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 have:

• seen how division is used to help solve probability problems
• used tables as data structures used to count outcomes and to compute probabilities
• seen how trees are a type of data structure

Standards

The activities and discussions in this lesson address the following NCTM standards:

Algebra

Use mathematical models to represent and understand quantitative relationships

• model and solve contextualized problems using various representations, such as graphs, tables, and equations

Understand and apply basic concepts of probability

• understand and use appropriate terminology to describe complementary and mutually exclusive events
• use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations
• compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models

Links to other standards.

Student Prerequisites

• Arithmetic: Students must be able to:
  • use division to count outcomes in probability problems
  • 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

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

Key Terms

• combinatorics
• experimental probabiltiy
• probability
• superscript
• theoretical probabiltiy

Lesson Outline

1. Focus and Review

   Remind students of what they learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:

2. Objectives

   Let the students know what they will be doing and learning today. Say something like this:

   • Today, class, we will learn how to use division to solve probability problems.
   • We are going to use the computers to help us, but please do not turn your computers on until I ask you to. I want to show you a little about this activity first.

3. Teacher Input
   • Explain the Racing Game with Two Dice, which will introduce the concept of data structures and computing particular probabilities.

4. Guided Practice

   • Have the students begin with the Racing Game with Two Dice 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 its own way of randomly choosing which players make one or two steps.

   • Lead a discussion based on Tables and Combinatorics, discussing tables as data structures.
   • Have a discussion about Divisibility as it can be used in probability. The discussion is based on the Two Dice and a Table activity.
   • Lead the discussion Tree as a data structure.

     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.

5. Independent Practice
   • Have the students work alone or in small groups and 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

   • You may wish to bring the class back together for a discussion of the findings. Once the students have been allowed to share what they found, summarize the results of the lesson.

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.