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Taking a Chance

Instructor: Paul Wm. Rahmoeller

• B.S. in Ed., M.Ed.
• Jefferson Junior High School, Columbia, MO
• MSA 1985-2000

Course Description

This course will introduce scholars to the mathematics of the predictable and the unpredictable. We will learn how mathematical models are developed and used to predict outcomes in politics, contests, advertising, and science. Scholars will be actively engaged in experiments, surveys, data analysis, and games. There will be an emphasis on problem formulation and problem solving in a cooperative learning environment.

Rationale for inclusion in a program for gifted students.

Many students are not exposed to Probability and Statistics beyond the eighth grade due to curricular offerings at their schools. The study of Probability provides a venue for investigating a number of important and useful algorithms and processes that are otherwise not covered. Some topics in Probability also mirror the history of mathematics development so there are many connections that can be made.

Probability topics can often be introduced intuitively, taught by guiding students through development of their own techniques, and connected to other mathematics by applying the same formulas discovered by students to new areas. This allows the student to create their own mathematics in a way that is similar to the historical development of the same mathematics. Perhaps for the first time, students can understand what a practicing mathematician does without having to know math beyond first year Algebra.

Statistics is a field often avoided by students unless it is specifically listed as a graduation requirement for advanced degrees. Understanding statistical analysis and methods empowers people to make their own judgments and reach their own conclusions about original research, political polls, demographics, and even simple surveys. Statistics influence sports, entertainment, policy development, and many other areas that touch our lives daily, but statistics are understood by only a very small minority of our population.

Major Topics Covered

• Sample Spaces
• Classical Probability
• Empirical Probability
• Odds
• Tree diagrams
• Combinations and Permutations
• Pascal’s Triangle
• Correlation Coefficient
• Simpson’s Paradox
• Chi squared test for significance
• Problem Solving Techniques
• Series and Sequences
• Probability in Genetics
• Mathematical modeling
• Sampling Techniques
• Surveys
• Spreadsheets as a tool
• Conditional Probabilities
• Graphs and Tables
• Standard Normal Curve
• y = nekt growth/decay formula
• Null Hypothesis testing

Learning Objectives

Students will be able to...

• construct sample spaces for simple events. such as choosing one from a group. Lists, charts, tree diagrams, geometric models, and Pascal’s triangle are all used for sample spaces at some time.
• construct two-, three-, and multidimensional sample spaces for repetitions of simple events such as coin tosses and choosing one from a group.
• use a sample space to calculate or find the probability of events.
• state the difference between classical and empirical probability.
• record and gather data from surveys, experiments, or sets of observations.
• arrange data in the form of graphs, tables, or charts by hand or using a calculator or computer so that it is in readable form.
• perform the calculations necessary to perform a chi-squared test of significance for a set of data.
• read and interpret the appropriate tables or charts to reach reasonable conclusions based on statistical tests.
• explain the use of the Null Hypothesis in statistics.
• predict the probability that a set of data belongs to a particular population using statistical analysis.
• justify the use of a certain models for analyzing a set of data.
• explain the concept of random sampling and give five concerns and possible remedies for obtaining non-random samples.
• use a spreadsheet or computer program to aid in the calculation of statistical analysis and/or graphing.
• evaluate their own performance on unit-related activities.

Prerequisite knowledge needed

• Algebra I or equivalent
• Formulas for Area and Perimeter of standard figures
• Curiosity and Persistence are appreciated, taught, and encouraged

Primary source materials

• Mosteller, Kruskal, Link, Pieters, Rising. Statistics by Example, Exploring Data, (Also, Detecting Patterns, Finding Models, and Weigning Chances). (Four books) Addison Wesley Publishing Company
• Dalton, LeRoy C. Algebra in the Real World. Dale Seymour Publications
• Milton and Corbet. “Strategies in Yahtzee: An Exercise in Elementary Probability.” The Mathematics Teacher, Dec, 1982, Vol. 5, No. 9.
• Triola, Mario F. Elementary Statisitics. Menlo Parl CA. The Benjamin/Cummings Publishing Company, Inc., 1980

