**Title**

Discovering Dominoes (Anne Smelker)

**Goals**

(1) Students will explore counting techniques, permutations, and combinations.

(2) Students will develop their combinatorial reasoning and learn to count without actually counting or applying formulas to count.

**Abstract**

This activity will allow students the opportunity to develop their abilities to count various outcomes and explore different methods and principles to counting. This activity develops the foundation to building strong combinatorial reasoning skills and the ability to apply the tools of counting, permutations, and combinations, with the reasoning process (not just applying formulas) while developing problem solving and critical thinking skills.

**Problem Statement**

A domino is a rectangle formed by two congruent squares. Each square contains an orderly pattern of "pips" or dots representing a number from zero through six. Explore and develop methods and strategies to solve the following problems involving dominoes.

**Instructor Suggestions**

(1) Discuss the problem statement with the students.

(2) Encourage students to organize the situations with dominoes.

(3) Have students work in small groups to explore and investigate the problems.

(4) As a class, have each group share and discuss their methods and/or strategies they developed to solve the problems.

(5) Discuss student's methods and/or strategies, how they developed their methods, what data organization they used, and how their methods/strategies relate to counting, permutations, combinations, and graph theory

**Materials**

"Discovering Dominoes" 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, Combinatorial Reasoning, Permutations, Combinations, Graph Theory

*Related Mathematics Concepts*

Probability, Algorithms

**NCTM Standards Addressed**

Problem Solving, Communication, Reasoning, Connections, Algebra, Geometry, Discrete Mathematics

**Colorado Model Content Standards Addressed**

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 combinatorial reasoning.

**Further Investigation
**

**Variations/Comments**

Try other board games such as Triominoes or other related games.

**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.