Title
Rain or Shine (Anne Smelker)
Goals
(1) Students will apply their logic skills to solve a discrete probability problem while developing and exploring Markov Chains.
(2) Students will develop and apply their understanding of discrete probability and combinations.
(3) Students will explore and develop an understanding and application of Markov Chains.
Abstract
This activity will allow students the opportunity to apply their logic skills to solve a discrete probability problem while developing and exploring Markov Chains. Students will utilize their knowledge of probability and graphs to construct a Markov Chain. This activity develops students' ability to use diagrams as a means of problem representation, logic skills to solve problems, and their understanding of probability. The activity will be an introduction to using Markov Chains in order to solve problems that involve a finite number of events or states that change over time.
Problem Statement
The weather during the Smedleys' vacation was strange.
It was cloudy on 13 different days, but it was never cloudy for an entire day.
Cloudy mornings were followed by clear afternoons.
Cloudy afternoons were preceded by clear mornings.
There were 11 clear mornings and 12 clear afternoons in all.
Utilize the above information to solve the following problems.
Instructor Suggestions
(1) Discuss the problem statement with the students.
(2) Have students work in small groups to explore and investigate the problems.
(3) As a class, have each group share and discuss their methods and/or strategies they developed to solve the problems.
(4) Discuss student's methods and/or strategies, how they developed their methods, what data organization they used, and how their methods/strategies relate to logic, discrete probability and Markov Chains.
Materials
"Rain or Shine" activity sheet
Time (Possibly two days depending on level of students)
Introduction of Problem Statement (5 minutes), Group Work (20 minutes), Presentations/Class Discussion (20 minutes)
Mathematics Concepts
Discrete Mathematics Concepts
Logical Reasoning, Discrete Probability, Markov Chains
Related Mathematics Concepts
Venn Diagrams, Solving Equations, Deductive Reasoning
NCTM Standards Addressed
Problem Solving, Communication, Reasoning, Connections, Algebra, Geometry, Discrete Mathematics, Probability
Colorado Model Content Standards Addressed
Problem Solving Techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This activity could be integrated into an Algebra or Geometry class when introducing Markov Chains, exploring probability, or developing logical reasoning.
Further Investigation
Variations/Comments
Portions of this activity may be eliminated to produce an excellent logical reasoning problem to utilize in less advanced courses which have not been introduced to probability
References/Resources
Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete Mathematics Through Applications. New York: W.H. Freeman and Company.
Kenney, M. J., and Hirsch, C.R. (Eds.). (1991). Discrete Mathematics Across the Curriculum, K-12. Reston, VA: National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: Author.
Schadler, R. (1984). Geometry Problems: One Step Beyond. Dale Seymour Publications.