Supplementary materials used

• NCTM 1906 Association Drive, Reston, VA 22091
• “Probability: Quantifying Chance” in Student Math Notes, NCTM Bulletin, May 1985.
• Math Disks, NCTM May 1984.
• CURRICULUM AND EVALUATION STANDARDS FOR SCHOOL MATHEMATICS
• PATTERNS AND FUCTIONS
• DEALING WITH DATA AND CHANCE
• DATA ANALYSUS AND STATISTICS
• NCTM 1906 Association Drive, Reston, VA 22091
• NEW TOPICS FOR SECONDARY SCHOOL MATHEMATICS
• Data Analysis, 1988
• Geometric Probability, 1988
• Numerous articles from The Mathematics Teacher, including, but not limited to:
  • Hildreth, David J. “Do Baseball Positions Correspond with a Player’s Race?”, April 1996
  • Exploring Measurements (1994), Exploring Surveys and Information From Samples (1987),
  • Exploring Data (1995), and Exploring Probability (1987)
• Dale Seymour Publications - Quantitative Literacy Series.
• Jones, Rich, Thornton, and Day, Investigating Probability and Statistics Using the TI-82 Graphing Calculator. 1996 Addison-Wesley Publishing Company
• University resources used
  • University computing resources and Internet access.
  • University Reactor, Columbia.
  • Mathemtics Library, UMC.

Accomplishing the Objectives

Each of the objectives deals with a skill, mathematical idea, or a mathematical model for some situation. Most classroom activities, therefore, involve practicing a skill, learning a new skill, building some mathematical model, examining and analyzing a formula, performing an experiment and collecting data, or checking to see how closely a model fits a real situation. There are a few activities that are exploratory in nature. An emphasis is put on finding the questions a particular piece of mathematics was designed to help answer, and then recreating or exploring the mathematics in order to actually answer the proposed question.

Students usually work in groups of 2 to 4 of their own choosing and design. Students have continual access to graphing calculators and usually teach each other how to use them in new ways. Students also have the limited availability of spreadsheet programs that are linked to graphing programs.

Sample Activities

• Counting the number of candies by color in a bag of candy and using a x2 test to determine if the candy company’s mixing process was working well at the time the given bag was filled.
• Tossing coins 20 times per day by each student and recording the results on an overhead graph to get a visual representation of a binomial distribution. As the number of repetitions increases over the three weeks, students can see how the data seems to more closely approximate the theoretical probabilities. This activity is usually used as a class opener at the beginning of class or right after a break.
• Using an object (such as a small paper cup) to demonstrate that some objects do not have theoretical probabilities that are easily predicted. Students first try to predict how the cup will land when dropped, open end up, open end down, or sideways. Usually there is little agreement on a theoretical probability. After each student performs the experiment about 100 times, they are ready to discuss empirical probabilities and how they differ from theoretical probabilities.
• Choosing an object several times from a group of similar objects with and without replacement. Creating a sample space for the theoretical probabilities, and predicting what the population characteristics are without being able to see the entire population at the same time. We do this with 50 candies in a sealed container where the student can only select one candy at a time and observe it’s color. After fifty observations, students may guess at how many of each color are in their set. If they are incorrect (which the always have been) then they must do 50 more observations. At this time they may guess again at how many of each color they have in their sealed container. Students are told how many of the color numbers are correct, but not which ones are correct. They must continue this process until they correctly predict the number of each color in their container. We record the data for each student and also how many repetitions they needed before they could tell what was in their container. This gives students some practical experience with sampling with replacement.
• Roll number cubes and count the number of times each of the six sides lands face up. We then construct sample spaces for 1, 2, 3, 4, 5, and 6 dimensions that model the probabilities for number cubes. We use several different mathematical models for these sample spaces, developing our own shortcuts and algorithms as we proceed. Eventually we use patterns, calculator functions, and spreadsheets to do our number crunching.

We “play” with Pascal’s triangle in 2 dimensions, and begin a development of 3 and 4 dimensional analogs to Pascal’s triangle. We use patterns, calculator functions, algebra, and geometric models to help us.

We learn a dance which illustrates the differences between combinations and permutations. This helps in our work with several mathematical models.