Abstract

The activity and two discussions of this lesson connect probability and geometry. The discussion on polyhedra 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

Activities

• Spinner
• Adjustable Spinner

Standards

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

• Probability
• Geometry

Key Terms

• sector
• degrees
• right angle
• estimate

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
• Spinner and Adjustable Spinner require either computer access or a set of materials for building spinners for each group of students.
• The From Geometry to Probability discussion refers to protractors for measuring angles, so each group of students should have a protractor.

Lesson Outline

1. The Spinner and the Adjustable Spinner activities 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.

2. The Probability and Geometry discussion shows that sometimes knowledge of geometry is required to answer probability questions.

   Using spinners, physical or virtual, from the Spinner and the Adjustable Spinner activities, 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?

3. The polyhedra discussion connects probability and geometry through construction of dice with various numbers of sides.

   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: equilateral triangles, 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.

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 and the Adjustable Spinner activities 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.