Discrete Mathematics Overview - Loudoun County Public ... Web viewCompare and contrast the recursive process and the Fibonacci sequence . ... Present a word problem where students

DISCRETE MATHEMATICSCURRICULUM GUIDE

Overview

Loudoun County Public Schools2016-2017

(Additional curriculum information and resources for teachers can be accessed through CMS and VISION)

Discrete Mathematics Overview

Loudoun County Public Schools Discrete Mathematics Curriculum Guide 2016-2017 2

Semester OverviewQuarter 3 Quarter 4ElectionsDM.8

Fair DivisionDM.7

ApportionmentDM.9

Paths and circuitsDM.2DM.3DM.4DM.5DM.1DM.11If time permits:DM.10DM.12DM.6DM.13

24 blocks 19 blocks

Quarter 3Number of Blocks Topics and Essential Questions

Optional Instructional Resources

24 blocks total for

Quarter 3

DM.8 The 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.

OBJECTIVES: The student will be able to:1. Create a preference schedule from the total number of ballots for any given

election2. Determine the winner of an election using plurality, majority, Borda Count,

Pairwise Comparison, and Plurality with Elimination 3. Compare and contrast the different voting procedures such as winning by

plurality, majority, Borda Count, Pairwise Comparison, and Plurality with Elimination

4. Recognize the notation for weighted voting system and be able to define the essential vocabulary such as quota, player, dictator, and dummy

5. Compare and contrast the differences between weighted voting and preferential voting.

6. Calculate the power distribution that exists in a weighted voting system of Banzhaf

7. Calculate the power distribution that exists in a weighted voting system of Shapley-Shubik

8. Study the applications of Banzhaf and Shapley-Shubik

Students will create a mock election on a particular topic with different options for people to choose from. The entire class will participate in each other surveys. A preference table will be created based on the data collected and the students will have to perform the different voting procedures to determine a winner. Voter Project Write Up




DM.7 The 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.

OBJECTIVES: The student will be able to:1. Investigate and describe situations using continuous division of infinitely

divisible set using the following Fair Division schemes: divider chooser, lone divider, lone chooser, last diminisher, and method of markers.

2. Investigate and describe situations involving discrete division using method of sealed bids.

3. Compare and contrast the differences between the two types of Fair Division schemes

4. Solve fair division problems that consist of n individuals or players who must partition some set of goods, s, into n disjoint

Students will create their own cake/pizza/or cookie that must have two flavors. It must be drawn and clearly labeled. Then a value system problem will be written and solved on a separate piece of paper.Value System Project

Students will be given card stock with different objects. Then each of the students will decide how much each of the items is worth. The method of sealed bids will be performed to determine who wins each of the items.

Candy can be lined up in a line and students will place their markers according to what pieces of candy they would like to have. The method of markers will be used to determine the allotment of candy.




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

OBJECTIVES: The student will be able to:1. Calculate the standard divisor using the total population and the total

number of seats available to apportion 2. Determine each state’s standard quota based on the standard divisor

that was calculated. 3. Find the lower and upper quota based on the quota that was

calculated. 4. Study the Apportionment methods: Hamilton, Jefferson, Adams,

Webster, and Huntington-Hill 5. Compare and contrast each of the different methods and the benefits

of using each one. 6. Apply each of the methods of apportionments to specific problems to

determine the allocation of seats. 7. Determine the relationship between salary caps and apportionment

Five states will be selected based on certain criteria in order to create a country. Students will have to decide how many seats should be divided amongst the different states. Each of the different methods of apportionment will be used to determine how many seats each state should receive based on the size of the population. Multiple Microsoft products will be used such as Excel, Publisher, PowerPoint, and Word. H:\Discrete Math\Chapter 4\Chapter 4 Activity.pdf

DM.2 The 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.

OBJECTIVES: The student will be able to:1. Explore the differences between a path and a circuit2. Determine if a graph has an Euler Path or Circuit and list it 3. Apply the Euler Circuit Algorithm to solve optimization problems

Provide different scenarios to students where they have to graphical representations


Quarter 3


Number of Blocks Topics and Essential Questions


19 blocks total for

Quarter 4

DM.2 The 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

OBJECTIVES: The student will be able to:1. Determine if a graph has a Hamilton Path or Circuit, and find it. 2. Count the number of Hamilton Circuits for a complete graph with n

vertices. 3. Determine the number of edges in a complete graph 4. Compare and contrast the differences between Euler and Hamilton 5. Define the different types of algorithms that exist; optimal,

inefficient, efficient, and approximate. 6. Calculate the optimal solutions to graphs and charts that have

Hamilton Circuits using one of the following algorithms: Brute-Force, Nearest Neighbor, Repetitive Nearest Neighbor, and Cheapest Link.

