Table of Contents
Jossey-Bass Teacher
Title Page
Copyright Page
About the Authors
About This Book
Alignment to National Council of Teachers of Mathematics Standards
How to Use This Resource
Part One - Implementing Projects in the Math Class
Chapter One - Overview of projects in the math class
Your Role
Supporting the Standards of the National Council of Teachers of Mathematics
Strategies for Problem Solving
Creating Your Own Projects
Chapter Two - Managing projects in your math class
Structuring Your Class
Creating and Maintaining a Positive Environment
Teaching Suggestions
Individual and Team Conferences
The Value of Cooperative Problem Solving
Organizing Teams
The Importance of Sharing
Writing in Math Class
Using Technology with Math Projects
Technology Training
Math Projects and the Internet
Chapter Three - Assessing math projects
Observation Logs
Point Systems for Project Assessment
Evaluating Writing
Part Two - The Projects
Section One - Math and Science
Project 1 - The benefits of recycling
Project 2 - Endangered species: Can they be saved?
Project 3 - Charting your calories
Project 4 - How many calories do you burn each day?
Project 5 - It is only natural
Project 6 - Designing a flower bed
Project 7 - Buying a class aquarium
Project 8 - The school’s new lunch program
Project 9 - What is the weather?
Project 10 - A flight to Mars
Section Two - Math and Social Studies
Project 11 - A great mathematician
Project 12 - An election poll
Project 13 - Landmarks
Project 14 - Creating a scale map
Project 15 - A game: What date is that?
Project 16 - An interview to math’s past
Project 17 - Rating consumer products
Project 18 - Lands of ethnic origin: A statistical potpourri
Project 19 - A land of immigrants
Project 20 - The games people play
Project 21 - Back to the past
Section Three - Math and Language
Project 22 - Becoming the experts
Project 23 - Great debates
Project 24 - The mathematics publishing company
Project 25 - Rating math Web sites
Project 26 - Fictional numbers: Writing a story
Project 27 - A mathematical autobiography
Project 28 - Puzzling
Project 29 - Lending a math hand
Project 30 - Sharing the math word
Project 31 - Keeping a math journal
Project 32 - Math portfolios
Section Four - Math and Art and Music
Project 33 - Making a math poster
Project 34 - Creating a logo
Project 35 - I wanna be like Escher
Project 36 - The Plus and Minus comic strip
Project 37 - Numbers and songs
Project 38 - The math in music
Project 39 - Making three-dimensional octahedra and classroom decorations
Project 40 - Creating a greeting card for Math Awareness Month
Project 41 - The geometry and art of architecture
Project 42 - Designing a quilt pattern
Section Five - Math and Sports and Recreation
Project 43 - Choosing a membership plan at a health club
Project 44 - Equipment for the school’s workout room
Project 45 - Comparing sports superstars
Project 46 - Math and the big game
Project 47 - Your unique exercise program
Project 48 - The big dance
Project 49 - The numbers game
Project 50 - Planning a sundae party
Project 51 - Going on vacation
Section Six - Math and Life Skills
Project 52 - Making a budget
Project 53 - A floor plan of my room
Project 54 - The costs of pets
Project 55 - Maintaining a math class Web site
Project 56 - Selecting a sound system using the Internet
Project 57 - Buying a car
Project 58 - What is on the test?
Project 59 - Checks and balances
Project 60 - Math in my life: An assessment
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Jossey-Bass Teacher
Jossey-Bass Teacher provides K-12 teachers with essential knowledge and tools to create a positive and lifelong impact on student learning. Trusted and experienced educational mentors offer practical classroom-tested and theory-based teaching resources for improving teaching practice in a broad range of grade levels and subject areas. From one educator to another, we want to be your first source to make every day your best day in teaching. Jossey-Bass Teacher resources serve two types of informational needs—essential knowledge and essential tools.
Essential knowledge resources provide the foundation, strategies, and methods from which teachers may design curriculum and instruction to challenge and excite their students. Connecting theory to practice, essential knowledge books rely on a solid research base and time-tested methods, offering the best ideas and guidance from many of the most experienced and well-respected experts in the field.
Essential tools save teachers time and effort by offering proven, ready-to-use materials for in-class use. Our publications include activities, assessments, exercises, instruments, games, ready reference, and more. They enhance an entire course of study, a weekly lesson, or a daily plan. These essential tools provide insightful, practical, and comprehensive materials on topics that matter most to K-12 teachers.


