Additional Project Ideas

• Money Problems
  Math can make you money! If you understand some basic math, you can make good decisions about how to keep, spend, and use your hard earned dollars. Try an experiment comparing the same balance in different types of bank accounts. How much better is a savings account than a checking account? What difference does the interest rate make? Which is better, an account that earns compound or simple interest? Can you compare the short and long term costs of borrowing money compared to saving the cash for a purchase? (ING, 2006) Use mathematical arguments to answer classic questions like, "Which is more: one million dollars, or one penny the first day, double that penny the next day, then double the previous day's pennies and so on for a month?" (Dr. Math, 2006)
  • Dr. Math, 2006. "Doubling Pennies," The Math Forum, Universtiy of Drexel School of Education. [accessed: 5/1/06] http://mathforum.org/dr.math/faq/faq.doubling.pennies.html
  • ING, 2006. "Planet Orange," sponsored by ING Direct. [accessed: 5/1/06] http://www.orangekids.com/

• Scale Models
  Many industries rely on scale models to develop new products and designs. Architects, industrial designers, artists, clothing designers, and car manufacturers all use scale models. Each model is built to a scale that relates the actual object to the model through a ratio. Can you determine a formula for constructing a scale model? You can use your formula to make a model of your house, school, neighborhood, or town (CUBE, 2002). You can make scale models of the Wright Brothers aircraft designs from 1900-1903. (Storm and Benson, 2003) You can also do the opposite, blow a model up into a life size object. Can you determine a formula to convert a model into life size? You can use your formula to evaluate if a model is realistic. Try comparing toy cars, dolls, dinosaurs, or action figures to life size objects. Are the models realistic? (McCoy, 2004; Fassett and Millbyer, 2006)
  • CUBE, 2002. "CUBE Lesson Plans," The Center for Understanding the Built Environment (CUBE). [accessed: 5/1/06] http://www.cubekc.org/lessons.html
  • Fassett, C. and Millbyer, L. 2006. "Dinosaur Activity for Proportions," Oswego City School District Regents Exam Prep Center. [accessed: 5/1/06] http://regentsprep.org/Regents/math/scale/Tdino.htm
  • McCoy, L. P., 2004. "Project Potpourri: Implementing Standards-Based Instruction," Wake Forest University. [accessed: 5/1/06] http://www.wfu.edu/~mccoy/mprojects.pdf
  • Storm, R., and Benson, T. 2003. "Models of the Wright Brothers' Aircraft (1900-1903)," Glenn Research Center (GRC), National Aeronautics and Space Administration (NASA). [accessed: 5/1/06] http://wright.nasa.gov/ROGER/models.htm

• Data Models
  Math is used by many different types of scientists to model phenomenon and evaluate data from an experiment. By building mathematical models scientists can understand how different physical, chemical, and biological processes are affected by different variables. The most important tools are: making a graph to give a visual representation of the relationships between your variables and making an equation to give a way of computing the relationships between your variables. Find a source of data, either from an online database or from your own experiment, and make a mathematical model. Try to make a graph and an equation for your data. How good is it at predicting additional results? Is the relationship linear, dynamic, or exponential? Is your data positively or negatively correlated? Are there any outliers and how do you decide if you should get rid of them? Can you show how to use a graph to evaluate sources of error in an experiment? Can you use your model to propose a hypothesis or theory for the relationship? (Wattenburg, 1998)

  Wattenburg, F., 1998. "Mathematical Modeling in a Real and Complex World," Mantana State University, Department of Mathematics. [accessed: 5/1/06] http://www.math.montana.edu/frankw/ccp/modeling/topic.htm

