described by naming in turn the vertices visited in traversing it. Adjacent edges share a common vertex. A vertex can appear on the path more than once. An edge can be part of a path only once. The number of edges in the path is called the length of the path.

The number of edges that have a specific vertex as an endpoint is know as the degree (or valence) of that vertex. In the figure below the degree of vertex A is 4.

In a connected graph, any two vertices can be joined by a path, called an edge. It is possible to travel from any vertex to any other vertex along consecutive edges of the graph. The following is an example of a connected graph.

Example 1: B F

C

A E G

D

(continued)

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http://en.wikipedia.org/wiki/Vertex_(graph_theory)


Curriculum Information Essential Questions and UnderstandingsTeacher Notes and Elaborations

TopicGraphs



Teacher Notes and Elaborations (continued)If a graph is not connected, it is referred to as disconnected. A graph that is disconnected is made up of pieces that are by themselves connected. These pieces are called the components of the graph. The following is an example of a disconnected graph made up of two components.

Example 2: B F

C

A E G

D

A digraph or directed graph is a graph in which each edge has an arrow indicating the direction of the edge. Such directed edges are appropriate when the relationship is “one-sided” which limits the traversal possibilities rather than symmetric (e.g., one-way streets as opposed to regular streets).

A complete graph is a graph in which each pair of vertices is adjacent. For example the following is two different representations of the same complete graph.

Example 3: Example 4: A

A B

D E B

C

D C E

Graphs can be represented in several ways. The diagrams such as the ones above is one way. Another way to represent the information is to list the set of vertices and the set of edges.

Vertices ={A, B, C, D, E} Edges = {AB, AC, AD, AE, BC, BD, BE, CD, CE, DE}

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DISCRETE MATHEMATICS CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS(continued)


TopicGraphs



Teacher Notes and Elaborations (continued)A third way to represent the information is with an adjacency matrix. In mathematics and computer science, an adjacency matrix is a means of representing vertices and edges of a graph. A 5 × 5 matrix in which both the rows and columns correspond to the vertices A, B, C, D, and E can be used to represent Examples 3 or 4 from the previous page. If an edge exists between vertices, a 1 appears in the corresponding position in the matrix; otherwise a 0 appears.

A B C D E A 0 1 1 1 1B 1 0 1 1 1C 1 1 0 1 1D 1 1 1 0 1E 1 1 1 1 0

Planarity comes from the term planar graph. In graph theory, a planar graph is a graph that can be embedded in a plane so that no two edges intersect except at a vertex. Graphs can be planar (of or in a geometric plane) or nonplanar (a graph that cannot be drawn without edges that intersect within a plane).

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http://en.wikipedia.org/wiki/Graph_theory

http://en.wikipedia.org/wiki/Planar_graph

http://en.wikipedia.org/wiki/Computer_science

http://en.wikipedia.org/wiki/Mathematics


Curriculum Information Resources Sample Instructional Strategies and Activities

TopicGraphs

Virginia SOL DM.1

Text:Excursions in Modern Mathematics, 5th edition, Tannenbaum, Pearson Prentice Hall

For All Practical Purposes, COMAP, Freeman Press

Discrete Mathematics through Applications, Crisler, Fisher, and Foelich, Freeman Press

PWC Mathematics Websitehttp://pwcs.mathschoolfusion.us/

Virginia Department of Education Website http://www.doe.virginia.gov/instruction/mathematics/index.shtml

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http://www.doe.virginia.gov/instruction/mathematics/index.shtml


http://www.pwcsmath.com/




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DISCRETE MATHEMATICS CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSTopicGraphs

Virginia SOL DM.2The student will solve problems through investigation and application of circuits, cycles, Euler Paths, Euler Circuits, Hamilton Paths, and Hamilton Circuits. Optimal solutions will be sought using existing algorithms and student-created algorithms.

Unit of StudyCircuits, Trees, and Paths

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Determine if a graph has an Euler

Circuit or Path, and find it. Determine if a graph has a Hamilton

Circuit or Path, and find it. Count the number of Hamilton Circuits

for a complete graph with n vertices. Use the Euler Circuit algorithm to solve

optimization problems.

