Probability and Geometry (elementary)

Abstract

This lesson introduces students to the idea that probability and geometry are sometimes linked. Students will practice what they already know about geometry to learn more about probability.

Objectives

Upon completion of this lesson, students will:

  • have practiced calculating probability
  • understand how geometry can help solve probability problems

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 3

  • Statistics, Data Analysis, and Probability

    • 1.0 Students conduct simple probability experiments by determining the number of possible outcomes and make simple predictions

Grade 4

  • Statistics, Data Analysis, and Probability

    • 2.0 Students make predictions for simple probability situations

Seventh Grade

  • Statistics and Probability

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

Grades 3-5

  • Data Analysis and Probability

    • Understand and apply basic concepts of probability

  • Geometry

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

5th Grade

  • Data Analysis and Probability

    • The student will demonstrate through the mathematical processes an understanding of investigation design, the effect of data-collection methods on a data set, the interpretation and application of the measures of central tendency, and the application of basic concepts of probability.

4th Grade

  • Data Analysis and Probability

    • Standard 4-6: The student will demonstrate through the mathematical processes an understanding of the impact of data-collection methods, the appropriate graph for categorical or numerical data, and the analysis of possible outcomes for a simple event.

5th Grade

  • Data Analysis & Probability

    • The student will understand and apply basic statistical and probability concepts in order to organize and analyze data and to make predictions and conjectures.

  • Geometry

    • The student will develop an understanding of geometric concepts and relationships as the basis for geometric modeling and reasoning to solve problems involving one-, two-, and three-dimensional figures.

4th Grade

  • Data Analysis & Probability

    • The student will understand and apply basic statistical and probability concepts in order to organize and analyze data and to make predictions and conjectures.

  • Geometry

    • The student will develop an understanding of geometric concepts and relationships as the basis for geometric modeling and reasoning to solve problems involving one-, two-, and three-dimensional figures.

Grade 5

  • Probability and Statistics

    • 12. The student describes and predicts the results of a probability experiment.

3rd Grade

  • Probability and Statistics

    • 3.23 The student will investigate and describe the concept of probability as chance and list possible results of a given situation.

4th Grade

  • Probability and Statistics

    • 4.19.b The student will determine the probability of a given simple event, using concrete materials.

5th Grade

  • Probability and Statistics

    • 5.18 The student will, given a problem situation, collect, organize, and display a set of numerical data in a variety of forms, using bar graphs, stem-and-leaf plots, and line graphs, to draw conclusions and make predictions.

Student Prerequisites

  • Technological: Students must be able to:
    • perform basic mouse manipulations such as point, click and drag
    • use a browser for experimenting with the activities
  • Geometry: Students must be able to:
    • understand the basics of angles
  • Arithmetic: Students must be able to:
    • work with fractions
    • understand the relationship between fractions and percentages

Teacher Preparation

  • Access to a browser
  • Pencil and paper
  • 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.

Key Terms

estimate

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

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

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 and its relationship to geometry.
    • 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

    • Lead a discussion about how probability and geometry are related.

  4. Guided Practice

    • Open the Spinner Game in your browser and explain how to work the applet.
    • Ask students which color you're more likely to spin.
    • Ask students how you could make one color more likely.
    • Open the Adjustable Spinner in your browser and explain how to work the applet.

  5. Independent Practice

  6. Closure

    • Discuss what the students discovered through the exploration questions.
    • Ask students how this applies to dice.

Alternate Outline

If only one computer is available, this lesson can be rearranged in the following ways:

  • Have students construct spinners with different sized sections 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.
  • Give extra time with the applets for students who are having trouble understanding.

Suggested Follow-Up

Advanced students might follow-up this lesson with the Probability and Geometry Lesson.