• Topologies
  What do knots, maps, mazes, driving directions, and doughnuts have in common? The answer is topology, a branch of mathematics that studies the spatial properties and connections of an object. Topology has sometimes been called rubber-sheet geometry because it does not distinguish between a circle and a square (a circle made out of a rubber band can be stretched into a square) but does distinguish between a circle and a figure eight (you cannot stretch a figure eight into a circle without tearing) (Wikipedia contributors, 2006). A common joke is that topologists are people who don't know the difference between a coffee cup and a doughnut. A project in topology can have many forms. Can "Euler's Solution" help you efficiently run your errands? Can you figure out the number of possible routes to get to school? What is the minimum number of colors needed to color in a U.S. map so that no two states that are touching have the same color? Use topology to untangle knots or to discover knots that cannot be untangled. Use topology to solve mazes, draw circuit diagrams, make phylogenetic trees, or fold origami! The possibilities are endless... (Britton, 2006)
  • Britton, J., 2006. "R-U-B-B-E-R Geometry (Topology)," Camosun College, Victoria, BC, Canada. [accessed: 5/1/06] http://britton.disted.camosun.bc.ca/jbrubbergeom.htm
  • Wikipedia contributors, 2006. "Topology," Wikipedia, The Free Encyclopedia. [accessed 5/20/06] http://en.wikipedia.org/w/index.php?title=Topology&oldid=59314342

• Fractals
  A fractal is, "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced/size copy of the whole" (Mandelbrot, 1982). There are many different fractal patterns, each with unique properties and typically named after the mathematician who discovered it. A fractal increases in complexity as it is generated through repeated sets of numbers called iterations. There are many interesting projects exploring fractal geometry that go beyond the Exploring Fractals Science Buddies project listed above. For example, can you show that the perimeter of the Koch Fractal increases infinitely even though the area is finite? (Lanius, 2004) Can you use fractal geometry to investigate and model objects from nature? Try matching up a fractal pattern with the natural geometry of clouds, ferns, flowers, shells, or trees (Frame and Mandelbrot, date unknown). Can you program a java applet to draw different fractals? (Devaney, 1999)
  • Devaney, R.L., 1999. "Java Applets," Boston University, Dept. of Mathematics and Statistics. [accessed: 5/1/06] http://math.bu.edu/DYSYS/applets/index.html
  • Frame, M. and Mandelbrot, B. B. Date unknown. "Fractal Geometry Panorama," Mathematics Department, Yale University. [accessed: 5/1/06] http://classes.yale.edu/Fractals/Panorama/
  • Lanius, C., 2004. "The Koch Snowflake," Rice University, Department of Mathematics. [accessed: 5/1/06] http://math.rice.edu/~lanius/frac/koch.html
  • Mandelbrot, B., 1982. The Fractal Geometry of Nature. New York, NY: W. H. Freeman & Co.

• Origami
  How do you turn a 2-dimentional piece of paper into a 3-dimentional work of art? Origami, the classical art of Japanese paper folding, is loaded with mathematical themes and concepts. What are the common folds in origami, and how do they combine to create 3-dimentional structure? Can you classify different types of origami into classes based upon the types of folds they use? Can you show Kawasaki's Theorem, that if you add up the angle measurements of every other angle around a point, the sum will be 180? (Andersen, 2004) Can you show that Huzita's Axioms, a set of seven rules of paper folding, are true? (Wikipedia contributors, 2006)
  • Andersen, E. M., 2004. "Origami and Math," Paperfolding.com [accessed: 5/1/06] http://www.paperfolding.com/math/
  • Wikipedia contributors, 2006. "Huzita-Hatori axioms," Wikipedia, The Free Encyclopedia. [accessed 5/1/06] http://en.wikipedia.org/w/index.php?title=Huzita-Hatori_axioms&oldid=56396768

• Playing Games
  Almost all of the games we play are based on math in some way or another. Card games, board games, and computer games are designed using statistics, probabilities, and algorithms. Begin by reading about games and game theory. Then you can choose your favorite game and investigate the mathematical principles behind how it works. Can combinatorial game theory help you to win two-player games of perfect knowledge such as go, chess, or checkers? (Weisstein, 2006; Watkins, 2004) In a multi-player game like Dots and Boxes, does the first player have an advantage? Can you formulate a winning strategy against an opponent? In single-player games can you beat the game with an algorithm? Can you invent your own game using mathematical concepts? Can you describe the math behind traditional or cultural games? (McCoy, 2004) Artificial Intelligence: Teaching the Computer to Play Tic-Tac-Toe.
  • McCoy, L. P., 2004. "Project Potpourri: Implementing Standards-Based Instruction," Wake Forest University. [accessed: 5/1/06] http://www.wfu.edu/~mccoy/mprojects.pdf
  • Watkins, J., 2004. Across the Board: The Mathematics of Chessboard Problems. Princeton, NJ: Princeton University Press.
  • Weisstein, E.W., 2006 "Games." From MathWorld--A Wolfram Web Resource. [accessed: 5/1/06] http://mathworld.wolfram.com/topics/Games.html

