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| {{Multiple issues|orphan=October 2008|howto=October 2008|notability=October 2008}}
| | Hello, dear friend! I am Consuelo. I smile that I could join to the entire world. I live in Italy, in the AP region. I dream to check out the different nations, to obtain acquainted with interesting people.<br>xunjie 森馬とスミスバーニー衣装上場企業の2つの中国のパフォーマンスは良くありません。 |
| | 価格戦争と綿花価格大規模な環境で、 |
| | トニーの下着ブランド美：女性の健康のための調節可能な下着を中心に設計された製品のメンテナンスや修理の両方を節約する神の守護ビーナスベイ美容ランジェリーブランドの美しさだけでなく、 [http://cambridgetoafrica.org/fileshare/images/tomford.html �ȥ�ե��`�� ���饹 �����`] 竹繊維の多種多様な製品を生産するための処理竹繊維織物を通じ、 |
| | 古代アンティークヒスイのような非現在のこの気持ちのように、 |
| | 紫色のテクスチャアンテプリマブラックとジェイソンウーの近くから、 [http://www.rheintalverlag.ch/newsletter/gaga.php gaga milano �rӋ] シャオ風水ロング衣類のために2012春と夏の新しいドラゴンの年をテーマにした一連の設計：鳳ロングパラダイス。 |
| | 我々は気にクマの子供服ブランドのファッション製品の賑やかなクロスを開いたベア子供服ブランドのファッションストアで賑やかなクロスを開設 - コアコンセプトとして、 |
| | 誰もがそれをチェックアウトしたかった場所になった後の町は歴史痴漢の350年を持っています。[http://www.rheintalverlag.ch/newsletter/gaga.php �����ߥ�� ���n] この国を救うためにノーと言った愛国中国の人々のビバは、 |
| | 自身のスタイリングさわやか。 |
| | 私たちはあなたにエキサイティングなファッションのインスピレーションの多くをもたらすと信じています。 |
| | として記載されている「世界で喜び気ままな子どもたちの完全な。 [http://aphroditeinn.gr/webmail/mime/li/shop/mall/nb.html �˥�`�Х�<br><br>���˩`���`] |
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| '''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.<ref name="Kitts, N page 462">Kitts, N: "Using Homemade Algebra Tiles to Develop Algebra and Pre algebra Concepts", page 462. MATHEMATICS TEACHER, 2000.</ref> '''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.<ref name="Kitts, N page 463">Kitts, N: "Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts", page 463. MATHEMATICS TEACHER, 2000.</ref> 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".<ref>Stein, M: Implementing Standards-Based Mathematics Instruction", page 105. Teachers College Press, 2000.</ref>
| | my homepage - [http://3kydd.com/bio/pdfs/dr.martens.php クリスチャンルブタン ブーツ] |
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| ==Physical attributes==
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| The algebra tiles are made up of small squares, large squares, and rectangles. The [[1 (number)|number one]] is represented by the small square, which is also known as the unit tile. The rectangle represents the [[Variable (mathematics)|variable]] x and the large square represents x<sup>2</sup>. 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 s<sup>2</sup>, 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 x<sup>2</sup>. 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 [[Factorization|factoring]] and multiplying [[polynomials]].<ref name="Kitts, N page 462" />
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| 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.<ref name="Kitts, N page 462" /> 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 x<sup>2</sup> 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]].<ref>[http://www.eaieducation.com/525010.html Overhead Projector Algebra Tiles]</ref>
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| Algebra tiles can be made. 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.<ref>{{cite web|url=http://www.teachervision.fen.com/algebra/printable/6192.html |title=Algebra Tiles Printable (6th - 8th Grade) - TeacherVision.com |publisher=Teachervision.fen.com |date= |accessdate=2013-07-22}}</ref> 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.<ref>[http://www.regentsprep.org/regents/math/ALGEBRA/teachres/ttiles.htm Homemade Algebra Tiles]</ref> Algebra tiles can also be made for the [[overhead projector]] by cutting the shapes out of colored plastic report covers.<ref name="Kitts, N page 463" />
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| [[Virtual manipulatives for mathematics|Virtual Algebra Tiles]] are available from The National Library of Virtual Manipulatives,<ref>[http://nlvm.usu.edu/en/nav/vlibrary.html]</ref> the [[Ubersketch]] and in the sample files that ship with [[The Geometer's Sketchpad]].''
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| ==Uses==
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| ===Adding integers===
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| ===Subtracting integers===
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| Algebra tiles can also be used for subtracting [[integers]]. A person can take a problem such as <math>6-3=?</math> and begin with a group of six unit tiles and then take three away to leave you with three left over, so then <math>6-3=3</math>. Algebra tiles can also be used to solve problems like <math>-4-(-2)=?</math>.get if you had the problem <math>-4+2</math>. Being able to relate these two problems and why they get the same answer is important because it shows that <math>-(-2)=2</math>. 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 <math>5-8</math>. 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.<ref name="phschool.com">{{cite web|url=http://www.phschool.com/professional_development/teaching_tools/pdf/using_algebra_tiles.pdf |title=Prentice Hall School |publisher=Phschool.com |date= |accessdate=2013-07-22}}</ref>
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| ===Multiplication of integers===
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| [[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 [[divisor|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.
