Developing a Mathematics Project

Types of Projects
Choosing a Project
Research Methods
Construction Materials and Methods
Suggested Topics (Grades 6–8)

Calculating Instruments, Measuring Devices, and Computing Methods
Numeration Systems
Number Bases
Sets
Statistics and Probability
Paper Folding
Curve Stitching
Optical Illusions
Motion Geometry
Geometry
Functions
Patterns and Art
Games and Puzzles
Calendar and Time
Topology, Knots, and Surfaces
Algebra

Excerpts from Mathematics Projects Handbook, Fourth Edition, by Glenn D. Allinger, Lyle E. Andersen, Cynthia S. Thomas, Karma G. Nelson, and Adrien L. Hess. Copyright © 1977, 1982, 1989, 1999 by THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS.
To order a copy, go to the NCTM store.

A mathematics project consists of all the effort expended in solving a problem, exploring an idea, researching a hypothesis, or applying a mathematical principle from beginning to end—that is, the initial planning, the study, the exhibit, and the written or audiovisual report.

Types of Projects

Projects are usually of three general types. They are:

1. documentation projects, which involve finding out information about a subject, a person, or a discovery made by someone else and reporting on the findings;

2. documentation/experimentation projects, which go beyond documentation and include the answering of a question or proving of a conjecture or hypothesis; and

3. experimentation projects, which involve conducting, analyzing, and reporting on a carefully controlled, inductive experiment or the development of a deductive proof.

Choosing a Project

The best projects grow out of students' interests; therefore, they should be chosen by the students themselves. The selected projects should express the students' broadest knowledge and highest skills, and they should be enjoyable to do.

Teachers' suggestions, mathematics club meetings, demonstrations, lists of previous projects, and so on help stimulate students' interest. Today's advances in technology have created numerous new opportunities that students can use when looking for projects. Searching the World Wide Web might lead to exciting opportunities involving topics such as space travel or newly created mathematics (i.e. dynamical systems and chaos theory), which may be applied to real-world problems.

Research Methods

Once a specific idea or problem has been chosen, it should be thought through carefully. The successive steps should be enumerated and possible pitfalls noted. A written record should be kept at every stage of the project.

When the preliminary plans are made, the student should begin a comprehensive reading program to broaden his or her understanding of the possibilities and limitations of the proposed project. Extensive use should be made of Web searches, books, and periodicals from all available sources.

Because accurate notes should be kept of all readings, observations, suggestions, and speculations, paper and pencil or a computer should be handy at all times. The notes should include a complete record of sources.

As the exploration of a topic develops, plans can be discussed with other people. Discussing an idea with someone else often gives a new and clearer perspective. Parents, teachers, professional mathematicians, scientists and engineers, community professionals, and other students can often make constructive comments, criticisms or suggestions.

In all research, experimentation, and study, the student should keep in mind the last step in the preparation of the exhibit—the completion of the written report. If all ideas (original or not), guesses (good or bad), and measurements, sketches, and readings are recorded in a notebook as the project develops, the final report will be much easier to write. Specific guidelines are available for competitions. The final report, however, will include the following:

• Title
  Project name, student's name, school, grade

• Statement of the Problem or Investigation
  The essentials of the investigation or research must be clearly stated.

• Table of Contents
  List each component of the paper and the page number where it begins. The table of contents is completed after everything else is done and the pages are numbered.

• Introduction
  One paragraph that gives an overview of the project. This paragraph should give the reader a brief synposis of the project. It contains the hypothesis and an explanation of why the student chose this topic and what he or she hoped to find or prove.

• Background research
  Include what was found in the published materials. Use popular sources such as newspapers and magazines as well as Web sites, books, journals, technical materials, and interviews.

• Procedure
  The procedure, methods, and materials for investigation or experiment may be communicated as lists or written in paragraph form. Drawings may help make the information more clear. Tell exactly what was done. Show all the information found, including the steps that initially led to errors.

• Results
  What are the findings? Are the results consistent with commonly held beliefs? Briefly summarize the results. Be specific. Do not generalize.

