(a) General requirements. Students shall be awarded
onehalf to one credit for successful completion of this course. Prerequisite:
Algebra II.
(b) Introduction.
(1) The desire to achieve educational excellence is
the driving force behind the Texas essential knowledge and skills
for mathematics, guided by the college and career readiness standards.
By embedding statistics, probability, and finance, while focusing
on fluency and solid understanding, Texas will lead the way in mathematics
education and prepare all Texas students for the challenges they will
face in the 21st century.
(2) The process standards describe ways in which students
are expected to engage in the content. The placement of the process
standards at the beginning of the knowledge and skills listed for
each grade and course is intentional. The process standards weave
the other knowledge and skills together so that students may be successful
problem solvers and use mathematics efficiently and effectively in
daily life. The process standards are integrated at every grade level
and course. When possible, students will apply mathematics to problems
arising in everyday life, society, and the workplace. Students will
use a problemsolving model that incorporates analyzing given information,
formulating a plan or strategy, determining a solution, justifying
the solution, and evaluating the problemsolving process and the reasonableness
of the solution. Students will select appropriate tools such as real
objects, manipulatives, paper and pencil, and technology and techniques
such as mental math, estimation, and number sense to solve problems.
Students will effectively communicate mathematical ideas, reasoning,
and their implications using multiple representations such as symbols,
diagrams, graphs, and language. Students will use mathematical relationships
to generate solutions and make connections and predictions. Students
will analyze mathematical relationships to connect and communicate
mathematical ideas. Students will display, explain, or justify mathematical
ideas and arguments using precise mathematical language in written
or oral communication.
(3) In Discrete Mathematics for Problem Solving, students
are introduced to the improved efficiency of mathematical analysis
and quantitative techniques over trialanderror approaches to management
problems involving organization, scheduling, project planning, strategy,
and decision making. Students will learn how mathematical topics such
as graph theory, planning and scheduling, group decision making, fair
division, game theory, and theory of moves can be applied to management
and decision making. Students will research mathematicians of the
past whose work is relevant to these topics today and read articles
about current mathematicians who either teach and conduct research
at major universities or work in business and industry solving realworld
logistical problems. Through the study of the applications of mathematics
to society's problems today, students will become better prepared
for and gain an appreciation for the value of a career in mathematics.
(4) Statements that contain the word "including" reference
content that must be mastered, while those containing the phrase "such
as" are intended as possible illustrative examples.
(c) Knowledge and skills.
(1) Mathematical process standards. The student uses
mathematical processes to acquire and demonstrate mathematical understanding.
The student is expected to:
(A) apply mathematics to problems arising in everyday
life, society, and the workplace;
(B) use a problemsolving model that incorporates analyzing
given information, formulating a plan or strategy, determining a solution,
justifying the solution, and evaluating the problemsolving process
and the reasonableness of the solution;
(C) select tools, including real objects, manipulatives,
paper and pencil, and technology as appropriate, and techniques, including
mental math, estimation, and number sense as appropriate, to solve
problems;
(D) communicate mathematical ideas, reasoning, and
their implications using multiple representations, including symbols,
diagrams, graphs, and language as appropriate;
(E) create and use representations to organize, record,
and communicate mathematical ideas;
(F) analyze mathematical relationships to connect and
communicate mathematical ideas; and
(G) display, explain, and justify mathematical ideas
and arguments using precise mathematical language in written or oral
communication.
(2) Graph theory. The student applies the concept of
graphs to determine possible solutions to realworld problems. The
student is expected to:
(A) explain the concept of graphs;
(B) use graph models for simple problems in management
science;
(C) determine the valences of the vertices of a graph;
(D) identify Euler circuits in a graph;
(E) solve route inspection problems by Eulerizing a
graph;
(F) determine solutions modeled by edge traversal in
a graph;
(G) compare the results of solving the traveling salesman
problem (TSP) using the nearest neighbor algorithm and using a greedy
algorithm;
(H) distinguish between realworld problems modeled
by Euler circuits and those modeled by Hamiltonian circuits;
(I) distinguish between algorithms that yield optimal
solutions and those that give nearly optimal solutions;
(J) find minimumcost spanning trees using Kruskal's
algorithm;
(K) use the critical path method to determine the earliest
possible completion time for a collection of tasks; and
(L) explain the difference between a graph and a directed
graph.
