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© 2003 by
Cobb County School District
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Scantron Corporation.
All Rights Reserved.
Made with
Curriculum Designer by
Scantron Corporation
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Cobb County School District |
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Mathematics (Updated July 2003) |
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Mathematics - Discrete Mathematics |
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Goals and Descriptions |
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PS1: NUMBER SENSE
The learner will be able to
reason and draw conclusions from numerical information.
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M.DISC.1.1 Number Forms: Apply/Notation
The learner will be able to
apply mathematical notations appropriately.
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M.DISC.1.2 Proofs: Counterexamples
The learner will be able to
devise counterexamples.
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M.DISC.1.3 Argument: Validity
The learner will be able to
follow a valid argument.
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M.DISC.1.4 Arguments: Evaluate/Validity
The learner will be able to
evaluate the validity of an argument.
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M.DISC.1.5 Proofs: Create
The learner will be able to
create elementary, valid arguments, including direct and indirect proofs and proofs by mathematical induction.
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M.DISC.1.6 Social Choice: Use/Fair Division
The learner will be able to
obtain problem solutions through the use of fair-division algorithms.
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M.DISC.1.7 Set Theory: Explain/Recognize
The learner will be able to
explain sets with suitable notation, including set-builder notation, and recognize finite and infinite sets.
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M.DISC.1.8 Set Theory: Identify Relations
The learner will be able to
recognize basic relations among sets and/or appropriately use words like subset, equality, proper subset, superset, universe, and/or empty set, through the use of Venn Diagrams.
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M.DISC.1.9 Sets: Operations
The learner will be able to
use operation on sets such as intersection, union, complement, difference, and/or symmetric difference.
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M.DISC.1.10 Set Theory
The learner will be able to
define, identify, represent, and/or use commutative laws, associative laws, distributive laws, and/or DeMorgan's Laws and create basic proofs with the aid of Venn Diagrams.
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M.DISC.1.11 Set Theory: Determine
The learner will be able to
find power sets and/or Cartesian products of sets.
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M.DISC.1.12 Mathematical Concepts: Form
The learner will be able to
develop mathematical definitions and/or express generalizations derived through explorations.
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M.DISC.1.13 Mathematical Concepts: Questions
The learner will be able to
ask clarifying and extending questions regarding mathematical concepts.
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M.DISC.1.14 Analyzing Problems: Conjectures
The learner will be able to
create and/or test conjectures.
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M.DISC.1.15 Mathematical Concepts: Relationships
The learner will be able to
ponder and/or clarify mathematical concepts and relationships.
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M.DISC.1.16 Mathematical Information: Read
The learner will be able to
read mathematical presentations with understanding.
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M.DISC.1.17 Math Concepts: Illustrate/Orally
The learner will be able to
illustrate mathematical concepts orally.
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M.DISC.1.18 Math Concepts: Illustrate/Written
The learner will be able to
illustrate mathematical concepts in written form.
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M.DISC.1.19 Mathematical Concepts: Notation
The learner will be able to
recognize the economy, power, and elegance of mathematical notation as well as its part in the formulation of mathematical ideas.
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M.DISC.1.20 Mathematical Connections: Identify
The learner will be able to
identify connections among mathematical concepts.
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M.DISC.1.21 Mathematical Connections: Apply
The learner will be able to
apply connections among mathematical concepts.
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M.DISC.1.22 Mathematical Connections: Disciplines
The learner will be able to
recognize connections between mathematics and other disciplines.
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M.DISC.1.23 Mathematical Connections: Disciplines
The learner will be able to
apply connections between mathematics and other disciplines.
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M.DISC.1.24 Mathematical Concepts: Decisions
The learner will be able to
figure out election results through the application of various procedures including preference schedules for plurality, Borda, Condorcet, or other ranking methods.
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M.DISC.1.25 Mathematical Concepts
The learner will be able to
recognize paradoxes and apply weighted voting and power indices in addition to Arrow's fairness criteria to find out what is meant by Arrow's Impossibility Theorem.
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M.DISC.1.26 Mathematical Concepts
The learner will be able to
recognize methods of apportionment, and the paradoxes that may exist within them, and/or gain familiarity with the Balinski and Young Impossibility Theorem.
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PS2: OPERATIONS AND THEIR RELATIONSHIPS
The learner will be able to
reason and draw conclusions from numerical information.
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M.DISC.2.1 Real Numbers: Examine
The learner will be able to
inspect the real number system and define its various subsets (positive integers P, natural numbers N, integers Z, rational numbers Q, irrational numbers, and real numbers R, apply interval notation and the division algorithm, as well as divisibility properties.