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| ===Modeling and simplifying algebraic expressions===
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| Modeling algebraic expressions with algebra tiles is very similar to modeling [[addition]] and [[subtraction]] of integers using algebra tiles. In an expression such as <math>5x-3</math> 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 <math>4x+5-2x-3</math> you can combine the positive and negative x tiles and unit tiles to form zero pairs to leave you with the expression <math>2x+2</math>. 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.<ref name="phschool.com" />
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| 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 <math>3(x+1)=3x+3</math> then we would make three sets of one unit tile and one x tile and then combine them together to see if would have <math>3x+3</math>, which we would.<ref>[http://www.regentsprep.org/rEGENTS/math/realnum/Tdistrib.htm ]{{dead link|date=July 2013}}</ref>
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| ===Solving linear equations===
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| Manipulating algebra tiles can help students solve [[linear equations]]. In order to solve a problem like <math>x-6=2</math> 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 <math>x=8</math>.<ref name="phschool.com" /> You can also use the [[subtraction]] property of equality to solve your [[linear equation]] with algebra tiles. If you have the equation <math>x+7=10</math>, 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 <math>x=3</math>.<ref name="Kitts, N page 464">Kitts, N: "Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts", page 464. MATHEMATICS TEACHER, 2000.</ref> There are programs online that allow students to create their own [[linear equations]] and 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 [[TeacherTube]] 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]
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| ===Solving linear systems===
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| 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.
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| ===Multiplying polynomials===
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| 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 x<sup>2</sup>, which is the answer.
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| [[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 [[Integer factorization|factors]] are the [[length]] and [[width]].<ref>Stein, M: Implementing Standards-Based Mathematics Instruction", page 98. Teachers College Press, 2000.</ref> As with the [[monomials]], you set up the sides of the rectangle to be the [[Integer factorization|factors]] and then you fill in the rectangle with the algebra tiles.<ref>Stein, M: Implementing Standards-Based Mathematics Instruction", page 106. Teachers College Press, 2000.</ref> This method of using algebra tiles to multiply [[polynomials]] is known as the area model<ref>Larson R: "Algebra 1", page 516. McDougal Littell, 1998.</ref> 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 x<sup>2</sup> tiles, five positive x tiles, and two positive unit tiles. So the solution is 2x<sup>2</sup>+5x+2.
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| ===Factoring===
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| 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<sup>2</sup> tile, three positive x tiles, and two positive unit tiles. You form the rectangle by having the x<sup>2</sup> tile in the upper right corner, then you have two x tiles on the right side of the x<sup>2</sup> tile, one x tile underneath the x<sup>2</sup> 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 [[Integer factorization|factors]] are <math>x+1</math> and <math>x+2</math>.<ref name="Kitts, N page 464" /> In a sense this is the reverse of the procedure for multiplying [[polynomials]].
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| ===Completing the square===
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| The process of [[completing the square]] can be accomplished using algebra tiles by placing your x<sup>2</sup> 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 x<sup>2</sup>+6x you start off with one positive x<sup>2</sup> tile and six positive x tiles. You place the x<sup>2</sup> tile in the upper left corner and then you place three positive x tiles to the right of the x<sup>2</sup> tile and three positive unit x tiles under the x<sup>2</sup> tile. In order to fill in the square we need nine positive unit tiles. we have now created x<sup>2</sup>+6x+9, which can be factored into <math>(x+3)(x+3)</math>.<ref>{{cite web|author=Donna Roberts |url=http://www.regentsprep.org/Regents/math/algtrig/ATE12/completesq.htm |title=Using Algebra Tiles to Complete the Square |publisher=Regentsprep.org |date= |accessdate=2013-07-22}}</ref>
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| ==References==
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| {{reflist|2}}
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| ==Sources==
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| * Kitt, Nancy A. and Annette Ricks Leitze. "Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts." ''MATHEMATICS TEACHER'' 2000. 462-520.
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| * Stein, Mary Kay et al., ''IMPLEMENTING STANDARDS-BASED MATHEMATICS INSTRUCTION''. New York: Teachers College Press, 2000.
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| * Larson, Ronald E., ''ALGEBRA 1''. Illinois: McDougal Littell,1998.
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| == External links ==
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| * [http://www.assessmentservices-edu.com/Algebra-tiles.aspx Algebra tile manipulatives]
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| [[Category:Mathematics education]]
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Hello, dear friend! I am Consuelo. I smile that I could join to the entire world. I live in Italy, in the AP region. I dream to check out the different nations, to obtain acquainted with interesting people.
xunjie 森馬とスミスバーニー衣装上場企業の2つの中国のパフォーマンスは良くありません。
価格戦争と綿花価格大規模な環境で、
トニーの下着ブランド美：女性の健康のための調節可能な下着を中心に設計された製品のメンテナンスや修理の両方を節約する神の守護ビーナスベイ美容ランジェリーブランドの美しさだけでなく、 [http://cambridgetoafrica.org/fileshare/images/tomford.html �ȥ�ե��`�� ���饹 �����`] 竹繊維の多種多様な製品を生産するための処理竹繊維織物を通じ、
古代アンティークヒスイのような非現在のこの気持ちのように、
紫色のテクスチャアンテプリマブラックとジェイソンウーの近くから、 [http://www.rheintalverlag.ch/newsletter/gaga.php gaga milano �rӋ] シャオ風水ロング衣類のために2012春と夏の新しいドラゴンの年をテーマにした一連の設計：鳳ロングパラダイス。
我々は気にクマの子供服ブランドのファッション製品の賑やかなクロスを開いたベア子供服ブランドのファッションストアで賑やかなクロスを開設 - コアコンセプトとして、
誰もがそれをチェックアウトしたかった場所になった後の町は歴史痴漢の350年を持っています。[http://www.rheintalverlag.ch/newsletter/gaga.php �����ߥ�� ���n] この国を救うためにノーと言った愛国中国の人々のビバは、
自身のスタイリングさわやか。
私たちはあなたにエキサイティングなファッションのインスピレーションの多くをもたらすと信じています。
として記載されている「世界で喜び気ままな子どもたちの完全な。 [http://aphroditeinn.gr/webmail/mime/li/shop/mall/nb.html �˥�`�Х�
���˩`���`]
my homepage - クリスチャンルブタン ブーツ