NCTM Series Navigating through Navigating through Probability in Grades 9-12 AATM State Conference September 27, 2008 Shannon Guerrero Asst Professor, Math Education Northern Arizona University AATM Newsletter Editor

NCTM Navigation Series  NCTM published series that offers activities, ideas & materials to roadmap implementation of Principles & Standards  Chapters organized around major concepts of particular NCTM content strand being targeted

NCTM Navigation Chapters  Each chapter begins with discussion of ensuing content & activities – provides big picture of NCTM strand & associated activities  Each activity consists of 3 elements – Engage, Explore, & Extend  Each component contains suggested materials, questions to pose, & teaching tips

Contents of the CD-ROM  Introduction  Table of Standards and Expectations, Number and Operations, Pre-K-12  Applets  Binomial Distribution Simulator  Geometric Distribution Simulator  Random Number Generator  Adjustable Spinner  Blackline Masters and Templates  Readings from Publications of the National Council of Teachers of Mathematics and Other Sources

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

Why study probability? Chance is all around us – we do not live in a totally deterministic world. Chance is all around us – we do not live in a totally deterministic world. People must make decisions in the face of uncertainty. People must make decisions in the face of uncertainty. Probability is indispensable for analyzing data; data are indispensable for estimating probabilities. Probability is indispensable for analyzing data; data are indispensable for estimating probabilities.

“Big Ideas” of Probability with High School Students  Probability as long-run relative frequency  Determining probability through an analysis of outcomes  Independence and conditional probability  Moving from sample spaces to probability distributions  Expected value as “average” behavior in the long run

Questions to Ponder How could we begin to answer?  What is the chance of an auto accident occurring at a complicated intersection?  What is the chance that a boy will develop hair loss if his father did?  What is the chance that a baseball player will get two hits in three consecutive batting attempts?  What is the chance that the stock market will go up three days in a row?

Chapter 1 – Probability as Long-Run Relative Frequency  Simulation can be useful tool for estimating probabilities  The cumulative stabilization of the relative frequency of an outcome in a large number of trials makes it a good estimate of the probability of the outcome.

Chapter Two – Sample Spaces  Students should be able to describe sample spaces such as the set of possible outcomes when four coins are tossed.  High school students should learn to identify mutually exclusive, joint, and conditional events.  All students should understand how to compute the probability of a compound event.

Chapter 3 – Independence and Conditional Probabilities  All students should understand the concepts of conditional probability and independent events.  Two outcomes are independent if the occurrence of one does no change the probability of the other.  Students can use the sample space to answer conditional probability questions.

Chapter 4 – Probability Models and Distributions  Students can gather simulated data to investigate the probability of an event.  Students can examine the declining probabilities of strings of outcomes from repeated trials.  All of the activities in this chapter direct students to run multiple trials with random numbers.

Chapter 5 – Expected Value  Expected value – the average value in the long run  All students should compute and interpret the expected value of random variables in simple cases.  Life insurance companies use expected- value calculations to determine premiums that they can expect to attract customers while ensuring a profit.