How Can Your Faucet Save Water?

Objective

In this experiment you will test several water saving products to measure how much they reduce the flow of water through the faucet.

Introduction

How much water does your family use? Trying to use less water is important because water is a limited resource. Learning about water conservation issues and water saving tips can help you use less water. Many water saving tips focus on your habits and actions, like not letting the water run while you brush your teeth. But there are also many things that can be done to save water that have to do with how a structure is built.

The plumbing fixtures in your house can be water wasters or savers, depending upon the products being used. One simple way to reduce water use in your home is to install low-flow faucets or aerators in your kitchen and bathroom sinks. These products save water by reducing the water flow and adding air to the water stream so that less water comes out of the tap over the time you have it on. These products are usually simple to install and can be bought at the hardware store.

In this experiment you can test several water saving products to see which ones do the best job saving water. You will do this by measuring the flow of water in gallons per minute (GPM).

Terms, Concepts and Questions to Start Background Research

To do this type of experiment you should know what the following terms mean. Have an adult help you search the internet, or take you to your local library to find out more!

• water use and conservation
• flow rate
• gallons per minute (GPM)
• low-flow faucet
• aerator

Questions

• How much water can a low-flow faucet or aerator save?
• Which type of product works best?
• Can you estimate the potential savings for a home, school, or large office building?

Bibliography

• Learn all about water conservation from this website developed by the City of Tampa, Florida. Learn about the issues, become a "Water Ambassador", or read award winning projects. They even offer a special "Drinking Water Award" to the best water conservation project each year! Check it out:
  City of Tampa, 2006. "Water Education and Conservation," Tampa, FL. [accessed June 23, 2006] http://www.tampagov.net/dept_Water/conservation_education/
• AWWA, 2006. "Water Wiser," Denver, CO: American Water Works Association (AWWA). [accessed June 23, 2006] http://www.awwa.org/waterwiser/
• This site has a java applet you can use to make printable, color graphs of your data:
  NCES, 2006. "Create a Graph," National Center for Education Statistics (NCES) U.S. Dept. of Education. [accessed March 3, 2006] http://nces.ed.gov/nceskids/createagraph/

Materials and Equipment

• several faucet aerator attachments from different manufacturers
• kitchen or bathroom sink
• large collection bowl (not too big to fit in the sink though!)
• a large measuring cup
• stopwatch
• masking tape
• tools (a wrench and pliers)

Experimental Procedure

1. Shop for low-flow faucet aerators from the hardware store. Choose at least 4 different water saving products.
2. Read the instructions and have an adult available to help you install the hardware on a kitchen or bathroom faucet.
3. Install the first faucet product by following the manufacturer's instructions. After installing each aerator you will test it for water use before installing the next product. You should do at least three separate tests for each product and then average your results to get better data.
4. Test each faucet product after installing it by running water for 10 seconds from the faucet into a collection bowl. To keep the flow of water constant, mark where you turn the faucet with a piece of masking tape so that you turn it to the same place each time. To keep the time constant, set a stop watch to 10 seconds and turn off the water when it beeps.
5. Pour the water from the collection bowl into a large measuring cup and record the amount of water in a data table:

   Product	Volume		Time		Gallons per Minute (GPM)
   	Ounces	Gallons	Seconds	Minutes

6. You collected water for 10 seconds, but usually the amount of water flow is given in gallons per minute. So you will need to do some calculations to convert your measurements.
7. First, convert the volume measurement. Since you measured in ounces, and there are 128 ounces in a gallon, then divide your answer by 128 to get your measurement in gallons.
8. Now convert the time from seconds to minutes. Since there are 60 seconds in a minute, if you divide your answer by sixty it will be in minutes.
9. To calculate the rate of flow in gallons per minute (GPM) divide the measurement of volume in gallons by the measurement of time in minutes.
10. After you have all of the products measured in gallons per minute, make a graph of your data. You can make your graph by hand, or use a site like Create A Graph to make your graph on the computer.
11. Compare the results using your graph. Which products worked the best?

