Illinois State University Mathematics Department

 MAT 409: Topics in Algebra and Combinatorics for K-8 Teachers

## Session 1: 13 June 2016

Combinatorics in the Real World
• Acquaintances?
• A high school teacher's discovery (text: p 6)
• Testing Positive . . . What Does That Mean?
• Three Facts to Consider
• Approximately one-half of 1% of the US population ages 13 and older is HIV infected.
• One version of an HIV test yields 1.5% false positives (test is positive, but in reality the person tested is not HIV infected) and 0.3% false negatives (test is negative, but in reality the person tested is  HIV infected)
• A person from the general US population tests positive for HIV infection. What's the probability the person actually is HIV infected?

Perfect Covers of Chess Boards
• Description and Exploration
• An Exemplary Combinatorics Problem Situation

The Pigeonhole Principle
• An Initial Problem
• Generalizing the Principle
• A Concluding Problem

Summation and Product Notation

Basic Counting Strategies: Did not get to this today!

Course Information

Assignment #1

• Preface (pp 1-4)
• Chapter 1 (pp 6-12)
• Chapter 2 Section 1 (pp 15-20)
• Learning from Others (pp 23-25)

Problems

• Chapter 2 Section 1 (pp 20-22): 2,4,6,8,10,12,one of 15-17
• Chapter 2 Section 2 (pp 28-29): 2,3,4,5,6; 11 Not due next time!
• Chapter 2 Section 3 (pp 34-35): 2,3,5,7,8 Not due next time!

Early Send-Ins

• By 9:32 pm tonight (Mon 6/13/16), email to me (day@ilstu.edu; write MAT 409 Assignment #1 as subject line) your responses to:
• Ch 2 Section 1 (pp 20-21): one of 2, 4, or 6; one of 8 or 10

#### Perfect Covers of Chess Boards

We will explore the problem of finding the perfect covers of an n-by-m rectangular chess board with dominoes measuring 2-by-1 units. A perfect cover means that there are no gaps or overlaps when we cover the board entirely with the dominoes.

We first consider two questions:

1. What restrictions are there on n and m?
2. How many ways are there to cover an n-by-m board?

We will work in small groups to determine the restrictions on m and n and encourage multiple ways to justify our results. We will then consider the second question, again exploring the strategies used among the groups.

To extend the problem, we can look at a pruned 8-by-8 chess board. Does a perfect cover exist for such a board?

Another extension is to consider perfect covers of a j-by-k-by-l rectangular prism, using a 1-by-1-by-2 domino-like prism.

We'll carry out an activity designed to remix the student groups and to encourage you to get to know your colleagues. We will next return to the perfect cover problem and use our exploration to identify fundamental components of a problem-solving approach to combinatorics problems:

• Can a task be completed?
• How so?
• In how many ways?
• What are some extensions and variations to the problem?

In using these key questions to exemplify the type of investigations that will underscore course activities, we will emphasize the need to justify our efforts as we progress in solving a problem. We will talk about the need to consider or search for elegant and creative ways to approach problems.