NCTM SeriesNCTM Series NavigatingNavigating throughthrough

Probability Probability inin Grades 9-12 Grades 9-12

AATM State ConferenceAATM State Conference

September 27, 2008September 27, 2008

Shannon GuerreroShannon Guerrero

Asst Professor, Math EducationAsst Professor, Math Education

Northern Arizona UniversityNorthern Arizona University

AATM Newsletter EditorAATM Newsletter Editor

Shannon.Guerrero@nau.eduShannon.Guerrero@nau.edu

NCTM Navigation SeriesNCTM Navigation Series

NCTM published series that offers NCTM published series that offers activities, ideas & materials to roadmap activities, ideas & materials to roadmap implementation of Principles & Standardsimplementation of Principles & Standards

Chapters organized around major Chapters organized around major concepts of particular NCTM content concepts of particular NCTM content strand being targetedstrand being targeted

NCTM Navigation ChaptersNCTM Navigation Chapters

Each chapter begins with discussion of Each chapter begins with discussion of ensuing content & activities – provides big ensuing content & activities – provides big picture of NCTM strand & associated picture of NCTM strand & associated activitiesactivities

Each activity consists of 3 elements – Each activity consists of 3 elements – Engage, Explore, & ExtendEngage, Explore, & Extend

Each component contains suggested Each component contains suggested materials, questions to pose, & teaching materials, questions to pose, & teaching tipstips

Contents of the Contents of the CD-ROMCD-ROM IntroductionIntroduction Table of Standards and Expectations, Number and Table of Standards and Expectations, Number and

Operations, Pre-K-12Operations, Pre-K-12 AppletsApplets

Binomial Distribution SimulatorBinomial Distribution Simulator Geometric Distribution SimulatorGeometric Distribution Simulator Random Number GeneratorRandom Number Generator Adjustable SpinnerAdjustable Spinner

Blackline Masters and TemplatesBlackline Masters and Templates Readings from Publications of the National Council Readings from Publications of the National Council

of Teachers of Mathematics and Other Sourcesof Teachers of Mathematics and Other Sources

Why Study Probability?Why Study Probability?

What are the “big idea” of What are the “big idea” of probability for 9-12 grade students?probability for 9-12 grade students?

Why study probability?Why study probability?

• Chance is all around us – we do not live in Chance is all around us – we do not live in a totally deterministic world.a totally deterministic world.

• People must make decisions in the face of People must make decisions in the face of uncertainty.uncertainty.

• Probability is indispensable for analyzing Probability is indispensable for analyzing data; data are indispensable for estimating data; data are indispensable for estimating probabilities.probabilities.

““Big Ideas” of Probability with High Big Ideas” of Probability with High School StudentsSchool Students

Probability as long-run relative frequencyProbability as long-run relative frequency Determining probability through an Determining probability through an

analysis of outcomesanalysis of outcomes Independence and conditional probabilityIndependence and conditional probability Moving from sample spaces to probability Moving from sample spaces to probability

distributionsdistributions Expected value as “average” behavior in Expected value as “average” behavior in

the long runthe long run

Questions to PonderQuestions to PonderHow could we begin to answer?How could we begin to answer?

What is the chance of an auto accident occurring at a What is the chance of an auto accident occurring at a complicated intersection?complicated intersection?

What is the chance that a boy will develop hair loss if What is the chance that a boy will develop hair loss if his father did?his father did?

What is the chance that a baseball player will get two What is the chance that a baseball player will get two hits in three consecutive batting attempts?hits in three consecutive batting attempts?

What is the chance that the stock market will go up What is the chance that the stock market will go up three days in a row?three days in a row?

Chapter 1 – Probability as Long-Run Chapter 1 – Probability as Long-Run Relative FrequencyRelative Frequency

Simulation can be useful tool for estimating Simulation can be useful tool for estimating probabilitiesprobabilities

The cumulative stabilization of the relative The cumulative stabilization of the relative frequency of an outcome in a large number frequency of an outcome in a large number of trials makes it a good estimate of the of trials makes it a good estimate of the probability of the outcome.probability of the outcome.

Chapter Two – Sample Spaces Chapter Two – Sample Spaces Students should be able to describe sample Students should be able to describe sample

spaces such as the set of possible outcomes spaces such as the set of possible outcomes when four coins are tossed.when four coins are tossed.

High school students should learn to identify High school students should learn to identify mutually exclusive, joint, and conditional mutually exclusive, joint, and conditional events.events.

All students should understand how to All students should understand how to compute the probability of a compound event.compute the probability of a compound event.

Chapter 3 – Independence and Chapter 3 – Independence and Conditional ProbabilitiesConditional Probabilities

All students should understand the All students should understand the concepts of conditional probability and concepts of conditional probability and independent events.independent events.

Two outcomes are independent if the Two outcomes are independent if the occurrence of one does no change the occurrence of one does no change the probability of the other.probability of the other.

Students can use the sample space to Students can use the sample space to answer conditional probability questions.answer conditional probability questions.

Chapter 4 – Probability Models and Chapter 4 – Probability Models and Distributions Distributions

Students can gather simulated data to Students can gather simulated data to investigate the probability of an event.investigate the probability of an event.

Students can examine the declining Students can examine the declining probabilities of strings of outcomes from probabilities of strings of outcomes from repeated trials.repeated trials.

All of the activities in this chapter direct All of the activities in this chapter direct students to run multiple trials with random students to run multiple trials with random numbers.numbers.

Chapter 5 – Expected ValueChapter 5 – Expected Value

Expected value – the average value in the Expected value – the average value in the long runlong run

All students should compute and interpret All students should compute and interpret the expected value of random variables in the expected value of random variables in simple cases.simple cases.

Life insurance companies use expected-Life insurance companies use expected-value calculations to determine premiums value calculations to determine premiums that they can expect to attract customers that they can expect to attract customers while ensuring a profit.while ensuring a profit.