VI. Probability and Discrete Mathematics

1. Probability
2. Discrete Mathematics

Additional Links to General Mathematics Sites

• Mathematics Games and Student Activities
• Mathematics Lessons Plans
• Interactive Learning and Problem Solving Activities
• Word Searches and Puzzles in Mathematics
• Miscellaneous Sites

High School

1. Benchmark 1. "Derive and use formulas for calculating permutations and combinations."
   1. Calculate Your Chance of Winning the Lottery - Discrete Mathematics (VI.2.HS.1)
      This site allows students to input information to determine their odds of winning the lottery. This site includes a "real world" model for understanding those odds.
   2. Counting Techniques Activity - Discrete Mathematics (VI.2.HS.1)
      This activity focuses on the application of permutation and combinations to an actual lottery. Students are asked to individually determine the total possible tickets and then work in groups to develop a scheme to guarantee winning the lottery. It is intended to be used after the students have been introduced to the concepts of counting, permutations and combinations. This is a teacher lesson plan.
   3. DMP Home Page - Discrete Mathematics (VI.2.HS.1,3,4,5&6)
      This discrete math page has many activities for permutation and combinations, recurrence and interation, and graph theory.
   4. Winning the Lottery - Discrete Mathematics (VI.2.HS.1)
      A hands-on activity simulating playing the lottery where students derive a formula for the probability of winning the game.
2. Benchmark 2. "Use sets and set relationships to represent algebraic and geometric concepts."
   1. Lesson on Deductive Arguments - Discrete Mathematics (VI.2.HS.2)
      This site thoroughly explains Venn Diagrams. It teaches the students how to use Venn Diagrams, gives them an example, and then has them work through a bunch of examples. This web site is a great way for students to work with logic and learn how to use Venn Diagrams to solve problems. This is a student activity.
   2. Venn Diagrams - Discrete Mathematics (VI.2.HS.2&3)
      Interactive Cross curricular site. Student has to pick region of a Venn Diagram that a set belongs in. It uses examples in science as well as math.
3. Benchmark 3. "Use vertex-edge graphs to solve network problems such as finding circuits, critical paths, minimum spanning trees and adjacency matrices."
   1. Colorful Mathematics - Discrete Mathematics (VI.2.HS.3&6)
      Colorful Mathematics is a series of educational software presenting advanced mathematical concepts to 9-12 students in a game-oriented approach. The five games offered utilize simple coloring and/or drawing techniques to illustrate some state of the art mathematical concepts from graph theory and computer science. This site is an excellent introduction to vertex-edge graphs in discrete mathematics. Teachers are provided with sample discussion questions to use with the students. This is a student activity.
   2. DMP Home Page - Discrete Mathematics (VI.2.HS.1,3,4,5&6)
      This discrete math page has many activities for permutation and combinations, recurrence and interation, and graph theory.
   3. Graph Theory - Discrete Mathematics (VI.2.HS.3&5)
      A complete lesson plan with printable handouts and student activities. Concepts presented involve routing and networking.
   4. Graph Theory Tutorials - Discrete Mathematics (VI.2.HS.3&5)
      This is the home page for a series of short interactive tutorials introducing the basic concepts of graph theory. These tutorials are created using the Web Tutor so that most of the pages of this tutorial require that students pass a quiz before continuing to the next page, while others ask for a written comment. The Web Tutor must be able to keep track of student's progress, so they will need to register for each of these courses by pressing the [REGISTER] button on the bottom of the first page of each tutorial. This is an excellent site for students to learn about graph theory and analyzing different algorithms.
   5. Matrices Help Relationships: An Airline Problem - Discrete Mathematics (V1.2.HS..3&6)
      This site provides an example of how matrices can be used to solve an airline problem.
   6. Networks and Matrices - Discrete Mathematics (VI.2.HS.3)
      This site provides an example of using networks and matrices.
   7. Robot Sketcher - Discrete Mathematics  (VI.2.HS.3,5&6)
      This is a student interactive site.  In this investigation, you can build compound arms having multiple joints of two types: one that rotates and is typical of rotating motors and one that slides and is typical of hydraulic lifts. Students have two questions to answer/explore as they manipulate the robot arms and explore the patterns that can be produced.
   8. Speedy Delivery: Service Woes - Discrete Mathematics (VI.2.HS.3&6)
      Students become familiar with basic elements of a network by finding the shortest route for a delivery truck. The website is not interactive, but contains worksheets, case studies and teacher notes.
   9. Venn Diagrams Self Test - Discrete Mathematics (VI.2.HS.2&3)
      This is a Venn Diagram self test. The set is given in union/intersection format and the student is encouraged to shade in a copy of the diagram on paper, then the answer is given by moving the mouse over the problem.
4. Benchmark 4. "Analyze and use discrete ideas, such as induction, iteration and recurrence relations."
   1. Chaos in the Classroom - Discrete Mathematics (VI.2.HS.4)
      This site goes through a series of steps using chaos theory to construct Sierpinski's Triangle. The is a student activity.
   2. DMP Home Page - Discrete Mathematics (VI.2.HS.1,3,4,5&6)
      This discrete math page has many activities for permutation and combinations, recurrence and interation, and graph theory.
   3. Fire! Probability and Chaos - Discrete Mathematics (VI.2.HS.4)
      This lesson utilizes concepts of probability, graphing and graph interpretation, mean, and variance in analyzing a simulation of a forest fire. A complete lesson plan is included and a student interactive applet is provided giving the number of iterations performed.
   4. Induction and Deduction - Discrete Mathematics (VI.2.HS.4)
      There is a discussion of what is inductive and deductive reasoning. There is a quiz at the end. This site has six tutorials that one can use. This is a student activity.
   5. Investigating Elimination of Medicine from the Body - Discrete Mathematics (VI.2.HS.4)
      These investigations have students use graphs and tables that illustrate the use of iteration, recursion,and algebra to model and analyze the changing amount of medicine in an athlete's body.
   6. Exploring Fractals - Discrete Mathematics (VI.2.HS.4)
      This site is a student activity. It is set up for students to lead themselves through a lesson on fractals. There is a lot of explanation for the students to read. The students learn about iteration, dimension, and self-similarity with fractals. There are lots of great pictures and links to other places to get information. Once students have learned all of the basics, they are lead through some investigations. Students using this site would need to know how to do logarithms. This site is recommended for upper level math students.
5. Benchmark 5. "Describe and analyze efficient algorithms to accomplish a task or solve a problem in a variety of contexts, including practical, mathematical and computer-related situations."
   1. DMP Home Page - Discrete Mathematics (VI.2.HS.1,3,4,5&6)
      This discrete math page has many activities for permutation and combinations, recurrence and interation, and graph theory.
   2. Graph Theory - Discrete Mathematics (VI.2.HS.3&5)
      A complete lesson plan with printable handouts and student activities. Concepts presented involve routing and networking.

