Transcript Let`s Do Algebra Tiles

Let’s Work With
Algebra Tiles
Algebra Tiles
Algebra tiles can be used to model
operations involving integers.
 Let the small green square represent
+1 and the small pink square
represent -1.

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The green and pink 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 pink rectangle represent 1 x or -x .
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The green and red rods are additive
inverses of each other.
Algebra Tiles
Let the green square represent x2. The pink
square represents -x2.
 As with integers, the green squares and the
pink squares 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.
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Practice with Integers
Algebra tiles can be use to model
adding, subtracting, multiplying, and
dividing real numbers.
 Remember, if you don’t have enough
of something, you can add “zero
pairs”

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Modeling Addition/Subtraction
-2 – 3
 Start with -2

Take away positive 3. But wait, I don’t
have 3 so I must add zero pairs!
 Now remove positive 3
 You are left with -5
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Modeling Addition/Subtraction
-1 + 3
 Start with -1

Now, add positive 3
 Now, you must remove any “zero
pairs”
 You are left with 2.

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You try
-5 + 3
 -1 + 4
 4 – -2
 2 + -3
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Modeling Multiplication
2 x -3
 This means I need 2 rows of -3
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Which is -6
 This could also mean “the opposite of
3 rows of two”
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You try
1 x -4
 3 x -3
 -2 x 4
 -2 x -2
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Modeling Division
2

6

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The “6” is called the dividend. The “2” is called the divisor.
Place the “2” outside, and then line up the 6 inside.

You answer is what fits on top, which is 3. (called the quotient)
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You Try
4/2
 8 / -4
 -6 / 3
 3 / -1
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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.
 To solve equations with polynomials.
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Modeling Polynomials
2x2
4x
3 or +3
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More Polynomials
2x + 3
4x – 2
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Adding 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.
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Adding Polynomials
(2x + 4) + (x + 1) = 3x + 5
Combine like terms to get three
x’s and five positive ones
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Adding Polynomials
(3x – 1) – (2x + 4)
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Now remove 2x and remove 4. But WAIT, I
don’t have 4 so I must add zero pairs.
Now remove 2x and remove 4
And you are left with x - 5
You Try
(2x – 1) + (x + 2)
 (x + 3) + (x – 2)
 (2x – 1) – (x + 5)
 (3x + 5) + (x – 1)
 (4x – 3) – (3x – 2)
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Adding Polynomials
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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)
Distributive Property
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Multiplying a monomial to a polynomial
3(x – 2) = 3x - 6
Distributive Property
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-2(x - 4) = -2x + 8
You try
4 (x + 2)
 2 (x – 3)
 -2 (x + 1)
 -2 ( x – 1)
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Multiplying Polynomials
(x + 2)(x + 3)
Fill in each section of the
area model
Combine like terms
x2 + 2x + 3x + 6 = x2 + 5x + 6
Multiplying Polynomials
(x – 1)(x +4)
Fill in each section of the
area model
Make Zeroes or
combine like terms
and simplify
x2 + 4x – 1x – 4 = x2 + 3x – 4
You Try

(x + 2)(x – 3)

(x – 2)(x – 3)
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(x – 1) ( x + 4)

(x – 3) (x – 2)
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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 – 5x + 6
= (x – 2)
x–3
You Try
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x2 + 7x +6
x+1
2x2 + 5x – 3
x+3
x2 – x – 2
x–2
x2 + x – 6
x+3
Factoring Polynomials
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Factoring is the process of writing a
polynomial as a product.
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.
Factoring Polynomials
3x + 3 = 3 · (x + 1)
2x – 6 = 2 · (x – 3)
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You Try
Factor 4x – 2
 Factor 3x + 6
2
 Factor x  2 x
2
2x
x
 Factor
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Factoring Polynomials
x2 + 6x + 8
= (x + 2)(x + 4)
Factoring Polynomials
x2 – 5x + 6 = (x – 2)(x – 3)
Remember: You must
form a RECTANGLE
out of the polynomial
Factoring Polynomials
x2 – x – 6 = (x + 2)(x – 3)
This time the polynomial
doesn’t form a rectangle,
so I have to add “zero
pairs” in order to form a
rectangle.
You Try
x2 + 3x + 2
 x2 + 4x + 3
 x2 + x – 6

x2 – 1

x2 – 4

2x2 – 3x – 2
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Solving Equations
We can use algebra tiles to solve
equations.
 Whatever you do to one side of the
equal sign, you have to do to the other
to keep the equation “balanced”.
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Solving Equations
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3x + 4 = 2x – 1
First build each side of the equation
=
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Now remove 2x from each side.
Next, remove 4 from each side. But wait, I don’t
have 4 so I must add “zero pairs”
Remove 4 from each side
You are left with x = -5
Solving Equations
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4x + 1 = 2x + 7
First, build each side of the equation
=
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Next, remove 2x from each side.
Remove 1 from each side.
Now divide each side by 2.
Your result is x = 3.
You Try
2x + 3 = x – 2
 x – 4 = 2x + 1
 3x + 1 = x – 5
 8x – 2 = 6x + 4
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Credits

Adapted by Marcia Kloempken, Weber High School
from David McReynolds, AIMS PreK-16 Project