| (a) General requirements. Students shall be awarded
one credit for successful completion of this course. Prerequisite:
Algebra II. This course is recommended for students in Grades 11 and
12.
(b) Introduction.
(1) Career and technical education instruction provides
content aligned with challenging academic standards and relevant technical
knowledge and skills for students to further their education and succeed
in current or emerging professions.
(2) The Science, Technology, Engineering, and Mathematics
(STEM) Career Cluster focuses on planning, managing, and providing
scientific research and professional and technical services, including
laboratory and testing services, and research and development services.
(3) Discrete Mathematics for Computer Science provides
the tools used in most areas of computer science. Exposure to the
mathematical concepts and discrete structures presented in this course
is essential in order to provide an adequate foundation for further
study. Discrete Mathematics for Computer Science is generally listed
as a core requirement for Computer Science majors. Course topics are
divided into six areas: sets, functions, and relations; basic logic;
proof techniques; counting basics; graphs and trees; and discrete
probability. Mathematical topics are interwoven with computer science
applications to enhance the students' understanding of the introduced
mathematics. Students will develop the ability to see computational
problems from a mathematical perspective. Introduced to a formal system
(propositional and predicate logic) upon which mathematical reasoning
is based, students will acquire the necessary knowledge to read and
construct mathematical arguments (proofs), understand mathematical
statements (theorems), and use mathematical problem-solving tools
and strategies. Students will be introduced to discrete data structures
such as sets, discrete functions, and relations and graphs and trees.
Students will also be introduced to discrete probability and expectations.
The six strands include creativity and innovation; communication and
collaboration; research and information fluency; critical thinking;
problem solving, and decision making; digital citizenship; and technology
operations and concepts.
(4) Students are encouraged to participate in extended
learning experiences such as career and technical student organizations
and other leadership or extracurricular organizations.
(5) 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) Creativity and innovation. The student develops
products and generates new understanding by extending existing knowledge.
The student is expected to:
(A) model algorithms and real-world situations using
formal tools of symbolic logic;
(B) model computer science problems by using graphs
and trees; and
(C) calculate the probabilities of events and expectations
of random variables for such problems as games of chance.
(2) Communication and collaboration. The student communicates
and collaborates with peers to contribute to his or her own learning
and the learning of others. The student is expected to:
(A) convert spoken language statements to appropriate
statements in propositional logic;
(B) explain basic terminology of sets, functions, and
relations;
(C) state the definition of the Master theorem;
(D) use the context of a particular application to
interpret the meaning derived when computing the permutations and
combinations of a set;
(E) interpret associated operations and terminology
in context; and
(F) define and provide examples of logical equivalence,
normal forms, validity, and modus ponens/modus tollens.
(3) Research and information fluency. The student locates,
analyzes, processes, and organizes data. The student is expected to:
(A) construct truth tables for negation, conjunction,
disjunction, implication, biconditional, and bit operators; and
(B) use truth tables to demonstrate propositional relations.
(4) Critical thinking, problem solving, and decision
making. The student uses appropriate strategies to analyze problems
and design algorithms. The student is expected to:
(A) analyze practical examples using appropriate models
of sets, functions, and relations;
(B) compare and contrast tautology, contradiction,
and contingency as related to propositional equivalences;
(C) compare and contrast examples and use of counterexamples,
contrapositions, and contradictions;
(D) describe the appropriate use and limitations of
predicate logic;
(E) apply formal methods of symbolic propositional
and predicate logic;
(F) use formal logic proofs and logical reasoning to
solve problems;
(G) outline the basic structure of proofs, including
direct, indirect, contradiction, induction, existence, and constructive
proofs;
(H) compare and contrast the types of problems best
satisfied by direct, indirect, contradiction, induction, existence,
and constructive proofs;
(I) relate mathematical induction to recursion and
recursively defined structures;
(J) compare and contrast weak, strong, and structural
induction, including when each is most appropriately used and examples
of each;
(K) compare and contrast dependent and independent
events;
(L) use recurrence equations to analyze algorithms
and other practical problems;
(M) use counting techniques to analyze algorithms and
other practical problems;
(N) apply probability tools to solve problems; and
(O) define, compare, and contrast simple graphs, multigraphs,
and directed and undirected graphs using definitions, properties,
and examples, including special cases.
(5) Digital citizenship. The student explores and understands
safety, legal, cultural, and societal issues relating to the use of
technology and information. The student is expected to:
(A) model ethical acquisition and use of digital information;
(B) demonstrate proper digital etiquette, responsible
use of software, and knowledge of acceptable use policies; and
(C) investigate how the concepts of discrete mathematics
are related to relevant problems and significant questions.
(6) Technology operations and concepts. The student
understands technology concepts, systems, and operations as they apply
to computer science. The student is expected to:
(A) perform operations associated with sets, functions,
and relations;
(B) apply basic counting principles, including cardinality
and the pigeonhole principle;
(C) apply appropriate precedence when using logical
operators;
(D) use appropriate strategies, including De Morgan's
Laws, to identify propositional equivalences;
(E) identify and appropriately use predicates, existential
and universal quantifiers, and valid arguments;
(F) identify possible applications of proofs, including
evaluating algorithmic complexity;
(G) state and appropriately use the product and sum
rules;
(H) compute permutations and combinations of a set;
(I) solve a variety of basic recurrence equations;
(J) apply the binomial theorem to independent events;
(K) apply Bayes' theorem to dependent events;
(L) demonstrate transversal methods for trees and graphs;
and
(M) relate graphs and trees to data structures, algorithms,
and counting.
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