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2025-02-23T21:30:49Z

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{{Short description|Type of mathematical manipulative}}
{{POV|date=April 2021}}

'''Algebra tiles''', also known as '''Algetiles''', or '''Variable Blocks''', 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.<ref name="Kitt, N">Kitt 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="Kitt, N" /> 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 name="Stein, M">Stein 2000.</ref>
==Examples==

===Solving linear equations using addition===

The [[linear equation]] <math>x-8=6</math> can be modeled with one positive <math>x</math> 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.<ref name="Kitt, N" /> For example, eight positive unit tiles can be added to both sides.<ref name="Kitt, N" /> Zero pairs of unit tiles are removed from the left side, leaving one positive <math>x</math> tile. The right side has 14 positive unit tiles, so <math>x=14</math>.

<gallery mode="packed">
File:Algebra tile x-6=2.jpg|Algebra tile model of <math>x-6=2</math>
File:Algebra tile solving x-6=2 using addition.jpg|Algebra tile model of <math>x-6+6=2+6</math>
File:Algebra tile x=2.jpg|Algebra tile model of <math>x=8</math>
</gallery>

===Solving linear equations using subtraction===

The equation <math>x+7=10</math> can be modeled with one positive <math>x</math> 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 <math>x</math> tile on the left side and three positive unit tiles on the right side, so <math>x=3</math>.<ref name="Kitt, N" />

<gallery mode="packed">
File:Algebra tile x+7=10.jpg|Algebra tile model of <math>x+7=10</math>
File:Algebra tile solving x+7=10 using subtraction.jpg|Algebra tile model of <math>x=3</math>
</gallery>

===Multiplying polynomials===
When using algebra tiles to multiply a [[monomial]] by a [[monomial]], the student must 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 one multiplies [[integers]] using algebra tiles. Once the sides of the rectangle are represented by the algebra tiles, one would then try to figure out which algebra tiles would fill in the rectangle. For instance, if one had x×x, the only algebra tile that would complete the rectangle would be x2, which is the answer.

[[Multiplication]] of [[binomial (polynomial)|binomial]]s is similar to [[multiplication]] of [[monomials]] when using the algebra tiles . Multiplication of [[binomial (polynomial)|binomial]]s can also be thought of as creating a rectangle where the [[Integer factorization|factors]] are the [[length]] and [[width]].<ref name="Stein, M" /> As with the [[monomials]], one would set up the sides of the rectangle to be the [[Integer factorization|factors]] and then fill in the rectangle with the algebra tiles.<ref name="Stein, M" /> 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 [[binomial (polynomial)|binomial]]s with each other. An example of multiplying [[binomial (polynomial)|binomial]]s is (2x+1)×(x+2) and the first step the student would take is set up two positive x tiles and one positive unit tile to represent the [[length]] of a rectangle and then one 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===
[[File:Algebra tile factoring.jpg|thumb|Algebra tile model of <math>x^2+3x+2</math>]]

In order to factor using algebra tiles, one has to start out with a set of tiles that the student combines into a rectangle, this may require the use of adding zero pairs in order to make the rectangular shape. An example would be where one is given one positive x2 tile, three positive x tiles, and two positive unit tiles. The student forms the rectangle by having the x2 tile in the upper right corner, then one has 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 [[Integer factorization|factors]] are <math>x+1</math> and <math>x+2</math>.<ref name="Kitt, N" /> In a sense this is the reverse of the procedure for multiplying [[polynomials]].

==References==
{{reflist|2}}

==Sources==
* Kitt, Nancy A. and Annette Ricks Leitze. "Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts." ''MATHEMATICS TEACHER'' 2000. 462-520.
* [[Mary Kay Stein|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 ==
* [http://nlvm.usu.edu/en/nav/vlibrary.html The National Library of Virtual Manipulatives]

[[Category:Mathematical manipulatives]]
[[Category:Algebra education]]

Algebra tile - Revision history

imported>Red Aviation at 21:30, 23 February 2025