Playing, Analyzing, and Designing Games to Explore Probability
Target Audience:
This Project Based Instruction unit is intended for a regular seventh grade mathematics class.
Project Description:
In this three to four week project, students will explore the concepts of probability by first playing and analyzing simple games of chance. Students will start by thinking about games of chance vs. games of skill. In the first week, students will design a spinner and then invent a simple game to go with their spinner for their first taste of designing a game. They will proceed to find the experimental and theoretical probabilities and develop ideas about whether or not a game is “fair”. As the students analyze commercial games of chance, they will be asked to relate what they learn to the evolving design of their own game. See calendar and final product description / rubric at the end of this document for a more detailed description.
Driving Question:
How can students use probability strategies to work in groups of three or four to design and evaluate games of chance in the process of organizing activities for a Game Day for their school?
Overall Goals:
This Project Based Instruction project is designed to immerse students in authentic probability situations. Students will perform typical exercises used to teach probability and to develop the required mathematical skills used in the study of probability, but they will also participate in several activities throughout the project designed to connect to their individual experiences in order to extend the meaning and understanding of probability concepts.
Project Objectives:
Objective 1: Throughout the unit, students will learn to determine and interpret experimental and theoretical probabilities of real-life events (mostly playing games) using physical and mathematical models to represent the probabilities.
Objective 2: As new concepts are explored and skills are acquired, students will be challenged to design several games of chance by employing these different concepts and skills. Students will utilize mathematical concepts of probability in the design and testing of their games that will help them determine “fairness”, test the probabilities of different outcomes, and assign values to different outcomes (depending on the difficultly of achieving the outcome.)
Objective 3: Students will use the vocabulary of probability in their evaluations of commercial games and their own game design and will be very comfortable with moving back and forth from fraction (0 – 1), decimal (0.0 – 1.0), and percent (0 – 100%) calculations of probability. Students will review geometrical concepts of area and make the connection between
Objective 4: As students become more versed in the conventions and language of probability and statistics, it will inform their critical reading and thinking skills. The students will begin to understand that the way a set of data is displayed influences its interpretation.
Objective 5: Throughout the unit, students (in groups of three) will have had the opportunity to demonstrate their understanding of the concepts of probability in the analysis of design of their own games of chance (at least two). The unit will culminate with the students presenting their “creations” to their peers (along with posters that illustrate the rules and chances of winning the game) in the form of a Game Day Extravaganza. The students will completely plan, organize, and execute the activity as a class.
Assessment:Throughout the project, students will document their individual processes in project notebooks, which will help them organize the knowledge and materials in their study of the concepts of probability. The project notebooks will include a project journal where the students will answer a short question or two (posed by the teacher) every day as well as compile pertinent information about the project. The journal and notebook will provide formative assessment opportunities for the teacher(s) and self-assessment opportunities for the students. There will be a unit test at the end of the unit (before the Game Day Extravaganza) which will assess the standards addressed in the unit. One day will be spent reviewing the project discussing how the standards relate to the project. Students will write a final reflection with specific focus on the math that was explored.
Rationale:
Students begin exploring the concepts of probability and statistics in middle school and though the activities used to illustrate the concepts might be “hands-on”, they are often not very relevant to students (such as flipping a coin one hundred times to see how many times it lands on heads or predicting how many red marbles will be chosen from a bag full of colored marbles.) These activities can provide valuable lessons in the mechanics of working with probabilities, but for students to “develop an appreciation for the pervasive use of probability in the real world” (NCTM Standards), it is important that they participate in explorations of probability that are more relevant to their experiences. Students need to connect with the material.
Background:
On the first day of the unit, instructor will review prior knowledge that will be utilized in the unit. Students will be expected to be confident in their manipulation (simplification and other operations) and understanding of fractions (whole vs. part). This unit will re-emphasize the relationship of fraction to decimal to percent. The basic ideas of statistics (mean, median, mode) will also be reviewed.
The unit will be standards based focused on the seventh grade probability and statistics TEKS as detailed below. The breakdown of the organization is shown in the concept map below:
Standards:
TEKS addressed:
Review (6th grade TEKS)
6.9 Probability and statistics. Thestudent uses experimental andtheoretical probability to makepredictions.The student is expected to:
- construct sample spaces using lists and tree diagrams; and
- find the probabilities of a simple event and its complement and describe the relationship between the two.
6.10Probability and statistics. Thestudent uses statistical representationsto analyze data.
The student is expected to:
- select and use an appropriate representation for presenting and displaying different graphical representations of the same data including line plot, line graph, bar graph, and stem and leaf plot;
- identify mean (using concrete objects and pictorial models), median, mode, and range of a set of data;
- sketch circle graphs to display data; and
- solve problems by collecting, organizing, displaying, and interpreting data.