Variations

• Calculate the weekly, monthly, and yearly savings of using a faucet aerator. Ask your parent for a water bill and find out how much they spend on each gallon of water. Then use your results to calculate the potential savings. Do the potential savings fall short of, meet, or exceed the initial cost? How long does it take to recover the initial cost with savings?
• Should you shower or take a bath? Try a similar experiment to see if a shower or bath will use more water. You can also find low-flow shower heads to test with a similar experiment. What is the best way to bathe and conserve water?
• To flush or not to flush, that is the question! Test these water saving household toilet tips. How much water does your toilet use for each flush? How much water could you save by flushing every other time? Can sinking a brick in your tank help reduce the water used?

Credits

Sara Agee, Ph.D., Science Buddies

Can the Color of Your House Reduce Your Energy Bill?

Objective

In this experiment you will investigate if the color of a structure affects the temperature inside the structure when in different environments.

Introduction

How does the color of your house affect the temperature inside of it? Consider this explanation by Jill Morton at ColorMatters.com:

"When summer comes to your Hemisphere, can color reduce the energy consumption in your home or business? Consider this: Would you be cooler wearing a light shirt or a dark shirt on a hot sunny day? If your science teacher or parents didn't convince you, the scientific fact is that white reflects the radiant energy rays of the sun and black absorbs them.

The same principle has a significant impact on a house. The hotter the roof, the hotter the rooms below. Light colored roofing and /or reflective coatings are like a white shirt for your house because they reflect radiant energy. Even a light gray hue is better than a blue or green." (Morton, 2006)

In this experiment you will test this idea by painting shoe boxes with light, medium, or dark colors to model painted houses. Then you will put the boxes in warm and cool environments and measure the temperature inside each box. Will color matter?

Terms, Concepts and Questions to Start Background Research

To do this type of experiment you should know what the following terms mean. Have an adult help you search the internet, or take you to your local library to find out more!

• surface temperature
• indoor temperature
• outdoor temperature
• reflection
• absorption

Questions

• How different are the surface, indoor, and outdoor temperatures of a structure?
• How do colors that reflect or absorb light affect the indoor temperature of a structure?
• Which colors are best for saving energy in hot or cold climates?

Bibliography

• This project idea was adapted from a science fair project by William S. in Mrs. Hannemann's 3rd grade class at Williams Elementary in Rockledge, Florida:
  S., William, 2003. "Does the color of your house affect the temperature inside of it?" Williams Elementary, Rockledge, Florida. [accessed June 23, 2006]
  http://www.energywhiz.com/3-5/SCIFAIR/2003WilliamS.htm
• Morton, J. L., 2006. "Color and Energy Matters," ColorMatters.com [accessed June 23, 2006]
  http://www.energywhiz.com/
• This site has a java applet you can use to make printable, color graphs of your data:
  NCES, 2006. "Create a Graph," National Center for Education Statistics (NCES) U. S. Dept. of Education. [accessed March 3, 2006] http://nces.ed.gov/nceskids/createagraph/

Materials and Equipment

• 3 shoe boxes
• 4 thermometers
• heat lamp
• large tray
• ice
• rock salt
• plastic bag (preferably white)

Experimental Procedure

1. Collect 3 shoe boxes of the same size.
2. Paint one box white, one box gray, and one box black. Allow the paint to dry completely.
3. Put a thermometer in each box, place the boxes on a table and place another external thermometer on the table.
4. Record the "Starting Temperature" of each thermometer in a data table:

   Box	Color	Starting Temperature (oC)	Room Temperature (oC)	Heated Temperature (oC)	Cooled Temperature (oC)
   none	none
   1	White
   2	Gray
   3	Black

5. Put the lids on the boxes and leave at room temperature for 30 minutes.
6. Take each lid off and quickly record the "Room Temperature" in the data table.
7. Put the lids back on and place a heat lamp above the boxes to simulate a warm, sunny day.
8. Arrange the boxes underneath the lamp so that they are all equally distant from the light source. This will be a control to be sure that one box is not getting more light or heat than another box because it is closer to the light source. Put the last thermometer in the center of the boxes to measure the external temperature of the environment.
9. Leave the boxes and the external thermometer under the heat lamp for 30 minutes.
10. Take each lid off and quickly record the "Heated Temperature" in the data table.
11. Were the temperatures the same or different? If the temperatures were different, which box heated up the most or the least?
12. Put the lids back on and keep the heat lamp above the boxes but add a tray of ice beneath the boxes to simulate a sunny winter day. Make the tray of ice by sprinkling a layer of ice with rock salt and covering with a white plastic bag to keep the boxes dry.
13. Arrange the boxes underneath the lamp on the ice tray so that they are all equally distant from the light source and put the thermometer in the center of the boxes.
14. Leave the boxes and the external thermometer on the ice tray under the heat lamp for 30 minutes.
15. Take each lid off and quickly record the "Cooled Temperature" in the data table.
16. Were the temperatures the same or different? If the temperatures were different, which box stayed warmest? Which box cooled off the most?
17. After you have recorded all the temperatures, make a graph of your data. You can make your graph by hand, or use a site like Create A Graph to make your graph on the computer.
18. Compare the results using your graph. Which colors worked the best to reduce temperature changes in which conditions?