• Playing Music
  Music has many mathematical elements in it: rhythm, pitch, scale, frequency, interval, and ratio. There are many ways to turn these elements into a science fair project. You can investigate how the scale is based upon a special type of number sequence called a Harmonic Series. Another scale used by Bach, called the "Well-Tempered-Scale" or the "Equal-Tempered-Scale", is based upon a series. How are these mathematical series and ratios related to notes, chords, intervals, and octaves? You can show how these concepts are applied to generating discreet notes and pitches with different instruments. You can also investigate rhythm and time in music: can you find a way to represent your favorite rhythms numerically? (Rusin, 2004)

  Rusin, D., 2004. "Mathematics and Music," Northern Illinois University, Department of Mathematical Sciences. [accessed: 5/1/06] http://www.math.niu.edu/~rusin/uses-math/music/index.html

• Solving Logic Problems
  You may know Lewis Carroll as the author of Alice in Wonderland, but did you know that in real life he was a mathematician who studied symbolic logic and logical reasoning? How can math help you solve Lewis Carroll's Logic Game? (Bogomolny, 2006) How are algorithms for solving the game Sudoku similar to solving a logic problem? (Hayes, 2006) For the super-advanced mathematical genius, try to evaluate currently available, logic-based computational tools, or design a better one! (SharpToolBox.com, 2006)
  • Bogomolny, A., 2006. "Lewis Carroll's Logic Game," Cut The Knot Software, Inc. [accessed: 5/1/06] http://www.cut-the-knot.org/LewisCarroll/index.shtml
  • Hayes, B., 2006. "Unwed Numbers: The mathematics of Sudoku, a puzzle that boasts "No math required!" American Scientist Online: Vol 94, number 1, page 12. [accessed: 5/1/06] http://www.americanscientist.org/template/AssetDetail/assetid/48550?&print=yes
  • SharpToolBox.com, 2006. "Mathematics, Logic, Artificial Intelligence, Neural Networks, Constraint programming, Rule-based Engines," MadGeek, SharpToolBox.com. [accessed: 5/1/06] http://sharptoolbox.com/categories/math-logic-ai-rules

• Magic Squares
  A magic square is an arrangement of numbers from 1 to n² in an n x n matrix. In a magic square each number occurs exactly once such that the sum of the entries of any row, column, or main diagonal is the same. You can make several magic squares and investigate the different properties of the square. Can you make an algorithm for constructing a Magic Square? Can you show that the sum of the entries of any row, column, or main diagonal must be n(n²+1)/2? Are there any other hidden properties of a Magic Square? Show the differences between special instances of the Magic Square, like the Lo Shu, Durer, Ben Franklin, or Sator Magic Squares. Can magic squares be constructed in 3 dimensions? You can also investigate other shapes, like magic circles and stars (Alejandre, 2006; Pickover, 2002). Or test the question, "Is there really no math in Sudoku?" (Hayes, 2006)
  • Alejandre, S., 2006. "Magic Squares," The Math Forum, University of Drexel School of Education. [accessed: 5/1/06] http://mathforum.org/alejandre/magic.square.html
  • Hayes, B., 2006. "Unwed Numbers: The mathematics of Sudoku, a puzzle that boasts "No math required!" American Scientist Online: Vol 94, number 1, page 12. [accessed: 5/1/06] http://www.americanscientist.org/template/AssetDetail/assetid/48550?&print=yes
  • Pickover, C. A., 2002. The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures Across Dimensions. Princeton, NJ: Princeton University Press.

