Abstract

The activity and two discussions of this lesson connect probability and geometry. The Polyhedra discussion leads to platonic solids, and the Probability and Geometry discussion leads to connections between angles, areas and probability. The subtle difference between defining probability by counting outcomes and defining probability by measuring proportions of geometrical characteristics is brought to light.

Objectives

Upon completion of this lesson, students will:

• have practiced calculating probability
• have seen how geometry can help solve probability problems
• have learned about platonic solids

Standards

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

Data Analysis and Probability

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

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

• understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects

• draw geometric objects with specified properties, such as side lengths or angle measures
• use visual tools such as networks to represent and solve problems

Links to other standards.

Student Prerequisites

• Arithmetic: Students must be able to:
  • use addition, multiplication and division in solving probability problems
  • work with fractions in solving probability problems
• 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
• Spinner Worksheet
• If desired, the Polyhedra discussion can be demonstrated with the students using:
  1. cardboard or plastic forms of equilateral triangles, squares, and regular pentagons to trace on paper. If a set of pre-cut paper figures consisting of 30-40 triangles, 10-15 squares, and 15-20 pentagons is available, then forms and scissors are unnecessary.
  2. scissors to cut the paper
  3. scotch tape to put the polyhedra together
• The Spinner Game and the Adjustable Spinner Game require either computer access or a set of materials for building spinners for each group of students.
• The Angles and Expected Value Discussion: from geometry to probability discussion refers to protractors for measuring angles, so each group of students should have a protractor.

Key Terms

• estimate
• experimental probabilty
• probability
• random number generator
• theoretical probabiltiy

Lesson Outline

1. Focus and Review

   Remind students what has been 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:

   • Who has ever watched the game wheel of fortune?
   • Have you ever noticed when they put the $10,000 space on the wheel it is significantly smaller than the rest of the spaces?
   • Do you think size of the space affects whether or not you will land on the space?

2. Objectives

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

   • Today, class, we are going to begin learning about probability.
   • We are going to use the computers to learn about probability, 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
   • The Probability and Geometry discussion shows that sometimes knowledge of geometry is required to answer probability questions.

4. Guided Practice
   • Have the students work with the Spinner Game and the Adjustable Spinner Game to demonstrate probability concepts using spinners.

     Each student or group of students can construct a spinner or use the software to construct a "virtual spinner." Conducting multiple experiments with the spinners, students can determine experimentally the chances of selecting each sector, and compare these chances.

     If students use physical spinners, they will have to tally the results of the experiments by hand. Each group of students can use the Spinner Experiments Table for that.

   • Using spinners, physical or virtual, from the Spinner Game and the Adjustable Spinner Game, groups of students can discuss how to find the exact probability of selecting each sector on their spinner, and then compare their findings with experimental data from the Spinner Game. The following questions can help the students:
     1. What features of the spinner (e.g., size, color of sectors, etc.) make a difference for the probability, and what features do not make a difference?
     2. How can we decide which of the two sectors has a better chance to be selected? Can we do it without cutting the spinner and superimposing the sectors?

   • The Polyhedra discussion connects probability and geometry through construction of dice with various numbers of sides.

5. Independent Practice
   • Have students construct their own dice. We can loosely call a die a 3-D object that can land in several different ways when it is rolled on a flat surface. Most people are familiar with six-sided dice. The following activities and questions can be interesting to individual students or to groups of students:

     1. Come up with a way to construct a "die" that has as many sides as you want, starting from 3: 3, 4, 5, 100, ... Hint: pencil.

     2. Using the following rules, try to construct various dice:

     • You can use polygons of only one type: either equilateral triangles or squares or regular pentagons
     • Each vertex of the die has the same number of sides connected to it. In practice, you can start from forming one vertex out of several polygons. Their number will be dictated by geometry (3, 4 or 5 for triangles, 3 for squares, 3 for pentagons). Then attach the same number of sides to the remaining vertices, finishing the polyhedron.

     The dice that can be constructed this way are called platonic solids.

     3. Can you construct a platonic solid type die out of regular hexagons? Why or why not? Try it.

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.

• Have students construct spinners out of several different materials, and then compare the results they obtain. Which materials or designs produce spinners that produce more truly "random" results? Compare the results of many spins with these spinners with the computer-generated results from the Spinner Game and the Adjustable Spinner Game to show students the advantage of using a computer model to produce accurate results.
• Use Buffon's Needle as an additional example of the connection between probability and geometry.
• Have groups of students read the two discussions in this lesson and prepare presentations for their classmates that explain the content of the discussions.

Suggested Follow-Up

After these discussions and activities, the students will have an understanding of how geometry can be used to solve probability problems. The next lesson, Conditional probability and probability of simultaneous events, leads to a deeper consideration of the related mathematics and to acquiring new tools for solving problems, namely the ideas and formulas connected with conditional probability and probability of simultaneous events.