Variations

• What about colors other than black, white, and gray? You can make this project more difficult by increasing the number of colors you choose. Most major paint manufacturers can tell you the Light Reflectance Value (LRV) of any color paint chip. White reflects 80% of the light and black reflects 5%. You can conduct your test for a series of colors with different LRV values. Will the temperature increase or decrease with the LRV number of the paint color? Which colors should you choose for hot or cold climates? What about climates with four full seasons?
• House color is only one aspect of house design. Does the material you choose on the outside of your house make a difference? In this experiment you used a painted shoe box, a paper product, to simulate a painted house. What about real materials used to cover houses? Try the experiment using different materials: adobe, stucco, wood, siding, metal, etc. Which materials are the coolest or the warmest? Use a light and dark colored version of each material. Do some materials have more extreme temperature differences than others? Which materials are the most stable against fluctuations? Which type of climate is suited best by each material?
• Another way to save energy is to use insulation, a material that is placed inside the walls of a structure to keep the indoor temperature from fluctuating. Compare different materials used for insulating walls: fiberglass, paper. Try a similar experiment by wrapping the shoe box with each material. Which materials provide the most stable indoor environment? Look up the R-factor for each material. Does it correlate with your results?

Credits

Sara Agee, Ph.D., Science Buddies

This project idea was adapted from a science fair project by William S. in Mrs. Hannemann's 3rd grade class at Williams Elementary in Rockledge, Florida: S., William, 2003. "Does the color of your house affect the temperature inside of it?" Williams Elementary, Rockledge, Florida. [accessed June 23, 2006] http://www.energywhiz.com/3-5/SCIFAIR/2003WilliamS.htm

Pure Mathematics

Wikipedia defines mathematics as "the study of quantity, structure, space and change." With a definition like that, it's easy to see why mathematics is often called "the language of science." Mathematics is essential for analyzing and communicating scientific results, and for stating scientific theories in a way that is clear, succinct, and testable.

If you don't find something that grabs your interest, keep checking back, because we'll be adding new projects in other areas of math!

• The Birthday Paradox
• Throwing You Some Curves: Is Red or Blue Longer?
• The Effects of Card Counting on a Simple Card Game
• What Makes a Team's Winning Percentage Deviate from the Pythagorean Relationship?
• Exploring Fractals
• Additional Project Ideas

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.

Exploring Fractals

Objective

This project has three objectives:

1. Explore. You should explore the complex patterns generated at the edges of the Mandelbrot set, and the relationship between a point in the Mandelbrot set and the corresponding Julia set.
2. Understand. You will need to understand how an image of the Mandelbrot set is generated, and how the image of a Julia set corresponding to a point in the Mandelbrot set is generated. You should also understand the relationship between different points in the Mandelbrot set image and the characteristics of the Julia sets associated with each type of point.
3. Explain. You should be able to explain in your own words what you learned in numbers 1 and 2.

Introduction

What are fractals? Here's an often-used example. Imagine that you want to measure the length of a coastline. Pick any coastline you want, but for this example we'll use the coast of California. There are different ways you could go about making the measurement. For instance, you could take measurements from a map, or you could set off on a family vacation to drive along the whole coast and measure it that way. The first idea is certainly easier, so let's consider that one first.