(3) Planning and scheduling. The student uses heuristic
algorithms to solve realworld problems. The student is expected to:
(A) use the list processing algorithm to schedule tasks
on identical processors;
(B) recognize situations appropriate for modeling or
scheduling problems;
(C) determine whether a schedule is optimal using the
critical path method together with the list processing algorithm;
(D) identify situations appropriate for modeling by
bin packing;
(E) use any of six heuristic algorithms to solve bin
packing problems;
(F) solve independent task scheduling problems using
the list processing algorithm; and
(G) explain the relationship between scheduling problems
and bin packing problems.
(4) Group decision making. The student uses mathematical
processes to apply decisionmaking schemes. The student analyzes the
effects of multiple types of weighted voting and applies multiple
voting concepts to realworld situations. The student is expected
to:
(A) describe the concept of a preference schedule and
how to use it;
(B) explain how particular decisionmaking schemes
work;
(C) determine the outcome for various voting methods,
given the voters' preferences;
(D) explain how different voting schemes or the order
of voting can lead to different results;
(E) describe the impact of various strategies on the
results of the decisionmaking process;
(F) explain the impact of Arrow's Impossibility Theorem;
(G) relate the meaning of approval voting;
(H) explain the need for weighted voting and how it
works;
(I) identify voting concepts such as Borda count, Condorcet
winner, dummy voter, and coalition; and
(J) compute the Banzhaf power index and explain its
significance.
(5) Fair division. The student applies the adjusted
winner procedure and Knaster inheritance procedure to realworld situations.
The student is expected to:
(A) use the adjusted winner procedure to determine
a fair allocation of property;
(B) use the adjusted winner procedure to resolve a
dispute;
(C) explain how to reach a fair division using the
Knaster inheritance procedure;
(D) solve fair division problems with three or more
players using the Knaster inheritance procedure;
(E) explain the conditions under which the trimming
procedure can be applied to indivisible goods;
(F) identify situations appropriate for the techniques
of fair division;
(G) compare the advantages of the divider and the chooser
in the dividerchooser method;
(H) discuss the rules and strategies of the dividerchooser
method;
(I) resolve cakedivision problems for three players
using the lastdiminisher method;
(J) analyze the relative importance of the three desirable
properties of fair division: equitability, envyfreeness, and Pareto
optimality; and
(K) identify fair division procedures that exhibit
envyfreeness.
(6) Game (or competition) theory. The student uses
knowledge of basic game theory concepts to calculate optimal strategies.
The student analyzes situations and identifies the use of gaming strategies.
The student is expected to:
(A) recognize competitive game situations;
(B) represent a game with a matrix;
(C) identify basic game theory concepts and vocabulary;
(D) determine the optimal pure strategies and value
of a game with a saddle point by means of the minimax technique;
(E) explain the concept of and need for a mixed strategy;
(F) compute the optimal mixed strategy and the expected
value for a player in a game who has only two pure strategies;
(G) model simple twobytwo, bimatrix games of partial
conflict;
(H) identify the nature and implications of the game
called "Prisoners' Dilemma";
(I) explain the game known as "chicken";
(J) identify examples that illustrate the prevalence
of Prisoners' Dilemma and chicken in our society; and
(K) determine when a pair of strategies for two players
is in equilibrium.
(7) Theory of moves. The student analyzes the theory
of moves (TOM). The student uses the TOM and game theory to analyze
conflicts. The student is expected to:
(A) compare and contrast TOM and game theory;
(B) explain the rules of TOM;
(C) describe what is meant by a cyclic game;
(D) use a game tree to analyze a twoperson game;
(E) determine the effect of approaching Prisoners'
Dilemma and chicken from the standpoint of TOM and contrast that to
the effect of approaching them from the standpoint of game theory;
(F) describe the use of TOM in a larger, more complicated
game; and
(G) model a conflict from literature or from a reallife
situation as a twobytwo strict ordinal game and compare the results
predicted by game theory and by TOM.
