Title
Prairie Dogs (Eric Knuth)
Goals
(1) Students will generalize number patterns and produce recurrence expressions.
(2) Students will develop an understanding of the usefulness of closed form expressions in recurrence relations.
Abstract
This activity develops students' ability to use diagrams as a means of problem representation, generalize number patterns, and write a recurrence expression. The activity will be an introduction to using the method of finite differences for finding closed form solutions.
Problem Statement
Ten prairie dogs need to connect all their burrows to one another in order to be sure that they can evade their enemy, the ferret. Assume they build each tunnel so that they have direct access to each of the other burrows. How many tunnels do they need to build? How many tunnels would be needed if there had been fifty prairie dogs?
Instructor Suggestions
(1) Describe the problem situation to the students.
(2) Encourage students to organize their data.
(3) Separate students into small groups to investigate the problem.
(4) Each group should present its results to the class.
(5) Discuss students' solutions in terms of their data organization, the methods used in finding a solution, and ideas for finding a closed form for the recurrence relation.
Materials
Prairie Dogs handout
Time
Problem introduction (5 minutes), group work (25 minutes), small group presentation (10 minutes), class discussion (10 minutes)
Mathematics Concepts
Discrete Mathematics Concepts:
recursion, finite differences
Related Mathematics Concepts:
patterns, sequences
NCTM Standards Addressed
Problem Solving, Communication, Reasoning, Connections, Algebra, Geometry, Discrete Mathematics
Colorado Model Content Standards Addressed
Algebraic Techniques (2), Geometric Techniques (4),Problem Solving Techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This activity could be integrated into (1) an Algebra class as an introduction for students as they begin to study the method of finite differences; (2) a Geometry class investigating problems isomorphic to this problem (e.g., number of diagonals in a polygon with n sides); (3) an Advanced Algebra class examining recursive relationships.
Further Investigation
Provide other problems in which students must develop a closed form expression for a number sequence (e.g., "handshake" problem).
Variations/Comments
References/Resources
Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete mathematics through applications. New York: W. H. Freeman and Company.
Hart, E., Maltas, J., & Rich, B. (1990). Teaching discrete mathematics in grades 7-12. Mathematics Teacher, 83(5), 362-367.
Kenney, M. J., & 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.