Mathematics, Grade 9 - 12, Discrete Mathematics, 2010 NUMBER AND QUANTITY 1.) Analyze topics from elementary number theory, including perfect numbers and prime numbers, to determine properties of integers. (Alabama) 2.) Determine characteristics of sequences, including the Fibonacci sequence, the triangular numbers, and pentagonal numbers. (Alabama) Example: Write a sequence of the first 10 triangular numbers and hypothesize a formula to find the nth triangular number. 3.) Use the recursive process and difference equations to create fractals, population growth models, sequences, series, and compound interest models. (Alabama) 4.) Convert between base ten and other bases. (Alabama) ALGEBRA 5.) Determine results of operations upon 3 x 3 and larger matrices, including matrix addition and multiplication of a matrix by a matrix, vector, or scalar. (Alabama) 6.) Analyze determinants and inverses of 2 x 2, 3 x 3, and larger matrices to determine the nature of the solution set of the corresponding system of equations, including solving systems of equations in three variables by echelon row reduction and matrix inverse. (Alabama) 7.) Solve problems through investigation and application of existence and nonexistence of Euler paths, Euler circuits, Hamilton paths, and Hamilton circuits. (Alabama) Example: Show why a 5 x 5 grid has no Hamilton circuit. a. Develop optimal solutions of application-based problems using existing and student-created algorithms. (Alabama) 8.) Apply algorithms, including Kruskal's and Prim's, relating to minimum weight spanning trees, networks, flows, and Steiner trees. (Alabama) a. Use shortest path techniques to find optimal shipping routes. (Alabama) 9.) Determine a minimum project time using algorithms to schedule tasks in order, including critical path analysis, the list-processing algorithm, and student-created algorithms. (Alabama) GEOMETRY 10.) Use vertex-coloring techniques and matching techniques to solve application-based problems. (Alabama) Example: Use graph-coloring techniques to color a map of the western states of the United States so no adjacent states are the same color, including determining the minimum number of colors needed and why no fewer colors may be used. 11.) Solve application-based logic problems using Venn diagrams, truth tables, and matrices. (Alabama) STATISTICS AND PROBABILITY 12.) Use combinatorial reasoning and counting techniques to solve application-based problems. (Alabama) Example: Determine the probability of a safe opening on the first attempt given the combination uses the digits 2, 4, 6, and 8 with the order unknown.                 Answer: The probability of the safe opening on the first attempt is 1/24. 13.) Analyze election data to compare election methods and voting apportionment, including determining strength within specific groups. (Alabama)