Lets Do Algebra Tiles

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Lets 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

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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 flip-side) represent -1.

• The yellow and red squares are additive inverses
  of each other.

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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 flip-side) represent -1 x
  or -x .

• The green and red rods are additive inverses of
  each other.

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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.
• Dont use cancel out for zeroes use zero pairs
  or add up to zero

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Addition of Integers

• 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

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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)
• Make zeroes to get two positives
• (3) (-4)
• Make three zeroes to get one negative
• After students have seen many examples of
  addition, have them formulate rules.

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-1
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Subtraction of Integers

• Subtraction can be interpreted as take-away.
• 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

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Subtraction of Integers

• (5) (2)
• Take away two positives
• To get three positives
• (-4) (-3)
• Take away three negatives
• To get one negative

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-1
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Subtracting Integers

• (3) (-5)
• Add five zeroes Take away five negatives
• To get eight positives
• (-4) (1)
• Add one zero Take away one positive
• To get five negatives

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-5
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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

• 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

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Multiplication of Integers

• The counter indicates how many rows to make. It
  has this meaning if it is positive.
• (2)(3)
• (3)(-4)

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Two groups of three positives
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Three groups of four negatives
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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
• (-2)(3)
• (-3)(-1)

-6

• Two groups of three

• Opposite of
• To get six negatives

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)
• Divide into two equal groups
• (-8)/(2)
• Divide into two equal groups

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Division of Integers

• A negative divisor will mean take the opposite
  of. (flip-over)
• (10)/(-2)
• Divide into two equal groups
• Find the opposite of
• To get five negatives

-5
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Division of Integers

• (-12)/(-3)
• After students have seen many examples, have them
  formulate rules.

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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)
• Three Groups of x to get three xs
• Three groups of 2 to get 6

3x 32
3x 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 xs and five
positives
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More Polynomials

• 3x 1 2x 4
• This process can be used with problems containing
  x2.
• (2x2 5x 3) (-x2 2x 5)
• (2x2 2x 3) (3x2 3x 2)

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Substitution

• 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

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Substitution

• 3 2x
• 3 2(4)
• 3 8
• 11

let x 4
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Solving Equations

• 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.
• 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.
• 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 xs are the same as five negatives
• Divide both sides into two equal partitions
• Two and a half negatives is the same as one x

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Solving Equations
-1 -1

• 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
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

• 2 x 3 x 5

- x - x
x 3 -5
-3 - 3
x -8

• Two xs 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 (102)(103)
• Multiplication using the array method allows
  students to see all four sub-products.

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Multiplication using Area Model

• (12)(13) (102)(103)
• 100 30 20 6 156

10 x 3 30
10 x 10 102 100
10 x 2 20
2 x 3 6
10 x 2 20
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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
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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 3x 4
x2
4x
1x
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
• 2x 6

(x 1)
3
(x 3)
2
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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 1

(x 6)
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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

• 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/mat
  h06_07/nsmedia/tools/Algebra_Tiles/Algebra_Tiles.h
  tml
• National Library of Virtual Manipulatives
  http//nlvm.usu.edu/en/nav/topic_t_2.html

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Resources

• David McReynolds
• AIMS PreK-16 Project
• Noel Villarreal
• South Texas Rural Systemic Initiative
• Jo Ann Mosier Roland ODaniel
• Collaborative for Teaching and Learning
• Partnership for Reform Initiatives in Science and
  Mathematics