# Algebra tile

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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.[1] 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.[1] 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".[2]

## Physical attributes

Examples of algebra tiles

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 ${\displaystyle x}$; and the large square represents ${\displaystyle x^{2}}$. The side of the ${\displaystyle x^{2}}$ tile is equal to the length of the ${\displaystyle x}$ tile. The width of the ${\displaystyle x}$ tile is the same as the side of the unit tile. Additionally, the length of the ${\displaystyle x}$ 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 ${\displaystyle x}$ tile, or a negative and a positive ${\displaystyle x^{2}}$ tile) which together form a sum of zero.[1]

## Uses

### Adding integers

Adding Integers is the best place to start when you want to get used to the idea of representing numbers with a quantity of tiles. Any integer can be represented by using the same number of tiles in the correct color. For example for a 6 you could select six yellow tiles. For -3 you would select 3 red tiles. The tiles are usually double sided with yellow on one side and red on the other. This allows the student to grasp the powerful concept of "taking the opposite" A negative simply means the opposite. So one yellow tile is positive one and the opposite (flip it over) is negative one. This is idea comes in handy when dealing with a - (-2) Start with two negative ones (red side) and the extra negative means take the opposite or flip them over. - (-2) = 2.

When adding tiles think of combining the quantities together. if you are adding 2 + 3 you combine 2 yellow tiles with 3 yellow tiles and combined you have 5 yellow tiles. The same idea works for combining negative numbers. If you are adding -3 + -1 you combine 3 red tiles with 1 red tile to get 4 red tiles. -3 + -1 = -4

When you add positive numbers to negative numbers using algebra tiles you need to bring in the idea of "elimination" or "zero pairs" every time you add a positive one to a negative one they eliminate each other resulting in a zero. This is true for any number of tiles. As long and you have the same quantity and opposite sign they will eliminate each other (or create a zero pair). For example if you add -5 + 7 you will combine five red tiles with seven yellow tiles. You can match the red and yellow tiles up one at a time to eliminate 5 of the yellow tiles and you will be left with 2 yellow tiles and no red tiles. -5 + 7 = 2.

If you start with more yellow tiles than red, your answer will be positive. If you start with more red tiles than yellow, your answer will be negative.

One more example: -5 + 2. You are combining 5 red tiles with 2 yellow tiles. The 2 yellow tiles will eliminate (or form a zero pair) with 2 of the red tiles leaving 3 red tiles behind. -5 + 2 = -3

### Subtracting integers

Algebra tiles can also be used for subtracting integers. A person can take a problem such as ${\displaystyle 6-3=?}$ and begin with a group of six unit tiles and then take three away to leave you with three left over, so then ${\displaystyle 6-3=3}$. Algebra tiles can also be used to solve problems like ${\displaystyle -4-(-2)=?}$.get if you had the problem ${\displaystyle -4+2}$. Being able to relate these two problems and why they get the same answer is important because it shows that ${\displaystyle -(-2)=2}$. Another way in which algebra tiles can be used for integer subtraction can be seen through looking at problems where you subtract a positive integer from a smaller positive integer, like ${\displaystyle 5-8}$. Here you would begin with five positive unit tiles and then you would add zero pairs to the five positive unit tiles until there were eight positive unit tiles in front of you. Adding the zero pairs will not change the value of the original five positive unit tiles you originally had. You would then remove the eight positive unit tiles and count the number of negative unit tiles left. This number of negative unit tiles would then be your answer, which would be -3.[3]

### Multiplication of integers

Multiplication of integers with algebra tiles is performed through forming a rectangle with the tiles. The length and width of your rectangle would be your two factors and then the total number of tiles in the rectangle would be the answer to your multiplication problem. For instance in order to determine 3×4 you would take three positive unit tiles to represent three rows in the rectangle and then there would be four positive unit tiles to represent the columns in the rectangle. This would lead to having a rectangle with four columns of three positive unit tiles, which represents 3×4. Now you can count the number of unit tiles in the rectangle, which will equal 12.

### Modeling and simplifying algebraic expressions

Modeling algebraic expressions with algebra tiles is very similar to modeling addition and subtraction of integers using algebra tiles. In an expression such as ${\displaystyle 5x-3}$ you would group five positive x tiles together and then three negative unit tiles together to represent this algebraic expression. Along with modeling these expressions, algebra tiles can also be used to simplify algebraic expressions. For instance, if you have ${\displaystyle 4x+5-2x-3}$ you can combine the positive and negative x tiles and unit tiles to form zero pairs to leave you with the expression ${\displaystyle 2x+2}$. Since the tiles are laid out right in front of you it is easy to combine the like terms, or the terms that represent the same type of tile.[3]