Key VocabularycircuitcycleEuler CircuitEuler PathHamilton CircuitHamilton Pathoptimizationpathunicursal tracing

Essential Questions What is the difference between a path, a cycle, and a circuit? What is the difference between Euler and Hamilton’s approaches to a graph? What is a real-world application for using an Euler circuit or path? What is a real-world application for using a Hamilton circuit or path? How is an Euler circuit used to solve an optimization problem?

Essential Understandings Euler’s Theorem states: If G is a connected graph and all its valences are even, then G

has an Euler Circuit. Pairs of routes (circuits) correspond to the same Hamilton Circuit because one route can

be obtained from the other by traversing the vertices in reverse order.

There are Hamilton Circuits.

A multigraph is connected if there is a path between every pair of vertices.

Teacher Notes and ElaborationsA path is a connected sequence of edges showing a route on the graph that starts at a vertex and ends at a vertex; a path is usually described by naming in turn the vertices visited in traversing it. Thus, a path can also be thought of as describing a sequence of adjacent edges – a trip along the edges of the graph. Whereas a vertex can appear on the path more than once, an edge can be part of a path only once.

The number of edges in the path is called the length of the path.

A circuit has the same definition as a path, with the additional requirement that is must start and end at the same vertex. A vertex can appear on the path more than once. An edge can be part of path only once. In a circuit, the path must start and end at the same vertex.

A cycle is a path that starts and ends at the same vertex and does not use any edge or vertex more than once.

In unicursal tracings, a drawing can be traced without lifting the pencil or retracing any of the line. A closed unicursal tracing ends in the same place it starts. Open unicursal tracings don’t end back at the starting place.

(continued)


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DISCRETE MATHEMATICS CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSTopicGraphs

Virginia SOL DM.2The student will solve problems through investigation and application of circuits, cycles, Euler Paths, Euler Circuits, Hamilton Paths, and Hamilton Circuits. Optimal solutions will be sought using existing algorithms and student-created algorithms.


Teacher Notes and Elaborations (continued)Example: This graph has many paths, circuits, and cycles. The following lists two possible examples of each.

- A, B, E, D: This is a path from vertex A to vertex D, consisting of edges AB, BE, and ED. The length of this path is 3.

- A, C, B, E, E, D: This path of length 5 is possible because of the loop at E. A

- A, C, B, E, D, A: This is a circuit because it starts and ends at the same vertex. B The length of the circuit is 5.

- E, E: This is a circuit of length 1.

E C - A, C, B, A: This is a cycle. It does not use any vertex or edge more than once.

- A, D, E, B, A: This is a cycle.

D

An Euler Circuit is a circuit that travels through every edge of a graph, and the starting and ending vertex are the same. This is essentially the same as a closed unicursal tracing of the graph.

An Euler Path is a connected path that travels through every edge of a graph once and only once, starting at one odd vertex and ending at the other odd vertex. The length of an Euler path is the number of edges in the graph. In order to be an Euler graph, it must have only one pair of odd vertices.

A Hamilton Circuit is a circuit that must visit each vertex of the graph once and only once (except at the end, where it returns to the starting vertex).

A Hamilton Path is a path that starts at one vertex then, passes through every vertex of the graph once and only once, and ends at a different vertex.

The difference between a Hamilton circuit and an Euler circuit (Hamilton Path and Euler Path) is the word vertex vs. edge.

Optimization is the determination of the best solution to solving a graph. Often times the graph’s solution is subject to constraints of the problem.


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TopicGraphs

Virginia SOL DM.2






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DISCRETE MATHEMATICS CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSCurriculum Information Essential Knowledge and Skills

Key VocabularyEssential Questions and Understandings

Teacher Notes and ElaborationsTopicGraphs

Virginia SOL DM.3The student will apply graphs to conflict-resolution problems, such as map coloring, scheduling, matching, and optimization. Graph coloring and chromatic number will be used.


The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Model projects consisting of several

subtasks, using a graph. Use graphs to resolve conflicts that

arise in scheduling.