You'll have to decide on which map to use, and also how to measure the squiggly coastline on that map. Let's say you decide to use a map of the continental United States, and for following the squiggly coastline you're going to use a pair of dividers (like a compass for drawing circles, but with two points instead of a point and a pencil). You use the map scale to adjust the points of your dividers to span 100 km, then you "walk" the dividers along the coastline on the map—from just north of Tijuana, Mexico, to just south of Brookings, Oregon—counting the steps your dividers make. You multiply the number of steps by 100 km, and you've got your answer, right? But wait, along the way you noticed that, with a spacing of 100 km, your dividers didn't quite follow all of the promontories and bays accurately. So you reset the dividers to 50 km, and measure again. This time, you get a bigger number. Which one is right? And then you start thinking, "Hmm, maybe to be really accurate, I need a more detailed map." So you get a map of just California, and you measure yet again, and get an even larger number. You begin to realize that the amount of coastline you measure depends on the size of your ruler.

Fractals are mathematical functions that behave in a similar way. As you "zoom in" on a fractal boundary, you find that each new level of magnification continues to show complex structure. No matter how much you magnify, you never reach a smooth line. Here, for example, are four successive magnifications of regions from a famous fractal, the Mandelbrot set. Starting from the upper left and moving clockwise, each step is a 10-fold magnification of the yellow box region in the previous image.

What makes this so interesting is that the complexity arises from simple mathematical expressions. For example, the images of the Mandelbrot set and Julia sets are generated using the function

z_n+1 = f (z_n) = z_n² + c (Equation 1)

Equation 1 produces a series of numbers (z₀, z₁, z₂, z₃,...). To get the next number in the series (z_n+1), you take the number you have now (z_n), square it, and a constant (c). You keep on doing this, plugging in your newly-generated number to get the next one (this is called "iterating" the function). All you need to get started are two values: z₀ and c. For the Mandelbrot set, z₀ varies and c is fixed; for Julia sets, z₀ is fixed, and c varies.

Probably the hardest part of this whole project is this: the variable z in Equation 1 is a complex number, but don't let that scare you. If you understand how to graph an (x, y) value, you can understand complex numbers (see the Bibliography for more information). For now, just think of each z_n as a pair of numbers (x, y).

So how does Equation 1 end up producing the amazingly complicated Mandelbrot set images? Here's how it works. Remember that Equation 1 produces a series of numbers, and that each series begins with a value for z₀ (c is fixed at 0 for the Mandelbrot set). How each series behaves depends on what you pick for z₀. For values far from zero, the series rapidly approaches infinity. For values closer to zero, the series can remain bounded. In fact, the Mandelbrot set is the set of all z₀ values for which Equation 1 does remain bounded. To generate the images shown above, the computer program treated the rectangular viewing area as if it was a small piece of the plane. So each pixel in that region corresponds to an (x, y) coordinate. For each pixel, that (x, y) value is plugged in for z₀ and Equation 1 is iterated. If the resulting series remains bounded, that pixel is colored black. If the resulting series heads off towards infinity, that pixel is assigned a color (or gray value, as in the examples above).

How does the computer program know if the series is heading off towards infinity? Each time the program iterates Equation 1, it checks to see if the resulting value is inside or outside of a circle of radius 2.0, with its center at the origin. If the value is inside this circle (called the "escape" region—you'll see why in a second), the program goes on to calculate the next value. If the value is outside the circle, it means the series is going to head off towards infinity (it's escaped!), so the program stops iterating Equation 1, colors the pixel in, and moves on to the next one. The color assigned to a pixel whose series has "escaped" depends on how many iterations (steps) it took before the series left the escape region. Every pixel whose series escaped in 1 step is assigned color 1; every pixel whose series escaped in 2 steps is assigned color 2, and so on. One last thing: the program mustn't keep iterating forever, so a limit is set on how many times to iterate for each pixel. If the series hasn't escaped before the limit is reached, the pixel is colored black. It's a simple procedure, but it produces some amazingly complicated images.

Here are some examples for four different starting points (circled in yellow). The white lines show the progression of points in the series. In the first row are points within the Mandelbrot set. For these particular points, an endless, repeating sequence resulted (the iterations end up back at the starting point). In the second row are points in the colored region of the image, thus we know that the corresponding series "escaped." The white lines show how many steps it took. For points further from the edge of the Mandelbrot set, it takes fewer iterations before the series "escapes" (see image at lower left). For points right on the edge of the set, it generally takes a large number of iterations before the series "escapes" (see image at lower right). You can explore this yourself with the Fractal Microscope program (see Bibliography), using its "Orbits" feature.

