Here is a challenging problem for anyone with an interest in geometry. This project requires background research to solve it, but it is an excellent illustration of visual thinking in mathematics.
Figure 1 below shows a series of circles (iC₁, iC₂, iC₃, ..., iC₃₀), inscribed inside an arbelos. What is an arbelos? The arbelos is the white region in the figure, bounded by three semicircles. The diameters of the three semicircles are all on the same line segment, AC,…
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Math_p011

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Time Required

Average (6-10 days)

Prerequisites

Good grasp of Euclidean geometry, a firm understanding of how to construct a mathematical proof, determination

Want to send coded messages to your friends? Can you write a simple letter-substitution encryption program in JavaScript? How easy is it to break the simple code? Can you write a second program that "cracks" the letter-substitution code? Investigate other encryption schemes. What types of encryption are least vulnerable to attack?
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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,…
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Can computer games be used to educate? Pick a game that incorporates educational material. There are a variety of educational computer games on the market in the areas of math, history, science, and language, as well as many simpler versions free on the Internet. Make up a test based on the subject area covered by the computer game. Have your volunteers take the test before and after playing the game. Does playing the computer game improve people's score on the test?
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Research the famous collapse of the Tacoma Narrows suspension bridge.
What lessons were learned about the potentially damaging effects of wind on bridges? What structures stabilize a bridge against wind forces? Build models and use a wind tunnel to test your hypothesis.
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This a straightforward, but interesting, project in geometry. It is a good first proof to try on your own. You should be able to figure it out by yourself, and you'll gain insight into a basic property of circles.
Figure 1 below shows a semicircle (AE, in red) with a series of smaller semicircles (AB, BC, CD, DE, in blue) constructed inside it. As you can see, the sum of the diameters of the four smaller semicircles is equal to the diameter of the large semicircle. The area of the larger…
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Math_p010

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Time Required

Very Short (≤ 1 day)

Prerequisites

Must understand the concept of a mathematical proof

Visit your local landfill to collect dumping statistics by watching the types of trash people bring to the dump. Identify problem areas and types of waste that are commonly brought to the dump. Propose new uses for common landfill problems like tires, construction materials, plastics, appliances, and computer parts. Measure packaging materials of some common products. Invent ways to reduce the amount of packaging and increase the use of recyclable materials like popcorn or corn-based packing…
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Here's an interesting project idea with a variation that combines computer science, physics and music. You'll need a piano in a quiet room, a microphone and a computer with digital sound recording and analysis software. The project shows you how you can make a piano string start vibrating without hitting its key. You can record the sounds on the computer, and use sound analysis software to measure the frequencies of the induced vibrations. For more details see: . Be sure to check out the…
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This is an interesting geometry project that goes back to the time of Archimedes, the famous Greek mathematician. You can combine this mathematical project with computer science and take this ancient problem into the twenty-first century with a dynamic diagram using the geometry applet.
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Math_p018

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Time Required

Short (2-5 days)

Prerequisites

You should either currently be taking or have already completed a first course in geometry. You must understand the concept and method of a mathematical proof.

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…
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