Key Vocabularychromatic numberdegree of vertices (valence)optimization

Essential Questions What are the various methods involved in the optimization of scheduling and map

coloring? When using vertex coloring describe when a conflict cannot be resolved?

Essential Understandings Every planar graph has a chromatic number that is less than or equal to four (the four-

color-map theorem). A graph can be colored with two colors if and only if it contains no cycle of odd lengths. Two regions can be the same color if the regions share a common vertex but not a

common edge. The chromatic number of a graph cannot exceed one more than the maximum number of

degrees of the vertices of the graph.

Teacher Notes and ElaborationsProblems are called coloring problems because the labels placed on the vertices of the graphs are referred to as colors. The process of labeling the graph is called coloring the graph, and the minimum number of labels, or colors that can be used is known as the chromatic number of the graph. Every planar graph has a chromatic number that is less than or equal to four (the four-color-map theorem).

The degree of a vertex (valence) is the number of edges at that vertex (a loop contributes twice toward the degree). An odd vertex is a vertex of odd degree; an even vertex is a vertex of even degree.

A

B

E C

D

(continued)

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DISCRETE MATHEMATICS CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSCurriculum Information Essential Questions and Understandings

Teacher Notes and ElaborationsTopicGraphs

Virginia SOL DM.3The student will apply graphs to conflict-resolution problems, such as map coloring, scheduling, matching, and optimization. Graph coloring and chromatic number will be used.


Teacher Notes and Elaborations (continued)In the diagram on the previous page, vertex A has degree 3, vertex B has degree 4, vertex C has degree 3, vertex D has degree 2 and vertex E has degree 4 (because of the loop). All together, the graph has 2 odd vertices and 3 even vertices therefore it can be traced.

Optimization is the determination of the best solution to solving a graph. Often times the graph’s solution is subject to constraints of the problem.

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TopicGraphs

Virginia SOL DM.3






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DISCRETE MATHEMATICS CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSThis page is intentionally left blank.



TopicGraphs

Virginia SOL DM.4The student will apply algorithms, such as Kruskal’s, Prim’s, or Dijkstra’s, relating to trees, networks, and paths. Appropriate technology will be used to determine the number of possible solutions and generate solutions when a feasible number exists.


The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Use Kruskal’s Algorithm to find the

shortest spanning tree of a connected graph.

Use Prim’s Algorithm to find the shortest spanning tree of a connected graph.

Use Dijkstra’s Algorithm to find the shortest spanning tree of a connected graph.

Key VocabularyDijkstra’s AlgorithmKruskal’s Algorithmminimum spanning treenetworkpathPrim’s Algorithmspanning treessubgraphtree

Essential Questions What are the best methods in determining minimum spanning trees? What are real-world applications of spanning trees?

Essential Understandings A spanning tree of a connected graph G is a tree that is a subgraph of G and contains

every vertex of G.

Teacher Notes and ElaborationsA network is created when all points are connected so that one can go from any point to any other point.

A subgraph uses only some of the edges of a connected graph.

A subgraph that includes every one of the vertices of an original graph and only some of the edges is called a spanning subgraph. A spanning subgraph should also be connected and should not contain any circuits.

A tree is a graph that is connected and has no cycles.

Ex. Tree Ex. Not a Tree (contains a cycle) A F

D

B D A C E C G

E B

A spanning tree is a subgraph that includes all the vertices of the graph. For example:Given Graph G

A E

B C D

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DISCRETE MATHEMATICS CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS (continued)


TopicGraphs



Teacher Notes and Elaborations (continued)The three spanning trees from graph G are:

A E A E A E

B C D B C D B C D Any graph whose edges have numbers attached to them is called a weighted graph. The numbers are called the weights of the edges. A spanning tree with the least total weight is called a minimum spanning tree.

Given the following weighted graph: Removing edge AB from the first circuit and edge FG from the second circuit results in one of many spanning trees:

E 3 F 3 E F

A 4 6 A 4 3 3

5 C 9 D G C 9 D G 7 7 B 4 5 B 4 5

I 8 H I 8 H

Given the same weighted graph, to find the minimum spanning tree remove edge BC from the first circuit and edge IH from the second circuit because they have the highest values in their respective circuits.

