Algebra tiles are hands-on manipulatives for visual learning of various algebraic concepts. Algebra tiles consist of square and rectangular shapes to concretely model and manipulate abstract algebraic equations and expressions. A set of Algebra Tiles typically includes:
|1||Small square with all sides 1-unit long (area is 1)|
|x||Rectangle with a 1-unit long side and an x-unit long side (area is x square units)|
|x²||Large square with all sides x-units long (area is x²)|
|y||Rectangle with a 1-unit long side and an y-unit long side (area is y square units)|
|y²||Large square with all sides y units long (area is y²)|
|xy||Large rectangle with an x-unit long side and a y-unit long (area is xy)|
To reinforce the idea that x and y are variables, their lengths are typically not multiples of the unit length. Algebra tiles typically have different colors on each side. One color represents a positive value, and the other color represents a negative color. While all sets of Algebra Tiles include 1 and x tiles, only some include y, y², and xy tiles. An Algebra Tiles Frame is an orthogonal bracket that helps to create arrays of tiles as well as to place tiles along the horizontal and vertical axes to clearly see the array length.
Virtual Algebra Tiles include a number of other features to do things that are cumbersome or impossible with physical Algebra Tiles. For example, Brainingcamp's Algebra Tiles Manipulative also performs an animation when positive values are dropped on negative values so that students can visualize the concept of a zero pair. The virtual Algebra Tiles also easily snap together and include background grids and mats for specialized applications. Labels can to turned on so that students can more explicitly see which tiles represent each of the unit, x, x², y, y², and xy values. Brainingcamp's virtual Algebra Tiles also let students resize the length of the x, x², y, y², and xy tiles to reinforce the idea that variables represent an unknown or undetermined value.
The addition 2 + (-7) can be modeled by creating a group of 2 positive unit tiles and a group of 7 negative unit tiles. Drag opposite signed tiles on top of each other to make zero pairs. When no more zero pairs can be made there are 5 negative unit tiles remaining, so 2 + (-7) equals -5.
The subtraction -2 – (-4) can be modeled by taking 4 negative units from two negative unit tiles. Start by placing 2 negative tiles. Next add zero pairs until there are enough negative tiles to remove 4 negative tiles. There are 2 positive tiles remaining, so -2 – (-4) equals 2.
To perform the multiplication 3 × 2 using an Algebra Tiles frame, model 3 along the vertical axis and 2 along the horizontal axis. On the other side of the axis create a rectangle with a height of 3 and a width of 2. It takes 6 unit tiles to create the rectangle so 3 × 2 equals 6.
To simplify 5x + 4 – 2x – 2, combine or group like terms. Create as many zero pairs as possible by combining positive and negative x tiles for each group of like terms.
To solve 3x + 2 = 8, model 3x + 2 on one side of an equals sign and 8 on the other side. To isolate the variable x on one side of the equals sign, start by adding 2 negative unit tiles to both sides. Make zero pairs by combining positive and negative unit tiles. To isolate x from the equation 3x = 6, divide the workspace into 3 rows and distribute the tiles into 3 equal groups. With an x isolated in each group there are 2 tiles on the opposite side of each row, so x = 2 is a solution to the equation 3x + 2 = 8.
Solving Systems of Equations
To solve the system of equations x + y = 8 and x – y = 4, split the workspace into two rows. Model the equation x + y = 8 in the top row and the equation x – y = 4 in the bottom row. Isolate the variable in the equation x + y = 8 row by adding a negative y tile on each size and creating zero pairs. The isolated variable x equals 8 – y. So in the bottom row, substitute the x tile with 8 – y and simplify to create the equation 8 – 2y = 4. To isolate for y in the bottom row, add 8 negative unit tiles to each side in the row and make zero pairs. Next create two equal groups in the bottom row to find that y equals 2. Now return to the top equation and substitute y with 2 positive unit tiles to create the equation x = 8 – 2. Make zero pairs in the top row to find that x equals 6. So the solution to the system of equations is x = 6 and y = 2.
Add and Subtract Polynomials
To perform the addition (2x² + 3x – 4) + (x² - 2x + 3), start by modeling the terms of the polynomials. Drag opposite signed tiles on top of each other to make zero pairs. The solution, when no more zero pairs can be made, is 3x² + x – 1. To perform the subtraction (2x² + 3x – 4) - (x² - 2x + 3), add the inverse of the second polynomial to the first polynomial and mode (2x² + 3x – 4) + (-x² + 2x - 3). After simplifying and creating zero pairs, the solution is x² + 5x – 7.
To perform the multiplication (2x + 3)(x+1), use a frame to create an array. Model (2x + 3) along the vertical axis and (x+1) along the horizontal axis. On the other side of the axis create a rectangle with a height of (2x + 3) and a width of (x+1). The rectangle can be filled with two x² tiles, five x tiles, and three unit tiles, so (2x + 3)(x+1) equals 2x² + 5x + 3.
To perform the division (x² + 5x + 4)/(x + 1), use a frame to create an array. Model the divisor along the vertical axis. Model the dividend with a rectangle with height equal to the length of the divisor. Find the quotient by placing tiles along the horizontal axis equal in length to the width of the rectangle. There is 1 x tile and 4 unit tiles along the horizontal axis, so (x² + 5x + 4)/((x + 1) equals (x + 4).
To factor x² + 7x + 12, start with the largest tile, x². Next add the seven x tiles beside the x² tile. Finally place the 12 unit tiles. If the result does not create a rectangle, rearrange the x tiles and unit tiles. When a rectangle has been created, place tiles along the vertical and horizontal axes equal in length to the height and width of the rectangle. The factors of x² + 7x + 12 equal the length and width of the rectangle, or (x + 4) and (x + 3).
Completing the Square
To complete the square x²+ 6x start with the largest tile, x². Next add six x tiles beside the x² tile. To complete a square area add 9 positive unit tiles. The sides of the squares are both (x + 3) long, so the square models x²+ 6x + 9 and the factors are (x + 3)(x + 3).