The foundation of good science is accurate measurement. An experiment in which four students measure four objects of different sizes provides an introduction to concepts of accuracy, averages, absolute differences and percent differences. This project is suitable for any student with the ability to divide and find percents.
Middle school students with some knowledge of right triangles can learn basic trigonometric ratios with a more sophisticated measurement activity. The student measures off a fairly large distance -- several hundred feet -- from a building. She then uses an inclinometer, a device to measure angles of elevation and depression, to find the angle between that spot and the top of the building. Because a right triangle forms between the ground, the building and the angle between the top of the building and the point on the ground, the student can use right triangle trigonometry to find the height of the building.
The SNAP Foundation lists a number of non-competitive and problem-based projects for math fairs. Activities include "Reverse the Fish" that requires students to change the direction of a fish made of toothpicks by moving as few pieces as possible. Another project requires students to interchange red and green chips by moving red chips only to the right and green chips only to the left. They are suitable for young children because they rely on manipulation of objects rather than a prior knowledge of mathematics and emphasize the development of logical thinking and spatial reasoning.
The Tower of Hanoi is an ancient mathematical puzzle consisting of three pegs, one of which contains six rings of different sizes and arranged according to size, with the largest disc on the bottom and the smallest disc on top. The object is to move the rings to another peg without ever having a larger disc rest on top of a smaller one. This puzzle will help middle and high school students learn to find mathematical patterns.
Elementary school students can investigate the relationship between perimeter and area by creating polygons with the same perimeter and computing their areas. Suppose the perimeter is 12. The polygons would be an equilateral triangle with sides of length 4, a square with sides of length 3, a pentagon with sides of length 2.4 and a hexagon with sides of length 2. Students would compute the area of each polygon and determine how the area changes with the number of sides. They would also determine which shape produces the greatest area.
Middle school students can examine the different techniques for computing the value of pi to any number of decimal places and determine which method or methods are more accurate.
A former U.S. president, James. A. Garfield, developed an elegant proof of the Pythagorean Theorem that involved forming a trapezoid from two right triangles and using elementary algebra to obtain the equation a^2 + b^2 = c^2. Middle and high school students who know how to square binomials might investigate this proof and compare it to other proofs of this theorem from ancient times until the present.
Introduce elementary school students to probability by teaching them how to increase their chances of winning at "Go Fish." Young children can understand that if they have one ace in their hand and three jacks that their chances of getting a card from an opponent is better if they ask for an ace.
Slightly older children can count the different colors of M & Ms in five or more bags of candy and use the data to construct histograms, bar graphs and pie graphs.
Since many games depend on probability, middle school students can develop strategies for card counting that can increase their chances of winning at Hi-Lo. Hi-Lo is a simple game that requires guessing whether the next card played will be higher or lower than the previous card. These probabilities change with every card played. Learning a method to estimate probabilities and improve the chances of winning the game will provide an excellent background in probability theory.