Discrete Mathematics Project

Counting Techniques Activity


Batting Orders (Kim Kendrick)


(1) Students will continue to explore the concept of permutations and combinations and apply the Fundamental Counting Property to solve a real world problem.

(2) Students will compare solutions as conditions vary within the problem and be able to justify their solutions in sentence format.

(3) Students will work individually to arrive at solutions and then work in small groups in order to arrive at mutual solutions and be able to explain, and justify how they arrived at this solution to the rest of the class.


This activity requires students to have a basic knowledge of how to select and arrange objects. The students are asked to apply their knowledge to determine the number of batting arrangements possible for a softball team and compare solutions as the data varies and the conditions change. This assignment includes numerical as well as written solutions and explanations.

Problem Statement

Let students know that in this situation they are the manager of a softball league. They have not decided on a particular batting order for the next game, and they must determine the total possible arrangements. Because both men and women participate on the team, the league rules state that at least two women must play at any given time. They will be required to apply this new criteria, find the solution(s) and compare them to the previous answer. This comparison must be made in clear and concise sentence format.

Instructor Suggestions

(1) Set the stage by discussing the "Problem Statement" (see above) with the class.

(2) Distribute the "Batting Order" activity sheet (see attachment) and allow the students to individually read the activity.

(3) When all of the students are finished reading, have each of the students complete the assignment individually.

(4) After all of the students have finished, have them form small-groups and compare their solutions. The group must come to a consensus on the solutions.

(5) When the small-groups are finished, have a spokesperson from each group share their solutions and explain their method and reasoning involved in arriving at those solutions.

(6) The teacher should provide closure to the activity by showing the correct solutions and explaining the methods and reasoning required to obtain those solutions if this was not done in the presentation period.


"Batting Order" activity sheet, chalk board, calculators


Introduction of problem statement (5 minutes), Individual work (20 minutes), Small group work (10-15 minutes), Presentation of small-group work (10-15 minutes), Large group discussion (5-15 minutes)

Mathematics Concepts

Discrete Mathematics Concepts:

Counting techniques (permutations and combinations)

NCTM Standards Addressed

Problem Solving, Communication, Reasoning, Connections (within mathematics and across disciplines), Arrangements, Discrete Mathematics.

Colorado Model Content Standards Addressed

Algebraic Techniques (2), Problem Solving Techniques (5), Linking Concepts and Procedures (6)

Curriculum Integration

This activity will be used in the first year of the Integrated Math Program as a separate module. It could also be used in a traditional Algebra I class when discussing series or in an Algebra II class when first introducing and discussing probability.


This lesson requires students to compare and justify solutions they have obtained. This may confuse and frustrate students if they have not arrived at various solutions. They may need some extra monitoring to make sure they are staying on track.


Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete Mathematics Through Applications. New York: W.H. Freeman and Company.

Last updated January 16, 1997