# MiddleSchoolPortal/Whats The Chance: Concepts of Probability

<keywords content="middle school mathematics games prediction outcomes opportunity theory" />

<metadescription content="This free, standards-based, online publication, developed for middle school mathematics teachers, supports teaching probability by describing and linking to inquiry-based lessons and activities." />

### What's the Chance? Probability Concepts - Introduction

Probability offers one of those rare intersections where classroom mathematics crosses middle school experience. The problem scenarios investigated at this level often start with a game—as did theoretical probability itself. Students find that many of the phenomena they encounter in game playing have predictable outcomes. To reach that conclusion, they need opportunities to consider data they generate, noting patterns that emerge and comparing their results with those predicted by theory.

## Contents

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The activities, lesson plans, and project ideas selected for this publication offer such opportunities, not only in the context of games, since other situations can be analyzed in the light of probability. Even though several resources include, or even rely on, an online simulation, they can usually be adapted well to classrooms without computer access. The selected resources include background information for your own review of the concepts. The final section considers the place of probability in the NCTM Standards.

We hope these resources offer worthwhile opportunities for your class to consider, "What's the chance that . . . ?"

### Background Information for Teachers

Until fairly recently, probability was not considered a middle school topic. You may feel that you have not had as much practice in teaching concepts of probability as you have in teaching other math topics. These professional resources, designed for teachers, offer review of the material as well as ideas for the classroom.

Probability In this online workshop, you can investigate probability by exploring games of chance, probability models, tree diagrams, random events, and even the binomial probability model. Included in this session are video clips of teachers as they work through the problems, hands-on activities, and an online simulation for exploring these abstract concepts. Instruction on each idea is clearly set out through hands-on problems as well as diagrams and text explanation. Practice problems with solutions are included. You will also find links to online articles written for teachers. This is one session in the free online course: Learning Math: Data Analysis, Statistics, and Probability.

### Projects

These projects complement the resources found in Activities by involving students in deeper thinking about probability over a longer period of time. Each offers background information for the teacher, ideas for classroom discussion, and classroom materials needed (or how to make them). Most important, they draw students into scenarios that pose intriguing questions: In a family of five, is the chance of having all girls greater than the chance of having, say, three girls and two boys? Which slot on a Plinko board will give the highest payoff over time? And, what does probability have to do with winning games?

The Smithville Families This multi-faceted unit easily converts to a class project. It begins with the number sequences of Pascal's triangle and goes on to discovery of their relationship to theoretical probability. Students first generate several rows of the famous triangle, noting its internal patterns. Then they consider questions relating to the Smithville families, which each has five children: What is the total number of possible girl/boy combinations in a five child family? And what is the theoretical probability of each combination? And how do these questions relate to Pascal’s triangle?

Plinko : probability from a game show This is another lesson plan that could profitably expand into a project. Besides all directions and needed diagrams, the resources include an online simulation of Plinko, a game of chance, as well as directions on how to make your own Plinko board. Playing the game, students gather data for investigation into experimental probability. Discussions also open to theoretical probability, including tree diagrams and counting paths that give insight into why game shows arrange the payoffs the way they do.

Data Management: A Look at Leisure Activities Among the lessons in this unit on analyzing data are lessons that require students to consider probability in the context of game playing. They must come up with game strategies based on their calculations of experimental and theoretical probabilities. The entire unit could make up a class project that covers many topics of data gathering, display, and analysis.

### Activities for the Classroom

These activities involve students in solving intriguing problems, each in the context of a basic probability concept. Some use virtual simulations in which students gather data from a thousand throws of a die or tosses of a coin in mere moments, allowing them to concentrate on the essentials of the problem. All are adaptable to the classroom setting, with or without a computer at hand.

Simulations and online interactive activities

Box model (grades 6-8) An extremely versatile applet! In a box are shown the numbers 1-15, each a different color. Users can select which of these numbers, and how many of each, to add to the pot. As many random draws as 10000 can be made from the pot in just a few seconds. A chart shows the results of the simulated drawing with replacement from the numbers in the pot, as well as the theoretical probability of the drawing. This applet can be used to simulate flipping a coin or tossing a die or picking colored marbles from a hat.

Understanding Experimental Probability Students can choose one of six available spinners, figure the theoretical probability of the needle landing on each color of the spinner, then spin it hundreds of times in a second to compare the experimental probability. Or they can work with two regular 6-sided number cubes or even design their own number cubes, and again experiment with the "what should happen" and the "what actually happened."

Adjustable Spinner Another experiment with theoretical and experimental probability! Students create an interactive spinner with one to twelve sectors; the area of each segment is shown as a percentage in a table. As students spin, they see how many times the needle actually landed on each segment, also shown as a percentage, and compare the two probabilities.

Rocket launch probability The applet simulates the launch of a three-stage rocket; a successful launch requires that all three stages pass tests before takeoff. Students set the probability of passing at each stage on a scale from 0 to 100 percent. After each launch attempt, success or failure is reported, and the overall success rate is given as a percent and as the cumulative number of successful and failed launches. This applet is unique in that it can be used to observe a multistage event, each stage with a different probability of success.

Racing Game with Two Dice Two [or more] players each roll a die, and the lucky player moves one step to the finish. Students can decide which rolls win, the length of the race, and how many steps the winner of the roll takes to the finish line. They can also see a second how many times each player would win out of 1000 games. Playing the game is easy; the trick is explaining why one player, over the long haul, wins so often.