• Extensions
  List any new research ideas that this project suggested. Investigations often raise more questions than are answered.

Construction Materials and Methods

Selecting and using materials is an important factor in the success of an exhibit. For a project that requires a computer, a great amount of the software is already available, and an electronic presentation (e.g. PowerPoint audiovisual computer-assisted presentation) would likely enhance the project.

Developing a mathematical model that provides an explanation of the mathematics is strongly encouraged. The exhibit must be designed with several things in mind. From the standpoint of its maker, the exhibit should be—

• functional: It should tell the story of the study and research, and any displayed material should be in harmony with this objective;

• practical: The number of components, the time needed to set them up, and any limits on the contents of the exhibits must be considered;

• portable: The exhibit should be compact, durable, and easy to transport.

Suggested Topics (Grades 6–8)

Calculating Instruments, Measuring Devices, and Computing Methods

The construction and use of calculating instruments and measuring devices is important and worthwhile. However, the mathematical principles being modeled by the device or instrument are even more important. If the student can apply the principle to a new situation or put a measuring device to a new use, the result becomes even more worthwhile.

Devices and instruments such as the soroban, the abacus, the slide rule, and Napier's rods or bones can serve as mathematical models, provide interesting histories about the development of computation tools, and be used in project applications. Some commonly used tools such as the compass, ruler, Vernier calipers, and protractor also have fascinating histories.

Interesting projects can also be developed by learning about the inventor and how the invention was used. John Napier's bones or rods provide us with an excellent example.

• Are Napier's rods still commonly used?

• Where is the abacus still used?

• Is the calculator faster than the abacus?

Students are often intrigued by calculation methods that vary from traditional algorithms. A student might develop a project on alternative algorithms for division.

• Why do scratch, doubling and summing, and lattice methods of multiplication work?

• What are the "Russian peasant" and the "finger method" for multiplication? Why do they work?

• What contributions did Babbage, von Neumann, Kemeny, and Papert make with regard to computers?

• What is the Greenwood method of division?

• How do computers work?

Numeration Systems

Numeration systems are a fertile source of topics for projects. The rudiments of many of the ideas may already be familiar to the students, but the theory behind them and their applications may intrigue them.

A number is a concept of an idea. That idea can be expressed as a numeral using different kinds of symbols. Numerous number systems have been developed through the ages (i.e., Roman numerals, the Egyptian numeration system, and cuneiform are a few examples). The Hindu-Arabic number system that we use originated about A.D. 600.

• What are some ancient methods of recording results of counting?

• What is the Euclid algorithm?

• How are Morse code, Braille, semaphore, and sign language related to numbers, numerals, and number systems?

• Can you create your own number system?

• What is a symmetric numeration system?

Number Bases

Number bases other than base 10 are valuable for students to investigate. Students can create addition and multiplication tables and learn about the field properties for a number system.

The basis for all digital computers is electronic circuits that have only two states—either on or off—which correspond to the binary (base 2) numbers 1 and 0. Even computers that use a hexadecimal system (base 16) are extensions of the binary principle.

• Can negative numbers be used for bases?

• Can fractions be used with bases other than base 10 (e.g. base 2)?

• Can base-10 calculating devices and strategies be used with other number bases?

• What base systems are used in Nim or in a set of cards to tell a person's age?

• Can you build a base-12 calendar?

Sets

The notion of sets, or collections of objects, is nearly as ancient as the human race. It was not until the late 19th century that mathematicians concluded that sets are an important basic mathematical concept.

• What are sets. and how can they be applied to number systems and operations on number systems (arithmetic)?

• How are set theory and electric circuitry related?

• How do the prime numbers relate to the Fibonacci numbers?

• How is set notation used in geometry?

• How are Venn diagrams used to solve problems?

Statistics and Probability

All biological and physical phenomena lend themselves to data collection and the subsequent application of statistics. A study involving one or both of these concepts can be somewhat elementary, or it can be quite complex and extend over several years. Data used in statistics must be obtained by measuring, recording, or weighing various objects and phenomena.