The classic algorithm for finding the shortest path between two vertices in a digraph is Dijkstra’s Algorithm.1. Label the starting vertex S and circle it. Examine all edges that have S as an endpoint. Darken the edge with the shortest length and

circle the vertex at the other endpoint of the darkened edge.2. Examine all uncircled vertices that are adjacent to the circled vertices in the graph.3. Using only circled vertices and darkened edges between the vertices that are circled, find the lengths of all paths from S to each

vertex being examined. Choose the vertex and the edge that yield the shortest path. Circle this vertex and darken this edge. Ties are broken arbitrarily.

4. Repeat steps two and three until all vertices are circled. The darkened edges of the graph form the shortest routes from S to every other vertex in the graph.

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(continued)


TopicGraphs



Teacher Notes and Elaborations (continued)Kruskal’s Algorithm finds the minimum spanning tree in any connected weighted graph, but in such a way that no edge forms a circuit. This works by building the solution, one edge at a time. At each step, choose the “cheapest” edge available that does not close any circuits.

1. Find the “cheapest” edge in the graph. If there is more than one, choose one at random. Mark it in red (or any other color).2. Find the next cheapest edge in the graph. If there is more than one, choose one at random. Mark it in red.3. Find the next cheapest unmarked edge in the graph that does not create a red circuit. If there is more than one, choose one at

random. Mark it in red.4. Repeat the previous step until the red edges span every vertex of the graph. The red edges form a minimum spanning tree of the

graph.

Prim’s Algorithm is a well-known algorithm for finding the minimum spanning tree of a weighted graph. It builds a minimum spanning tree T by expanding outward in connected links from some vertex. One edge and one vertex are added at each stage. The edge added is the one of least weight that connects the vertices already in T with those not in T, and the vertex is the endpoint of this edge that is not already in T.

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TopicGraphs

Virginia SOL DM.4






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TopicRecursion and Optimization

Virginia SOL DM.5The student will use algorithms to schedule tasks in order to determine a minimum project time. The algorithms will include critical path analysis, the list-processing algorithm, and student-created algorithms.


The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Specify in a digraph the order in which

tests are to be performed. Identify the critical path to determine

the earliest completion time (minimum project time).

Use the list-processing algorithm to determine an optimal schedule.

Create and test scheduling algorithms.

Key Vocabularycritical pathdigraph (directed graph)Earliest Start Time (EST)Latest Start Time (LST)list-processing algorithmminimum project time

Essential Questions What is critical path analysis? What is the difference between the earliest start time and the latest start time for the

same vertex? What are real-world examples of critical path analysis?

Essential Understandings Critical path scheduling sometimes yields optimal solutions.

Teacher Notes and ElaborationsA digraph or directed graph is a graph in which each edge has an arrow indicating the direction of the edge. Such directed edges are appropriate when the relationship is “one-sided” rather than symmetric (e.g., one-way streets as opposed to regular streets). It consists of two finite sets of vertices and a set of directed edges, where each is associated with an ordered pair of vertices called its endpoints.

An optimal priority list gives a priority list that gives a decent schedule. One commonly used strategy is to do the longer jobs first and leave the shorter jobs for last. This translates into writing the priority list by listing the tasks in decreasing order of processing times, with longest first, second longest next, and so on. This is referred to as the decreasing-time list and combined with the priority list model the decreasing-time algorithm. This is not however always the best method. The task should not be prioritized by how long it takes to execute it, but rather by the sum total of all tasks that lie ahead of it or, the greater the total amount of work lying ahead of a task, the sooner that task should be started. This is outlined by the critical path concept.

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(continued)


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DISCRETE MATHEMATICS CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSTopicRecursion and Optimization

Virginia SOL DM.5The student will use algorithms to schedule tasks in order to determine a minimum project time. The algorithms will include critical path analysis, the list-processing algorithm, and student-created algorithms.


