= Algebra Tiles =
Algebra tiles are known as
mathematical manipulativesthat 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 prisoninmates for their General Educational Development(GED) tests. [Kitts, N: "Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts", page 462. MATHEMATICS TEACHER, 2000.] Algebra tiles allow both an algebraic and geometric approach to algebraic concepts. They give studentsanother way to solve algebraic problems other than just abstract manipulation. [Kitts, N: "Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts", page 463. MATHEMATICS TEACHER, 2000.] The National Council of Teachers of Mathematics( NCTM) recommends a decreased emphasis on the memorization of the rules of algebraand the symbol manipulation of algebrain their "Curriculum and Evaluation Standards for Mathematics". According to the NCTM1989 standards " [r] elating models to one another builds a better understanding of each". [Stein, M: Implementing Standards-Based Mathematics Instruction", page 105. Teachers College Press, 2000.]
The algebra tiles are made up of small squares, large squares, and rectangles. The
number oneis represented by the small square, which is also known as the unit tile. The rectangle represents the variablex and the large square represents x2. The lengthof the side of the large square is equal to the lengthof the rectangle, also known as the x tile. When visualizing these tiles it is important to remember that the areaof a square is s2, which is the length of the sides squared. So if the lengthof 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 lengthas 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 factoringand multiplying polynomials. [Kitts, N: "Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts", page 462. MATHEMATICS TEACHER, 2000.]
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. [Kitts, N: "Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts", page 462. MATHEMATICS TEACHER, 2000.] Having the two colors on both sides allows for more numbers to be represented with a fewer number of 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 alright to use, it is just important to have a least two colors to denote positive and negative. Some commercially made algebra tiles can be purchased for the
overhead projector. These are made out a plastic translucent material. [http://www.eaieducation.com/525010.html Overhead Projector Algebra Tiles]
Algebra tiles can also be made at
homeinstead of buying them commercially. Templates for the algebra tiles can be found online, [http://www.teachervision.fen.com/algebra/printable/6192.html Algebra tile template] , which can be printed and then cut out. [ [http://www.teachervision.fen.com/algebra/printable/6192.html] ] Once the shapes are cut out of the printer paper they can be used to cut out algebra tiles from card stockor Foamies, which are foam-like materials, about 1/8- inchthick. [http://www.regentsprep.org/regents/math/ALGEBRA/teachres/ttiles.htm Homemade Algebra Tiles] Algebra tiles can also be made for the overhead projectorby cutting the shapes out of colored plastic report covers. [Kitts, N: "Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts", page 463. MATHEMATICS TEACHER, 2000.]
Algebra tiles can be used for adding
integers. [Kitts, N: "Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts", page 463. MATHEMATICS TEACHER, 2000.] To demonstrate this ability you can consider the problem . In order to solve this problem using algebra tiles a person would group two of the positive unit tiles together and then group three of the positive unit tiles together to represent separately 2 and 3. In order to represent the person would then combine their two groups together. Once this step is complete the person can then count that together there are 5 unit tiles, so . Since adding a number with the negative of that number gives you zero, for instance , adding a negative unit tile and a positive unit tile will also give you zero. When you add a positive tile and a negative tile it is known as the zero pair. In order to show that any integerplus its negative is zeroa person can physically represent this concept through algebra tiles. Let us take the example used earlier where . A person would fist lay out two negative unit tiles and then two positive unit tiles, which would then be combined into two sets of zero pairs. These two sets of zero pairs would then be equal to zero. [ [http://www.phschool.com/professional_development/teaching_tools/pdf/using_algebra_tiles.pdf] ] Understanding zero pairs allows you to also add positive and negative integers that are not equal. An example of this would be , where you would group seven negative unit tiles together and then four positive unit tiles together and then combine them. Before you count the number of tiles that you now have you would have to create zero pairs and then remove them from you final answer. In this example you would have four zero pairs which would remove all of the positive unit tiles and you would be left with three negative unit tiles, so .
