Middle School Mathematics Teachers

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Middle School Mathematics Teachers
CircleInstitute for Mathematics EducationNov.
3, 2009
Using Algebra Blocks for Teaching Middle School
Mathematics Exploring a Variety of Topics

• Cynthia Anhalt
• canhalt_at_math.arizona.edu

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Algebra Blocks

• Figure out the pieces.

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Identifying the pieces
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Algebra Expression Mat


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Using algebra blocks for combining similar terms
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Combining Similar Terms

• 5x 3y 4 - 2 4x - y
• x2 2y -2 5 2x2 - 3y
• 2xy - x -3 3y2 - 2x2 5 (-xy) 2x
• 4x 3x2 2xy (-2x) x2

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Using algebra blocks for multiplying binomials
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Consider (x y) (x y)

• What is the common solution that students choose
  on the AIMS exam?
• y X2 y2
• Why is y x2 2xy y2 not understood as the
  solution by so many students?
• FOIL is sometimes taught as a procedure without
  conceptual understanding. Do you agree or
  disagree? Why?
• Would LOIF or OILF work? Most 7-12th grade
  students dont know.
• What is the underlying concept?
• multiplication of two binomials
• When FOIL is taught as a procedure, most
  students have a difficult time multiplying (x y
  3) (x 2y). Why do you suppose?

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Use the area model of multiplication with the
algebra blocks for multiplying the two binomials
(x y) (x y)
(x y) (x y) x2 xy xy y2 x2 2xy
y2
xy
x y
xy
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Use the algebra blocks to show (2x y) (x y
4)
2x2 2xy 8x xy y2 4y
By combining similar terms 2x2 3xy 8x y2
4y
xy
xy
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Create your own multiplication of binomials
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Using algebra blocks for teaching perimeter
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Determine the perimeter
P 4x 2y
P y 5 (y -1) P 2y 4
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Determine the perimeter
P 1 x (y-1) 1 (y-1) 3 1 x 2x
2y 4
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Determine the perimeter
x2
xy
y
x
P x 1 1 y x (y-x) 1 x 1 (x-1)
1 (y-1) 1 (y-x) x x P 4x 4 4y
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Create your own shape for the class to find the
perimeter
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Algebra Equation Mat

- -

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Equations solve for x
2(x-3) -4

- -

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Equation solve for x

• 2(x-3) -4

- -
2x -6 -4
6 6
2x 2
_ _ 2 2

x 1
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Equation solve for y

- -

• 5(y-4) 10

5y -20 10
20 20
5y 30

_ _ 5 5
y 6
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Create your own equation for the class to solve
for x

- -

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Discussion

• What do you suppose are the benefits of using the
  algebra blocks?
• What do you suppose are the drawbacks of using
  algebra blocks?
• Other comments?

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How do the algebra blocks interface with what
NCTM advocates in teaching mathematics?

• Consider the NCTM process standards Explain the
  potential of the NCTM process standards in
  teaching with algebra blocks.
• Connections
• Representation
• Communication
• Problem Solving
• Reasoning Proof
• How does using the algebra blocks impact students
  who are ELL?

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Thank you for sharing your evening with me.

• Cynthia Anhalt
• canhalt_at_math.arizona.edu

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How can algebraic thinking begin in the
elementary grades?

• How can the elementary mathematics curriculum can
  be taught with algebraic thinking as a goal?
• How can base-10 blocks be used to aid in teaching
  multiplication that leads to the distributive
  property?

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Multiplication Models
What models would represent 2 x 4?

• Set model

2 groups of 4 discrete objects
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Consider the area model for multiplication of
whole numbers

• 3 x 4 12

4
3
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Base 10 Blocks

• 3 x 12

How can we convert these sets into an area model?
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Area model using Base 10 Blocks
3 x 12 36
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Consider the area model to make the connection of
arithmetic to algebraic thinking

• 12 x 13 (10 2) (103)

100 20 30 6
Using base-10 blocks, we have
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Show the area model of 14 x 23 with base 10
blocks
(104)(203) 200803012 322
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Discussion

• How do these ideas promote algebraic thinking in
  the elementary mathematics classrooms or in the
  middle grades?
• How can you incorporate a variety of models of
  mathematical representation into your teaching?

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Thank you for sharing you evening with me.

• Cynthia Anhalt
• canhalt_at_math.arizona.edu