Law of cosines

Algebra tiles are known as 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".

Physical attributes

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.

Commercially made algebra tiles are usually made from plastic and have one side of one color and the other side of another color. the difference in the color is supposed to denote one side that is positive and one side that is negative. Traditionally, one side is red to represent the negative and one side is green to represent the positive. Having the two colors on both sides allows for more numbers to be represented with the same tiles. It also makes it easier to change positives to negatives when performing a procedure such as multiplying a positive and a negative number. There are some tiles where the positive x and x² tile will be the same color, but the positive unit tile is a different color. This representation is still all right to use, it is just important to have a least two colors to denote positive and negative. Translucent plastic algebra tiles can be bought for the overhead projector.

Algebra tiles can be made. Templates for the algebra tiles can be found online,Algebra tile template, which can be printed and then cut out. Once the shapes are cut out of the printer paper they can be used to cut out algebra tiles from card stock or Foamies, which are foam-like materials, about 1/8-inch thick. Algebra tiles can also be made for the overhead projector by cutting the shapes out of colored plastic report covers.

Virtual Algebra Tiles are available from The National Library of Virtual Manipulatives, the Ubersketch and in the sample files that ship with The Geometer's Sketchpad.

Uses

Subtracting integers

Algebra tiles can also be used for subtracting integers. A person can take a problem such as $6-3=?$ and begin with a group of six unit tiles and then take three away to leave you with three left over, so then $6-3=3$. Algebra tiles can also be used to solve problems like $-4-(-2)=?$.get if you had the problem $-4+2$. Being able to relate these two problems and why they get the same answer is important because it shows that $-(-2)=2$. 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 $5-8$. 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.

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 $5x-3$ 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 $4x+5-2x-3$ you can combine the positive and negative x tiles and unit tiles to form zero pairs to leave you with the expression $2x+2$. 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.

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 $3(x+1)=3x+3$ then we would make three sets of one unit tile and one x tile and then combine them together to see if would have $3x+3$, which we would.

Solving linear equations

Manipulating algebra tiles can help students solve linear equations. In order to solve a problem like $x-6=2$ you would first place one x tile and six negative unit tiles in one group and then two positive unit tiles in another. You would then want to isolate the x tile by adding six positive unit tiles to each group, since whatever you do to one side has to be done to the other or they would not be equal anymore. This would create six zero pairs in the group with the x tile and then there would be eight positive unit tiles in the other group. this would mean that $x=8$. You can also use the subtraction property of equality to solve your linear equation with algebra tiles. If you have the equation $x+7=10$, then you can add seven negative unit tiles to both sides and create zero pairs, which is the same as subtracting seven. Once the seven unit tiles are subtracted from both sides you find that your answer is $x=3$. There are programs online that allow students to create their own linear equations and manipulate the algebra tiles to solve the problem. Solving Linear Equations Program This video from TeacherTube also demonstrates how algebra tiles can be used to solve linear equations. Teacher Tube Solving Equations

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. 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. This method of using algebra tiles to multiply polynomials is known as the area model 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

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 $x+1$ and $x+2$. 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 $(x+3)(x+3)$.

References

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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.

External links

• Algebra tile manipulatives