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What are different ways that
you can teach students
multiplication of 2 binomials?
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Why do students
struggle with
factoring?
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Using
Algebra Tiles for
Student
Understanding
Examining the Tiles
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Colors and shapes
= 1,
= x,
= -1,
= -x,
= x2
Model the following expressions
+3
-2x
x–4
x2 + 3x - 2
= -x2
Zero Pairs
Called zero pairs because they are
additive inverses of each other.
 When put together, they model zero.

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Model
Simplify
Initial illustration:
before combining
terms
Final illustration
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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Polynomials
Represent each of the given
expressions with algebra tiles.
 Draw a pictorial diagram of the
process.
 Model the symbolic expression -2x + 4
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Modeling Polynomials
2x2 + 3
4x – 2
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Add/Subtract Polynomials
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This process can be used with
problems containing x2. Model each
of the following using tiles.
(2x2 + 5x – 3) + (-x2 + 2x + 5) = x2 + 7x + 2
(2x2 – 2x + 3) – (3x2 + 3x – 2) = -x2 - 5x + 5
Verify your solutions graphically.
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Multiplying Polynomials
Algebra tiles can be used to multiply
polynomials.
 Use tiles and frame to represent the
problem. The factors will form the
dimensions of the frame.
(vertical) and (horizontal)
 The product will form a rectangular
array inside frame. “Area Model”
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Multiplication using
“Area Model”
(2)(3) =
Place 2 sm. squares on the vertical and 3 sm.
squares on the horizontal
Fill in the interior of the area model
with appropriate algebra tiles to
form a rectangular array.
2x3=6
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Multiplying Polynomials

What are all the different ways that you
can demonstrate multiplication of two
binomials?
Tiles
 Box method
 Distributive Property (aka rainbow or foil)
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Multiplying Polynomials
(x )(x + 3)
Fill in each section
of the area model
Algebraically
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x2 + x + x + x = x2 + 3x
Multiplying Polynomials
(x + 2)(x + 3)
Fill in each section
of the area model
Algebraically
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x2+ 2x+ 3x + 6
= x2+ 5x + 6
Multiplying Polynomials
(x – 1)(x + 4)
Fill in each section
of the area model
Make zero pairs or
combine like terms
and simplify
x2 + 4x – 1x – 4 = x2 + 3x – 4
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Multiplying Polynomials
Use the tile frame and tile pieces to
model the product of each problem
below. Think about how you will
connect the algebraic procedure to
the model. Verify your solution using
the box method and distributive
property.
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(2x + 3)(x – 2)
(x – 2)(x – 3)
Virtual Algebra Tiles
http:..media.mivu.org/mvu_pd/a4a/homework/applets_applet_home.html
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Dividing Polynomials
Algebra tiles can be used to divide
polynomials.
 Use tiles and frame to represent the
problem. Dividend should form the
rectangular array inside the frame.
Divisor will form one of the dimensions
(one side) of the frame (vertical side).
 Be prepared to use zero pairs in the
dividend.
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Dividing Polynomials
= (x + 3)
2x2 + 6x
2x
Now, determine the
tiles that would
represent the horizontal
axis: quotient (answer)
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Dividing Polynomials
x2 + 7x + 6
x+1
= (x + 6)
Now, determine
the tiles that
would represent
the horizontal
axis: quotient
(answer)
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Dividing Polynomials
Practice dividing polynomials using tiles.
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2x2 + 5x – 3
x+3
x2 – x – 2
x–2
x2 + x – 6
x+3
Factoring Polynomials
Algebra tiles can be used to factor
polynomials. Use tiles and the frame
to represent the problem.
 Use the tiles to form a rectangular
array inside the frame. (area model)
 Be prepared to use zero pairs to fill in
the array.
 Draw pictures.
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Factoring
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When using tiles…..
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Big squares can't touch little squares.
Little squares must all be together.
Only equal length sides may touch.
You may not lay two equally sized tiles of
different colors next to each other.
Use all of the pieces to make a rectangle.
Once you have correctly arranged the tiles
into a rectangle, the factors of the quadratic
are the length and width of the rectangle.
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Factoring Polynomials
3x + 3 = 3 · (x + 1)
2x – 6 = 2 · (x – 3)
Note the two are positive,
this needs to be developed
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Factoring Polynomials
x2 + 6x + 8 = (x + 2)(x +4)
x2 + 4x + 2x + 8
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Factoring Polynomials
x2 – 5x + 6 = (x – 2)(x – 3)
x2 - 3x - 2x + 6
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Factoring Polynomials
x2 + 5x + 8 = prime
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Factoring Polynomials
x2 – x – 6 = (x + 2)(x – 3)
x2 - 3x + 2x - 6
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Factoring Polynomials
x2 – 9
= (x + 3)(x – 3)
x2 - 3x + 3x - 9
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Factoring Polynomials
2x2 + x – 6 = (2x - 3)(x + 2)
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2x2 - 3x + 4x - 6
Factoring Polynomials
2x2 + 3x – 4 = prime
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Virtual Algebra Tiles
http:..media.mivu.org/mvu_pd/a4a/homework/applets_applet_home.html
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Factoring Polynomials
Practice factoring
x2 + x – 6
x2 – 4
4x2 – 9
2x2 – 3x – 2
3x2 + x – 2
-2x2 + x + 6
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