The distributive property is modeled through the algebra tiles by demonstrating that a(b+c)=(a×b)+(a×c). You would want to model what is being represented on both sides of the equation separately and determine that they are both equal to each other. If we want to show that ${\displaystyle 3(x+1)=3x+3}$ then we would make three sets of one unit tile and one x tile and then combine them together to see if would have ${\displaystyle 3x+3}$, which we would.[4]

### Solving linear equations using addition

The linear equation ${\displaystyle x-8=6}$ can be modeled with one positive ${\displaystyle x}$ 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.[1] For example, eight positive unit tiles can be added to both sides.[1] Zero pairs of unit tiles are removed from the left side, leaving one positive ${\displaystyle x}$ tile. The right side has 14 positive unit tiles, so ${\displaystyle x=14}$.

### Solving linear equations using subtraction

The equation ${\displaystyle x+7=10}$ can be modeled with one positive ${\displaystyle x}$ 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 ${\displaystyle x}$ tile on the left side and three positive unit tiles on the right side, so ${\displaystyle x=3}$.[1]

### Solving linear systems

Linear systems of equations may be solved algebraically by isolating one of the variables and then performing a substitution. Isolating a variable can be modeled with algebra tiles in a manner similar to solving linear equations (above), and substitution can be modeled with algebra tiles by replacing tiles with other tiles.

### Multiplying polynomials

When using algebra tiles to multiply a monomial by a monomial you first set up a rectangle where the length of the rectangle is the one monomial and then the width of the rectangle is the other monomial, similar to when you multiply integers using algebra tiles. Once the sides of the rectangle are represented by the algebra tiles you would then try to figure out which algebra tiles would fill in the rectangle. For instance, if you had x×x the only algebra tile that would complete the rectangle would be x2, which is the answer.

Multiplication of binomials is similar to multiplication of monomials when using the algebra tiles . Multiplication of binomials can also be thought of as creating a rectangle where the factors are the length and width.[2] As with the monomials, you set up the sides of the rectangle to be the factors and then you fill in the rectangle with the algebra tiles.[2] This method of using algebra tiles to multiply polynomials is known as the area model[5] and it can also be applied to multiplying monomials and binomials with each other. An example of multiplying binomials is (2x+1)×(x+2) and the first step you would take is set up two positive x tiles and one positive unit tile to represent the length of a rectangle and then you would take one positive x tile and two positive unit tiles to represent the width. These two lines of tiles would create a space that looks like a rectangle which can be filled in with certain tiles. In the case of this example the rectangle would be composed of two positive x2 tiles, five positive x tiles, and two positive unit tiles. So the solution is 2x2+5x+2.

### Factoring

Algebra tile model of ${\displaystyle x^{2}+3x+2}$

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 x2 tile, three positive x tiles, and two positive unit tiles. You form the rectangle by having the x2 tile in the upper right corner, then you have two x tiles on the right side of the x2 tile, one x tile underneath the x2 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 ${\displaystyle x+1}$ and ${\displaystyle x+2}$.[1] In a sense this is the reverse of the procedure for multiplying polynomials.

### Completing the square

The process of completing the square can be accomplished using algebra tiles by placing your x2 tiles and x tiles into a square. You will not be able to completely create the square because there will be a smaller square missing from your larger square that you made from the tiles you were given, which will be filled in by the unit tiles. In order to complete the square you would determine how many unit tiles would be needed to fill in the missing square. In order to complete the square of x2+6x you start off with one positive x2 tile and six positive x tiles. You place the x2 tile in the upper left corner and then you place three positive x tiles to the right of the x2 tile and three positive unit x tiles under the x2 tile. In order to fill in the square we need nine positive unit tiles. we have now created x2+6x+9, which can be factored into ${\displaystyle (x+3)(x+3)}$.[6]

## References

1. Kitt 2000.
2. ^ a b c Stein 2000.
3. ^ a b "Prentice Hall School" (PDF). Phschool.com. Archived from the original (PDF) on 2012-02-12. Retrieved 2013-07-22.
4. ^ [1] Archived May 16, 2008, at the Wayback Machine
5. ^ Larson R: "Algebra 1", page 516. McDougal Littell, 1998.
6. ^ Donna Roberts. "Using Algebra Tiles to Complete the Square". Regentsprep.org. Archived from the original on 2013-08-18. Retrieved 2013-07-22.

## Sources

• Kitt, Nancy A. and Annette Ricks Leitze. "Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts." MATHEMATICS TEACHER 2000. 462-520.
• Stein, Mary Kay et al., IMPLEMENTING STANDARDS-BASED MATHEMATICS INSTRUCTION. New York: Teachers College Press, 2000.
• Larson, Ronald E., ALGEBRA 1. Illinois: McDougal Littell,1998.