Discrete Match: Grades 9, 10, 11, 12

Number and Quantity

Algebra

Functions

Geometry

Data Analysis, Statistics and Probability

Election theory

Number bases

Binomial theorem and Pascal's Triangle

Logic and Boolean algebra

Iterative and recursive functions

Estimation

Graph theory

Permutations and combinations

Counting and combinatorial reasoning

Use election theory techniques to analyze election data.

Investigate and describe weighted voting and the results of various election methods. Both very standard and less well-known techniques will be studied and compared; these may include approval and preference voting as well as plurality, majority, run-off, sequential run-off, Borda count, and Condorcet winners.

Use fair division techniques to solve apportionment problems.

Understand various bases as used in computer science and numerical data transmission, especially base 2, base 8, and base 12.

Expand the understanding of place value to include numbers written in various numerical systems and in various bases.

Use base 2 arithmetic to understand checksums in data transmission.

Convert numbers between bases, especially multi-digit numbers.

Compare ancient numeral systems in various bases to base 10 and base 8 numerals.

Perform familiar arithmetic processes in base 2, base 8, and base 12.

Use the binomial theorem to expand powers of binomials

Build the binomial theorem using graphics/pyramid design; interpret it for both a multivariable binomial expansion and a variable and numeric binomial expansion.

Represent, apply, and describe relationships among the binomial theorem, Pascal's triangle, and combinations.

Construct and describe patterns in Pascal's triangle.

Develop the symbols and properties of Boolean algebra; connect Boolean algebra to standard logic.

Construct truth tables to determine the validity of an argument.

Analyze basic electrical networks; compare the networks to Boolean Algebra configurations.

Develop electrical networks and translate them into Boolean Algebra equations.

Represent and analyze functions by using iteration and recursion.

Use iteration and recursion to model and solve problems.

Use iterative and recursive thinking to solve a variety of contextual problems.

Create and analyze iterative geometric patterns, including fractals.

Describe, analyze, and create iterative procedures and recursive formulas by using technology such as computer software and graphing calculators.

Put numbers in perspective through estimation, comparison, and scaling.

Apply estimation techniques in solving Fermi-type problems.

Apply estimation techniques to data given in a variety of ways.

Use graph theory to model and solve contextual problems.

Use vertex-edge graphs to model and solve a variety of problems related to paths, circuits, networks, and relationships among a finite number of objects.

Apply graphs to conflict-resolution problems, such as map coloring, scheduling, matching, and optimization.

State a variety of map color programs. Develop solutions to examples of maps and discuss the final resolution of the four-color problem.

Discuss the different interpretations of the four-color problem and the validity of a computer proof.

Interpret the Binomial Theorem to solve coloring problems and numerical problems.

Represent, analyze, and apply permutations and combinations.

Design and interpret simple experiments using tree-diagrams, permutations, and combinations.

Use reasoning and formulas to solve counting problems in which repetition is not allowed and in which ordering does not matter.

Apply counting principles to probabilistic situations involving equally likely outcomes.

Solve counting problems by using Venn diagrams and show relationships modeled by the Venn diagram.

Apply the Law of Large numbers to contextual situations.

Recognize the difference between continuous and discrete situations.

Apply appropriate counting techniques in discrete situations.

Derive basic combinatorics identities by counting the same sets two different ways to get a basic identity.

Use combinatorial reasoning to construct proofs as well as solve a variety of problems.

Informally prove the classical identity C(n,k) = C(n-1,k-1) +C(n-1,k) for integers n and k with 0 < k < n.

Connect Pascal's triangle and probability to solve problems.

Discuss the various examples and consequences of innumeracy; consider poor estimation, improper experimental design, inappropriate comparisons, and scientific notation comparisons.

Prove the sum of the first n integers adds up to n(n+1)/2 in three different manners.