DISCRETE MATHEMATICS
CURRICULUM GUIDE
Overview
Loudoun County Public Schools
2015-2016
(Additional curriculum information and resources for teachers can be accessed through CMS and VISION)
Discrete Mathematics Overview
Quarter 3
Elections
DM.8
Semester Overview
Quarter 4
Fair Division
DM.7
Apportionment
DM.9
24 blocks
Paths and circuits
DM.2
DM.3
DM.4
DM.5
DM.1
DM.11
If time permits:
DM.10
DM.12
DM.6
DM.13
20 blocks
Number
Of Blocks
Topics and Essential Questions
24 blocks
total
for Qu 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 election
2. 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 ShapleyShubik
8. Study the applications of Banzhaf and Shapley-Shubik
Optional Instructional
Resources

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
Number
Of Blocks
Optional Instructional
Resources
Topics and Essential Questions
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.
Number
Of Blocks
Optional Instructional
Resources
Topics and Essential Questions
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.

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

Provide different scenarios
to students where they
have to graphical
representations
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
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 circuit
2. Determine if a graph has an Euler Path or Circuit and list it
3. Apply the Euler Circuit Algorithm to solve optimization problems
Number
Of Blocks
20 blocks
total
for Qu 4
Topics and Essential Questions
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.
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 length
3. 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.
Optional Instructional
Resources
 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.
Number
Of Blocks
Topics and Essential Questions
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 tree
2. 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.
Optional Instructional
Resources
 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 path analysis, the listprocessing 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
outdegree
2. Specify in a digraph the order in which tasks are to be performed
3. Model projects consisting of several subtasks using a graph
4. 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 schedule
8. Create and test scheduling algorithms
A list of chores with the
time it takes to 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.
Number
Of Blocks
Topics and Essential Questions
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 problems
4. Investigate and describe valence and connectedness
5. Determine whether a graph is planar or nonplanar
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 sort
2. Describe and apply a coding algorithm: ISBN numbers, UPC codes, zip codes,
and banking codes
3. Use bubble sort to order elements of an array by comparing adjacent elements
4. Merge two sorted lists into a single sorted list
Optional Instructional
Resources
(con’t from previous)
Number
Of Blocks
Topics and Essential Questions
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 situations
2. Model arithmetic and geometric sequences and series recursively
3. Compare and contrast the recursive process, and create fractals
4. 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 growth
6. Compare and contrast the recursive process and the Fibonacci sequence
7. Find the recursive relationship that generates the Fibonacci sequence
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 set
3. Use Venn diagrams to codify and solve logic problems
4. Use matrices as arrays of data to solve logic problems
Optional Instructional
Resources

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.
Number
Of Blocks
Topics and Essential Questions
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 inequalities
3. Identify feasibility region of a system of linear inequalities with no more than
four constraints
4. Identify the coordinates of the corner points of a feasibility region
5. Find the maximum or minimum value of the system
6. Describe the meaning of maximum or minimum value in terms 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 order
2. 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
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.
Optional Instructional
Resources
 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.


Plan, conduct, and
analyze investigations
dealing with probability.
Explain the combinatoric
formulas and give
examples why the
formulas work.
If time permits:
Number
Of
Blocks
Optional Instructional
Resources/Activities
Topics and Essential Questions
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 situations
9. Model arithmetic and geometric sequences and series
recursively
10. Compare and contrast the recursive process, and
create fractals

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.
11. 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
growth
13. Compare and contrast the recursive process and the
Fibonacci sequence
14. Find the recursive relationship that generates the
Fibonacci sequence
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
problems
8. 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.

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.
OBJECTIVES: The student will be able to:
7. Solve linear programming problems
8. Model real-world problems with systems of linear
inequalities
9. Identify feasibility region of a system of linear
inequalities with no more than four constraints
10. Identify the coordinates of the corner points of a
feasibility region
11. Find the maximum or minimum value of the system
12. Describe the meaning of maximum or minimum
value in terms 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 order


Plan, conduct, and
analyze investigations
dealing with probability.
Explain the combinatoric
formulas and give
examples why the
formulas work.
7. 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 problems
9. 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.
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)