Teacher Notes and Elaborations (continued)The critical path of a project digraph is the longest path from Start to End. The total processing time for the critical path is called the critical time. It is possible to use the concept of critical paths to generate good schedules. Tasks are prioritized in decreasing order of critical times. The priority list when the tasks are listed in decreasing order of critical times is called the critical-path list. The process of creating a schedule using the critical-path list as the priority list is called the critical path-algorithm.

vacuuming dusting 1 2 3

running dishwasher 4 5

doing laundry

This is a practical example of a digraph involving a cleaning service and the tasks involved in the job. Some tasks can be done simultaneously shortening the length of time needed to complete the entire job. By shortening the time required, the company optimizes the number of jobs that can be completed in one day.

By assigning times to each task as well as noting prerequisites needed a list-processing algorithm occurs.

The earliest-start time (EST) is the earliest that an activity can begin if all the activities preceding it begin as early as possible.

If an activity is not on the critical path, it is possible for it to start later than its earliest-start time and not delay the project. The latest a task can begin without delaying the project’s minimum completion time is know as the latest-start time (LST) for the task.

The earliest completion time for a project is also referred to as the minimum project time.


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Virginia SOL DM.5






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Teacher Notes and ElaborationsTopicRecursion and Optimization

Virginia SOL DM.6The student will solve linear programming problems. Appropriate technology will be used to facilitate the use of matrices, graphing techniques, and the Simplex method of determining solutions.

Unit of StudyLinear Programming

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Model real world problems with

systems of linear inequalities. Identify the feasibility region of a

system of linear inequalities with no more than four constraints.

Identify the coordinates of the corner points of a feasibility region.

Find the maximum or minimum value of the system.

Describe the meaning of the maximum or minimum value in terms of the original problem.

Key Vocabularyconstraintscorner pointfeasibility regionlinear programming modelmaximum valueminimum valueoptimal solutionSimplex method

Essential Questions What is linear programming? What are the constraints, maximums and minimums, of linear programming? What is meant by the feasibility region? How is the maximum value of a profit line determined? What is a situation in which minimization is used to find the optimal solution? What is a situation in which maximization is used to find the optimal solution?

Essential Understandings Linear programming models an optimization process. A linear programming model consists of a system of constraints and an objective

quantity that can be maximized or minimized. Any maximum or minimum value for a system of inequalities will occur at a corner

point of a feasibility region.

Teacher Notes and ElaborationsRunning a profitable business requires a careful balancing of resources. A manager must choose the best use of these resources. Often the range of possible choices can be described by a set of linear inequalities called constraints. In most situations the number of alternative solutions to the constraints is so great that it is hard to find the best one. Linear programming provides mathematical methods for finding the best solution.

Decision making in management science requires that a minimum or a maximum of a linear function be found.

Linear programming is a management science technique that helps a business allocate the resources it has on hand to make a particular mix of products that will maximize profit. Linear programming is a tool for maximizing or minimizing a quantity, typically a profit or a cost, subject to constraints.

A linear programming model consists of two ingredients1. objective function (an equation in three variables that describes a quantity that must

be maximized or minimized); and2. constraints (linear inequalities that incorporate any limitations/restrictions that

affect the quantity that is to be maximized/minimized).

Simplex method is the standard method of solving a linear programming problem that proceeds by pivoting to produce a finite sequence of basic feasibility points corresponding to vertices. The simplex method makes it possible to find the best corner point by evaluating only a tiny fraction of all the corners.

(continued)

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Virginia SOL DM.6The student will solve linear programming problems. Appropriate technology will be used to facilitate the use of matrices, graphing techniques, and the Simplex method of determining solutions.

Unit of StudyLinear Programming

Teacher Notes and Elaborations (continued)A feasibility region a set of all possible solutions to a linear programming problem. The feasible region contains corner points which are used to determine any maximum or minimum value for the system of inequalities. These points are used in the objective function to determine the optimal solution. The shape of a feasible region for a linear programming problem has some important characteristics, without which the corner point principle would not work:

1. The feasible region is a polygon in the first quadrant, where both x ≥ 0 and y ≥ 0. This is because the minimum constraints require that both x and y be nonnegative.

2. The region is a polygon that has neither dents nor holes.

Maximum and minimum values are often referred to as extreme values. A maximum or minimum value of a linear expression , if it exists, will occur at a corner point (point of intersection of the two linear equations) of the feasible region.