Handout a map of the school and have the students determine the shortest route to take to get to each of their classes.

The distances between various cities will be given and students will have to create a concert tour as if they were rock stars. The shortest route will be chosen using one of the algorithms they have learned. A PowerPoint presentation will be given describing their concert tour, how much it cost, and which algorithm they used to come up with the tour.

Students will be different worksheets to determine the maximum numbers of colors are needed to color a pattern.

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

OBJECTIVES: The student will be able to: 1. Determine every planar graph has a chromatic number that is less

than or equal to four based on the four-color-map theorem 2. Discover a graph can be colored with two colors if and only if it

contains no cycle of odd length3. Comprehend the chromatic number of a graph cannot exceed one

more than the maximum number of degrees of vertices of a graph 4. Apply the four-color- map theorem to multiple maps.



DM.4 The 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.

OBJECTIVES: The student will be able to:1. Define the following vocabulary words: tree, spanning tree, shortest

network, network, minimum spanning tree2. Use Kruskal’s Algorithm to find the shortest spanning tree of a

connected graph. 3. Use Prim’s Algorithm to find the shortest spanning tree of a

connected graph 4. Use Dijkstra’s Algorithm to find the shortest spanning tree of a

connected graph.

Students will have to determine how to construct a road network connecting many towns. Two towns must be connected by one road. One of the algorithms will be applied to determine the minimum spanning tree.

DM.5 The student will use algorithms to schedule tasks in order to determine a minimum project time. The algorithms will include critical

A list of chores with the time it takes to

path analysis, the list-processing algorithm, and student-created algorithms.

OBJECTIVES: The student will be able to:1. Determine the degree of each of the vertices as well as the indegree

and outdegree2. Specify in a digraph the order in which tasks are to be performed 3. Model projects consisting of several subtasks using a graph4. Create and solve scheduling problems using decreasing time,

backflow, and critical path algorithms 5. Identify the critical path to determine the earliest completion of time

(minimum project time)6. Apply critical path scheduling to yield optimal solutions. 7. Use list-processing algorithm to determine an optimal schedule8. Create and test scheduling algorithms

complete the task will be given to students. The chores must be completed in a certain order. Students will have to determine how long it will take to schedule the chores in the least amount of time possible with the given order.



DM.1 The 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)

OBJECTIVES: The student will be able to:1. Find the valence of each vertex in a graph. 2. Define the essential vocabulary associated with graph theory;

adjacent, degree, paths, circuits, connected graphs, and bridges. 3. Represent the vertices and edges of a graph as an adjacency matrix,

and use the matrix to solve problems4. Investigate and describe valence and connectedness5. Determine whether a graph is planar or nonplanar

(continued from previous)

DM.11 The student will describe and apply sorting algorithms and coding algorithms used in sorting, processing, and communicating information. These will include

a) bubble sort, merge sort, and network sort; and b) ISBN, UPC, zip, and banking codes.

OBJECTIVES: The student will be able to:1. Describe select, and apply sorting algorithms: Bubble-sort, merge

sort, and network sort2. Describe and apply a coding algorithm: ISBN numbers, UPC codes,

zip codes, and banking codes3. Use bubble sort to order elements of an array by comparing adjacent

elements4. Merge two sorted lists into a single sorted list



DM.10 The student will use the recursive process and different equations with the aid of appropriate technology to generate:

a) Compound interest;b) Sequences and series;c) Fractals; d) Population growth models; and e) The Fibonacci sequence

OBJECTIVES: The student will be able to:1. Use finite differences and recursion to model compound interest and

population growth situations2. Model arithmetic and geometric sequences and series recursively 3. Compare and contrast the recursive process, and create fractals4. Discover that a fractal is a figure whose dimensions is not a whole

number and they are self-similar 5. Solve problems showing linear and exponential growth6. Compare and contrast the recursive process and the Fibonacci

sequence 7. Find the recursive relationship that generates the Fibonacci sequence

Students will research how much a particular car costs and finance through a bank. They will also have to determine how much the car will depreciate each year.

DM.12 The student will select, justify, and apply an appropriate technique to solve a logic problem. Techniques will include Venn diagrams, truth tables, and matrices.

OBJECTIVES: The student will be able to:1. Study two-valued (Boolean) algebra which serves as a workable

method for interpreting the logical truth and falsity of compound statements

2. Understand how Venn diagrams provide pictures of topics in set theory, such as intersection and union, mutually exclusive sets, and

the empty set3. Use Venn diagrams to codify and solve logic problems4. Use matrices as arrays of data to solve logic problems



DM.6 The 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.