Do fractals have any uses besides generating intriguing images? As you might expect, there are applications in computer graphics. Fractals can be used to generate more realistic-looking objects from simple algorithms. Fractals have also been used in cell-phone antenna design (see Bibliography). One of the interesting things discoveries in mathematics (and a lot of scientific discoveries, for that matter) is that it can take awhile before someone realizes how to apply the new knowledge. Sometimes the ideas have to percolate through to completely different areas of science before someone realizes "Aha, this is just what I've been looking for to help me solve..." If you find something interesting, go ahead and study it. Do your background research to see how your ideas fit with what is already known. Do your work well, and publish your results. If you think it's interesting, chances are, someone else will find your results interesting, too. Your discoveries may even help someone solve a problem that you don't even know about.

Something that you can have fun exploring is the relationship between a point in the Mandelbrot set and the Julia set corresponding to that point. Remember that both sets come from the same equation, but with different starting values. For the Mandelbrot set, z₀ varies and c is fixed, while the reverse is true for Julia sets. The Bibliography has several different programs you can use to explore how the sets are related, and the Experimental Procedure will help you get started.

Terms, Concepts and Questions to Start Background Research

To do a project on fractals, you should do research that enables you to understand the following terms and concepts:

• fractal
• iteration
• complex number
• Mandelbrot set
• Julia sets

Bibliography

• Three different programs to explore Mandelbrot and Julia sets:
  • The Fractal Microscope lets you zoom in on the Mandelbrot set, explore Julia sets corresponding to points in the Mandelbrot set, and investigate the "escape" behavior of points in and around the Mandelbrot set with its "Orbits" feature:
    http://www.shodor.org/master/fractal/software/mandy/index.html
  • The National Library of Virtual Manipulatives (NLVM) has an applet that helps you explore the relationship between the Mandelbrot set and Julia sets:
    http://nlvm.usu.edu/en/nav/frames_asid_136_g_3_t_3.html
  • Professor David Joyce, Department of Mathematics and Computer Science at Clark University, has written an applet to explore the Mandelbrot set. It has a nice grayscale display option, and also the ability to vary the shape of the "escape" region. You can also explore different generating functions with this applet:
    http://aleph0.clarku.edu/~djoyce/julia/MandelbrotApplet.html
• For an introduction to complex numbers, see:
  http://www.clarku.edu/~djoyce/complex/
• For more on how fractals are generated, see the ThinkQuest site on Fractal Basics:
  http://library.thinkquest.org/3288/basic.html
• What are fractals?
  http://www.jracademy.com/~jtucek/math/fractals.html
• Uses of fractals:
  
  • http://www.jracademy.com/~jtucek/math/uses.html
  • Musser, George, Practical Fractals, Scientific American, July, 1999.

Materials and Equipment

• computer with Internet access
• Web browser
• printer

Experimental Procedure

1. First, do your background research and make sure that you understand how a computer program generates images of the Mandelbrot set. It will make it much easier for you to understand how to use the programs.
2. Use the Fractal Microscope and the NLVM applet to explore both the Mandelbrot set and the Julia sets. Your background research should give you some ideas of things to try to look for relationships between points in the Mandelbrot set and the corresponding Julia sets.
3. Read the instructions for each program. The NLVM applet is simpler, but both programs have controls to help you with your explorations. The instructions for the Fractal Microscope are very helpful. Keep thinking back to how the images are generated. With your understanding of how the algorithms work to make the images, combined with the information in the instructions, you should be able to understand each of the program controls.
4. Take notes as you explore. The Fractal Microscope has a Parameters window that you can open so you can keep track of where you are in the image. Write down the locations of interesting images, so that you can recreate them later.
5. You can also save images from the Fractal Microscope, and download them to your computer for printing. Collect images that show how Julia sets relate to different points in the Mandelbrot set. Make sure to keep track of the starting points.
6. For your display, you can have an image of the Mandelbrot set, surrounded by sample Julia set images, with arrows showing their c value on the Mandelbrot set.
7. Remember objective #3: you should be able to explain what you've learned in your own words. Practice by explaining your results to your parents or a friend.

Variations

• Another way to generate fractal images is called Iterated Function Systems (IFS). Instead of studying Mandelbrot and Julia sets, investigate how IFS-type fractals are generated. The Sierpinski Triangle, Von Koch snowflake, and various fern-like or tree-like fractal shapes are all well-known examples.

Credits

Andrew Olson, Science Buddies