AFM: Applications of Finite Mathematics

Applications of Finite Mathematics

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Logic, set theory, matrices, and combinatorics are the four pillars: propositional statements and truth tables, matrix operations and systems, counting with permutations and combinations, and probability models. Geometric proofs provide a framework for formal reasoning, and polynomial identities connect to discrete number theory.

AFM.1Represent logic statements in words, with symbols, and in truth tables, including conditional, biconditional, converse, inverse, contrapositive, and quantified statements.AFM.10Use the Pigeonhole Principle to solve counting problems.AFM.11Find patterns in application problems involving series and sequences, and develop recursive and explicit formulas as models to understand and describe sequential change.AFM.12Determine characteristics of sequences, including the Fibonacci Sequence, the triangular numbers, and pentagonal numbers.AFM.13Use the recursive process and difference equations to create fractals, population growth models, sequences, and series.AFM.14Use mathematical induction to prove statements involving the positive integers.AFM.15Develop and apply connections between Pascal’s Triangle and combinations.AFM.16Use vertex and edge graphs to model mathematical situations involving networks.AFM.17Solve problems involving networks through investigation and application of existence and nonexistence of Euler paths, Euler circuits, Hamilton paths, and Hamilton circuits. Note: Real-world contexts modeled by graphs may include roads or communication networks.AFM.18Apply algorithms relating to minimum weight spanning trees, networks, flows, and Steiner trees.AFM.19Use vertex-coloring, edge-coloring, and matching techniques to solve application-based problems involving conflict.AFM.2Represent logic operations such as and, or, not, nor, and x or (exclusive or) in words, with symbols, and in truth tables.AFM.20Determine the minimum time to complete a project using algorithms to schedule tasks in order, including critical path analysis, the list-processing algorithm, and student-created algorithms.AFM.21Use the adjacency matrix of a graph to determine the number of walks of length n in a graph.AFM.22Analyze advantages and disadvantages of different types of ballot voting systems.AFM.23Apply a variety of methods for determining a winner using a preferential ballot voting system, including plurality, majority, run-off with majority, sequential run-off with majority, Borda count, pairwise comparison, Condorcet, and approval voting.AFM.24Identify issues of fairness for different methods of determining a winner using a preferential voting ballot and other voting systems and identify paradoxes that can result.AFM.25Use methods of weighted voting and identify issues of fairness related to weighted voting.AFM.26Explain and apply mathematical aspects of fair division, with respect to classic problems of apportionment, cake cutting, and estate division. Include applications in other contexts and modern situations.AFM.27Identify and apply historic methods of apportionment for voting districts including Hamilton, Jefferson, Adams, Webster, and Huntington-Hill. Identify issues of fairness and paradoxes that may result from methods.AFM.28Use spreadsheets to examine apportionment methods in large problems.AFM.29Critically analyze issues related to information processing including accuracy, efficiency, and security.AFM.3Use truth tables to solve application-based logic problems and determine the truth value of simple and compound statements including negations and implications.AFM.30Apply ciphers (encryption and decryption algorithms) and cryptosystems for encrypting and decrypting including symmetric-key or public-key systems.AFM.31Apply error-detecting codes and error-correcting codes to determine accuracy of information processing.AFM.32Apply methods of data compression.AFM.4Determine whether a logical argument is valid or invalid, using laws of logic such as the law of syllogism and the law of detachment.AFM.5Prove a statement indirectly by proving the contrapositive of the statement.AFM.6Use multiple representations and methods for counting objects and developing more efficient counting techniques. Note: Representations and methods may include tree diagrams, lists, manipulatives, overcounting methods, recursive patterns, and explicit formulas.AFM.7Develop and use the Fundamental Counting Principle for counting independent and dependent events.AFM.8Using application-based problems, develop formulas for permutations, combinations, and combinations with repetition and compare student-derived formulas to standard representations of the formulas.AFM.9Use various counting techniques to determine probabilities of events.
Example Problems
Evaluate:
Given the following recursive formula for the sequence:

and


What is ?
A fair coin is tossed 12 times. Let be the number of tosses that land heads.
Write an expression that can be used to find .
How many unique ways are there to arrange the letters in the word BALLOON?
Ravi bought 4 newly released comic books and wants to line them up on his shelf in some order.

How many unique arrangements can Ravi make?
Goblins

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