OBJECTIVES: The student will be able to:1. Solve linear programming problems 2. Model real-world problems with systems of linear inequalities3. Identify feasibility region of a system of linear inequalities with no

more than four constraints4. Identify the coordinates of the corner points of a feasibility region5. Find the maximum or minimum value of the system6. Describe the meaning of maximum or minimum value in terms of the

original problem

Present a word problem where students will have to not only define the constraints but the objective function as well. Once the constraints are defined, students will graph the inequalities to determine the coordinates of the corner points of the feasibility region. The maximum or minimum will be calculated and put into context of the original problem.

DM.13 The student will apply the formulas of combinatorics in the areas of

a) The Fundamental (Basic) Counting Principle;b) Knapsack and bin-packing problems;c) Permutations and combinations; and d) The pigeonhole principle

OBJECTIVES: The student will be able to:1. Find the number of combinations possible when subsets for elements

are selected from a set of n elements without regard to order2. Use the Fundamental (Basic) Counting principle to determine the

number of possible outcomes of an event.3. Use the knapsack and bin-packing algorithms to solve real world

problems

Plan, conduct, and analyze investigations dealing with probability.

Explain the combinatoric formulas and give examples why the formulas work.

4. Find the number of permutations possible when r objects are selected from n objects are ordered

5. Use the pigeonhole principle to solve packing problems to facilitate proofs.

If time permits:Number

of Blocks Topics and Essential Questions

Optional Instructional Resources/Activities

DM.10 The student will use the recursive process and different equations with the aid of appropriate technology to generate:

f) Compound interest;g) Sequences and series;h) Fractals; i) Population growth models; and j) The Fibonacci sequence

OBJECTIVES: The student will be able to:8. Use finite differences and recursion to model compound interest and

population growth situations9. Model arithmetic and geometric sequences and series recursively 10. Compare and contrast the recursive process, and create

fractals11. Discover that a fractal is a figure whose dimensions is not a

whole number and they are self-similar 12. Solve problems showing linear and exponential growth13. Compare and contrast the recursive process and the Fibonacci

sequence 14. Find the recursive relationship that generates the Fibonacci

sequence

Students will research how much a particular car costs and finance through a bank. They will also have to determine how much the car will depreciate each year.

DM.12 The student will select, justify, and apply an appropriate technique to solve a logic problem. Techniques will include Venn diagrams, truth tables, and matrices.

OBJECTIVES: The student will be able to:5. Study two-valued (Boolean) algebra which serves as a workable

method for interpreting the logical truth and falsity of compound statements

6. Understand how Venn diagrams provide pictures of topics in set theory, such as intersection and union, mutually exclusive sets, and the empty set

7. Use Venn diagrams to codify and solve logic problems8. Use matrices as arrays of data to solve logic problems

DM.6 The 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.

OBJECTIVES: The student will be able to:7. Solve linear programming problems 8. Model real-world problems with systems of linear inequalities9. Identify feasibility region of a system of linear inequalities with no

more than four constraints10. Identify the coordinates of the corner points of a feasibility

region11. Find the maximum or minimum value of the system12. Describe the meaning of maximum or minimum value in terms

of the original problem

Present a word problem where students will have to not only define the constraints but the objective function as well. Once the constraints are defined, students will graph the inequalities to determine the coordinates of the corner points of the feasibility region. The maximum or minimum will be calculated and put into context of the original problem.

DM.13 The student will apply the formulas of combinatorics in the areas of

e) The Fundamental (Basic) Counting Principle;f) Knapsack and bin-packing problems;g) Permutations and combinations; and h) The pigeonhole principle

OBJECTIVES: The student will be able to:6. Find the number of combinations possible when subsets for elements

are selected from a set of n elements without regard to order7. Use the Fundamental (Basic) Counting principle to determine the

number of possible outcomes of an event.8. Use the knapsack and bin-packing algorithms to solve real world

problems9. Find the number of permutations possible when r objects are

selected from n objects are ordered 10. Use the pigeonhole principle to solve packing problems to

facilitate proofs.

Plan, conduct, and analyze investigations dealing with probability.

Explain the combinatoric formulas and give examples why the formulas work.

Additional information about the Standards of Learning can be found in the

VDOE Curriculum Framework

(click link above)

Additional information about math vocabulary can be found in the

VDOE Vocabulary Word Wall Cards

(click link above)

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

http://www.doe.virginia.gov/testing/sol/frameworks/mathematics_framewks/2009/framewk_discrete_math.pdf

Documents

Discrete Mathematics Overview - Loudoun County Public ... Web viewCompare and contrast the recursive process and the Fibonacci sequence . ... Present a word problem where students