About the Authors
Judith A. Muschla received her B.A. in mathematics from Douglass College at Rutgers University and is certified to teach grades K-12. She has taught mathematics in South River, New Jersey, for over twenty-five years. She has taught math at various levels at South River High School, from basic skills through pre-calculus. She has also taught at South River Middle School where, in her capacity as a team leader, she helped revise the mathematics curriculum to reflect the Standards of the National Council of Teachers of Mathematics, coordinated interdisciplinary units, and conducted mathematics workshops for teachers and parents. She was a recipient of the 1990-1991 Governor’s Teacher Recognition Program award in New Jersey and was named the 2002 South River Public School District Teacher of the Year. Along with teaching, she has been a member of the state Standards Review Panel for the Mathematics Core Curriculum Contents Standards in New Jersey.
Including this second edition of Hands-On Math Projects with Real-Life Applications, Judith and Gary Muschla have coauthored seven math books published by Jossey-Bass: The Math Teacher’s Book of Lists (1995; 2nd edition, 2005), Math Starters! 5- to 10-Minute Activities to Make Kids Think, Grades 6-12 (1999), Geometry Teacher’s Activities Kit (2000), Math Smart! Over 220 Ready-to-Use Activities to Motivate and Challenge Students, Grades 6-12 (2002), Algebra Teacher’s Activities Kit (2003), and Math Games: 180 Reproducible Activities to Motivate, Excite, and Challenge Students, Grades 6-12 (2004).
Gary Robert Muschla received his B.A. and M.A.T. from Trenton State College and taught in Spotswood, New Jersey, for more than twenty-five years. He spent many of his years in the classroom teaching mathematics at the elementary level. He has also taught reading and writing and is a successful author. He is a member of the Authors Guild.
He has written several resources for teachers, among them The Writing Teacher’s Book of Lists (1991; 2nd edition, 2004), Writing Workshop Survival Kit (1993; 2nd edition, 2005), English Teacher’s Great Books Activities Kit (1994), Reading Workshop Survival Kit (1997), Ready-to-Use Reading Proficiency Lessons and Activities, 4th Grade Level (2002), Ready-to-Use Reading Proficiency Lessons and Activities, 8th-Grade Level (2002), and Ready-to-Use Reading Proficiency Lessons and Activities, 10th-Grade Level (2003), all published by Jossey-Bass. He currently writes and serves as a consultant in education.

For Erin

We thank Michael J. Pfister, assistant superintendent of South River Public Schools, Kevin W. Kidney, principal of South River High School, Paul J. Coleman, assistant principal of South River High School, Geraldine Misiewicz, math supervisor, and our colleagues for their support of our writing.
Thanks also to Steve D. Thompson, our editor, for his support of our efforts to complete this second edition.
Special thanks to Caroline Fitzgerald and Geri Priest, who read the original manuscript and offered many helpful suggestions, to Jamie Egan for helping us better understand the mathematical aspects of music, and Colleen Duffey Shoup as well as Dover Clip-Art for the illustrations.
We greatly appreciate the help and encouragement of Susan Kolwicz, whose advice on the first edition enabled us to take our rough ideas and fashion them into a practical resource for teachers.
We are indebted to Sonia Helton, professor of education at the University of South Florida, for her insightful recommendations for the projects in this book.
We want to thank our daughter, Erin, who read the first draft of this new edition from the perspective of a young math teacher and caught several oversights and omissions.
And finally, we thank our students. In the end, they are why all of us are in this business.

About This Book
Appropriate for grades 6 through 12, Hands-On Math Projects with Real-Life Applications, Second Edition consists of two parts: Part One focuses on implementation and management, and Part Two contains sixty projects for your students. The projects support the Standards of the National Council of Teachers of Mathematics (NCTM), as well as meet the mandates of the No Child Left Behind Act that call for project-based learning, problem-solving strategies in mathematics, and the integration of technology in the classroom.
The new edition of this book retains the valued features of the first edition, while updating its relevance and extending its scope. All of the projects have been revised to reflect current trends, numerical data have been redone, and information on the use of technology has been greatly expanded. For many projects, students will use the resources of the World Wide Web to obtain information that they then use to solve the problem the project presents. Some projects have been replaced with new projects that hold greater applications for today, including Project 25, “Rating Math Web Sites,” Project 55, “Maintaining a Math Class Web Site,” and Project 56, “Selecting a Sound System Using the Internet.”
The new edition, like the original one, is designed for easy implementation. Each project stands alone and may be used with students of various grade levels and abilities, providing teachers with great flexibility for instruction.
To prepare students for the demands they will face in the workplace, math teachers must provide a classroom environment where students are challenged to solve real-life problems, where they may collaborate and share ideas, where they use calculators and computers, where they express their thoughts orally and in writing, and where they recognize that mathematics is not an isolated subject but is connected to other disciplines. The projects in this book will help you to achieve these goals.

