What is algebraic tiles?
==UsesAdding integers=== '''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 2+3=?. 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 2+3 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 2+3=5. Since adding a number with the negative of
that number gives you zero, for instance -2+2=0, 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 [[integer]] plus its negative
is [[zero]] a person can physically represent this concept through
'''algebra tiles'''. Let us take the example used earlier where
-2+2=0. 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
-7+4=?, 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 -7+4=-3. ===Subtracting
integers===
'''Algebra tiles''' can also be used for subtracting
[[integers]]. A person can take a problem such as 6-3=? and begin
with a group of six unit tiles and then take three away to leave
you with three left over, so then 6-3=3. '''Algebra tiles''' can
also be used to solve problems like -4-(-2)=?. 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 -4-(-2)=-2, which is also the same answer you would get
if you had the problem -4+2. Being able to relate these two
problems and why they get the same answer is important because it
shows that -(-2)=2. 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 5-8. 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===
[[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
[[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. ===Modeling and simplifying algebraic expressions===
Modeling algebraic expressions with '''algebra tiles''' is very
similar to modeling [[addition]] and [[subtraction]] of integers
using '''algebra tiles'''. In an expression such as 5x-3 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 4x+5-2x-3
you can combine the positive and negative x tiles and unit tiles to
form zero pairs to leave you with the expression 2x+2. 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===
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 3(x+1)=3x+3 then we would make three sets of
one unit tile and one x tile and then combine them together to see
if would have 3x+3, 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 x-6=2 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
x=8.[http://www.phschool.com/professional_development/teaching_tools/pdf/using_algebra_tiles.pdf]
You can also use the [[subtraction]] property of equality to solve
your [[linear equation]] with '''algebra tiles'''. If you have the
equation x+7=10, 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 x=3.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 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 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] ===Multiplying polynomials===
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 x2, which is the answer.
[[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 [[factors]] are the [[length]]
and [[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
[[factors]] and 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 [[polynomials]] is known
as the area modelLarson R: "Algebra 1", page 516. McDougal Littell,
1998. 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 x2
tiles, five positive x tiles, and two positive unit tiles. So the
solution is 2x2+5x+2. ===Factoring===
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 [[length]]and then one positive x tile
and two positive unit tiles for the [[width]]. This means that the
two [[factors]] are x+1 and x+2. 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 square]] can 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 square]]
you would determine how many unit tiles would be needed to fill in
the missing square. In order to [[complete the square]] of 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
(x+3)(x+3).[http://www.regentsprep.org/Regents/math/algtrig/ATE12/completesq.htm]
Jordan
Professor
Vivi
