Title
From Shapes to Handshakes (Anne Smelker)
Goals
(1) Students will generalize number patterns and produce recurrence and/or closed form expressions.
(2) Students will develop an understanding of the usefulness of closed form expressions in recurrence relations.
(3) Students will apply methods used to solve geometry problems to "real world" problems or applications.
Abstract
This activity will allow students the opportunity to develop their abilities to use diagrams as a means of problem representation, generalize number patterns, and write recurrence expressions. This activity develops a connection between analyzing patterns in geometric shapes and real world situations.
Problem Statement
Explore the geometric problems and develop methods to solve each of the situations. After developing a strategy and/or method to solving the problems, apply these methods to the real world situations and determine if the method(s) used are effective. Discuss in groups how the methods and/or strategies were developed, how to determine whether the methods and/or strategies were effective, and why or why not.
Instructor Suggestions
(1) Discuss the problem statement with the students.
(2) Encourage students to organize the situations.
(3) Have students work in small groups to explore and investigate the situations. Assign two situations from both the geometric scenarios and real world scenarios to each group.
(4) As a class, have each group share and discuss their methods and/or strategies they developed.
(5) Discuss student's methods and/or strategies, how they developed their methods, what data organization they used, ideas for finding a closed form for the recurrence relation, and how their approaches relates to solving the other problems.
Materials
"From Shapes to Handshakes" activity sheet
Time
Introduction of Problem Statement (5 minutes), Group Work (20 minutes), Presentations/Class Discussion (20 minutes)
Mathematics Concepts
Discrete Mathematics Concepts
Counting Techniques, Recursion, Iteration
Related Mathematics Concepts
Number 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 an Algebra or Geometry class when introducing number patterns and recursion
Further Investigation
Variations/Comments
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.
Serra, M. (1993). Discovering Geometry: An Inductive Approach. Berkeley, CA: Key Curriculum Press.