Alignment to National Council of Teachers of Mathematics Standards
The following table indicates the NCTM Standards addressed by the projects of this book. Checks indicate the specific standards with which each project aligns. For some projects, skills will vary depending on the material students create for the project.

How to Use This Resource
Hands-On Math Projects with Real-Life Applications, Second Edition is divided into two parts. Part One, “Implementing Projects in the Math Class,” contains three chapters devoted to classroom management. Part Two, “The Projects,” offers sixty math projects designed to enhance your math program.
Before assigning any of the projects, we recommend that you read through Part One in its entirety because the information it provides will help you to implement project activities in your class. After reading Part One, you may select those projects in Part Two that best support your program and satisfy the needs of your students.
In Part One, Chapter One provides an overview of how to incorporate math projects in your class, Chapter Two details a variety of specific classroom management techniques and suggestions, and Chapter Three offers several methods for evaluating the work of your students. Each of the chapters includes several lists that summarize information for various topics, making it easy for you to find the information you need. For example, you will find that your role expands when projects become part of your program. “The Teacher’s Role During Math Projects” outlines the many tasks you may assume when your students are engaged in project work.
Each chapter of Part One also includes several reproducibles for students that can be helpful in establishing the routines necessary for successful project work. For example, it is possible that many students may not have had much experience in working cooperatively to solve a complex problem. Distributing copies of “Rules for Working in Math Teams” highlights the behaviors that characterize effective teams. Knowing what is expected of them helps students to behave appropriately and achieve the goals you set out for them.
Part Two contains sixty projects divided into six major sections:
Section One, Math and Science
Section Two, Math and Social Studies
Section Three, Math and Language
Section Four, Math and Art and Music
Section Five, Math and Sports and Recreation
Section Six, Math and Life Skills
Although the breakdown is useful for planning interdisciplinary units or finding a project that ties in to another subject, each project stands alone. Each may be used to introduce, enhance, or conclude a unit or topic, or be used as a challenge, enrichment, or extra credit. Some projects may be used as ongoing activities—for example, Project 31, “Keeping a Math Journal,” or Project 32, “Math Portfolios.” Project 24, “The Mathematics Publishing Company,” shows students how to create and produce a mathematics magazine, which you may decide to publish regularly throughout the year.
Each project follows the same format. Information for the teacher is presented first: background, goals of the project, math skills that are covered, special materials and equipment that are needed, and development. This material is followed by the Student Guide, which provides strategies and suggestions on how students may solve the problem the project presents. Data sheets and worksheets provide students with additional information or a specialized work space. The student guides, data sheets, and worksheets, which are numbered according to each project, are reproducible for your convenience.
We suggest that you use this book as a resource, selecting projects you need to enhance your curriculum. The sixty projects offer a variety of real-life situations that will help your students to realize the relevance math has in their lives, while at the same time applying specific mathematics skills.
We trust that you will find this book to be a helpful resource as you encourage and support your students in their efforts to learn math. Our best wishes to you.

Part One
Implementing Projects in the Math Class

Chapter One
Overview of projects in the math class
In a well-run math class, computation, problem solving, and critical thinking are all taught. Instead of learning skills in isolation, students learn math in context where they can see how it is applied in real situations. In this way they come to recognize the importance of math in their own lives. The connection between math and the real world is a strong one. This is especially true of math classes in which projects are a significant part of the curriculum.
Filled with activity and enthusiasm, a successful project-oriented math class is a center of individual learning, collaboration, cooperation, and sharing. Students work alone, together, and with the teacher. Along with learning fundamental math skills, students learn to think logically, analyze data, make decisions, and solve multifaceted problems that arise out of real-life situations. Students thus use the skills they are learning in meaningful ways.

Your Role

Your role changes when your students work on math projects. Along with your traditional responsibilities of introducing concepts, demonstrating skills with example problems, and grading the work of your students, you will become a facilitator and promoter. The horizons of your teaching will expand. More of your time will be spent working directly with individuals and groups. As students work on solving problems, you will circulate around the room, offering advice and suggestions, asking questions that lead to insights or direction, and giving encouragement and praise. Sometimes you may simply monitor a group’s efforts or model appropriate behavior. Occasionally you may need to pull a group back on task. (See “The Teacher’s Role During Math Projects.”)
There are many ways you can incorporate projects into your curriculum. While following your text, you can easily provide regular project activities. You may build time for projects into your schedule, for example, a day or two each week, or do units on projects a few times a year. Some teachers introduce a multistep project and then give students time to work on it at the end of class over the next few days. No matter how you provide the time, you should be consistent. Students not only need sufficient time for working on projects, they need to know when they will be working on them. This information enables students to come to class prepared and ready to work.