Optimal solution is the best method for solving a problem. In some cases the goal may be to finish a job as quickly as possible. In other situations the objective might be to maximize profit or minimize cost. Optimization problems are those that maximize (such things as profit, yield, coverage) or minimize (such things as cost, time required, waste).

Sometimes the minimum value is the optimal solution.

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Virginia SOL DM.6






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TopicElection Theory and Fair Division

Virginia SOL DM.7The student will analyze and describe the issue of fair division (e.g., cake cutting, estate division). Algorithms for continuous and discrete cases will be applied.

Unit of StudyFair Division

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Investigate and describe situations

involving discrete division (e.g., estate division).

Use an algorithm for fair division for a group of indivisible objects.

Investigate and describe situations involving continuous division of an infinitely divisible set (e.g., cake cutting).

Use an algorithm for fair division of an infinitely divisible set.

Key Vocabularycontinuous fair divisiondiscrete fair divisionequal shareestate division algorithmfair divisionfair sharemixed fair division

Essential Questions What is meant by fair division? What is the difference between continuous and discrete objects? What is the difference between fair share and equal share? In cake cutting, which is the preferred role, the cutter or the chooser? Why?

Essential Understandings Group decision making combines the wishes of many to yield a single fair result. A fair division problem may be discrete or continuous. The success of the estate division algorithm requires that each heir be capable of placing

a value on each object in the estate. A fair division problem consists of n individuals (players) who must partition some set

of goods, s, into n disjoint sets.

Teacher Notes and ElaborationsGiven a set of goods to be divided, and a set of parties/players entitled to share the set of goods, a fair division is the ultimate goal. That is, to divide the goods into shares (one share for each party/player) in such a way that each party/player gets a fair share. A fair-division method is a set of rules defining how the game is to be “played”.

A division among n people is called fair if each person feels that he or she has received at

least of the set. Fair share is relative. What may be a fair share to P may not necessarily

be a fair share to a different player Q. It is also possible for a share to be a fair share to P, but not necessarily the one that P likes the best.

A division among n people is called equal if each person receives exactly of the set

(equal share).

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(continued)



Virginia SOL DM.7The student will analyze and describe the issue of fair division (e.g., cake cutting, estate division). Algorithms for continuous and discrete cases will be applied.

Unit of StudyFair Division

Teacher Notes and Elaborations (continued)Depending on the nature of the set of goods to be divided, a fair division game can be classified as one of three types: continuous, discrete, or mixed. In a continuous fair division game, the set of goods is divisible in infinitely many ways and shares can be increased or decreased by arbitrarily small amounts. Typical examples of continuous fair division games involve the division of land, a cake, a pizza, etc. A fair division game is discrete when the set of goods is made up of objects that are indivisible like a painting, a house, cars, boats, etc. A mixed fair division game is one in which some of the components are continuous and some are discrete. Dividing an estate consisting of a house, a collection of jewelry, and a parcel of land is a mixed fair division game.

Types of continuous fair division are: 1. The Divider-Chooser method/Cut-and-Choose method 2. The Lone-Chooser method/Inspection method3. The Last-Diminisher method/Moving Knife method

There are a number of widely used criteria for a fair division. Some of these conflict with each other but often they can be combined. The criteria described here are only for when each player is entitled to the same amount.

- A proportional or simple fair division guarantees each player gets his fair share. For instance if three people divide up a cake each gets at least a third by their own valuation.

- An envy-free division guarantees no-one will want somebody else's share more than their own. - An exact division is one where every player thinks everyone received exactly their fair share, no more and no less. - An efficient or Pareto optimal division ensures no other allocation would make someone better off without making someone

else worse off. The term efficiency comes from the economics idea of the efficient market. A division where one player gets everything is optimal by this definition so on its own this does not guarantee even a fair share.

- An equitable division is one where the proportion of the cake a player receives by their own valuation is the same for every player. This is a difficult aim as players need not be truthful if asked their valuation.

