**Algebra tiles Manipulative…..**

**Objectives:**

· To eliminate the frustration or anxiety involved with multiplying and factoring polynomials through the usage of Algebra Tiles.

· Understanding the concept of variables by naming tiles.

· Visualizing quadratic terms in polynomials as squares.

· Performing basic operations, such as addition and subtraction, on polynomials

· Factoring second-degree polynomials .

**INTRODUCTION :**

An old Chinese proverb states:

I hear, and I forget;I see, and I remember;I do, and I learn.

· They say mathematics is not a spectator sport, but by demonstrations, it can be done.

· In this mathematics model, representation of factorization of algebraic expressions using algebra tiles is done.

· That is how algebra manipulatives, i.e. algebra tiles make the difficult concept of "factoring second-degree polynomials" into a simple puzzle that’s fun. We have used a set of algebra tiles to factor the given polynomial. They help convert the abstract concept of polynomials into tangible objects - tiles.

**What are manipulatives?**

· Manipulatives are materials that are physically handled in order to see actual examples of heads - the tools for solving equations, visual representations of algebraic factoring, and a multitude of other skills and concepts. Manipulatives can introduce mathematical topics and reinforce conceptual understanding in powerful ways.

What are Algebra Tiles?

· They work by the concept that every rectangle has a length, width and area. The length and the width are the lengths of the sides of the rectangle in some unit. The area is how many squares of that unit it takes to cover the rectangle.The sides of the rectangle are the factors of the given polynomial.

**Nomenclature:**

Algebra tiles have a variety of other names:

Algetiles

Math tiles

Virtual tiles.

**History:**

The first use of algebraic manipulatives to illustrate algebraic ideas was by math educator Zoltan dienes who used base 10 blocks.The idea was powerful, and launche dthe idea of algebraic manipulatives.

· Mary Laylock improved on dieles’ model, by using multi base blocks. Instead of just working with base ten blocks, a trinomial factoring had to work on all bases. She also introduced upstairs representation of minus, which made it possible to represent a product involving minus like (x+1)(x-1) in a geometrically correct way.

· Peter Rasmussen used base +10, +5, +25 tiles for convenience. He also created the non commensurable x, which solved the prolem of false factorings encountered when using arithmetic blocks fo variables. His model of minus combined with Laylock’s idea with clor scheme. The tiles were only painted on one side, so that if a tile is turned over its unpainted side, it is considered negative.

· The algebra tiles of today are based on Rasmussen’s ground breaking model.

· Algebra tiles can be very beneficial in understanding the concept of "like terms" and combining of terms.

Application of Algebra Tiles:

· Multiplication of monomials and binomials

· Representation of quadratic polynomials.

· Solution of Linear equations in one variable.

· Solution of a quadratic equation

· Understanding integers.

· Addition and subtraction of polynomials.

We have prepared Algebra Tiles

• Square tiles of dimension 10 X 10 each, representing x^2.

• Rectangular tiles of dimension 10 X 1 each , representing x .

• Square tiles of dimension 1 X 1 each representing 1.

· Assumption:

We have assumed that

• In the rectangular tiles each of dimension x sq units the top side represents (+ x) and the bottom side represents (- x).

• Similarly in the square tiles each of dimension 1 sq unit the top side represents (+1) and the bottom side represents (-1).

·

· By splitting the middle term of the given polynomial we get the expression x^2 +3x +2x + 6.

· Place a square tile of dimension 10X10 representing x^2 .

· Add 3 tiles of dimension 10 X 1 each to any side of the tile x^2.The area of new shape formed represents x^2 +3x.

· Add 2 tiles of dimension 10 X 1 each to the side adjacent to the previous side. The area of new shape formed represents x^2 +3x+2x.

· Add 6 tiles of dimension 1 X 1 each to complete the rectangle. The area of new shape formed represents x^2 +3x+2x+6.

· Mary Laylock improved on dieles’ model, by using multi base blocks. Instead of just working with base ten blocks, a trinomial factoring had to work on all bases. She also introduced upstairs representation of minus, which made it possible to represent a product involving minus like (x+1)(x-1) in a geometrically correct way.

· Peter Rasmussen used base +10, +5, +25 tiles for convenience. He also created the non commensurable x, which solved the prolem of false factorings encountered when using arithmetic blocks fo variables. His model of minus combined with Laylock’s idea with clor scheme. The tiles were only painted on one side, so that if a tile is turned over its unpainted side, it is considered negative.

· The algebra tiles of today are based on Rasmussen’s ground breaking model.

**Utility:**

· They help in bridging the gap from the concrete to the abstract.· Algebra tiles can be very beneficial in understanding the concept of "like terms" and combining of terms.

Application of Algebra Tiles:

· Multiplication of monomials and binomials

· Representation of quadratic polynomials.

· Solution of Linear equations in one variable.

· Solution of a quadratic equation

· Understanding integers.

