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Probability

Summary

Students will begin to understand the concept of probability through hands on activities.


Materials

Attachments

Websites

Invitation to Learn

  • Paper
  • Container

Flip the Coin Activity:

Bag of Colors

Probability Posters

  • Colored chips
  • Dice
  • Colored marbles
  • Colored centimeter cubes
  • Small circles or tiles (numbered 1-20)
  • Spinners
  • Probability Recording Sheet
  • Poster board / chart paper

Additional Resources

Books

Probability, by Sarah Jane Brian; ISBN 0590373676


Background for Teachers

Students need to understand the correct terms and vocabulary while discussing probability. It is important for students to learn to distinguish between theoretical and experimental probability. Students can also be introduced to the mathematical formulas.

Theoretical probability:

  • The numerical measure of the likelihood that an event will happen or the ratio of the number of ways the event can occur to the total number of possibilities.
  • It is the fraction of times we expect an event to occur if the same experiment is repeated over and over.
  • It is the represented by the fraction:
    Number of ways the event can occur
    Total number of possible outcomes
  • Theoretical probability does not change.

Example:
What is the probability of getting a number less than 3 when tossing an ordinary dice? There are six possible outcomes: 1, 2, 3, 4, 5, or 6; all of which are equally likely to occur. Two of these, 1 and 2, are less than 3; so the theoretical probability of getting a number less than 3 is: 2/6 = 1/3.

Experimental probability:

  • The numerical measure of what actually happens in an experiment.
  • It is the fraction of times an event actually occurs when the same experiment is repeated over and over.
  • It is represented by the fraction:
    Number of actual outcomes
    Total number of possible outcomes
  • The experimental probability may vary from the theoretical probability, but the more times the experiment is repeated, the closer the experimental probability approaches the theoretical probability.


Instructional Procedures

Invitation to Learn

Provide four slips of paper for each student. Ask them to write their name on a paper each time they can answer "yes" to the following questions:

  • Do you have black hair?
  • Are you an only child in your family?
  • Is your birthday in January or July?
  • Is there the letter "w" found in your first, middle, or last name?

Have students place the pieces of paper that have their names on them in a container.

Ask students to predict whether they think their name will be chosen. Draw one slip of paper out of the container. Compare students' predictions with the actual results. Tell students that today they will learn how to use mathematics to make better predictions.

Instructional Procedures

Flip the Coin Activity:

Conduct the following activity as a class:

  • Trial 1: Hold up a coin and ask the students: "if I flip this coin one time, how many possible outcomes are there?" (2: heads or tails)
  • Trial 2: Now ask: If I flip it ten times how many times would you predict that I would get heads? (1/2 of 10 or 5 times.) Pass one coin and paper to record to each student and direct them to flip the coin ten times and record the results. Ask: Did your outcomes match your prediction? Collect samplings from several students and record on board or overhead chart, pointing out that there was some variance.
  • Trial 3: Have the students predict how many times they will get heads if they flip the coin 30 times. (1/2 of 30 or 15 times) Have them flip the coin 30 times and record the results. Ask: Did your outcomes match your prediction? Again collect samplings from several students and record on board or overhead chart.
  • Trial 4: Have the students repeat the experiment, this time flipping the coin 100 times. Again take class samplings and record.

    Class discussion:

  • Through guided questioning, lead students to an understanding of the difference between what they predicted would occur, Theoretical Probability, and what actually occurred, Experimental Probability, then place words on the board.
  • Also discuss the ways they used to record their results. Again through guided questioning, help students to determine the best ways to record results (e.g., tally marks, T-charts, boxes, or columns).Be sure to use vocabulary such as: event, likely, unlikely, possible, impossible, outcomes, theoretical probability, and experimental probability during discussion.
  • Introduce the Theoretical Probability formula:
    Number of ways the event can occur
    Total number of possible outcomes
     
    and the Experimental Probability formula:
    Number of actual outcomes
    Total number of possible outcomes
  • Lead the discussion to an understanding of the idea that the experimental probability may vary from the theoretical probability, but the more times the experiment is repeated, the closer the experimental probability approaches the theoretical probability. Use the classes total results to illustrate this concept.

Bag of Colors

  1. Put students into cooperative learning groups with no more than four to a group
  2. Give each group a bag of tiles and a recording sheet for each student
  3. With the class, go through the three steps in writing a Theoretical Probability, found at the top of the Bag of Colors Recording Sheet.<
    • Step 1: Count the number of red tiles.
    • Step 2: Count all of the tiles.
    • Step 3: Write a fraction-Theoretical Probability.
    • Allow each group time to write the Theoretical Probability fraction for the remaining three colors. Do a quick check to make sure they are correct.
  4. On their own, each group will fill out the chart and conduct their experiment.

Probability Posters

A collection of various manipulatives to be used in conducting experiments with probability such as:

  1. Put students into cooperative learning groups with no more than four to a group.
  2. Each group will be given a different set of manipulatives to conduct probability experiments.
  3. Each group will then prepare a short visual presentation of their experiment.
  4. Give them time to conduct experiments, record results, and prepare presentation.


Extensions

  • Put manipulatives from Probability Station in a center and have students continue conducting probability experiments with materials that they didn't use before. Use Probability Recording Sheet.
  • In a center, provide manipulatives for students to create new probability experiments and share with the class.

    Family Connections

  • Share experiments with family.
  • Look for ways at home where probability can be used.


Assessment Plan

  • Completion of Bag of Colors and Probability Recording Sheet.
  • Group presentation of poster or chart displaying results and findings using probability journal.
  • Write a journal entry about what they have learned about theoretical and experimental probability.


    Bibliography

    Rivero, V. (2006) let technology be your guide. American school board journal, November, p52-53.

    The author gives seven tips for integrating tools of technology to help bolster students' knowledge in math and science education in the classrooms and schools.

    Blessman, J., Myszczak, B. (2001). Mathematics vocabulary and its effect on student comprehension. ERIC Source (ED455112). Retrieved January 12, 2007, from http:// www.eric.ed.gov.

    In this action research project, interventions were used for improving fifth grade students' comprehension of mathematical vocabulary. The following were used: math journals, student-created math dictionaries, children's literature to introduce and reinforce mathematical concepts, graphic organizers, visual aids, and written explanations of open-ended word problems. These interventions resulted in an increase in comprehension and use of mathematical vocabulary in math performance and in communication of mathematical ideas.


Created: 07/16/2007
Updated: 02/01/2018
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