Supporting the Standards of the National Council of Teachers of Mathematics

The projects in this book support the Standards of the National Council of Teachers of Mathematics (NCTM), with specific emphasis on the following:
• The Problem-Solving Standard
• The Communication Standard
• The Connections Standard
• The Representation Standard
Problem solving is an essential part of learning mathematics. At its most basic, it requires students to find a solution to a problem without initially knowing what methods or procedures to use. As they seek a solution to the problem, students must draw on their own knowledge, experience, and skills. They may be required to assume various tasks, including conducting research, analyzing and organizing data, and drawing conclusions. In their efforts to discover answers, they reinforce previously learned skills, acquire new skills, and gain a greater understanding of mathematics. By presenting students with a variety of practical, engaging problems to solve, the projects contained in this book support the Problem-Solving Standard and foster the learning of valuable problem-solving skills.
The projects also support the Communication Standard. Because effective communication depends on clear thought and expression, communication encourages students to think critically, formulate their ideas, and express those
The Teacher’s Role During Math Projects
Since discovery is an important part of any project, you must encourage your students to assume much of the responsibility for their learning and progress. Your role changes. Along with your traditional duties, you will be spending some of your class time doing many of the following:
• Presenting multistep, critical thinking projects based on real-life situations
• Organizing and monitoring groups so that members work effectively together
• Modeling appropriate behavior and problem-solving skills
• Demonstrating to students what it is to be an enthusiastic problem solver by showing them how you are willing to tackle projects that at first may seem impossible
• Brainstorming with groups
• Guiding students in their research efforts
• Showing students that process is crucial to finding solutions
• Offering suggestions to solve problems
• Offering encouragement and applauding efforts
• Explaining that mistakes are merely stepping-stones to finding solutions and to learning
• Answering questions
• Helping students sort through their thoughts as they consider problem-solving strategies
• Showing students that various strategies may be used to solve the same problem
• Providing sufficient time for working on projects
• Monitoring student behavior and ensuring that classroom procedures are followed
• Keeping students on task
• Evaluating and assessing student progress
• Providing time for sharing results
ideas with mathematical precision. Communication gives students the opportunity to state their ideas, listen to the ideas of others, and compare them to their own, furthering their understanding of math. An important part of every project of this book is sharing results through formal and informal presentations.
Along with the Standards for Problem Solving and Communication, working on math projects supports the Connections Standard. Although students are often taught mathematical skills in isolation or in packets of information, mathematics is a broad, complex subject in which ideas are interconnected, extending throughout the field of math and to other disciplines. As they work on projects, students will find relationships among ideas that will broaden their understanding of problems and solutions, thereby gaining an appreciation of the scope of mathematics and how math is interwoven through all parts of society.
The projects of this book also support the Representation Standard. Mathematical ideas are represented with notations, symbols, and figures. Typical examples of representations include numbers, expressions, diagrams, and graphs. There are many more, of course. As students work on projects, they will express mathematical ideas as representations, which then become tools to explore, model, and develop mathematical concepts. An understanding of mathematical representations will serve students well in their continued study of math.
Perhaps one of the greatest benefits of using math projects in the classroom is how students must draw on numerous skills as they work toward solutions. When students work on math projects, they expand their view of math to real-life situations and develop skills that stretch well beyond the traditional curriculum. Such results support the Standards of the NCTM and enrich the mathematical experiences of students.