The estate division algorithm is a fair division procedure for any number of parties that begins by having each heir (independently) assign a dollar value (a “bid”) to the item or items to be divided to reflect the absolute worth or each object to that heir. The allocation resulting from this procedure leaves each heir feeling that he or she received a dollar value at least equal to his or her fair share (and often more so). It never requires the dividing or sharing of an object, but it may require that the heirs have a large amount of cash on hand. Each heir will not necessarily receive the same dollar amount but rather what they perceive as the same dollar amount.

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http://en.wikipedia.org/wiki/Equity_(economics)

http://en.wikipedia.org/wiki/Efficient_market

http://en.wikipedia.org/wiki/Economics

http://en.wikipedia.org/wiki/Pareto_optimal

http://en.wikipedia.org/wiki/Exact_division

http://en.wikipedia.org/wiki/Envy-free

http://en.wikipedia.org/wiki/Proportional_(fair_division)




Virginia SOL DM.7






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DISCRETE MATHEMATICS CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSTopicElection Theory and Fair Division

Virginia SOL DM.8The student will investigate and describe weighted voting and the results of various election methods. These may include approval and preference voting as well as plurality, majority run-off, sequential run-off, Borda count, and Condorcet winners.

Unit of StudyElection Theory

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Determine in how many ways a voter

can rank choices. Investigate and describe the following

voting procedures:- weighted voting;- plurality;- majority;- sequential (winners run off);- sequential (losers are eliminated);- Borda count; and- Condorcet winner.

Compare and contrast different voting procedures.

Describe the possible effects of approval voting, insincere and sincere voting, a preference schedule, and strategic voting on the election outcome.

Key Vocabularyapproval votingBorda countCondorcet winnerinsincere votingmajority ruleparadoxpluralitypreference schedulepreference voting run-offsequential run-offsincere votingstrategic votingweighted voting

Essential Questions What are the various voting methods? How are voting methods alike and how are they different? Is there any one best voting method? What are the effects of sincere and strategic voting? Which voting methods are most likely to produce a paradox situation? What issues can arise when there is a presidential election with three candidates?

Essential Understandings Historically, popular voting methods have often led to counterintuitive results. A candidate who wins over every other candidate in a one-on-one ballot is a Condorcet

winner. A Borda count assigns points in descending order to each voter’s subsequent ranking

and then adds these points to arrive at a group’s final ranking. To select a voting system is to compromise between the shortcomings inherent in each

system.

Teacher Notes and ElaborationsApproval voting is a method of electing one or more candidates from a field of several in which each voter submits a ballot that indicates which candidates he or she approves of. Winning is determined by the total number of approvals a candidate obtains. Under approval voting, each voter is allowed to give one vote to as many of the candidates as he or she finds acceptable. No limit is set on the number of candidates for whom an individual can vote. Voters show disapproval of other candidates simply by not voting for them.

In the event of a tie, a run-off (additional election that will break the tie to determine the winner) will be held.

A sequential run-off (also referred to as sequential pairwise voting) is a voting system for elections with several candidates in which one starts with an agenda and pits the candidates against each other in one-on-one contests (based on ballots that are preference lists), with losers being eliminated as one moves along the agenda.

Sincere voting is the submission of a ballot that represents a voter’s true preferences while strategic voting (also referred to as insincere voting) does not represent a voter’s true preferences.

(continued)


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DISCRETE MATHEMATICS CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSTopicElection Theory and Fair Division

Virginia SOL DM.8The student will investigate and describe weighted voting and the results of various election methods. These may include approval and preference voting as well as plurality, majority run-off, sequential run-off, Borda count, and Condorcet winners.


Teacher Notes and Elaborations (continued)Weighted voting assigns numerical values to each place on the ballot. Totals for the candidates are computed on the sum of all votes, the winner is determined by the highest sum.

A Borda count assigns points in descending order to each voter’s subsequent ranking and then adds these points to arrive at a group’s final ranking.

The best known and most commonly used method for finding a winner in an election is the plurality method. Essentially this method says that the candidate with the most first–place votes wins. The plurality method as many flaws and is usually a poor method for choosing the winner of an election when there are more than two candidates. Its main weakness it that it fails to take into consideration the voter preferences other than first choice. Essentially the plurality method can violate the basic requirement of fairness called the Condorcet criterion. A candidate who wins over every other candidate in a one-on-one ballot is a Condorcet winner. Another weakness of the plurality method is the ease with which election results can be manipulated by a voter or a block of voters through insincere voting. An insincere voter changes the true order of his or her preference in the ballot in an effort to influence the outcome of the election against a certain candidate.

