Math Club: Counting 101

[Photo by ThunderChild tm.]

The last couple of weeks, in Math Club, we’ve been learning to count. My new set of MathCounts students have never heard of combinatorics, so we started at the very beginning:

• Counting and Probability I by Keone Hon
• Counting and Probability by Jason Batterson

The Science of Counting

One of my favorite things about coaching MathCounts is that I learn as much as the kids. When I was in school, as far as the curriculum was concerned, discrete math didn’t exist. The first year I coached our math team, I struggled with this topic, but I’ve learned enough over the years to stay at least two steps ahead of my students.

I typed up a one-page summary of the basics for my students:

Counting and probability basics (pdf 57KB)
Permutations, combinations, probability, odds — what’s the difference? This tip sheet gives you just the facts, with an absolute minimum of factorial notation. Mix with plenty of problems to solve, stir well, and discuss.

For Further Study

In addition to Mr. Hon’s tutorial, we’ll be using several of these online resources from the amazing Art of Problem Solving website:

• Why Discrete Math Is Important
  “Because discrete math does not figure prominently in most states’ middle or high school “high-stakes” progress exams, and because it also does not figure prominently on college-admissions exams such as the SAT, it is often overlooked. However, discrete math has become increasingly important in recent years, for a number of reasons…”

• Correcting for Overcounting
  “Often in counting problems, the most convenient solution is to count too much, and then somehow correct for the overcounting…” From AoPS Introduction to Counting & Probability, chapter 3.

• Probability Using Areas
  “Point C is chosen at random atop a 5 foot by 5 foot square table. A circular disk with a radius of 1 foot is placed on the table with its center directly on point C. What is the probability that the entire disk is on top of the table (i.e. that none of the disk hangs over an edge of the table)?…” From AoPS Introduction to Counting & Probability, chapter 10.

• Constructive Counting and 1-1 Correspondences
  “The basic idea is that in order to count the number of a items in a certain set, we think about how we would construct an item belonging to that set. During the construction, we keep track of the number of choices that we have at each step…” From AoPS Intermediate Counting & Probability, chapter 4.

• Three Types of Probability
  “If you think about it, labels are a big key to the way we organize ideas… In short, labels help us organize — and organization simplifies problem solving! This article seeks to demonstrate the power of intelligent classification using types of probability as an example…”

• Counting in the Twilight Zone
  “In this chapter we will look at counting methods which go far beyond common sense, and thus allow the counting of far more interesting things…” From The Art of Problem Solving, Volume 2: and Beyond, chapter 17.

Links, Links Galore

Here are two more websites with links to tutorials and discrete math puzzles:

Student Guides to Combinatorics

Discrete Math

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Have more fun on Let’s Play Math! blog:

• Substitute Teacher Experiments with Combinatorics
• 2008 Mathematics Game
• Puzzle: Random Blocks
• Puzzle: Figuring Out Figurate Numbers
• That’s Mathematics