Strategies for Problem Solving

The math projects your students will be doing will require computation, analysis, problem-solving and critical thinking skills, and decision making. Since the type and nature of problems vary, there can be no set plan or step-by-step process they can use all the time. You should familiarize your students with various strategies that they can draw on as needed. Emphasize that strategies are methods or procedures that can be used alone or with other strategies. If a student should ask what strategy is best for solving a particular problem, a good answer is, “The one that works best for you.” You will likely find that different students will use different strategies to solve the same problem.
While some students may be quite adept at problem solving, many will need guidance, and you may wish to distribute copies of “Problem-Solving Strategies.” It is a guide that can help students get started in solving problems and keep them moving along.
There is also much you can do in regular lessons to help students acquire sound problem-solving skills that will be useful to them throughout their lives. See “Helping Students Develop Problem-Solving Skills” for a list of suggestions.
Problem-Solving Strategies
There are many ways to solve multistep problems. If you believe that there can be only one or two, you limit your options and reduce your chances of finding a solution. Following are some suggestions and strategies.
Before you begin seeking the solution:
• Make sure that you understand the problem. This may require rereading it several times.
• Be sure you understand the question and what answers you are seeking.
• Look for “hidden” questions.
• Find the important information that the problem provides, and eliminate information that is not essential. (Sometimes problems contain facts that you do not need.)
• Supply any missing information. You may need to research and analyze data.
• Make sure that you understand any special facts, data, or units of measurement.
As you seek a solution, consider all of these strategies:
• Look for patterns, relationships, connections, sequences, or causes and effects.
• Use guess and check (also called trial and error). Choose a place to start, try a solution, and see if it works. If it does not, try another.
• Organize your facts and information in a list. Sometimes this exercise can show relationships that you might otherwise overlook.
• Construct a table or chart. This is another way of identifying relationships.
• Think logically. Look for sequence and order.
• Rely on common sense. Some answers simply are not possible. Do not waste time pursuing them.
• Sketch or draw a model to help you visualize the problem.
• Simplify the problem by breaking it down into manageable steps. Solve a sub-problem that leads to the solution of a bigger problem.
• Look at the problem from different angles.
• Estimate. Rounding numbers can make it easier to find a solution. Using whole numbers rather than fractions may help you to see operations more clearly.
• Act the problem out.
• Keep notes of your attempted solutions. This will reduce the chances that you will repeat steps that do not move you forward.
• Periodically review your notes and attempts at solutions. By rechecking what you have done, you might see something you overlooked.
• Do not give up. The persistent problem solver finds solutions.
When you believe you have found the answer:
• Double-check your work.
• Be certain that you used all necessary information.
• Recheck your calculations.
• Be sure that your answer is logical.
Helping Students Develop Problem-Solving Skills
You can help your students learn critical thinking and problem-solving skills by doing the following:
• Present students with real-life problems to which they can relate.
• Offer problems that have multiple solutions and can be solved through several strategies.
• Encourage your students to try various strategies in solving problems.
• Organize students into cooperative teams.
• Encourage students to brainstorm for ideas that might lead to solutions.
• Give problems that have missing information and too much information. Such problems require students to supply and eliminate data.
• Give problems that tie in to other subjects.
• Encourage students to keep logs or notes of their efforts at solving difficult problems.
• Encourage students not to give up; persistence is a major factor in successful problem solving.
• Require students to write explanations of how they solved problems.
• Remind students to always check answers for logic and accuracy.
• Encourage discussion and the sharing of solutions.
A vital part of any project is the sharing of solutions and results at the end of the activity. When results are shared, students have the opportunity to hear other viewpoints, learn about other methods used to solve problems, and realize that others may have experienced some of the same stumbling blocks they did. Not only does this help reduce an individual’s feelings that he or she is the only one having trouble, it also helps build a sense of class community and problem-solving camaraderie.
Sharing may be oral through presentations using technology such as interactive whiteboards or Microsoft PowerPoint, or written in the form of logs or reports. Thus, speaking and writing become essential components of your math class.
Perhaps the biggest factor that holds many students back from becoming good problem solvers is a lack of confidence. Many students doubt that they can solve complex problems and give up with little effort. Explain to your students that problem-solving skills come with practice. Just like anything else—learning to play a musical instrument, excelling at gymnastics, playing computer games—the more they work at solving problems, the better they will become. Distribute copies of “What It Takes to Become a Top Problem Solver” to highlight some of the characteristics that successful problem solvers share. The list can serve as a guide, detailing traits and attitudes that your students should strive to acquire throughout the year.

Creating Your Own Projects

While this book provides projects that require various steps and strategies, you may eventually wish to create projects of your own, designed specifically for your students. Material for math projects is all around you. As you develop projects, keep in mind the following points, which will help ensure that your projects are stimulating and exciting to your students:
• Base your projects on real-life situations that are meaningful to your students.
• Design projects that capture the interest of your students.
• Make sure that your students possess the mathematical skills to solve the problems they will encounter in your projects.
• Develop projects that require analysis, critical thinking, and decision making.
• Create projects that require students to formulate a plan to find a solution.
For suggestions where you can find material from which to create projects for your students, see “Sources for Developing Math Projects.”
What It Takes to Become a Top Problem Solver
Top problem solvers share many of the same traits. You can become a successful problem solver. All it takes is practice. The more problems you solve, the more skilled you will become. Try to make the following traits part of your personality.
Good problem solvers are:
• Confident that they can solve just about any problem.
• Persistent in solving problems.
• Willing to try different strategies to solve problems.
• Able to find important information and eliminate unimportant facts.
• Able to recognize patterns, relationships, and connections.
• Able to look at a problem from various viewpoints.
• Open to new ideas.
• Willing to make notes to keep track of their attempts at solutions.
• Able to draw on other experiences in the solving of problems.
• Able to use logic and common sense.
Sources for Developing Math Projects
Good material for creating your own math projects is all around you. The following sources are particularly useful.
• Your math text likely contains sections such as “Challenges” that offer interesting facts or situations that you can easily turn into fine projects. Some texts have sections of data banks that provide information that can serve as the basis for projects.
• The Internet offers vast information on countless topics. Information can be easily found by conducting a simple search by topic.
• National and local newspapers contain an assortment of valuable information. Charts and tables can be especially helpful.
• Regional and national magazines are good sources of information for projects.
• Almanacs and other reference books can provide unusual and interesting data on countless topics.
• Major events at school can be your springboard for creating projects. Use field day, homecoming, the Valentine’s Day dance, or the prom to capture the interest of your students.
• Books of math puzzles and games frequently offer a wealth of ideas for projects.
• Consult with your colleagues and develop projects that include two or more subject areas. Science and social studies in particular share many topics with mathematics.


