Title
Colorado Rockies: Winning Streaks and Losing Streaks (Dan Snook)
Goals
1. Students will develop and use a transition matrix and calculate a stable state for given information.
2. Students will use the stable state vector to make predictions.
Abstract
Students will investigate tendencies of the Colorado Rockies baseball team. They will use matrix
algebra to make predictions about the future winning percentages of the team.
Problem Statement
Given the tendency that the Rockies follow a victory with another victory 70% of the time, and follow a loss with another loss 60% of the time, students will draw conclusions about their season record. They will investigate outcomes based on winning or losing the opening game, and make a prediction as to the final regular season record.
Instructor Suggestions
1. Have students look at baseball standings in the newspaper, to make sure they understand the notation of a won-loss record, and that the winning percentage is listed as a 3 digit decimal.
2. Offer suggestions as to how to align the data in the transition matrix.
3. Make sure students are correctly inputting data in the matrix mode of the graphing calculators.
4. Discuss why winning or losing the first game does not affect the stable state.
Materials
"Colorado Rockies" worksheets, graphing calculators, dry erase board
Time
Checking standings from the newspaper (5 minutes), Problem introduction (5 minutes), Small group work (20 minutes), Discussion of results (10 minutes)
Mathematics Concepts
Discrete Mathematics Concepts
Markov Chains, Initial State Matrix, Transition Matrix, Stable State, Absorbing State
Related Mathematics Concepts
Percentages, Matrix Operations
NCTM Standards Addressed
Problem Solving, Reasoning, Communication, Connections, Algebra, Discrete Math
Colorado Model Content Standards Addressed
Number Sense (1), Algebraic Techniques (2), Data Collection and Analysis (3), Problem Solving Techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This activity would work well in a Pre-Algebra or Algebra I class after matrix operations of addition and multiplication have been taught. It is also a useful activity while teaching the use of the TI-83's ability to solve matrices.
Further Investigation
It may be of interest to determine why the stable state percentages are equal to 4/7 wins and
3/7 losses. Also, the class could look at the possibility of starting the transition matrix after the
All-star break (when teams have already played half of their games) or the problem could be expanded to look at homefield advantages.
Variations/Comments
It is important that students know how to do basic matrix calculations. However, it will also be obvious to a student that while it is simple to do the calculation D0T, there must be an easier way to calculate D0 T10 besides grinding out all of the multiplications. Thus, the graphing calculators should be used as a tool, not a crutch.
References/Resources
Colorado Model Mathematics Standards Task Force. (1995) Colorado model content standards for mathematics.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author:
Crisler, Nancy, Fisher, Patience, Froelich, Gary. (1994) Discrete Mathematics Through Applications. New York: W.H. Freeman and Company