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M.DISC.2.2 Logic: Concepts
The learner will be able to
define, identify, and/or represent a proposition, its converse, and/or its contrapositive.
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M.DISC.2.3 Proofs: Illustrate/Counterexamples
The learner will be able to
illustrate the applications of basic counterexamples.
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M.DISC.2.4 Proofs: Construct
The learner will be able to
create basic proofs through the use of elementary examples taken from arithmetic.
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M.DISC.2.5 Logic: Truth Tables/Complete
The learner will be able to
complete truth tables for basic logical connectives and write truth values of basic propositions.
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M.DISC.2.6 Logic: Tautology/Contradict
The learner will be able to
state and/or represent the definitions of tautology and/or contradiction.
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M.DISC.2.7 Logic: Negation
The learner will be able to
negate logical statements.
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M.DISC.2.8 Logic: Circuits
The learner will be able to
draw logic circuits with AND, OR, and NOT gates.
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M.DISC.2.9 Odd/Even: Construct/Basic/Proofs
The learner will be able to
construct basic proofs about even and odd numbers.
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M.DISC.2.10 Number Forms: Representing
The learner will be able to
write integers (presented in base ten) in any other base, emphasizing base two, and conversely.
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M.DISC.2.11 Logic: Recognize/Hypothesis
The learner will be able to
recognize the hypothesis and the conclusion of a variety of English constructed sentences.
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M.DISC.2.12 Logic: Translate
The learner will be able to
translate back and forth between English sentences and logical expressions.
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M.DISC.2.13 Logic: NAND/NOR
The learner will be able to
recognize and/or use NAND and NOR.
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M.DISC.2.14 Multiples: Determine/LCM
The learner will be able to
find least common multiples.
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M.DISC.2.15 Prime Factorization: Find
The learner will be able to
find the prime factorization of a number.
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M.DISC.2.16 Factors: Greatest Common
The learner will be able to
find greatest common factors.
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PS3: PATTERNS, RELATIONS, AND FUNCTIONS
The learner will be able to
apply and analyze functions, relations, and patterns in abstract and real world situations.
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M.DISC.3.1 Algorithms: Apply
The learner will be able to
use algorithms to determine minimal spanning trees and/or shortest paths in weighted graphs.
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M.DISC.3.2 Networks: Study/Graph/Models
The learner will be able to
study networks such as systems of roads, pipelines, and airline routes through the use of graphs as models.
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M.DISC.3.3 Problem Solving: Various Concepts
The learner will be able to
obtain problem solutions involving the concepts of connectedness, completeness, bipartiteness, planarity, and/or graph coloring.
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M.DISC.3.4 Tree: Minimal Spanning
The learner will be able to
determine the minimal spanning tree for a graph.
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M.DISC.3.5 Matrices: Create/Represent
The learner will be able to
create matrices to represent graphs.
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M.DISC.3.6 Matrices: Recognize
The learner will be able to
identify the representation of a graph in matrix form.
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M.DISC.3.7 Matrices: Study
The learner will be able to
apply the utility of an adjacency matrix to analyze connectivity properties of graphs and/or diagrams.
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M.DISC.3.8 Problem Solving: Population
The learner will be able to obtain solutions to population growth and/or
control problems through the use of the Leslie Model.
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M.DISC.3.9 Set Theory: Determine/Closure
The learner will be able to
determine if a set is closed with respect to a given operation.
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M.DISC.3.10 Set Theory: Operations
The learner will be able to
become familiar with several operations on a variety of sets, including the arithmetic operations on the set and subset of real numbers, set operations on P(S), logical operations on propositions, and 2X2 matrix operations.
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M.DISC.3.11 Set Theory: Properties
The learner will be able to
find properties of an operation that is defined on a given set.
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M.DISC.3.12 Set Theory: Operations
The learner will be able to
decide which of the properties (commutative, associative, existence on identify, existence of inverses) hold for particular sets and operations.
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M.DISC.3.13 Algebraic Concepts: Apply/Boolean
The learner will be able to
present logic circuits in an algebraic structure through the application of Boolean axioms.
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M.DISC.3.14 Algebraic Concepts: Use/Boolean
The learner will be able to
use Boolean Algebra to prove equivalence relations between logic circuits.
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M.DISC.3.15 Graph Theory: Eulerian/Hamiltonian
The learner will be able to
recognize properties of graphs as Eulerian or Hamiltonian circuits and/or paths.