Without question, math projects offer many benefits to students. Perhaps most important, when students work on authentic problems, they see how the math skills they are learning may be applied to the real world. Math projects open the door to bringing other subjects and disciplines into the math class, and students quickly recognize that math is interwoven through many parts of their lives. Math projects also give students the opportunity to work together cooperatively, share their experiences, and celebrate the solving of problems that might be overwhelming for one person to manage. Furthermore, when students collaborate on a project, students of all abilities have the chance to contribute to the solution. Everyone has a part to play; everyone has a role to fill; everyone can be a contributor to and a sharer in success.

Chapter Two
Managing projects in your math class
A successful math class in which projects are an important part of the course of study is the result of effective planning and management. Along with teaching required material, you must provide meaningful projects that have real-life applications. These are not easy charges. As students work on projects, they will be engaged in various tasks: they will need to consider different strategies, gather and analyze information, confer with each other, manipulate models, perform calculations, and test possible solutions. All this requires an environment that promotes vigorous inquiry, encourages students to assume the responsibility for their learning, and supports both individual and group activities.

Structuring Your Class

Math projects provide students a chance to use various skills in solving authentic problems. Since projects often reach beyond the math class, they offer an excellent way to broaden the scope of your curriculum and introduce exciting new activities to your teaching. There are many ways you can incorporate projects into your classes. Perhaps the easiest is to select projects that support the unit you are teaching. For example, if you are studying a unit in geometry, Project 6, “Designing a Flower Bed,” in which students work with rectangles, squares, circles, and scale, will be useful. If you are teaching a unit on data analysis, Project 12, “An Election Poll,” will supplement your instruction.
Projects used to enhance a unit can be built into your daily schedule. We suggest that you take a class period to introduce and begin the project. Explain the project; distribute any materials students might need; organize teams; and give students twenty minutes or so to plan, brainstorm potential strategies, and get started.
After this introductory period, resume your regular lessons, reserving about twenty minutes at the end of each period for students to continue working on the project. Providing students with time at the end of class eliminates the need for them to arrange to meet outside class (although sometimes students become so involved with a project that they meet on their own). This also allows you to assign the ordinary amount of homework and continue moving forward with the unit. Since not all teams will finish at the same time, groups that finish early may work on extensions of the project, write in math journals, or simply do homework. When every team is done, you should schedule a period for sharing results.
Another way to incorporate projects into your classes is to periodically set aside time for them. After you complete a unit, take three or four days to work on a project. Some teachers prefer this method because it gives students a break from the routines of the class but does not interfere with the general curriculum.
Perhaps you will decide to build projects into your schedule. You might reserve every Tuesday and Friday for working on projects. This plan has the advantage of establishing a regular schedule that ensures time for project work. Since the projects in this book stand alone, each can be used at any time during the year and still provide the benefits of using a variety of skills in meaningful contexts.
Some projects, especially those for individuals, may be completed at home. You might give students a choice of these projects as either assignments or bonus activities. Although the projects are completed on the students’ time, you should provide class time for sharing. Responses to their results are important to students.
When selecting projects for your students, be sure to consider their needs and abilities. Never assign projects that require skills your students have not yet mastered. Students will find such projects to be a frustrating struggle, and any new skills they acquire will be offset by the negative emotions they come to feel for math.
Without question, math projects offer students several benefits. They permit students to use many skills in solving various problems, give students a chance to take ownership of their mathematics learning, and help students to see the relevance between math and real life.

