Transcript Let`s Do Algebra Tiles

Let’s Do
Algebra Tiles
REL HYBIRD ALGEBRA RESEARCH PROJECT
Adapted from David McReynolds, AIMS PreK-16 Project
and Noel Villarreal, South Texas Rural Systemic Initiative
November , 2007
Algebra Tiles
Manipulatives used to enhance
student understanding of concepts
traditionally taught at symbolic level.
 Provide access to symbol
manipulation for students with weak
number sense.
 Provide geometric interpretation of
symbol manipulation.
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Algebra Tiles
Support cooperative learning, improve
discourse in classroom by giving
students objects to think with and talk
about.
 When I listen, I hear.
 When I see, I remember.
 But when I do, I understand.
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Algebra Tiles
Algebra tiles can be used to model
operations involving integers.
 Let the small yellow square represent
+1 and the small red square (the flipside) represent -1.
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The yellow and red squares are
additive inverses of each other.
Algebra Tiles
Algebra tiles can be used to model
operations involving variables.
 Let the green rectangle represent +1x
or x and the red rectangle (the flipside) represent -1 x or -x .
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The green and red rods are additive
inverses of each other.
Algebra Tiles
Let the blue square represent x2. The red
square (flip-side of blue) represents -x2.
 As with integers, the red shapes and their
corresponding flip-sides form a zero pair.
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Zero Pairs
Called zero pairs because they are
additive inverses of each other.
 When put together, they model zero.
 Don’t use “cancel out” for zeroes use
zero pairs or add up to zero
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Addition of Integers
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Addition can be viewed as “combining”.
Combining involves the forming and
removing of all zero pairs.
For each of the given examples, use
algebra tiles to model the addition.
Draw pictorial diagrams which show the
modeling.
Write the manipulation performed
Addition of Integers
(+3) + (+1) =
 Combined like objects to get four
positives
(-2) + (-1) =
 Combined like objects to get three
negatives
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Addition of Integers
(+3) + (-1) = +2
 Make zeroes to get two positives
(+3) + (-4) = -1
 Make three zeroes to get one negative
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After students have seen many examples of
addition, have them formulate rules.
Subtraction of Integers
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Subtraction can be interpreted as “takeaway.”
Subtraction can also be thought of as
“adding the opposite.” (must be extensively
scaffolded before students are asked to
develop)
For each of the given examples, use
algebra tiles to model the subtraction.
Draw pictorial diagrams which show the
modeling process
Write a description of the actions taken
Subtraction of Integers
(+5) – (+2) = +3
 Take away two positives
 To get three positives
(-4) – (-3) = -1
 Take away three negatives
 To get one negative
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Subtracting Integers
(+3) – (-5) = +8
Add five zeroes; Take away five negatives
To get eight positives
(-4) – (+1)= -5
Add one zero; Take away one positive
To get14 five negatives
Subtracting Integers
(+3) – (-3)=
After students have seen many
examples, have them formulate rules
for integer subtraction.
(+3) – (-3) is the same as 3 + 3 to get 6
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Multiplication of Integers
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Integer multiplication builds on whole
number multiplication.
Use concept that the multiplier serves as
the “counter” of sets needed.
For the given examples, use the algebra
tiles to model the multiplication. Identify the
multiplier or counter.
Draw pictorial diagrams which model the
multiplication process
Write a description of the actions performed
Multiplication of Integers
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The counter indicates how many rows to
make. It has this meaning if it is positive.
(+2)(+3) = +6
Two groups of
three positives
(+3)(-4) = +12
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Three groups of
four negatives
Multiplication of Integers
If the counter is negative it will mean “take
the opposite of.”
 Can indicate the motion “flip-over”, but be
very careful using that terminology
•Two groups of three
(-2)(+3) = -6
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•Opposite of
•To get six negatives
(-3)(-1) = +3
•Opposite of three
groups of negative one
•To get three positives
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Multiplication of Integers
After students have seen many
examples, have them formulate rules
for integer multiplication.
 Have students practice applying rules
abstractly with larger integers.
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Division of Integers
Like multiplication, division relies on
the concept of a counter.
 Divisor serves as counter since it
indicates the number of rows to
create.
 For the given examples, use algebra
tiles to model the division. Identify the
divisor or counter. Draw pictorial
diagrams which model the process.
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Division of Integers
(+6)/(+2) =
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Divide into two equal groups
(-8)/(+2) =
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Divide into two equal groups
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Division of Integers
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A negative divisor will mean “take the
opposite of.” (flip-over)
(+10)/(-2) = -5
Divide into two equal groups
 Find the opposite of
 To get five negatives
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Division of Integers
(-12)/(-3) = +4
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After students have seen many
examples, have them formulate rules.
Polynomials
“Polynomials are unlike the other
‘numbers’ students learn how to add,
subtract, multiply, and divide. They
are not ‘counting’ numbers. Giving
polynomials a concrete reference
(tiles) makes them real.”
David A. Reid, Acadia University
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Distributive Property
Use the same concept that was
applied with multiplication of integers,
think of the first factor as the counter.
 The same rules apply.
3(x + 2)
 Three is the counter, so we need
three rows of (x + 2)
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Distributive Property
3(x + 2)= 3·x + 3·2 = 3x + 6
Three Groups of x to get three x’s
 Three groups of 2 to get 6
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Modeling Polynomials
Algebra tiles can be used to model
expressions.
 Model the simplification of
expressions.
 Add, subtract, multiply, divide, or
factor polynomials.
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Modeling Polynomials
2x2
4x
3 or +3
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More Polynomials
Represent each of the given
expressions with algebra tiles.
 Draw a pictorial diagram of the
process.
