Algebra's Tiling

I don't think algebra tiles or blocks are a panacea for what ails student algebraists, but I do think they are powerful.  Even though as a mathematician I am pretty comfortable with symbolic reasoning, at heart I am a visual thinker.  Having a way to visualize algebra opens up many possibilities for learners, even when they had a good symbolic understanding beforehand.  The goal for me is not to replace symbolic manipulation, but support the concepts.

I designed the activity for students that may have not done a lot of investigation before.  So it starts with a lot of modeling, and then letting them try.  Students did an amazing job.

Even before doing the mental math (making the point that even number operations can be visualized, plus setting the context for the follow up activity), I asked the students to take a  look at the blocks to see what they noticed. Mr. Boeve had had the students play with the blocks the day before, which is an excellent idea.  They noticed corresponding dimensions, colors, different designs.  These were the tiles from Algebra Lab Gear, so there's 1/2 x blocks, 1/4 x blocks, 5 sticks and 25 sticks.

None of the students had seen the visual multiplication before, but they were willing to give it a try.  They made connections as to why the pieces represented what they did.  The verbal connection, x squared, was biggest, but then a few students recognized the x times x relation.  They picked up the symbol to picture representing quickly, and that gave an opportunity to talk about how there are many different ways to write things in math.  They had 2x+3+1x+2 and 2x +3 +x+2 and 3x+5.  We introduced what mathematicians call simplifying, which they connected to fractions.

I raised the problem of negatives - how could we show negatives, because algebra has a lot of those.  They thought we could have two color blocks, or use some kind of design.  How could we do it with the blocks we have?  Maybe we could separate the positives and negatives.  Nice thinking!

Sometimes I think of mini-lessons like these as equipping the students for problems.  The problem list offered practice, light extension and serious problems.  After 15 minutes to do their choice, which they loved having, we came back together and I asked if there were any they wanted to see me do?  They suggested problems, and if there was a student to explain them, they gave it a try.  One of the themes throughout was "give it a go."  I made sure to ask some students who had incomplete or incorrect thinking so we could talk about that, too.  Because the whole situation was different, it helped with making that safe.

In discussing the subtraction problems, they got to three separate ideas: taking away, zero pairs, and adding the opposite.  The students exploring these ideas were able to give reasons for why it made sense with the blocks.


We just worked with the document camera, but if you want virtual algebra tiles, there's the National Virtual Manipulative Library tiles, a two color version from the Michigan Virtual University, and NCTM's Illuminations Algebra Tiles.  None of them are ideal, but all are serviceable.  It's very possible to make homemade algebra tiles, and there was a good article about that in the Mathematics Teacher: "Algebra for All: Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts," Annette Ricks Leitze and Nancy A. Kitt, September 2000, Volume 93, Issue 6.

Thanks to Mr. Boeve and his classes for the nice opportunity!  Jill Beauchamp came along for experience with the algebra blocks, and she was a great support to the kids, so thanks to her, too.




Playing With Blocks

Photo credits: Eamonn @ Flickr

Algebra tiles Manipulatives

Today I am going to discuss about a mathematical model which we have prepared in our Planet Infinity.
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.
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) 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.