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Skills available for Alabama Precalculus standards

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FIN.1 Logical Reasoning

  • The validity of a statement or argument can be determined using the models and language of first order logic.

FIN.2 Advanced Counting

  • Complex counting problems can be solved efficiently using a variety of techniques.

    • 6 Use multiple representations and methods for counting objects and developing more efficient counting techniques.

    • 7 Develop and use the Fundamental Counting Principle for counting independent and dependent events.

      • 7.a Use various counting models (including tree diagrams and lists) to identify the distinguishing factors of a context in which the Fundamental Counting Principle can be applied.

    • 8 Using application-based problems, develop formulas for permutations, combinations, and combinations with repetition and compare student-derived formulas to standard representations of the formulas.

      • 8.a Identify differences between applications of combinations and permutations.

      • 8.b Using application-based problems, calculate the number of permutations of a set with n elements. Calculate the number of permutations of r elements taken from a set of n elements.

      • 8.c Using application-based problems, calculate the number of subsets of size r that can be chosen from a set of n elements, explaining this number as the number of combinations "n choose r."

      • 8.d Using application-based problems, calculate the number of combinations with repetitions of r elements from a set of n elements as "(n + r – 1) choose r."

    • 9 Use various counting techniques to determine probabilities of events.

    • 10 Use the Pigeonhole Principle to solve counting problems.

FIN.3 Recursion

FIN.4 Networks

  • Complex problems can be modeled using vertex and edge graphs and characteristics of the different structures are used to find solutions.

    • 16 Use vertex and edge graphs to model mathematical situations involving networks.

      • 16.a Identify properties of simple graphs, complete graphs, bipartite graphs, complete bipartite graphs, and trees.

    • 17 Solve problems involving networks through investigation and application of existence and nonexistence of Euler paths, Euler circuits, Hamilton paths, and Hamilton circuits.

      • 17.a Develop optimal solutions of application-based problems using existing and student- created algorithms.

      • 17.b Give an argument for graph properties.

    • 18 Apply algorithms relating to minimum weight spanning trees, networks, flows, and Steiner trees.

      • 18.a Use shortest path techniques to find optimal shipping routes.

      • 18.b Show that every connected graph has a minimal spanning tree.

      • 18.c Use Kruskal's Algorithm and Prim's Algorithm to determine the minimal spanning tree of a weighted graph.

    • 19 Use vertex-coloring, edge-coloring, and matching techniques to solve application-based problems involving conflict.

    • 20 Determine 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.

    • 21 Use the adjacency matrix of a graph to determine the number of walks of length n in a graph.

FIN.5 Fairness and Democracy

  • Various methods for determining a winner in a voting system can result in paradoxes or other issues of fairness.

    • 22 Analyze advantages and disadvantages of different types of ballot voting systems.

      • 22.a Identify impacts of using a preferential ballot voting system and compare it to single candidate voting and other voting systems.

      • 22.b Analyze the impact of legal and cultural features of political systems on the mathematical aspects of elections.

    • 23 Apply 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.

    • 24 Identify 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.

    • 25 Use methods of weighted voting and identify issues of fairness related to weighted voting.

      • 25.a Distinguish between weight and power in voting.

FIN.6 Fair Division

  • Methods used to solve non-trivial problems of division of objects often reveal issues of fairness.

    • 26 Explain 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.

    • 27 Identify 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.

    • 28 Use spreadsheets to examine apportionment methods in large problems.

FIN.7 Information Processing

  • Effective systems for sending and receiving information include components that impact accuracy, efficiency, and security.

    • 29 Critically analyze issues related to information processing including accuracy, efficiency, and security.

    • 30 Apply ciphers (encryption and decryption algorithms) and cryptosystems for encrypting and decrypting including symmetric-key or public-key systems.

      • 30.a Use modular arithmetic to apply RSA (Rivest-Shamir-Adleman) public-key cryptosystems.

      • 30.b Use matrices and their inverses to encode and decode messages.

    • 31 Apply error-detecting codes and error-correcting codes to determine accuracy of information processing.

    • 32 Apply methods of data compression.