Students could develop a survey, use a survey to create a prediction instrument, and represent the results with statistical representations (e.g. histograms, graphs, etc.). Students could also use and interpret statistical procedures such as scatterplots, box-and-whisker plots, and stem-and-leaf plots to represent their data. Investigations, which include carefully designed experiments, will help students see practical uses and possible misuses of statistics.

• Are measures of central tendency (mean, median, and mode) equivalent?

• What are measures of dispersion?

• Into what kind of statistical pattern do errors of measurement fall?

• Why are random numbers useful when simulating the likelihood of events?

• What are lines useful on scatterplots?

• What does the Law of Large Numbers imply about flipping a coin?

Paper Folding

Paper folding can be effectively used to develop or extend a project in mathematics at the middle school level. Paper folding and creasing can be used, with certain assumptions, to create any construction that can be made with traditional tools. The construction of regular polygons with strips of paper is intriguing. The method of building a "hexaflexagon" can be shown to unify algebra and geometry. This method can be used to find the area of a triangle.

• What assumptions are made when paper folding is used to construct geometric figures? Reference.

• What polygons and polyhedrons can be made by paper folding?

• How can paper folding be used to show certain mathematical fallacies?

• How can paper folding be extended to knot theory?

• How can some conjectures be proved with the help of paper-folding techniques?

Curve Stitching

Curves can be constructed with a series of line segments by using a straightedge, paper folding, string, or geometry visualization software. Curve stitching can also lead to the concepts of calculus and limits. For example, a line representing a function can be thought of as a collection of very short line segments, and the area under a curve can be pictured as a sum of rectangles that are infinitesimally narrow.

• How can curves be formed by straight lines?

• Can curve stitching be used to make three-dimensional models of curves?

• What is the envelope of a curve?

• How can paper folding be extended to knot theory?

• How do aborigines of Australia, Navajos and Eskimos of North America, and Batwa pygmies of Africa use string art in their cultures?

Optical Illusions

Seeing is believing, but sometimes our eyes deceive us! Some illusions have more than one explanation. As research and experiments are done, ideas about optical illusions change. Optical illusions are used in many ways. Illusions are an essential tool in some types of artwork. Special mirrors can produce optical illusions that make us laugh. Occasionally, illusions lead us to erroneous conjectures.

• Why do patterns cause illusions?

• Can optical illusions be extended to three dimensions?

• What tricks depend on optical illusions?

• How are optical illusions used in camouflaging, advertising, dress design, and architecture?

• Can you create "magic" using optical illusions?

Motion Geometry

Many people believe that the most useful everyday geometry involves slides, flips, turns, and combinations of these moves. In 1872, Felix Klein, a 23-year-old professor at the University of Erlangen, defined geometry as the study of properties of figures that do not change when a figure is transformed. This geometry is called motion of transformation geometry.

• What are everyday examples of motion geometry?

• How can the computer language called Logo and the movement of the turtle teach use about motions?

• How can motions be duplicated using the compass, straightedge, or turtle teach us about motions?

• How does a kaleidoscope use motion or transformation geometry?

• Can you create a four-line geometry?

Geometry

Ancient Babylonians used geometric principles to measure land areas. This is reflected in the name geometry, which literally means "land measure." The study of geometric shapes and their properties is an essential component of mathematics that is rich in concepts and applications.

The importance of geometry goes well beyond its everyday uses or even its many applications in architecture, engineering, and design. An understanding and appreciation of geometry is basic to understanding and appreciating mathematics.

• What are projective, descriptive, and differential geometries, and how are they similar and different?

• What regular n-gons tessellate the plane? Why?

• Why does Pick's Theorem work?

• Why are isoperimetric figures interesting?

• How can geometry be used to measure distances indirectly?

Discovering geometric theorems intuitively by using a calculator or computer geometry software package (e.g. Geometer's Sketchpad) has tremendous potential.

• What do you notice about the areas of the four small triangles that results when you connect the midpoints of the sides of a triangle?