Buffon's Needle Buffon's Needle involves dropping a needle on a lined sheet of paper, then counting the number of times the needle crosses a line. Remarkably, the probability of the needle landing on a line relates directly to the value of pi. Complete explanation, included for the teacher, involves trigonometry, but even without this in-depth explanation, this is an "Ah ha!" moment for your students!

Open-ended questions and hands-on activities

Bowl 'em over : does he have a chance? The initial question concerns averages: Given the scores of Helix’s first five games, what does he have to bowl in his last game to win the tournament? But then the question turns to the probability of his scoring that many points in the last game. Explained here are two suggested solutions, each from a different angle.

How could I send the check and not pay the bill? If you’re not paying attention to what you’re doing, what is the probability that you will put checks into the correct envelopes? Start with checks a, b, and c that have to go into envelopes A, B, and C. The solution is shown using three different approaches: a table, a tree diagram, and geometry.

Combination locks: I forgot the combination! How many combinations will I have to try? In this activity, students find the number of possible combinations for a lock. The combination uses three numbers, each from 0-39. Students first consider a simpler problem, a combination lock that uses only the numbers 1 to 3. This is thoroughly explained and illustrated with a tree diagram. The solution to the easier problem is then generalized to finding the number of possible arrangements for any combination lock. . Further challenges ask students to consider the possible combinations for other types of locks and the total number of phone numbers possible for an area code.

I win!: she always wins, it's not fair! The activity begins with playing a dice game but quickly moves to questioning whether or not the game is fair. The page offers a solution—and a challenge to students to change the game rules to make it fair!

Capture-Recapture: How Many Fish in the Pond? How do those wildlife experts estimate how many fish are in a pond? They use a method called capture-recapture, a statistical tool that students use in this activity to estimate the total number of fish in a pond, given the numbers of fish initially tagged and released, the tagged fish recaptured, and the total number of recaptured fish. This tool allows wildlife experts to make predictions, one objective of probability theory, about population size and growth.

### Lesson Plans

Probability theory began with a question related to a game—and so do the first three of these lessons. What probability means and how it affects outcomes can seem inconsequential to middle school students, but when they see its connection to real decisions in game playing, probability takes on real meaning. In these lessons, games serve to both motivate learning and connect the mathematics to actual experience.

The last of these resources is a set of connected lessons that cover the basic elements of probability for the middle school level. The lessons are designed to be used in a tutoring situation but can be valuable for the classroom as well.

Rescue Mission Game This two-day lesson introduces probability as well as forces used in flight. To guide a helicopter to stranded hikers on a mountaintop, students learn about: lift, drag, thrust, and gravity. Various spinners used in the game differ in the areas allotted to each of these forces. Which spinner should be used on the next turn? Before playing the game, students conduct a probability experiment with the spinners, tallying the results in tables and graphs. For each turn, they select the spinner with the greatest probability of helping them reach the lost hikers. An adventurous introduction to the basics of probability!

The Game of Skunk A practical use of mathematics is in decision making. Through Skunk, a game not only played but analyzed in the lesson, students consider "choice versus chance" and make decisions accordingly. Students roll dice to accumulate points by throwing several "good" rolls in a row, but they must decide when to stop before a "bad" roll wipes out their points. Probability comes into play as they try to create winning strategies.

Sticks and Stones Students play a game based on the Apache game "Throw Sticks," which was played at multi-nation celebrations. This is another practical application of the concepts of probability. Students collect data, examine the probable outcome of various moves, and use basic ideas of expected value to decide on game strategy.

Statistics and probability. Grades 6-8 Throughout this unit of seven lessons, students use hands-on learning activities to explore statistics and probability. Designed for mentoring situations at middle school level, the lessons focus on the essentials. For example, the probability activities introduce fairness in games and the computation of probability. Lessons include teaching guidelines as well as all handouts.

### SMARTR: Virtual Learning Experiences for Students

Visit our student site SMARTR to find related virtual learning experiences for your students! The SMARTR learning experiences were designed both for and by middle school aged students. Students from around the country participated in every stage of SMARTR’s development and each of the learning experiences includes multimedia content including videos, simulations, games and virtual activities. Visit the virtual learning experience on Probabiity.

### NCTM Standards

Data Analysis and Probability is one of the five content standards developed by the National Council of Teachers of Mathematics. NCTM advocates increased curricular emphasis on this subject in response to the staggering amount of data that reaches all of us in everyday life. In teaching students to deal with data, the Standards document observes that "ideas from probability serve as a foundation to the collection, description, and interpretation of data" (NCTM, p. 51).

To develop probabilistic thinking at the middle grades level, it is recommended that students have numerous opportunities to make predictions and test their conjectures. Computer simulations can generate large samples of data quickly (10,000 throws of two dice in only a few seconds), which allows students to concentrate on analysis of the data rather than be distracted by its collection. The NCTM Standards note, however, that students need to develop their thinking by carrying out actual experiments as well. In virtual as well as hands-on experiments, comparing prediction with actual outcomes can provoke the questions and discussion needed to explore inconsistencies and student misconceptions. We believe this collection of resources will both provoke such exploration and develop your students’ ideas of probability.

For a detailed outline of the expectations outlined by the National Council for the middle grades level, see http://standards.nctm.org/document/chapter6/data.htm.