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M.DISC.3.16 Tree: Definition
The learner will be able to
comprehend many different definitions of a tree.
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M.DISC.3.17 Tree: Analysis
The learner will be able to
obtain solutions to probability problems through the application of tree analysis and Markov's algorithm.
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M.DISC.3.18 Mathematical Modeling: Apply
The learner will be able to
apply the Leontlef input/output model of a given economy.
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M.DISC.3.19 Function/Relation: Define
The learner will be able to
make definitions of functions and/or relations, and/or determine, identify, and represent the concepts of domain, codomain, image, range, into, onto, and/or one-to-one.
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M.DISC.3.20 Relations: Properties
The learner will be able to
define, identify, and/or represent properties of relations including reflexive, transitive, symmetric, and/or antisymmetric.
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M.DISC.3.21 Equivalence: Identify/Relations
The learner will be able to
recognize equivalence relations.
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M.DISC.3.22 Order: Identify
The learner will be able to
recognize order relations.
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M.DISC.3.23 Functions: Recursive
The learner will be able to
identify recursively defined functions emphasizing recurrence relations and recursion.
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M.DISC.3.24 Operations: Binary
The learner will be able to
state the definition of and identify binary operations.
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M.DISC.3.25 Matrices: Identify/Apply
The learner will be able to
recognize and/or apply matrix properties.
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M.DISC.3.26 Matrices: Compute
The learner will be able to
perform addition, subtraction, multiplication, and scalar multiplication with matrices.
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M.DISC.3.27 Linear System of Equations:
The learner will be able to
obtain solutions (if they exist) to a linear system of equations.
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M.DISC.3.28 Compositions: Form
The learner will be able to
form the composition of functions.
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M.DISC.3.29 Inverses: Find/Relation/Fun
The learner will be able to
find the inverse of relations, particularly of functions.
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PS4: ALGEBRAIC MODELS IN PROBLEM SOLVING
The learner will be able to
use mathematical models to represent and understand quantitative relationships and change in real world situations.
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M.DISC.4.1 Problem Solving
The learner will be able to
solve problems that involve choosing suitable methods and tools, approximating, and evaluating the reasonableness of solutions.
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M.DISC.4.2 Problem Solving: Technology
The learner will be able to
solve problems that relate concepts to other concepts and to practical applications using appropriate technology.
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Data Analysis and Probability
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PS5: DATA ANALYSIS
The learner will be able to
analyze real world data collected using appropriate measures of central tendency and dispersion and select and use appropriate statistical methods to analyze real world data.
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M.DISC.5.1 Graphing: Theory Application
The learner will be able to
use graph theory methods in conjunction with activity analysis.
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M.DISC.5.2 Graphs: Examine/Structure
The learner will be able to
examine the structure of a graph and create and apply different displays.
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PS6: PROBABILITY
The learner will be able to
understand and apply concepts of probability.
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M.DISC.6.1 Permutations/Combinations: Apply
The learner will be able to
apply simple permutation and combination formulas.
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M.DISC.6.2 Combinations: Identify/Prove
The learner will be able to
recognize and/or prove elementary combinatorial identities through the application of combinatorial reasoning.
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M.DISC.6.3 Binomial Theorem: Apply/Relate
The learner will be able to
use the Binomial Theorem and/or associate it with Pascal's Triangle.
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M.DISC.6.4 Discrete Probability
The learner will be able to
obtain solutions to discrete probability problems through the application of addition and multiplication rules, independent events, conditional probabilities, and/or expected value.
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M.DISC.6.5 Mathematical Concepts
The learner will be able to
perform iterations with first order recurrence relations, and/or determine the closed form of a first order linear recurrence relation.
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M.DISC.6.6 Discrete Mathematics: Pigeon-hole
The learner will be able to
use the pigeon-hole principle.
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M.DISC.6.7 Discrete Mathematics
The learner will be able to
state and use the inclusion-exclusion principle.
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M.DISC.6.8 Mathematical Concepts: Iteration
The learner will be able to
use the process of iteration in many different scenarios.
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M.DISC.6.9 Algorithms: Analyze
The learner will be able to
analyze search and/or sort type algorithms.
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M.DISC.6.10 Counting Methods: Counting
The learner will be able to
use the fundamental counting principle.
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M.DISC.6.11 Mathematical Concepts: Solve
The learner will be able to
obtain solutions to second order linear recurrence relations (optional).
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й 2003 Cobb County School
District. All Rights Reserved
514 Glover Street | Marietta, Ga. 30060 | (770) 426-3300
Send your comments to Sue Brown
visitors since July 18, 2003
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