Creating and Maintaining a Positive Environment

Problem solving thrives in an environment in which people work on problems that have valid applications to life, feel free to risk making mistakes, and are encouraged to share their ideas. The best problem solving occurs in classes where students enjoy the freedom to pursue learning in their own way. The tone you set in your classroom, your expectations, and the procedures you maintain are the foundation for such an environment.
Since students usually rise, or fall, to a teacher’s expectations, always discuss your goals for the class with your students at the beginning of the year. Share with them how you intend to conduct the class, how the class will be organized, and what will be covered. Note what you expect from them.
For students to work efficiently on math projects, they need a classroom that is logistically comfortable for problem solving. Tables are ideal; however, if you do not have tables, you can push desks together. Either way, you should provide enough room between teams so that they can function as single entities without distractions from other groups. Along with enough space between teams, there should be enough work area for students to discuss possible strategies with each other, confer about data, manipulate and examine models, and work on calculations.
Support problem solving in whatever ways you can. Bulletin boards, corridor display cases, media center exhibits, and math fairs are just some ways to draw attention to your program. Always look for ways to highlight lists of problem-solving strategies, interesting articles about math, and the work of your students.
While there is much you can do to promote success in your classes, students too must strive to make the class beneficial. This is particularly true during group activities. Your students must be willing to accept more responsibility than is demanded by the traditional class. During project activities, they must remain focused on the tasks. Group work is not a time to talk about who might be named homecoming king or queen. Distributing copies of “The Responsibilities of the Math Student” to your students is an excellent way to share basic expectations for student behavior.
Unquestionably, student learning flourishes in a conducive environment. One of your most important tasks as a teacher is to create a classroom filled with enthusiasm, the spirit of inquiry, and the desire to learn. The best classes are founded on the spirit of cooperation and energetic intellectual pursuit, in which students believe that everyone can learn (and enjoy!) math. For a summary of characteristics of math classes that have a positive atmosphere, see “The Right Environment.”
The Responsibilities of the Math Student
A successful math class results when people work together to learn math. Accepting the following responsibilities is the first step to making this class worthwhile:
• Each day report to class on time and ready to work.
• Remember to bring your text, notebooks, pencils, calculators, and other materials to class.
• Pay attention in class, and ask questions when you do not understand something.
Everyone can learn math. Work hard, and finish your class work and homework.
• Work cooperatively with other students in groups. Share your ideas, and be willing to listen to the ideas of others.
• Try various strategies in solving problems.
• Remember that solving complicated, multistep problems takes time. Be persistent.
• Follow the classroom rules and procedures.
• Behave properly.
• Recognize the importance of math in your life.
The Right Environment
The following characteristics are found in math classes described as having a positive atmosphere:
• The goals of the class are high enough so that students have to work hard, but not so high that they feel frustrated with math and its applications.
• The classroom is built on openness, fresh ideas, and sharing.
• Teacher and students believe that everyone, regardless of gender and ethnicity, can learn math.
• Students’ work is prominently displayed.
• The classroom is designed to support inquiry and problem solving.
• The classroom is bright and cheerful.
• The classroom adheres to orderly procedures. Students maintain appropriate behavior and follow the classroom rules.
• Goals and objectives are clear to students.
• Classroom rules are fair and consistent.
• The grading system is reasonable and equitable.
• The teacher interacts with students and is a guide, nurturer, cheerleader, and provider of information.
• Teachers model problem-solving behavior and share with students their own enthusiasm for finding solutions.
• Math is connected to real-life problems and situations.
• Cooperation is encouraged.
• Enough time is provided for problem solving.
• Students are encouraged to consider and explain their reasoning during problem solving.
• Students are encouraged to use various strategies in solving problems. They come to recognize that the same problem may have many solutions.
• Sharing is encouraged, especially how students found solutions to problems.
• Calculators, computers, and other technologies are used regularly in class.
• Math is related to other subjects as much as possible.
• Manipulative materials are used whenever possible to show students relationships.
• Students learn the value of mathematics in their lives.
• Students and teachers become partners in learning mathematics.

Teaching Suggestions

While every teacher has his or her individual techniques and methods, we have found that the following plan is helpful in presenting projects and problem solving. It can be broken down into three parts: introduction, work time, and wrapping up.


Begin a math project by presenting the situation and problems that are to be solved. Offer examples, review any concepts or specific skills that students will need to solve the problems they will confront, and relate the project to real-life scenarios as much as possible. Encourage students to ask questions. Having a student paraphrase the project and what needs to be solved can be helpful in clarifying what everyone is to do.
Once students understand the project, distribute copies of student guide sheets and discuss the information presented there. Data sheets and additional materials, if any, should also be distributed. Having everything they need to begin helps students to see the full scope of the project.

