TX Math Standards (Discrete Mathematics for Problem Solving)

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

Use the adjusted winner procedure to resolve a dispute.

Explain how to reach a fair division using the Knaster inheritance procedure.

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

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

Identify situations appropriate for the techniques of fair division.

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

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

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

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

Identify fair division procedures that exhibit envy-freeness.

Recognize competitive game situations.

Represent a game with a matrix.

Identify basic game theory concepts and vocabulary.

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

Explain the concept of and need for a mixed strategy.

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

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

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

Explain the game known as "chicken".

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

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

Explain the concept of graphs.

Use graph models for simple problems in management science.

Determine the valences of the vertices of a graph.

Identify Euler circuits in a graph.

Solve route inspection problems by Eulerizing a graph.

Determine solutions modeled by edge traversal in a graph.

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

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

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

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

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

Explain the difference between a graph and a directed graph.

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

Explain how particular decision-making schemes work.

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

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

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

Explain the impact of Arrow's Impossibility Theorem.

Relate the meaning of approval voting.

Explain the need for weighted voting and how it works.

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

Compute the Banzhaf power index and explain its significance.

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

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.

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.

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

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

Analyze mathematical relationships to connect and communicate mathematical ideas.

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

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

Recognize situations appropriate for modeling or scheduling problems.

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

Identify situations appropriate for modeling by bin packing.

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

Solve independent task scheduling problems using the list processing algorithm.

Explain the relationship between scheduling problems and bin packing problems.

Compare and contrast TOM and game theory.

Explain the rules of TOM.

Describe what is meant by a cyclic game.

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

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.

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

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.