 Write the symbolic expression.
x+4
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More Polynomials
2x + 3
4x – 2
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More Polynomials
Use algebra tiles to simplify each of
the given expressions. Combine like
terms. Look for zero pairs. Draw a
diagram to represent the process.
 Write the symbolic expression that
represents each step.
2x + 4 + x + 2
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More Polynomials
2x + 4 + x + 1 = 3x + 5
Combine like terms to get
three x’s and five positives
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More Polynomials
3x – 1 – 2x + 4
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This process can be used with
problems containing x2.
(2x2 + 5x – 3) + (-x2 + 2x + 5)
(2x2 – 2x + 3) – (3x2 + 3x – 2)
Substitution
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Algebra tiles can be used to model
substitution. Represent original
expression with tiles. Then replace
each rectangle with the appropriate
tile value. Combine like terms.
3 + 2x
let x = 4
Substitution
3 + 2x =
3 + 2(4) =
3+8=
11
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let x = 4
Solving Equations
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Algebra tiles can be used to explain and justify the equation
solving process. The development of the equation solving
model is based on two ideas.
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Equivalent Equations are created if equivalent operations are
performed on each side of the equation. (Which means to use
the additon, subtraction, mulitplication, or division properties of
equality.) What you do to one side of the equation you must
do to the other side of the equation.
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Variables can be isolated by using the Additive Inverse Property
( & zero pairs) and the Multiplicative Inverse Proerty ( & dividing
out common factors). The goal is to isolate the variable.
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Solving Equations
x+2= 3
-2 -2
x =1
•x and two positives are the same as three positives
•add two negatives to both sides of the equation;
makes zeroes
•one x is the same as one positive
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Solving Equations
-5 = 2x
÷2 ÷2
2½ = x
•Two x’s are the same as five negatives
•Divide both sides into two equal partitions
•Two39and a half negatives is the same as one x
Solving Equations
1
 x
2
· -1 · -1
1
 x
2
•One half is the same as one negative x
•Take the opposite of both sides of the equation
•One half of a negative is the same as one x
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Solving Equations
x
 2
3
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3 •3
x = -6
•One third of an x is the same as two negatives
•Multiply both sides by three (or make both sides
three times larger)
•One half of a negative is the same as one x
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Solving Equations
2x+3=x–5
-x
-x
x + 3 = -5
+ -3 + - 3
x = -8
•Two x’s and three positives are the same as one x and five
negatives
•Take one x from both sides of the equation; simplify to get one x
and three the same as five negatives
•Add three negatives to both sides; simplify to get x the same as
eight negatives
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Solving Equations
3(x – 1) + 5 = 2x – 2
3x – 3 + 5 = 2x – 2
3x + 2 = 2x – 2
– 2 or + -2
3x = 2x – 4
-2x -2x
x = -4
“x is the same as four negatives”
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Multiplication
Multiplication using “base ten blocks.”
(12)(13)
 Think of it as (10+2)(10+3)
 Multiplication using the array method
allows students to see all four subproducts.
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Multiplication using “Area
Model”
(12)(13) = (10+2)(10+3) =
100 + 30 + 20 + 6 = 156
10 x 10 =
102 = 100
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10xx22==20
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10 x 3 = 30
2x3=6
Multiplying Polynomials
(x + 2)(x + 3)
Fill in each section of the
area model
Combine like terms
2 + 5x + 6
x2 + 2x
+
3x
+
6
=
x
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Multiplying Polynomials
(x – 1)(x +4)
Fill in each section of the
area model
Make Zeroes or
combine like terms
x2 + 4x – 1x – 4 = x2 + 3x – 4
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Multiplying Polynomials
(x + 2)(x – 3)
(x – 2)(x – 3)
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Factoring Polynomials
Algebra tiles can be used to factor
polynomials. Use tiles and the frame
to represent the problem.
 Use the tiles to fill in the array so as to
form a rectangle inside the frame.
 Be prepared to use zero pairs to fill in
the array.
 Draw a picture.
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Factoring Polynomials
3x + 3 = 3 · (x + 1)
2x – 6 = 2 · (x – 3)
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Factoring Polynomials
x2 + 6x + 8 = (x + 2)(x +4)
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Factoring Polynomials
x2 – 5x + 6 = (x – 2)(x – 3)
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Factoring Polynomials
x2 – x – 6 = (x + 2)(x – 3)
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Factoring Polynomials
x2 + x – 6
x2 – 1
x2 – 4
2x2 – 3x – 2
2x2 + 3x – 3
-2x2 + x + 6
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Dividing Polynomials
Algebra tiles can be used to divide
polynomials.
 Use tiles and frame to represent
problem. Dividend should form array
inside frame. Divisor will form one of
the dimensions (one side) of the
frame.
 Be prepared to use zero pairs in the
dividend.
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Dividing Polynomials
x2 + 7x +6
= (x + 6)
x+1
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Dividing Polynomials
x2 + 7x +6
x+1
2x2 + 5x – 3
x+3
x2 – x – 2
x–2
x2 + x – 6
x+3
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Conclusion
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Algebra tiles can be made using the Ellison
(die-cut) machine.
On-line reproducible can be found by doing
a search for algebra tiles.
Virtual Algebra Tiles at HRW
http://my.hrw.com/math06_07/nsmedia/tools
/Algebra_Tiles/Algebra_Tiles.html
National Library of Virtual Manipulatives
http://nlvm.usu.edu/en/nav/topic_t_2.html
Resources
David McReynolds
AIMS PreK-16 Project
 Noel Villarreal
South Texas Rural Systemic Initiative
 Jo Ann Mosier & Roland O’Daniel
Collaborative for Teaching and Learning
 Partnership for Reform Initiatives in Science
and Mathematics
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