Objectives

Probability

Given an experiment, define the probability of an event as the proportion of time the event is expected to occur as the experiment is repeated over and over.
Identify the outcomes of an experiment and represent them in a set S (outcome space).
Identify events as subsets of the outcome space.
Compute the probability of an event by computing the proportion of outcomes that result in the event's occurrence.
Explain that the relative frequency of an event calculated from an experiment does not always equal the probability the event will occur.

Relative Frequency

Simulate an experiment involving chance, interpret events, and obtain estimates of probability
Replicate an experiment and determine relative frequencies to obtain better and better approximations of the probability of an event.
Represent a relative frequency as a fraction, decimal, or percentage.
Explain that when an experiment is conducted multiple times, the relative frequency of an event is the proportion of time the event occurred.
Explain that the relative frequency of an event calculated from previous experiments does not guarantee what the relative frequency of the event will be in future experiments.
State that when an experiment is repeated more and more times, the relative frequency tends toward a given value.

Teaching Plan

DAY 1 (50-60 Minutes)

Introction to Probability (15-20 minutes)

The purpose of this section is to introduce students to probability and chance using intuitive examples that enable the students to reflect on the nature of probability.

Probability: Outcome Space and Events (10-15 minutes)

The purpose of this section is to help students understand the definitions of outcome space and events.

Probability: Equally Likely Outcomes (20-30 minutes)

The purpose of this section is to help students find the probability of an event in an experiment resulting in equally likely outcomes by (1) listing or otherwise quantifying the outcome space, (2) identifying the event as a subset of the outcome space, and (3) calculating the proportion of outcomes that result in the event's occurrence.

DAY 2 (50-60 Minutes)

Probability: Unequally Likely Outcomes (20 minutes)

The purpose of this section is to help students find or estimate the probability of an event in an experiment resulting in unequally likely outcomes.

Relative Frequency: The Long Run (30 minutes)

The purpose of this section is to (1) reinforce the long-run definition of probability by having students simulate chance processes over and over, observing how relative frequency approaches probability; (2) help students calculate the relative frequency of an event given the results of an experiment, and (3) help students understand the nature of chance and its relation to the long-run definition of probability.

DAY 3 (50-60 Minutes)

Experimenting to Find Probability (15-20 minutes)

The purpose of this section is to help students simulate experiments to estimate the probability of an event when the outcome space of the experiment is not entirely known.

Relative Frequency: Practice (15-20 minutes)

The purpose of this section is to present students with problems that can be solved using the principles addressed in the relative frequency sections. Some problems require students to simulate experiments by properly configuring a mathlet and using it to replicate the chance process described in the problem.

Have students complete the Practice section. Some of the activities in this section require students to configure mathlets to represent and simulate an experiment. Stand ready to assist students who struggle in determining how to properly set up the applets. (15-20 minutes)
When students have completed the eModule, have them complete the Quiz to assess their understanding of the concepts presented. (10-15 minutes)

Assessment

Use class discussions to gauge the progress of the class and to identify common misconceptions.
Walk around and observe individual students while they go through the lesson to identify individual progress and challenges.
Use the quiz for the purpose of summative assessment and grading.
If you identify problems in a particular assessment item or lesson activity, please email your finding to enlvm@math.usu.edu.

Credits

Lesson Design	Bob Heal, Joel Duffin, Neal Legler, Jeff LeMieux, Liz Parish
Web Development	Neal Legler
Applet Development	NLVM Team
Images	Denise Chandler, Neal Legler

Correlation to Standards

Correlation to NCTM Standards

Grades 6-8: Use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations.
Grades 6-8: Compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models.

Correlation to Utah Standards

Grade 6: V.2.a: Write the results of a probability experiment as a fraction, ratio, or percent between zero and one.
Grade 6: V.2.b: Compare experimental results with anticipated results.
Grade 7: 5.2.1: Conduct experiments to approximate the probability of simple events.
Grade 7: 5.2.2: Compare individual, small group, and large group results of an experiment.
Grade 7: 5.2.3: Write the results of a probability experiment as a ratio, decimal, or percent.
Grade 7: 5.2.5: Compute simple probabilities using methods such as lists, tree diagrams, or area models.
Prealgebra: 5.2.1: Conduct experiments to approximate the probability of simple events.
Prealgebra: 5.2.2: Recognize that results of an experiment more closely approximate the actual or theoretical probability of an event as the number of trials increases.
Prealgebra: 5.2.3: Derive the probability of an event mathematically, e.g., building a table or tree diagram, creating an area model, making a list, or using the basic counting principle.
Prealgebra: 5.2.4: Represent the probability of an event as a fraction, percent, ratio or decimal.

Correlation to Washington Standards

Grade 6: 1.4.2: Understand various ways to determine outcomes of events or situations.
Grade 7: 1.4.2: Understand and apply the procedures for determing the probabilities of multiple trials.
Grade 8: 1.4.2: Understand and apply the procedures for comparing theoretical probability and emperical results for independent or compound events.

Correlation to Maine Standards

Grades 5-8: 1: Find the probability of simple events and make predictions by applying the theories of probability.
Grades 5-8: 2: Explain the idea that probability can be represented as a fraction between and including zero and one.
Grades 5-8: 3: Use simulations to estimate probabilities.
Grades 5-8: 4: Find all possible combinations and arrangements involving a limited number of variables.

Using a computer projector, demonstrate how to go to the eNLVM website, go to your school and class, and login. (10 minutes). Go to the Probability and Relative Frequency eModule and, if necessary, demonstrate for students how to move between lessons and activities.
Have students login to the eNLVM, go to the Probability and Relative Frequency eModule, and complete the lesson entitled Introduction to Probability. (5-10 minutes)
You may wish to reinforce the concepts introduced in this section by giving additional examples of chance processes (i.e. shuffling a deck of cards, name drawings, lotteries, etc.). (Optional 5 minutes)

Have students complete the lesson entitled Outcome Space and Events. (20-30 minutes)
You may wish to reinforce the sections taught in this section through class discussion.

When students have finished the Outcome Space and Events section, have them work through the sections entitled Equally Likely Outcomes . (20-30 minutes)
Use any remaining class time to answer questions and work through some of the practice activities as a class. The Practice: Rolling Dice activity may be an especially good one to work through together. (5-10 minutes)

Have students and complete the section entitled The Long Run. Students may need additional support with this section as they learn to use the mathlets to simulate experiments. (20 Minutes)
Use the remaining class time to answer questions and simulate some of the experiments as a class. (5-10 minutes)

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