The word paradox is applied whenever there is a situation in which apparently logical reasoning leads to an outcome that seems impossible. Paradoxes underscore the fact that in apportionment, appearances can be deceiving – a seemingly simple and fair method can sometimes produce surprising and bizarre results.

A majority rule is a voting system for elections with two candidates (and an odd number of voters) in which the candidate preferred by more than half the voters is the winner.

Individual voters express their opinions through ballots. A ballot in which the voters are asked to rank the candidates in order of preference is called a preference ballot. A ballot in which ties are not allowed is called a linear ballot. When a preference ballot is used, different voters may have ranked the candidates in exactly the same way. So a logical way to organize the ballots is to group together identical ballots which is called a preference schedule for the election. It is the simplest and most compact way to summarize the results of preference voting.

One issue that may arise when there are three candidates running in an election are if one candidate can not win this spoils the election for a candidate who otherwise would win. This is known as the “spoiler problem”.


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Virginia SOL DM.8






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Teacher Notes and ElaborationsTopicElection Theory and Fair Division

Virginia SOL DM.9The student will identify apportionment inconsistencies that apply to issues such as salary caps in sports and allocation of representative to Congress. Historical and current methods will be compared.


The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Compare and contrast the Hamilton and

Jefferson methods of political apportionment with the Hill-Huntington method (currently in use in the U.S. House of Representatives) and the Webster-Wilcox method.

Solve allocation problems, using apportionment methods.

Investigate and describe how salary caps affect apportionment.

Key VocabularyAdams methodHamilton methodHill-Huntington methodJefferson methodWebster-Wilcox method

Essential Questions How is the number of representatives determined based on population? When does an increase in population result in a decrease in representation? Which election method(s) favor large states? Which election method(s) favor small states? Which election method does the United States House of Representatives most closely

follow?

Essential Understandings The apportionment of Congressional representative is based on the latest census.

Teacher Notes and ElaborationsThe Hamilton method is an apportionment algorithm, which assigns additional representative seats based on looking at the decimal portion of the quotas for the various groups. The group with the largest decimal remainder to their quota is given the first additional extra seat.

The Alabama Paradox occurs when an increase in the total number of seats to be apportioned causes a state to lose a seat in the House. This paradox is possible with the Hamilton method but not with divisor methods.

The Population Paradox occurs when an increase in a state’s population causes it to lose a seat while another state loses population (or increases population proportionally less) and gains a seat.

The New States Paradox occurs when a new state is added with its fair share of seats and this affects the number of seats due other states.

The Jefferson method is a divisor method, based on rounding all fractions down to the nearest integer. This method replaced the Hamilton method.

The Adams method was a mirror image of the Jefferson method except that instead of being based on modified lower quotas, it was based on modified upper quotas.

The Webster-Wilcox method is a divisor method, based on rounding all fractions the usual way.

The Hill-Huntington method is a divisor method that has been used to apportion the U.S. House of Representatives since 1940. The Hill-Huntington method is similar to the Webster-Wilcox method. The difference between these methods is in the cutoff point for rounding up or down. This method is based on the use of the geometric mean.

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http://www.ctl.ua.edu/math103/apportionment/paradoxs.htm#The%20New%20States%20Paradox

http://www.ctl.ua.edu/math103/apportionment/paradoxs.htm#The%20Population%20Paradox

http://www.ctl.ua.edu/math103/apportionment/paradoxs.htm#The%20Alabama%20Paradox




Virginia SOL DM.9






Mathematics Resourceswww.cut-the-knot.org/Curriculum/SocialScience/ApportionmentApplet.shtml

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http://www.cut-the-knot.org/Curriculum/SocialScience/ApportionmentApplet.shtml

http://www.cut-the-knot.org/Curriculum/SocialScience/ApportionmentApplet.shtml





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