• Around the World: The Geometry of Shooting Baskets
  Take shots at a set distance from the basket, but systematically vary the angle to the backboard. For a basic project: How do you think your success rate will vary with angle? Draw a conclusion from your experimental results. A bar graph showing success rate at different angles can help to illustrate your conclusion. For a more advanced project: Use your knowledge of geometry and basketball to come up with a mathematical expression to predict your success rate as a function of angle (measured as a percentage of your success rate with straight-on shots)? How well does your prediction agree with your actual results? (idea from Goodstein, 1999, pp. 103–105.)

  Goodstein, Madeline. Sports Science Projects: The Physics of Balls in Motion. Berkeley Heights, NJ: Enslow Publishers, 1999.

• Geometry of Goal-Scoring
  Block off one-third of a soccer net with a cone, 5-gallon bucket or some other suitable object. Shoot into the smaller side from a set distance, but systematically varying the angle to the goal line. Take enough shots at each angle to get a reliable sample. How does success vary with angle? For a basic project: How do you think your success rate will vary with angle? Draw a conclusion from your experimental results. A bar graph showing success rate at different angles can help to illustrate your conclusion. For a more advanced project: Use your knowledge of geometry and soccer to come up with a mathematical expression to predict your success rate as a function of angle (measured as a percentage of your success rate with straight-on shots)? How well does your prediction agree with your actual results? (idea from Gardner, 2000, pp. 108–110.)

  Gardner, Robert. Science Projects About the Physics of Sports. Berkeley Heights, NJ: Enslow Publishers, 2000.

• Statistical Significance: Using a t-Test
  Sunspot activity has been monitored continuously since about 1700. The historical data shows that sunspot activity rises and falls in a roughly 11-year cycle. This project shows you how you can use a spreadsheet program to perform both graphical and statistical analysis to look for patterns in cyclical data. You'll learn how to use a t-test, which is a measure of statistical signficance. Sunspot Cycles.

• Frequency Histograms
  Can you remember what the weather was like last week? Last year? Here's a project that looks at what the weather was like for over a hundred years. You'll use historical climate data to look at moisture conditions in regions across the contintental U.S. You'll use a spreadsheet program to calculate the frequency of different moisture conditions for each region and make graphs for comparison. Which part of the country has the most frequent droughts? The most frequent periods of prolonged rain? The most consistent precipitation? Here's one way to find out. Dry Spells, Wet Spells: How Common Are They?

• Relationships Between Variables: Using Correlation and Linear Regression
  Here's a project that combines sports and math. You'll learn how to use correlation analysis to choose the best team batting statistic for predicting run-scoring ability (Albert, 2003). You'll also learn how to use a spreadsheet to measure correlations between two variables. Which Team Batting Statistic Predicts Run Production Best?

