Imagine a symmetrical grid of nine points superimposed over the ball. Kicking the ball squarely on the center point imparts no spin, but kicking on any of the other points will impart spin on the ball. How will the resulting spin affect the trajectory of the ball for each of the 8 outer grid points? Kicking the ball with a sliding motion of the foot is another way to impart spin. Once you've made your predictions, you can set up to test them with a soccer ball, video camera and a tape…
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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!…
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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…
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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…
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Students who are mathematically inclined can use the student version of a program like MatLab or Mathematica to convert a digital image into numbers, then perform operations such as sharpening or special effects. This is a great way to learn about image processing algorithms.
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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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If you know or calculate the field of view for your camera, you can use it to measure distances and the height of almost anything. It's all a matter of basic trigonometry.
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Leaves grow in a different pattern than stems and shoots. They do not elongate along one axis, but instead spread out over time. Do all regions of the leaf grow equally? You can use markings on different regions of a growing leaf to see if the whole surface grows, or if growth is focused in a particular region, like the veins or edges of the leaf. If you look along a leafy stem, you may notice that leaves at different positions along the stem are different sizes. You can do an experiment to…
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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…
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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…
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