Work Time

As students work in teams, your task is to circulate around the room, offering help, encouragement, or simply observing. This is also a time to monitor and model student behavior.
Pay close attention that a team does not stray off the topic. If you see this happening, you might point out their mistake or nudge them in the right direction. However, avoid giving answers to any problems. If students feel that you will provide answers, they will be less inclined to do the hard thinking that will result in finding answers themselves. To encourage students to find their own answers, some teachers insist that they (the teachers) may be asked a question only after the question has been presented to the team and no one else is able to answer it.
As you observe students, you may find that a team has trouble starting. Sometimes this is caused by students’ not being able to focus the problem. Have students restate the problem and break it down into parts, concentrating their efforts to identify the most important facts. Teams may also have trouble finding strategies that will lead to solutions. In this case, suggest that teams brainstorm various strategies, and examine each one to see if it leads to a possible solution.
As you move around the room, be aware of the interactions of the members of each team. You will likely see that some groups work well together with everyone sharing ideas, others are dominated by one or two members, and some are just unmotivated. When a team is working well, leave it alone. Even offering a comment might disrupt its momentum. Remember that a project is a time for students to discover their own solutions. If a group is not working well, you should sit in on it and model appropriate behavior. Make sure that everyone is participating, and encourage team members to help each other. If necessary, for a time, assume the role of team leader to get things going, then gradually fade into the background as students begin to assume ownership of the project. With some teams, you may need to remind students of the proper procedures and behavior often, especially during the first few weeks of class.

Wrapping Up

Sharing is essential to the successful culmination of a math project. Discussing methods and results helps students to realize that some problems have multiple solutions that may be discovered through various strategies. This is an important lesson of authentic problem solving. In the real world, many problems have several solutions and can be solved in many ways. For more information on sharing, see “The Importance of Sharing” presented later in this chapter.

Individual and Team Conferences

As students work on projects, you will monitor the progress of the teams. In many cases, they will have questions, or you will need to discuss procedures, rules, or behavior. You will undoubtedly be conducting conferences with individuals or the entire team.
A conference does not have to be long; in fact, it may last only a minute or two. In most cases, it will be conducted at the students’ work area. The purpose of any conference is to help students better understand the project they are working on, as well as help them to improve their understanding of mathematics. Often you may find that students need you only to answer a simple question. In such instances, provide guidance and let them get back to work. If an individual or team seems stuck, use this as your starting point for the conference.
Focus any conference on a particular problem or skill. If you try to do too much, you will confuse students or provide them with too much information. Either way, you will end their efforts to solve the problem. During the conference, be sure to keep your tone positive, and offer specifics. You may need to point a team in the right direction to find more information, offer encouragement to the team that is about to give up, or assure a team that their efforts are worthwhile.
When you give praise, it should be genuine, because students can tell when it is not. Always avoid negative or sarcastic remarks, for these will only discourage students. The conference should be a time of help and support.

The Value of Cooperative Problem Solving

In many jobs, people work in teams, and the experience your students gain now as they work together on math projects will serve them not only in your class but in the future as well. Teamwork fosters inquiry and discussion, and students often learn more when working together than they do trying to solve a complicated problem alone. Cooperative learning also provides students with the opportunity to acquire valuable social skills.
When students work in teams, they are more likely to take an active role. It is easier for them to get involved because the team provides support to individuals. Seeing other team members struggling with the same problems helps students feel less intimidated about offering their thoughts, and many students who would not risk sharing ideas with the whole class usually will share with their team. Furthermore, when they offer suggestions toward the solution of a problem, they receive immediate feedback. This sharing frequently results in a free-wheeling give-and-take of mathematics that is as stimulating as it is useful.
As a team works on a math project, it becomes involved in various activities. Team members need to discuss and assign tasks, reflect on how to approach the problem, test strategies, gather and analyze data, reach solutions, and determine how to justify and share results. Teamwork helps build student confidence, promotes critical thinking, and results in a sense of ownership of the problem.

Organizing Teams

Random groups tend to make the best math teams, although you should reserve the right to make adjustments. Groups of four to six generally work well for complex projects. With fewer than four, it is sometimes hard to generate enough ideas, especially if one of the students is absent or is shy or quiet.
An easy way to make random pairings is to simply count down your roster in sets of five, assigning the numbers 1 to 5 to students. Then all of the number 1s in the class would be on team 1, all of the 2s on team 2, and so on. Before announcing the teams to the students, review them and make sure that you have a mix of students of high and low abilities, as well as a mix of gender and ethnicity. It is also often a good idea to avoid having best friends or students who do not get along on the same teams. Make any final changes before informing students about the groups.