Standards - Mathematics

2019 - Mathematics Course of Study Document
Mathematics (2019) Grade(s): 09-12 - Applications of Finite Mathematics

MA19.FM.16

Use vertex and edge graphs to model mathematical situations involving networks.

Unpacked Content

Knowledge

Students know:

  • How to construct a vertex and edge structure

Skills

Students are able to:

  • Determine what a vertex and an edge would represent in modeling a real-world problem.
  • Construct simple graphs, complete graphs, bipartite graphs, complete bipartite graphs, and trees..

Understanding

Students understand that:

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

Vocabulary

  • Graph
  • Vertex
  • Edge
  • Network
  • Complete graph
  • Bipartite graph
  • Tree
Mathematics (2019) Grade(s): 09-12 - Applications of Finite Mathematics

MA19.FM.17

Solve problems involving networks through investigation and application of existence and nonexistence of Euler paths, Euler circuits, Hamilton paths, and Hamilton circuits. Note: Real-world contexts modeled by graphs may include roads or communication networks.

Example: show why a 5x5 grid has no Hamilton circuit.

Unpacked Content

Knowledge

Students know:

  • How to make systematic lists to solve problems.

Skills

Students are able to:

  • Create a graph that models a given situation.
  • Apply an algorithm to find a Hamilton or Euler circuit or path in a graph.

Understanding

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.

Vocabulary

  • Degree of a vertex
  • Graph
  • Bipartite graph
  • Grid (a type of bipartite graph)
  • Vertex
  • Edge
  • Circuit (Euler, Hamilton)
  • Path (Euler, Hamilton)
  • Algorithm
Mathematics (2019) Grade(s): 09-12 - Applications of Finite Mathematics

MA19.FM.18

Apply algorithms relating to minimum weight spanning trees, networks, flows, and Steiner trees.

Example: traveling salesman problem

Unpacked Content

Knowledge

Students know:

  • Graphing procedures and properties.

Skills

Students are able to:

  • Model a problem using flows in networks.
  • Use technology or other tools to construct Steiner points.
  • Apply minimum weight spanning tree algorithms.

Understanding

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.

Vocabulary

  • Spanning tree
  • Minimum weight spanning tree
  • Network
  • Flow
  • Kruskal's algorithm
  • Prim's algorithm
  • Steiner tree
  • Steiner points
Mathematics (2019) Grade(s): 09-12 - Applications of Finite Mathematics

MA19.FM.19

Use vertex-coloring, edge-coloring, and matching techniques to solve application-based problems involving conflict.

Examples: Use graph-coloring techniques to color a map of the western states of the United States so that no adjacent states are the same color, determining the minimum number of colors needed and why no fewer colors may be used; use vertex colorings to determine the minimum number of zoo enclosures needed to house ten animals given their cohabitation constraints; use vertex colorings to develop a time table for scenarios such as scheduling club meetings or for housing hazardous chemicals that cannot all be safely stored together in warehouses.

Unpacked Content

Knowledge

Students know:

  • Graphing procedures and properties.

Skills

Students are able to:

  • Model application-based problems that may be solved using graph colorings.
  • Color the edges or vertices of a graph using the least number of colors so that no two adjacent vertices or edges are colored the same.
  • Interpret the coloring of the graph in terms of a solution for an application-based problem, such as scheduling committee meetings (vertex colorings) or class scheduling (edge-colorings).
  • Identify structures in a graph that require a minimum number of colors for a proper coloring.

Understanding

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.

Vocabulary

  • Vertex coloring
  • Matching techniques
  • Conflict graphs
  • Adjacent edges
  • Adjacent vertices
  • Odd wheel graph
  • Proper coloring

Aligned Learning Resources

Punctuation Dialogue Quick Check
Putting It Together
Quizizz Standard Deviation
Mathematics (2019) Grade(s): 09-12 - Applications of Finite Mathematics

MA19.FM.20

Determine the minimum time to complete a project using algorithms to schedule tasks in order, including critical path analysis, the list-processing algorithm, and student-created algorithms.

Unpacked Content

Knowledge

Students know:

  • Graphing procedures and properties.

Skills

Students are able to:

  • Model tasks of a project in a graph.
  • Identify critical paths using various algorithms.

Understanding

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.

Vocabulary

  • Graphs
  • Critical paths
  • List
  • processing algorithm
Mathematics (2019) Grade(s): 09-12 - Applications of Finite Mathematics

MA19.FM.21

Use the adjacency matrix of a graph to determine the number of walks of length n in a graph.

Unpacked Content

Knowledge

Students know:

  • How to form graphs.
  • How to determine walks and paths.
  • How to multiply matrices.

Skills

Students are able to:

  • Use a graph to create a matrix that shows the number of walks between any two vertices.
  • Use matrices to determine the number of walks of various lengths.

Understanding

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.

Vocabulary

  • Walk
  • Matrix
  • Adjacency matrix

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