• What is the sum of the interior angles of a triangle, quadrilateral, decagon, . . . , n-gon? Prove your conjecture.

Functions

A function is a basic concept. It is useful to describe it in several ways, including a collection of ordered pairs, points on a two-dimensional graph, or a specific formula stating a relationship.

• How is a piecewise function determined?

• Can all functions be represented with a diagram or a model?

• Are all relations functions?

• How did Galileo define functions for physical sciences?

• What can be expected if a nonlinear function is iterated so that the value for every iteration is included sequentially in the next one?

• Given a set of data points on a plane, can you find a function that contains every data point?

Patterns and Art

One of the earliest mathematical structures that students explore is patterning, usually through sounds and geometric shapes. That structure is extremely valuable when extending patterning to algebraic thinking. Mathematics is sometimes referred to as the science of patterns. Patterns in mathematics extend naturally into discussion and explorations of patterns in art and nature. The works of M.C. Escher can foster curiosity for many students. Reference. One example of patterning is the classic handshake problem. Patterning extends to symmetry and a rich source of projects can be found in the symmetry of chaos.

• What patterns are found in common algebraic manipulations?

• Analyze the wall patterns found in the Alhambra.

• How do the legal moves in a 15-puzzle correspond to permutations?

• How do Native American star quilts, Amish quilts, and West African Kente cloth compare?

• What are the figurate numbers, and how do they relate to dot patterns?

Games and Puzzles

Games have played a significant and colorful role in human history. Some of the oldest recorded games have been found on Egyptian tombs. Puzzles can be used to arouse students' interest in mathematical activities and encourage creative approaches to problem solving.

• What methods and strategies are used in games throughout the world?

• At what point do you know that you have won a game?

• Is there an advantage to being the first or last player?

• Can you develop a winning strategy for tic-tac-toe? Other games?

• What are the figurate numbers, and how do they relate to dot patterns?

Calendar and Time

All calendars do not look like the ones now used by people in the United States. An investigation into various historical calendars can lead into a study of the development of numbers.

• Why isn't our calendar divided into months with equal numbers of days?

• How did Pope Gregory XIII solve the leap-year problem?

• How did the Babylonians, Romans, and Egyptians keep track of days?

• How do African calendars compare?

• Are apparent solar time, mean solar time, and sidereal time equal?

Topology, Knots, and Surfaces

The study of topology is explorations into properties of geometric figures that are stretched, shrunk, twisted, or somehow distorted. Topology has been described as "rubber-sheet" geometry. Investigations into graph diagrams and knots and their surfaces will lead to very interesting projects. In 1736, Leonhard Euler used circuit tracing to solve the Königsberg Bridge problem, which had intrigued mathematicians for years. His methods of circuit tracing are used today in many mathematical explorations.

• What is a Möbius strip, a Klein bottle, or Jordan curve? Reference 1. Reference 2.

• How could the letters of the alphabet be sorted with respect to topology?

• What is the four-color theorem? Reference.

• How are maps and graph diagrams related?

• Does Euler's formula hold for any connected graph drawn on a sphere?

Algebra

Algebra includes five components: mathematical language; patterns, relations, and functions; multiple representations; modeling; and structure. These components share equal importance in the middle school. Mathematical language includes the use of variables and symbols. The concept of a function develops as one of the most important ideas as students learn to work with generalizations. A problem can be represented numerically, graphically, and symbolically. Mathematical modeling helps students make connections to real-world situations and includes concrete materials such as charts, tables, and simulations. Organizing algebra as a structure means that students think about how number systems are built from axioms such as the associative property.

• Are leveling payments cost effective?

• How do the length of a side and the area of a square relate?

• Analyze the resiliency of a ball's bounce.

• How can algebra be used to perform a mathematics trick?

• How fast can a group of people holding hands pass a squeeze?

• How can problems involving functions be solved numerically, graphically, and symbolically?

Additional Ideas

Check the MatheMUSEment articles about math in everyday life for additional project ideas.