Building a Visual Model for Equations and Polynomials

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Building a Visual Model for Equations and
Polynomials

• Jim Rahn
• LL Teach, Inc
• www.llteach.com
• www.jamesrahn.com
• james.rahn_at_verizon.net

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• Place the following tiles on your table. What
  does each represent?
• 1 green rectangle, 2 yellow squares, and 3 small
  red squares
• 2 red rectangles, 1 yellow small square, 2 red
  small squares, and 3 green rectangles
• 4 green rectangles, 2 red rectangles, 1 yellow
  small square, and 2 red small squares
• 2 Blue square, 1 red rectangle, 3 yellow small
  squares, 2 green rectangles, 1 red small square

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• Represent each of the following and then have
  simplify the expressions

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• Write down what pieces you would use to represent
  the following expression without building the
  picture and then describe your final answer.

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Adding expressions together

• First place expression one on the board. Then add
  to this board the expression two. What is the
  results?

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Now try to add these expressions by just
describing the tiles you would use
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Now try to subtract

• First place expression one on the board. Then
  try to remove or subtract expression two from the
  board. (Hint You might need some zero pairs to
  do this.)

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• Now try to subtract these expressions without
  using the tiles. Just describe what you would do

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Solving Equations with Algebra Models

• Set this picture up on your equation balance. By
  doing the same step to both sides of the
  equation, try to get the green rectangle by
  itself.
• What does one green rectangle equal? Describe
  your steps.
• Is there another set of steps you could use to
  find the value of one green rectangle?
• Does the value of the green rectangle make sense?

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• Set up this new picture and again solve for the
  value of the green tile.
• Is there more than one set of steps that you can
  use to find the value of the green rectangle?
• Does your answer make sense?

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• Set up this new picture and again solve for the
  value of the green tile.
• Is there more than one set of steps that you can
  use to find the value of the green rectangle?
• Does your answer make sense?

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• Use the algebra models to represent x 3 4 on
  the equation balance.
• Find the value of x by doing the same thing to
  both sides of the balance until you have the x
  (green rectangle) by itself.
• Does this value make sense?

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• Use the algebra models to represent 2x4 8.
• Find the value of x by doing the same thing to
  both sides of the balance until you have the x
  (green rectangle) by itself.
• Does this value make sense?

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• Use the algebra models to represent
• on the equation balance.
• Find the value of x by doing the same thing to
  both sides of the balance until you have the x
  (green rectangle) by itself.
• Does this value make sense?

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• Use the algebra models to represent
• on the equation balance.
• Find the value of x by doing the same thing to
  both sides of the balance until you have the x
  (green rectangle) by itself.
• Does this value make sense?

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• Use the algebra models to represent
• on the equation balance.
• Find the value of x by doing the same thing to
  both sides of the balance until you have the x
  (green rectangle) by itself.
• Does this value make sense?

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• Use the algebra models to represent
• on the equation balance.
• Find the value of x by doing the same thing to
  both sides of the balance until you have the x
  (green rectangle) by itself.
• Does this value make sense?

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• Use the algebra models to represent
• on the equation balance.
• Find the value of x by doing the same thing to
  both sides of the balance until you have the x
  (green rectangle) by itself.
• Does this value make sense?

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• Use the algebra models to represent
• on the equation balance.
• Find the value of x by doing the same thing to
  both sides of the balance until you have the x
  (green rectangle) by itself.
• Does this value make sense?

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Visualizing Multiplication with base 10 tiles

• Suppose we want to multiply 13 x 12.

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Multiplication with Algebra Tiles

• We will use the rectangle to represent
  multiplication
• Show 2x 1 with tiles.
• Show 2(x 1) with tiles.
• Arrange the shapes in a rectangle.
• What is the rectangles length?
• What is the rectangles width?
• What is the rectangles area?

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• Show 3x2 - x with tiles.
• Show 3(x2 - x) with tiles.
• How are the two pictures for 3x2 - x and 3(x2 -
  x) different?
• How are they the same?

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Setting Up a Multiplication Rectangle

• Along the outside we will place our factors to
  multiply.
• Fill in the rectangle with appropriate pieces.

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• Use the rectangular model to multiply x 1 by x
  2. Set up the dimensions along the top and the
  side. Then build the rectangle that has those
  dimensions.

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• What new piece did you use in the multiplication?
• How many different size pieces do you have in
  your rectangle?
• What is the name for each of these pieces?
• How many of each size piece do you have?
• What is the simplified answer for the
  multiplication of (x 1)(x 2)?

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• (x2)(x3)
• Set up the dimensions x2 and x 3 along the top
  and the side.
• Build the rectangle with those dimensions.
• How many different size pieces do you have in
  your rectangle?
• What is the name for each of these pieces?
• How many of each size piece do you have?
• What is the simplified answer for the
  multiplication of (x 2)(x 3)?

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Set up these multiplications

• Do you see any patterns?
• Can you predict how many x2 pieces will be in
  your answer?
• Can you predict how many x pieces will be in your
  answer?
• Can you predict how many units will be in your
  answer?

• (x3)(x1)
• (x2)(x5)
• (x2)(x4)

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Multiply these binomials

• Do you see any patterns?
• Can you predict how many x2 pieces will be in
  your answer?
• Can you predict how many x pieces will be in your
  answer?
• Can you predict how many units will be in your
  answer?

• (2x3)(x2)
• (x2)(3x5)
• (2x1)(x4)

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Can you multiply these binomials together without
the tiles?

• Think about the rectangle and what pieces will
  fit on the outside and what pieces will fit on
  the inside.
• (x1)(x4)
• (x7)(x5)
• (2x1)(x7)

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Lets Try Negative Numbers

• Think about the rectangle and what pieces will
  fit on the outside and what pieces will fit on
  the inside.
• (x-2)(x2)

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Can you multiply these binomials together without
the tiles?

• Think about the rectangle and what pieces will
  fit on the outside and what pieces will fit on
  the inside.
• (2x-1)(2x1)
• (x-3)(x3)
• (1-x)(1x)
• (1-2x)(12x)

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Multiply these

• (2x-1)(x1)

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Try some of these

• (3x2)(x-2)
• (2x1)(x-4)
• (1-x)(2x)

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How did Algebra Tiles Help You?

• Do you have a visual understanding for addition,
  subtraction, and multiplication of signed
  numbers?
• Do you have a visual understand for terms like 1,
  2x, and 3x2?
• Do you have a visual understanding for combining
  like terms?
• Do you have a visual understanding for adding or
  subtracting algebraic expressions?
• Do you have an understanding for importance of
  zero pairs?

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How did Algebra Tiles Help You?

• Do you have a visual image to connect (x3)(x2)
  with?
• Do you understand that (x3)(x2) will involve 4
  multiplications?
• Will (x3)(x2) have any zero pairs to combine?
• Will (x-3)(x2) have any zero pairs to combine?

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Building a Visual Model for Equations and
Polynomials

• Jim Rahn
• LL Teach, Inc
• www.llteach.com
• www.jamesrahn.com
• james.rahn_at_verizon.net