Ideas for teaching multiplication

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Ideas for teaching multiplication

• By Joe Hill

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Model 1 Balls in Cups

• One common way we think of multiplication is like
  putting balls in cups. For example, if I put 4
  balls in 5 cups, how any balls in all would I
  have?

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Heres a similar example

• If there are four tricycles in a room, how many
  wheels are there?

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This model can be shown as a tree diagram.

• For example, if three students each carry four
  books, how many books in all are there?

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In this model

• Note there are two distinct objects in each
  example (balls-cups, tricycles-wheels,
  students-books). In the multiplication, one
  factor is the number of items of the first object
  (balls, tricycles, students) and the other factor
  is the number of identical second items (cups,
  wheels, books)
• This model leads to an understanding of the
  Fundamental Counting Principle demonstrated in
  problems like this If you can select from 5
  types of sandwiches and 3 types of drinks, how
  many meals could you have consisting of a
  sandwich and a drink?
• It is not as useful in multiplication of
  fractions and decimals such as 1/2 x 1/3 or 0.2 x
  0.3. It also gives the student the false idea
  that the answer to a multiplication problem is
  always more than either factor.

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Model 2 Repeated addition

• 4 4 4 4 4 can be written as 5 x 4
• 6 6 6 can be written as 3 x 6

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In this model

• This differs from the last model in that there
  are not two distinct objects. Here one factor is
  the number to be repeatedly added and the other
  factor is the number of times it appears.
• This model ties in directly with skip counting
  which students do in the primary grades.
• Like the first model, it is not as useful in
  multiplication of fractions and decimals such as
  1/2 x 1/3 or 0.2 x 0.3. It also gives the
  student the false idea that the answer to a
  multiplication problem is always more than
  either factor.
• Another issue is that the commutative property
  isnt as obvious to the student. Does 3 3 3
  3 3 3 3 equal 7 7 7 ?

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Heres a skip counting/multiplication activity
that shows a connection to geometric patterns
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Repeatedly add 2sthat is, multiply by 2
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Repeatedly add 3sthat is, multiply by 3
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Here are the 4s
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What pattern do the 5s construct?
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Next are the 6swith a surprising result.
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What pattern will the 7s make?
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And the 8s
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Even the 9s...
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Model 3 The array model

• Think of multiplication as always creating a
  rectangle. For example, 4 x 5 means to create
  a rectangle 4 units long and 5 units tall. How
  many blocks would you have?

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Multiplication means to make a rectangle
Heres 2 x 7

• Heres 7 x 5

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TheMultiplicationGrid
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TheMultiplicationGrid
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This model helps in explaining our traditional
multplication algorithm
Consider 13 x 14
13 X 14 12 40 30 100 182
13 X 14 52 130 182
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In this model

• In this model, each factor is one of the
  dimensions of the rectangle.
• This model leads to an understanding of area
  calculations.
• With this model, it is easy to see the
  commutative property. For example, a 7 x 3
  rectangle has the same number of blocks in it as
  a 3 x 7 one.
• Unlike the other models, it can be useful in
  multiplication of fractions and decimals such as
  1/2 x 1/3 or 0.2 x 0.3. For 1/2 x 1/3, we just
  need to construct a rectangle with these
  dimensions (see next slide)
• It also leads the student to an understanding of
  algebra (see following slides)

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TheMultiplicationGrid
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Now lets review some Algebra...

• Multiply (x 1)(x 1)
• You were probably taught to use FOIL
• x(x) (x)1 1(x) 1(1)
• x2 2x 1

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Now lets see the same problem done as a
multiplication grid
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Another Algebra problem...

• Multiply (x 1)(x 2)
• Use FOIL again
• x(x) (x)2 1(x) 1(2)
• x2 3x 2

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Now lets see the same problem done as a
multiplication grid
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Review--three ways to look at multiplication

• As a cross product of two distinct items such
  as in this example When a bag of M Ms is
  shared among 7 students, each student got 5 M
  Ms. How many M Ms in all were there?
• As repeated addition such as in this example
  Count the number of fingers in all if there are
  six people in a room 10 10 10 10 10
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• As an array or rectangle. Each factor is a
  dimension of the rectangle. Here is 6 x 2