Algebra tiles
Algebra tiles are mathematical manipulatives that allow students to better understand ways of algebraic thinking and the concepts of algebra. These tiles have proven to provide concrete models for elementary school, middle school, high school, and college-level introductory algebra students. They have also been used to prepare prison inmates for their General Educational Development (GED) tests.Algebra tiles allow both an algebraic and geometric approach to algebraic concepts. They give students another way to solve algebraic problems other than just abstract manipulation. The National Council of Teachers of Mathematics (NCTM) recommends a decreased emphasis on the memorization of the rules of algebra and the symbol manipulation of algebra in their Curriculum and Evaluation Standards for Mathematics. According to the NCTM 1989 standards "[r]elating models to one another builds a better understanding of each".
The algebra tiles are made up of small squares, large squares, and rectangles. The number one is represented by the small square, which is also known as the unit tile. The rectangle represents the variable x and the large square represents x2. The length of the side of the large square is equal to the length of the rectangle, also known as the x tile. When visualizing these tiles it is important to remember that the area of a square is s2, which is the length of the sides squared. So if the length of the sides of the large square is x then it is understandable that the large square represents x2. The width of the x tile is the same length as the side length of the unit tile. The reason that the algebra tiles are made this way will become clear through understanding their use in factoring and multiplying polynomials. Additionally, to reinforce the fact that x is a variable and not a fixed number, the side length of the x tile is often not an integer multiple of that of the 1 tile (for instance, it may be between 5 and 6 times its length).
...
Wikipedia