Multiplying Integers: A Model for Teaching

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Multiplying IntegersA Model for Teaching

• Patty Norman
• and
• Logan Toone
• Davis School District Curriculum Department

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Common Models for Multiplication

• Algebraic Model - Multiplication as repeated
  addition of sets
• Example

3 x 5
3 x 5 5 5 5 15
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Common Models for Multiplication

• Geometric Model - Multiplication as the area of a
  rectangle
• Example

3 x 5
3 x 5 15
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A Comprehensive Model ?

• These two models for multiplication work
  primarily with whole numbers, which is great
• Is there a model that includes negative integers
  as well?

but
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A Model for All Integers

• We can map the possible combinations of positive
  and negative integers on a quadrant grid.

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A Model for All Integers

• We continue the model of multiplication as
  repeated addition of sets.

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A Model for All Integers
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In Other Words
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Convention for this Model

• The first number tells us how many sets.
• The second number tells us how many elements in
  each set.
• The commutative property would also allow us to
  work with five sets of three, but we chose this
  convention for the model.

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Quadrant 1Positive x Positive
Start with zero sets
15
Positives
Negatives
3 x 5 add 3 sets of positive 5
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Quadrant 2Negative x Positive
Start with zero equivalent (3 sets of negative 5
and 3 sets of positive 5)
-15
Positives
Negatives
(-3) x 5 subtract 3 sets of positive 5
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Quadrant 3Negative x Negative
Start with zero equivalent (3 sets of negative 5
and 3 sets of positive 5)
15
Positives
Negatives
(-3) x (-5) subtract 3 sets of negative 5
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Quadrant 4Positive x Negative
Start with zero sets
-15
Positives
Negatives
3 x (-5) add 3 sets of negative 5
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Summary