Seventh Grade Standards (7th grade TEKS)
7.1 Number, operation, and quantitative reasoning. The student represents and uses numbers in a variety of equivalent forms.
The student is expected to:
- convert between fractions, decimals, whole numbers, and percents mentally, on paper, or with a calculator.
7.10Probability and statistics. The student recognizes that a physical or mathematical model can be used to describe the experimental and theoretical probability of real-life events.
The student is expected to:
- construct sample spaces for simple or composite experiments.
- find the probability of independent events.
7.11 Probability and statistics. The student understands that the way a set of data is displayed influences its interpretation.
The student is expected to:
- make inferences and convincing arguments based on an analysis of given or collected data.
7.13Underlying processes and mathematical tools. The student applies Grade 7 mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school.
The student is expected to:
- identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with other mathematical topics;
- use a problem-solving model that incorporates understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness.
7.14Underlying processes and mathematical tools. The student communicates about Grade 7 mathematics through informal and mathematical language, representations, and models.
The student is expected to:
- communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models.
Final Product:
Design a Game of Chance
In a GROUP of 3 or 4 classmates, you will design a game of chance using the probability concepts explored in this unit. As a group, you will be responsible for the final product (game design) and a 1-pagereport that summarizes your findings. In preparation for the Game Day Extravaganza, you will prepare a poster advertising your game. All materials will be organized in a project notebook(one per team member). All group members will receive the same grade for the group part of the project.
Additionally, each student will submitan individual journal.
Instructions for Designing the Game
Step 1: Design a Game of Chance
The game should have a “theme”.
The game should be “fair”.
The game should be easy to set up and use. It should not require expensive equipment to make or maintain.
The game should take a relatively short time to play.
The rules for the game should be easily understood by people your age.
Step 2: Test Your Game
After your group has drafted a game design, you need to decide whether the game you have designed is reasonable (it has to be possible to win, but not too easy). Then you will need to try out your game. Your group should play the game several times until you can predict what will happen in the long run. Keep track of your trials, and document the process of making and testing the game in your journals.
Step 3: Present Your Game Design to Classmates for a Trial Game Day
Once you are satisfied that your game of chance is reasonable, you will present it to your classmates for a trial run. Your presentation should include two things: a mock-up of the game and an oral report describing the game including instructions and probabilities. (Give the experimental probability of winning the game that you found from playing the game several times. If possible, give the theoretical probability as well. For some games, such as tossing coins or drawing blocks from a container, finding the theoretical probability of winning is easy. For others, finding the theoretical probability may be too difficult. If you don’t give the theoretical probability of winning for your game, explain why you did not.) Your classmates will evaluate your game and provide feedback.
Step 4: Build the Game
With your group, build the final version of your game incorporating suggestions from your classmates to improve the design. With your game, include a set of rules that explains how the game is played, how much it costs to play (tokens), how a player wins, and how much a player wins (tokens).
Step 5: Make a Poster Advertising Your Game
The poster should be visually interesting.
The poster should state the name of the game and the name of the team that designed the game.
The poster should clearly state the rules of the game.
The poster should state the probability of achieving the different outcomes.
Design a Game of Chance - Rubric
GROUP RUBRIC
Points Awarded / Points Possible / Game Design / Report / due date ______10 / Presentation to Classmates
- Equal participation from all group members (3)
- Mock-up of game complete and playable(3)
- All group members prepared, enthusiastic, and professional (1)
- Each group member able to answer at least one question posed by classmates or teacher about the game design (3)
25 / Game Design
- Creative and well thought out design (10)
- Feedback from trial run and teacher comments utilized (5)
- Follow instructions of assignment (10)
7 / Poster
- Poster is neat and organized (2)
- Successfully demonstrates the math concepts(3)
- All required elements represented (2)
3 / Game Day Extravaganza
- Equal participation from all group members (3)
45 / Total Points for Group Part of Project
INDIVIDUAL RUBRIC
Points Awarded / Points Possible / Alternatives and Recommendations / due date ______12 / Project Notebook
- Notebook is neat and organized (3)
- Notebook contains all required materials (7)
10 / Journal
- Journal is neat & organized (3)
- Journal contains all required entries (7)
8 / 1-page Reflective Writes (2 – mid project & end of project)
- Reflectionshave been proofread – there should be no grammatical, spelling or punctuation errors (4)
- Thoughtful & meaningful (4)
30 / Total Points for Individual Part of Project
Student Total Grade: ______/75