Probability and Geometry

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 Addressed

Grade 6

  • Statistics and Probability

    • The student demonstrates a conceptual understanding of probability and counting techniques.

Grade 7

  • Statistics and Probability

    • The student demonstrates a conceptual understanding of probability and counting techniques.

Grade 8

  • Statistics and Probability

    • The student demonstrates a conceptual understanding of probability and counting techniques.

Grade 9

  • Geometry

    • The student demonstrates an understanding of geometric relationships.
    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

  • Statistics and Probability

    • The student demonstrates a conceptual understanding of probability and counting techniques.

Grade 10

  • Geometry

    • The student demonstrates an understanding of geometric relationships.
    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

  • Statistics and Probability

    • The student demonstrates a conceptual understanding of probability and counting techniques.

Seventh Grade

  • Statistics and Probability

    • Investigate chance processes and develop, use, and evaluate probability models.

Grades 6-8

  • Data Analysis and Probability

    • Understand and apply basic concepts of probability

Grades 9-12

  • Geometry

    • Use visualization, spatial reasoning, and geometric modeling to solve problems

Geometry

  • Geometry and Measurement

    • Competency Goal 2: The learner will use geometric and algebraic properties of figures to solve problems and write proofs.

Technical Mathematics I

  • Geometry and Measurement

    • Competency Goal 2: The learner will measure and apply geometric concepts to solve problems.

Technical Mathematics II

  • Geometry and Measurement

    • Competency Goal 1: The learner will use properties of geometric figures to solve problems.

Advanced Functions and Modeling

  • Data Analysis and Probability

    • Competency Goal 1: The learner will analyze data and apply probability concepts to solve problems.

Integrated Mathematics III

  • Geometry and Measurement

    • Competency Goal 2: The learner will use properties of geometric figures to solve problems.

6th Grade

  • Measurement

    • The student will demonstrate through the mathematical processes an understanding of surface area; the perimeter and area of irregular shapes; the relationships among the circumference, diameter, and radius of a circle; the use of proportions to determine
    • The student will demonstrate through the mathematical processes an understanding of surface area; the perimeter and area of irregular shapes; the relationships among the circumference, diameter, and radius of a circle; the use of proportions to determine unit rates; and the use of scale to determine distance.

7th Grade

  • Data Analysis and Probability

    • The student will demonstrate through the mathematical processes an understanding of the relationships between two populations or samples.

8th Grade

  • Measurement

    • The student will demonstrate through the mathematical processes an understanding of the proportionality of similar figures; the necessary levels of accuracy and precision in measurement; the use of formulas to determine circumference, perimeter, area, and
    • The student will demonstrate through the mathematical processes an understanding of the proportionality of similar figures; the necessary levels of accuracy and precision in measurement; the use of formulas to determine circumference, perimeter, area, and volume; and the use of conversions within and between the U.S. Customary System and the metric system.

5th Grade

  • Probability and Statistics

    • 5.17b The student will predict the probability of outcomes of simple experiments, representing it with fractions or decimals from 0 to 1, and test the prediction

8th Grade

  • Probability and Statistics

    • 8.11 The student will analyze problem situations, including games of chance, board games, or grading scales, and make predictions, using knowledge of probability.
    • 8.11 The student will analyze problem situations, including games of chance, board games, or

6th Grade

  • Measurement

    • 6.10 The student will estimate and then determine length, weight/mass, area, and liquid volume/capacity, using standard and nonstandard units of measure.

Textbooks Aligned

Grade Six

  • How Likely Is It?

    • Investigation Three: Using Spinners to Predict Chance

6th

  • Module 8 - MATH-Thematical Mix

    • Section 4: Geometric Probability
      • Reason for Alignment: This lesson shows how probability can be determined through the use of geometry. The discussion is well constructed for teachers, with illustrations and examples. This fits well with the last module of the book.

7th

  • Module 6 - Flights of Fancy

    • Section 2: Probability
      • Reason for Alignment: The Probability and Geometry lesson shows the relationship between geometry and probability through the use of spinners, area, degrees in a circle and other ideas. While not exactly in line with the idea of the target landing as in the textbook, it does provide more practice with probability skills.

8th

  • Module 8 - MATH-Thematical Mix

    • Section 4: Geometry and Probability
      • Reason for Alignment: This lesson shows the connection between probability and geometry. This is developed through several discussions. Two different spinner games are used as activities to illustrate how geometric probability is computed. This should tie to the work in this section of the textbook.

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 for experimenting with the activities

Teacher Preparation

  • 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

The best guess arrived at after considering all the information given in a problem

experimental probability

The chances of something happening, based on repeated testing and observing results. It is the ratio of the number of times an event occurred to the number of times tested. For example, to find the experimental probability of winning a game, one must play the game many times, then divide the number of games won by the total number of games played

probability

The measure of how likely it is for an event to occur. The probability of an event is always a number between zero and 100%. The meaning (interpretation) of probability is the subject of theories of probability. However, any rule for assigning probabilities to events has to satisfy the axioms of probability

random number generators

A device used to produce a selection of numbers in a fair manner, in no particular order and with no favor being given to any numbers. Examples include dice, spinners, coins, and computer programs designed to randomly pick numbers

theoretical probability

The chances of events happening as determined by calculating results that would occur under ideal circumstances. For example, the theoretical probability of rolling a 4 on a four-sided die is 1/4 or 25%, because there is one chance in four to roll a 4, and under ideal circumstances one out of every four rolls would be a 4. Contrast with experimental probability

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 Outline

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.