Course 1 Unit 4 - Graph Models
1st Edition

Graph Models is the fourth unit in the Contemporary Mathematics in Context program and first unit from the discrete mathematics strand. By the time students begin this unit, they will have developed the ability to make sense out of real-world data through the use of graphical displays and summary statistics. They will be able to recognize important patterns of change among related variables and represent those patterns using tables of numerical data, coordinate graphs, verbal descriptions, and symbolic rules. They will also have developed skill in using linear equations to model problems in situations which exhibit constant or nearly constant rate of change. (See the description of Course 1 Units.)

Unit Overview

This unit begins the study of vertex-edge graphs as mathematical models.

Unit Objectives
  • To use vertex-edge graphs to make sense of situations involving relationships among a finite number of elements - for example, conflict and prerequisite relationships
  • To gain experience in mathematical modeling by building and using vertex-edge graph models to solve problems in a variety of real-world settings
  • To develop the skill of algorithmic problem solving: designing, using, and analyzing systematic procedures for solving problems
  • To investigate and apply three powerful and widely used graph models: Euler paths, vertex coloring, and critical paths

Sample Overview

The sample material from Graph Models Lesson 2, "Managing Conflicts," follows a lesson introducing students to Euler paths and circuits and precedes a lesson exploring use of directed graphs for scheduling of large projects. See the unit table of contents below.

In this sample investigation, students build graph models to represent situations involving conflict, develop a coloring algorithm for the conflict, compare algorithms, and finally in the "Checkpoint" analyze their algorithms for strengths and weaknesses.

Instructional Design

Throughout the curriculum, interesting problem contexts serve as the foundation for instruction. As lessons unfold around these problem situations, classroom instruction tends to follow a common pattern as elaborated under Instructional Design.

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How the Discrete Mathematics Strand Continues

In Course 2, students continue their study of discrete mathematics in the first and fifth units. Unit 1, Matrix Models, develops student ability to use matrices and operations to represent and solve problems while connecting important mathematical ideas from several strands. Unit 5, Network Optimization, extends student ability to use vertex-edge graphs to represent and analyze real-world situations involving network optimization, optimal spanning networks, and shortest routes.

Unit 2, Modeling Public Opinion, in Course 3, develops student understanding of how public opinion can be measured using vote analysis methods, surveys, sampling distributions, the relationship between a sample and a population, confidence intervals, and margin of error. Also in Course 3, students study Discrete Models of Change which extends their ability to represent, analyze, and solve problems in situations involving sequential change and recursive change.

In Course 4, the unit Counting Models extends student ability to count systematically and solve enumeration problems, and develops understanding of, and ability to do, proof by mathematical induction.

Many of the mathematical concepts developed in the discrete mathematics strand are revisited in the other mathematical strands, thus enabling students to develop a robust, connected understanding of mathematics.

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