Abstract

The activities and discussions in this lesson are devoted to data structures and their applications to probability theory. Tables and trees are introduced, and some of their properties are discussed.

Objectives

Upon completion of this lesson, students will be able to:

• use division to help solve probability problems
• develop and use tables as data structures to determine outcomes and compute probabilities
• develop tree diagrams as a type of data structure

Activities

• Racing Game with Two Dice
• Two Dice and a Table

Standards

The activities and discussions in this lesson address the following Standard:

Probability and Statistics

• Compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models.

Key Terms

• combinatorics
• outcomes
• superscript
• logarithm

Student Prerequisites

• Arithmetic: Students must be able to:
  • use and apply the concept of divisibilty
  • use multiplication in working with data structures
• Technological: Students must be able to:
  • perform basic mouse manipulations such as point, click and drag
  • use a browser such as Netscape for experimenting with the activities

Teacher Preparation

• 1 gallon size plastic baggie
• 5-6 small objects such as marbles or M&M's or coins
• 2 6-sided dice
• Students will need:
  • access to a browser
  • pencil and paper
  • For the Racing Game with Two Dice: each group of students will need:
    1. two dice
    2. chips or other small game pieces of different colors
    3. the Racing Game field
    4. the Racing Game tally table
    5. the Racing Game Worksheet

Lesson Outline

1. Focus and Review
   • Remind students what they have learned about probability in previous lessons.

   • Motivate the idea of combinatorics with a plastic baggie of several colors of marbles, M&M's, or other small random objects. Let the class know what is in the baggie. Ask students "If I reach in the bag how many different possibilities are there of drawing any particular object from the baggie?" or something to that effect.

2. Objectives

   Let the students know what it is they will be doing and learning today. Lead in from the previous experiment. Say something like this:

   • "What we just did is a simple example of what is called 'combinatorics.' Any one of the different possbile combinations is called an 'outcome.' Today we will be studying combinatorics and outcomes."

3. Teacher Input
   • Show student the 2 dice and ask how many different ways are there to get the sum of the 2 dice to be 8. List the different combinations on the board. Reiterate that each one of these combinations is an outcome.

   • Lead a discussion based on Tables and Combinatorics, discussing tables as data structures.

4. Guided Practice
   • Briefly show and explain Racing Game with Two Dice, which will introduce the concept of data structures and computing the probability of particular outcomes. Several players "race to the finish" using the software or on paper. For every round, each player makes either one or two steps depending on the outcome of the roll of two dice.

   • Each group of students can come up with their own way of randomly choosing which players make one or two steps.

5. Independent Practice
   • Then have the students play the Two Dice and a Table activity, where students research tables as data structures and use tables to count outcomes and compute probabilities.

6. Closure
   • Hold a discussion about divisibility as it can be used in probability. The discussion is based on the Two Dice and a Table activity.

   • Conclude with a summarizing discussion, tree diagrams as data structures. This discussion introduces and develops the idea of trees as data structures. It is based on all the other parts of the lesson. Plan it as a "live" discussion where students have an opportunity to ask their own questions, because the topic tends to be interesting to many people and it can lead to various investigations in math and computer science.

Alternate Outlines

This lesson can be rearranged in several ways.

• Use the text in the Tables and Combinatorics discussion to prepare for a "live" discussion that can take place while students are using the Two Dice and a Table activity.
• Have students read the divisibility discussion independently, or use the text to prepare for a "live" discussion.

Suggested Follow-Up

After these discussions and activities, the students will have seen how data structures such as tables and trees can be used when solving probability problems. The next lesson, Expected Value, introduces and develops the idea of expected value. The discussion helps students investigate the definition and formula of expected value.