Resources

Sources for Additional Project Ideas

• Albert, Jim, 2003. Teaching Statistics Using Baseball. Washington, D.C.: The Mathematical Association of America.
• Alejandre, S., 2006. "Magic Squares," The Math Forum, University of Drexel School of Education. [accessed: 5/1/06] http://mathforum.org/alejandre/magic.square.html
• Andersen, E. M., 2004. "Origami and Math," Paperfolding.com [accessed: 5/1/06] http://www.paperfolding.com/math/
• Bogomolny, A., 2006. "Lewis Carroll's Logic Game," Cut The Knot Software, Inc. [accessed: 5/1/06] http://www.cut-the-knot.org/LewisCarroll/index.shtml
• Britton, J., 2006. "R-U-B-B-E-R Geometry (Topology)," Camosun College, Victoria, BC, Canada. [accessed: 5/1/06] http://britton.disted.camosun.bc.ca/jbrubbergeom.htm
• CUBE, 2002. "CUBE Lesson Plans," The Center for Understanding the Built Environment (CUBE). [accessed: 5/1/06] http://www.cubekc.org/lessons.html
• Devaney, R.L., 1999. "Java Applets," Boston University, Dept. of Mathematics and Statistics. [accessed: 5/1/06] http://math.bu.edu/DYSYS/applets/index.html
• Dr. Math, 2006. "Doubling Pennies," The Math Forum, Universtiy of Drexel School of Education. [accessed: 5/1/06] http://mathforum.org/dr.math/faq/faq.doubling.pennies.html
• Fassett, C. and Millbyer, L. 2006. "Dinosaur Activity for Proportions," Oswego City School District Regents Exam Prep Center. [accessed: 5/1/06] http://regentsprep.org/Regents/math/scale/Tdino.htm
• Frame, M. and Mandelbrot, B. B. Date unknown. "Fractal Geometry Panorama," Mathematics Department, Yale University. [accessed: 5/1/06] http://classes.yale.edu/Fractals/Panorama/
• Gardner, Robert. Science Projects About the Physics of Sports. Berkeley Heights, NJ: Enslow Publishers, 2000.
• Goodstein, Madeline. Sports Science Projects: The Physics of Balls in Motion. Berkeley Heights, NJ: Enslow Publishers, 1999.
• Hayes, B., 2006. "Unwed Numbers: The mathematics of Sudoku, a puzzle that boasts "No math required!" American Scientist Online: Vol 94, number 1, page 12. [accessed: 5/1/06] http://www.americanscientist.org/template/AssetDetail/assetid/48550?&print=yes
• ING, 2006. "Planet Orange," sponsored by ING Direct. [accessed: 5/1/06] http://www.orangekids.com/
• Lanius, C., 2004. "The Koch Snowflake," Rice University, Department of Mathematics. [accessed: 5/1/06] http://math.rice.edu/~lanius/frac/koch.html
• Mandelbrot, B., 1982. The Fractal Geometry of Nature. New York, NY: W. H. Freeman & Co.
• McCoy, L. P., 2004. "Project Potpourri: Implementing Standards-Based Instruction," Wake Forest University. [accessed: 5/1/06] http://www.wfu.edu/~mccoy/mprojects.pdf
• Pickover, C. A., 2002. The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures Across Dimensions. Princeton, NJ: Princeton University Press.
• Rusin, D., 2004. "Mathematics and Music," Northern Illinois University, Department of Mathematical Sciences. [accessed: 5/1/06] http://www.math.niu.edu/~rusin/uses-math/music/index.html
• SharpToolBox.com, 2006. "Mathematics, Logic, Artificial Intelligence, Neural Networks, Constraint programming, Rule-based Engines," MadGeek, SharpToolBox.com. [accessed: 5/1/06] http://sharptoolbox.com/categories/math-logic-ai-rules
• Storm, R., and Benson, T. 2003. "Models of the Wright Brothers' Aircraft (1900-1903)," Glenn Research Center (GRC), National Aeronautics and Space Administration (NASA). [accessed: 5/1/06] http://wright.nasa.gov/ROGER/models.htm
• Watkins, J., 2004. Across the Board: The Mathematics of Chessboard Problems. Princeton, NJ: Princeton University Press.
• Wattenburg, F., 1998. "Mathematical Modeling in a Real and Complex World," Mantana State University, Department of Mathematics. [accessed: 5/1/06] http://www.math.montana.edu/frankw/ccp/modeling/topic.htm
• Weisstein, E.W., 2006 "Games." From MathWorld--A Wolfram Web Resource. [accessed: 5/1/06] http://mathworld.wolfram.com/topics/Games.html
• Wikipedia contributors, 2006a. "Huzita-Hatori axioms," Wikipedia, The Free Encyclopedia. [accessed 5/1/06] http://en.wikipedia.org/w/index.php?title=Huzita-Hatori_axioms&oldid=56396768
• Wikipedia contributors, 2006b. "Topology," Wikipedia, The Free Encyclopedia. [accessed 5/20/06] http://en.wikipedia.org/w/index.php?title=Topology&oldid=59314342

Math Links

• There are many websites that explain basic probability theory. Here's a good one to get you started: http://library.thinkquest.org/11506/learn.html.
• If you'd like to read more about mathematics in general, try: http://en.wikipedia.org/wiki/Mathematics.

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