Standards - Mathematics

Students understand that:

• A conditional statement’s validity is based on the validity of its components.
• Truth tables must contain all possible assignments of true and false for each component.
• A statement is either true or false.

Students understand that:

• The validity of the simple statements that make up a compound statement determine the compound statement’s validity.

Students understand that:

• Complex situations including logic problems can be modeled using truth tables.
• Statements are logically equivalent if they have the same truth value for every possible assignment of true and false for each component.

Students understand that:

• Truth tables can be used to construct a valid argument or to determine the validity of an argument.
• In order for an argument to be valid, the form of the argument must be valid.

Students understand that:

• The contrapositive of a statement is logically equivalent to a statement.
• A statement can be shown to be true by the laws of logic by proving that its contrapositive is true.

Students understand that:

• Tree diagrams and other systematic methods can be used to count objects but may not be the most efficient method when counting large quantities.
• Recursive and explicit formulas can be developed from examining patterns in tree diagrams and systematic lists.

Students understand that:

• The Fundamental Counting Principle can be applied in contexts where an ordered list of events occur and there are a ways for the first event to occur, b ways for the second event to occur so the number of ways of the ordered sequence of events occuring is axb.

Students understand that:

• Permutation is an ordered selection of r distinct objects from a set of n objects.
• A combination is a selection of a set of r distinct unordered objects from a set of n objects.

Students understand that:

• Solving probability in a discrete setting requires first applying combinatorial reasoning and counting techniques to determine the size of the event of interest.
• Some events consist of a sequence of (or partition into) smaller events that may be independent or dependent.

Students understand that:

• If m>n and there are m pigeons (or any object) and n pigeonholes (or any position), there must be at least one pigeonhole with more than one pigeon.

Students understand that:

• The recursion process can be applied to many situations.
• A sequence lists the solutions of a set of related problems.
• Formulas can be hypothesized by identifying how the problems are related.

Students understand that:

• Models such as population growth should be recognized as recursively developed models.
• The recursion process can be applied to many situations.
• A sequence lists the solutions of a set of related problems.

Students understand that:

• Proof by induction is a way of proving statements that includes two steps.

Students understand that:

• Each row in Pascal’s triangle is the number of combinations of N take r where N is the row of the triangle starting with N=0 and r is the entry in the row from left to right.

Students understand that:

• Both the vertex and edge is used to represent some part of a real-world problem.

Students understand that:

• Graphs can be used to model real-world problems and Hamilton and Euler circuits and paths can provide solutions to such problems.
• An Euler circuit cannot exist in a graph with any odd degree vertices.
• An Euler path cannot exist in a graph without exactly two odd degree vertices.
• No known good algorithm has been established for finding a Hamilton path or circuit since no necessary and sufficient conditions for the existence of a Hamilton path or circuit have been identified.
• The graph must be connected in order for a Hamilton or Euler path or circuit to exist.

Students understand that:

• A spanning tree of a graph is the smallest subgraph.
• There are n-1 edges in a spanning tree of a graph with n vertices.
• Various algorithms are efficient methods for finding minimum weight spanning trees of a graph and shortest paths in a graph.
• Steiner points of a graph are vertices added to create a shortest spanning tree which connects the original vertices, using Euclidean distance as edge weights.
• Steiner points have degree 3, and the 3 edges form angles of 120 degrees.

Students understand that:

• -Techniques are used to minimize colors needed to color the vertices (edges) of a graph so that no two adjacent vertices (edges) are colored the same. -Real-world problems such as scheduling and conflict can be modeled with graphs and solved using the minimization of the number of colors.

Students understand that:

• Graphs can be used to model sequential tasks in a project.
• Critical paths identify the tasks that must be performed as soon as possible in order to minimize the time taken to complete the project.

Students understand that:

• Adjacency matrices can be used to determine the number of walks between any two vertices of varied lengths and is especially useful for calculating the number of walks when simple counting becomes too cumbersome.

Students understand that:

• There are a variety of voting systems other than those most frequently used systems and may provide advantages or disadvantages as compared to our current system.