User:Gbengle/sandbox

Physical attributes
Algebra tiles are made up of small squares, rectangles, and large squares. The small square, the unit tile, represents the number one; the rectangle represents the variable $$x$$; and the large square represents $$x^2$$. The side of the $$x^2$$ tile is equal to the length of the $$x$$ tile. The width of the $$x$$ tile is the same as the side of the unit tile. Additionally, the length of the $$x$$ tile is often not an integer multiple of the side of the unit tile.

The tiles consist of two colors: one to show positive values and another to show negative values. A zero pair is a negative and a positive unit tile (or a negative and a positive $$x$$ tile, or a negative and a positive $$x^2$$ tile) which together form a sum of zero.

Solving linear equations using addition
The linear equation $$x-8=6$$ can be modeled with one positive $$x$$ tile and eight negative unit tiles on the left side of a piece of paper and six positive unit tiles on the right side. To maintain equality of the sides, each action must be performed on both sides. For example, eight positive unit tiles can be added to both sides. Zero pairs of unit tiles are removed from the left side, leaving one positive $$x$$ tile. The right side has 14 positive unit tiles, so $$x=14$$.

Solving linear equations using subtraction
The equation $$x+7=10$$ can be modeled with one positive $$x$$ tile and seven positive unit tiles on the left side and 10 positive unit tiles on the right side. Rather than adding the same number of tiles to both sides, the same number of tiles can be subtracted from both sides. For example, seven positive unit tiles can be removed from both sides. This leaves one positive $$x$$ tile on the left side and three positive unit tiles on the right side, so $$x=3$$.

Factoring quadratic trinomials
In order to factor using algebra tiles you start out with a set of tiles that you combine into a rectangle, this may require the use of adding zero pairs in order to make the rectangular shape. An example would be where you are given one positive $$x^2$$ tile, three positive $$x$$ tiles, and two positive unit tiles. You form the rectangle by having the $$x^2$$ tile in the upper right corner, then you have two $$x$$ tiles on the right side of the $$x^2$$ tile, one $$x$$ tile underneath the $$x^2$$ tile, and two unit tiles are in the bottom right corner. By placing the algebra tiles to the sides of this rectangle we can determine that we need one positive $$x$$ tile and one positive unit tile for the length and then one positive $$x$$ tile and two positive unit tiles for the width. This means that the two factors are $$x+1$$ and $$x+2$$. In a sense this is the reverse of the procedure for multiplying polynomials.