Mathematics

Discrete Mathematics for Problem Solving

Texas Essential Knowledge and Skills (TEKS) · TEKS 2012

65 standards
  • Mathematical process standards (1)
    • DISCM.MATH.1.A

      Apply mathematics to problems arising in everyday life, society, and the workplace.

    • DISCM.MATH.1.B

      Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

    • DISCM.MATH.1.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.

    • DISCM.MATH.1.D

      Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

    • DISCM.MATH.1.E

      Create and use representations to organize, record, and communicate mathematical ideas.

    • DISCM.MATH.1.F

      Analyze mathematical relationships to connect and communicate mathematical ideas.

    • DISCM.MATH.1.G

      Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

  • Graph theory (2)
    • DISCM.MATH.2.A

      Explain the concept of graphs.

    • DISCM.MATH.2.B

      Use graph models for simple problems in management science.

    • DISCM.MATH.2.C

      Determine the valences of the vertices of a graph.

    • DISCM.MATH.2.D

      Identify Euler circuits in a graph.

    • DISCM.MATH.2.E

      Solve route inspection problems by Eulerizing a graph; October 2015 Update Page 25 §111.C. High School.

    • DISCM.MATH.2.F

      Determine solutions modeled by edge traversal in a graph.

    • DISCM.MATH.2.G

      Compare the results of solving the traveling salesman problem (TSP) using the nearest neighbor algorithm and using a greedy algorithm.

    • DISCM.MATH.2.H

      Distinguish between real-world problems modeled by Euler circuits and those modeled by Hamiltonian circuits.

    • DISCM.MATH.2.I

      Distinguish between algorithms that yield optimal solutions and those that give nearly optimal solutions.

    • DISCM.MATH.2.J

      Find minimum-cost spanning trees using Kruskal's algorithm.

    • DISCM.MATH.2.K

      Use the critical path method to determine the earliest possible completion time for a collection of tasks.

    • DISCM.MATH.2.L

      Explain the difference between a graph and a directed graph.

  • Planning and scheduling (3)
    • DISCM.MATH.3.A

      Use the list processing algorithm to schedule tasks on identical processors.

    • DISCM.MATH.3.B

      Recognize situations appropriate for modeling or scheduling problems.

    • DISCM.MATH.3.C

      Determine whether a schedule is optimal using the critical path method together with the list processing algorithm.

    • DISCM.MATH.3.D

      Identify situations appropriate for modeling by bin packing.

    • DISCM.MATH.3.E

      Use any of six heuristic algorithms to solve bin packing problems.

    • DISCM.MATH.3.F

      Solve independent task scheduling problems using the list processing algorithm.

    • DISCM.MATH.3.G

      Explain the relationship between scheduling problems and bin packing problems.

  • Group decision making (4)
    • DISCM.MATH.4.A

      Describe the concept of a preference schedule and how to use it.

    • DISCM.MATH.4.B

      Explain how particular decision-making schemes work.

    • DISCM.MATH.4.C

      Determine the outcome for various voting methods, given the voters' preferences.

    • DISCM.MATH.4.D

      Explain how different voting schemes or the order of voting can lead to different results.

    • DISCM.MATH.4.E

      Describe the impact of various strategies on the results of the decision-making process.

    • DISCM.MATH.4.F

      Explain the impact of Arrow's Impossibility Theorem.

    • DISCM.MATH.4.G

      Relate the meaning of approval voting.

    • DISCM.MATH.4.H

      Explain the need for weighted voting and how it works.

    • DISCM.MATH.4.I

      Identify voting concepts such as Borda count, Condorcet winner, dummy voter, and coalition.

    • DISCM.MATH.4.J

      Compute the Banzhaf power index and explain its significance.

  • Fair division (5)
    • DISCM.MATH.5.A

      Use the adjusted winner procedure to determine a fair allocation of property.

    • DISCM.MATH.5.B

      Use the adjusted winner procedure to resolve a dispute.

    • DISCM.MATH.5.C

      Explain how to reach a fair division using the Knaster inheritance procedure; Page 26 October 2015 Update.

    • DISCM.MATH.5.D

      Solve fair division problems with three or more players using the Knaster inheritance procedure.

    • DISCM.MATH.5.E

      Explain the conditions under which the trimming procedure can be applied to indivisible goods.

    • DISCM.MATH.5.F

      Identify situations appropriate for the techniques of fair division.

    • DISCM.MATH.5.G

      Compare the advantages of the divider and the chooser in the divider-chooser method.

    • DISCM.MATH.5.H

      Discuss the rules and strategies of the divider-chooser method.

    • DISCM.MATH.5.I

      Resolve cake-division problems for three players using the last-diminisher method.

    • DISCM.MATH.5.J

      Analyze the relative importance of the three desirable properties of fair division: equitability, envy-freeness, and Pareto optimality.

    • DISCM.MATH.5.K

      Identify fair division procedures that exhibit envy-freeness.

  • Game (or competition) theory (6)
    • DISCM.MATH.6.A

      Recognize competitive game situations.

    • DISCM.MATH.6.B

      Represent a game with a matrix.

    • DISCM.MATH.6.C

      Identify basic game theory concepts and vocabulary.

    • DISCM.MATH.6.D

      Determine the optimal pure strategies and value of a game with a saddle point by means of the minimax technique.

    • DISCM.MATH.6.E

      Explain the concept of and need for a mixed strategy.

    • DISCM.MATH.6.F

      Compute the optimal mixed strategy and the expected value for a player in a game who has only two pure strategies.

    • DISCM.MATH.6.G

      Model simple two-by-two, bimatrix games of partial conflict.

    • DISCM.MATH.6.H

      Identify the nature and implications of the game called "Prisoners' Dilemma".

    • DISCM.MATH.6.I

      Explain the game known as "chicken".

    • DISCM.MATH.6.J

      Identify examples that illustrate the prevalence of Prisoners' Dilemma and chicken in our society.

    • DISCM.MATH.6.K

      Determine when a pair of strategies for two players is in equilibrium.

  • Theory of moves (7)
    • DISCM.MATH.7.A

      Compare and contrast TOM and game theory.

    • DISCM.MATH.7.B

      Explain the rules of TOM.

    • DISCM.MATH.7.C

      Describe what is meant by a cyclic game.

    • DISCM.MATH.7.D

      Use a game tree to analyze a two-person game.

    • DISCM.MATH.7.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.

    • DISCM.MATH.7.F

      Describe the use of TOM in a larger, more complicated game.

    • DISCM.MATH.7.G

      Model a conflict from literature or from a real-life situation as a two-by-two strict ordinal game and compare the results predicted by game theory and by TOM. October 2015 Update Page 27 §111.C. High School.