      Graph Theory Tutorials - Discrete Mathematics (VI.2.HS.3&5)
      This is the home page for a series of short interactive tutorials introducing the basic concepts of graph theory. These tutorials are created using the Web Tutor so that most of the pages of this tutorial require that students pass a quiz before continuing to the next page, while others ask for a written comment. The Web Tutor must be able to keep track of student's progress, so they will need to register for each of these courses by pressing the [REGISTER] button on the bottom of the first page of each tutorial. This is an excellent site for students to learn about graph theory and analyzing different algorithms.

      Robot Sketcher - Discrete Mathematics  (VI.2.HS.3,5&6)
      This is a student interactive site.  In this investigation, you can build compound arms having multiple joints of two types: one that rotates and is typical of rotating motors and one that slides and is typical of hydraulic lifts. Students have two questions to answer/explore as they manipulate the robot arms and explore the patterns that can be produced.
6. Benchmark 6. "Use discrete mathematics concepts as described above to model situations and solve problems; and look for whether or not there is a solution (existence problems), determine how many solutions there are (counting problems) and decide upon a best solution (optimization problems)."
   1. Colorful Mathematics - Discrete Mathematics (VI.2.HS.3&6)
      Colorful Mathematics is a series of educational software presenting advanced mathematical concepts to 9-12 students in a game-oriented approach. The five games offered utilize simple coloring and/or drawing techniques to illustrate some state of the art mathematical concepts from graph theory and computer science. This site is an excellent introduction to vertex-edge graphs in discrete mathematics. Teachers are provided with sample discussion questions to use with the students. This is a student activity.
   2. DMP Home Page - Discrete Mathematics (VI.2.HS.1,3,4,5&6)
      This discrete math page has many activities for permutation and combinations, recurrence and interation, and graph theory.
   3. Matrices Help Relationships: An Airline Problem - Discrete Mathematics (V1.2.HS..3&6)
      This site provides an example of how matrices can be used to solve an airline problem.
   4. Robot Sketcher - Discrete Mathematics  (VI.2.HS.3,5&6)
      This is a student interactive site.  In this investigation, you can build compound arms having multiple joints of two types: one that rotates and is typical of rotating motors and one that slides and is typical of hydraulic lifts. Students have two questions to answer/explore as they manipulate the robot arms and explore the patterns that can be produced.
   5. Speedy Delivery: Service Woes - Discrete Mathematics (VI.2.HS.3&6)
      Students become familiar with basic elements of a network by finding the shortest route for a delivery truck. The website is not interactive, but contains worksheets, case studies and teacher notes.