Algebra tiles can also be used for subtracting
integers. A person can take a problem such as and begin with a group of six unit tiles and then take three away to leave you with three left over, so then . Algebra tiles can also be used to solve problems like . First you would start off with four negative unit tiles and then take away two negative unit tiles to leave you with two negative unit tiles. Therefore , which is also the same answer you would get if you had the problem . Being able to relate these two problems and why they get the same answer is important because it shows that . Another way in which algebra tiles can be used for integer subtractioncan be seen through looking at problems where you subtract a positive integerfrom a smaller positive integer, like . 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. [ [http://www.phschool.com/professional_development/teaching_tools/pdf/using_algebra_tiles.pdf] ]
Multiplication of Integers
Multiplicationof integerswith algebra tiles is performed through forming a rectangle with the tiles. The lengthand widthof your rectangle would be your two factorsand then the total number of tiles in the rectangle would be the answer to your multiplicationproblem. 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
additionand subtractionof integers using algebra tiles. In an expression such as 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 you can combine the positive and negative x tiles and unit tiles to form zero pairs to leave you with the expression . 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. [ [http://www.phschool.com/professional_development/teaching_tools/pdf/using_algebra_tiles.pdf] ]
Using the Distributive Property
distributive propertyis 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 then we would make three sets of one unit tile and one x tile and then combine them together to see if would have , which we would. [ [http://www.regentsprep.org/rEGENTS/math/realnum/Tdistrib.htm] ]
Solving Linear Equations
Manipulating algebra tiles can help students solve
linear equations. In order to solve a problem like 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 . [ [http://www.phschool.com/professional_development/teaching_tools/pdf/using_algebra_tiles.pdf] ] You can also use the subtractionproperty of equality to solve your linear equationwith algebra tiles. If you have the equation , 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 . [Kitts, N: "Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts", page 464. MATHEMATICS TEACHER, 2000.] There are programs online that allow students to create their own linear equationsand manipulate the algebra tiles to solve the problem. [http://my.hrw.com/math06_07/nsmedia/tools/Algebra_Tiles/Algebra_Tiles.html Solving Linear Equations Program] This video from Teacher Tube also demonstrates how algebra tiles can be used to solve linear equations. [http://www.teachertube.com/view_video.php?viewkey=7b93931b2e628c6e6244&page=&viewtype=&category= Teacher Tube Solving Equations]
When using algebra tiles to multiply a
monomialby a monomialyou first set up a rectangle where the lengthof the rectangle is the one monomialand then the widthof the rectangle is the other monomial, similar to when you multiply integersusing 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. Multiplicationof binomialsis similar to multiplicationof monomialswhen using the algebra tiles . Multiplication of binomialscan also be thought of as creating a rectangle where the factorsare the lengthand width. [Stein, M: Implementing Standards-Based Mathematics Instruction", page 98. Teachers College Press, 2000.] Like with the monomials, you set up the sides of the rectangle to be the factorsand then you fill in the rectangle with the algebra tiles. [Stein, M: Implementing Standards-Based Mathematics Instruction", page 106. Teachers College Press, 2000.] This method of using algebra tiles to multiply polynomialsis known as the area model [Larson R: "Algebra 1", page 516. McDougal Littell, 1998.] and it can also be applied to multiplying monomialsand binomialswith each other. An example of multiplying binomialsis (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 lengthof 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.
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
lengthand then one positive x tile and two positive unit tiles for the width. This means that the two factorsare and . [Kitts, N: "Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts", page 464. MATHEMATICS TEACHER, 2000.] In a sense this is the reverse of the procedure for multiplying polynomials.
Completing the Square
The process of
completing the squarecan 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 squareyou would determine how many unit tiles would be needed to fill in the missing square. In order to complete the squareof 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 . [ [http://www.regentsprep.org/Regents/math/algtrig/ATE12/completesq.htm] ]
* Kitt, Nancy A. and Annette Ricks Leitze. "Using Homemade Algebra Tiles to Develope 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.
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