· Addition and subtraction of polynomials.

**Description of Physical Model**

· Preparation:We have prepared Algebra Tiles

• Square tiles of dimension 10 X 10 each, representing x^2.

• Rectangular tiles of dimension 10 X 1 each , representing x .

• Square tiles of dimension 1 X 1 each representing 1.

· Assumption:

We have assumed that

• In the rectangular tiles each of dimension x sq units the top side represents (+ x) and the bottom side represents (- x).

• Similarly in the square tiles each of dimension 1 sq unit the top side represents (+1) and the bottom side represents (-1).

·

**PROCEDURE**

I)**Representation of x^2 +5x + 6**

· To represent this we need 1 square tile representing x^2 , 5 algebra tiles representing x and 6 algebra tiles representing 1.· By splitting the middle term of the given polynomial we get the expression x^2 +3x +2x + 6.

· Place a square tile of dimension 10X10 representing x^2 .

· Add 3 tiles of dimension 10 X 1 each to any side of the tile x^2.The area of new shape formed represents x^2 +3x.

· Add 2 tiles of dimension 10 X 1 each to the side adjacent to the previous side. The area of new shape formed represents x^2 +3x+2x.

· Add 6 tiles of dimension 1 X 1 each to complete the rectangle. The area of new shape formed represents x^2 +3x+2x+6.

II)

**Representation of x^2 -x - 6**

· To represent this we need 1 square tile representing x^2 , 5 algebra tiles representing x and 6 algebra tiles representing 1.

· By splitting the middle term of the given polynomial we get the expression x^2 -3x +2x - 6.

· Place a square tile of dimension 10X10 representing x^2 .

· Add 2 tiles of dimension 10 X 1 each to any side of the tile x^2.The area of new shape formed represents x^2 +2x.

· Subtract 3 tiles of dimension 10 X 1 each to the side adjacent to the previous side. The area of new shape formed represents x^2 +2x-3x.

· Subtract 6 tiles of dimension 1 X 1 each to complete the rectangle. The area of new shape formed represents x^2 +2x-3x-6.

.

III)

· By splitting the middle term of the given polynomial we get the expression x^2 -3x -2x + 6.

· Place a square tile of dimension 10X10 representing x^2 .

· Subtract 3 tiles of dimension 10 X 1 each to any side of the tile x^2.The area of new shape formed represents x^2 -3x.

· Add 6 tiles of dimension 1 X 1 each to get 2 tiles of dimension 10 X 1 each to the side adjacent to the previous side. The area of new shape formed represents x^2 -3x+6.

· Subtract 2 tiles of dimension 10 X 1 each to complete the rectangle. The area of new shape formed represents x^2 -3x+6-2x.

III)

**Representation of x^2 -5x + 6**

· To represent this we need 1 square tile representing x2 , 5 algebra tiles representing x and 6 algebra tiles representing 1.· By splitting the middle term of the given polynomial we get the expression x^2 -3x -2x + 6.

· Place a square tile of dimension 10X10 representing x^2 .

· Subtract 3 tiles of dimension 10 X 1 each to any side of the tile x^2.The area of new shape formed represents x^2 -3x.

· Add 6 tiles of dimension 1 X 1 each to get 2 tiles of dimension 10 X 1 each to the side adjacent to the previous side. The area of new shape formed represents x^2 -3x+6.

· Subtract 2 tiles of dimension 10 X 1 each to complete the rectangle. The area of new shape formed represents x^2 -3x+6-2x.

IV)

**Representation of x^2 +x -6**

· To represent this we need 1 square tile representing x^2 , 5 algebra tiles representing x and 6 algebra tiles representing 1.

· By splitting the middle term of the given polynomial we get the expression x^2 +3x -2x - 6.

· Place a square tile of dimension 10X10 representing x^2 .

· Add 3 tiles of dimension 10 X 1 each to any side of the tile x2.The area of new shape formed represents x^2 +3x.

· Subtract 2 tiles of dimension 10 X 1 each from the side adjacent to the previous side. The area of new shape formed represents x^2 +3x-2x.

· Subtract 6 tiles of dimension 1 X 1 each to complete the rectangle. The area of new shape formed represents x^2 +3x-2x-6.

·

**Observation**

· In the representation of x^2 +5x + 6 , a rectangle is formed whose sides are (x+3) and (x+2) which are the factors of it.

· In the representation of x^2 -x - 6 , a rectangle is formed whose sides are (x-3) and (x+2) which are the factors of it.

· In the representation of x^2 -5x + 6 , a rectangle is formed whose sides are (x-3) and (x-2) which are the factors of it.

· In the representation of x^2 +x - 6 , a rectangle is formed whose sides are (x+3) and (x-2) which are the factors of it.

**Result:**

Thus we have observed that in all the four cases a rectangle is formed whose sides are the factors of the given polynomial.

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