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Title: To navigate the PowerPoint presentation place the mouse arrow inside the box containing the slide and click on the mouse to advance the program. Or, use the up and down arrows on your keyboard


1
To navigate the PowerPoint presentation place
the mouse arrow inside the box containing the
slide and click on the mouse to advance the
program. Or, use the up and down arrows on your
keyboard
2
Multiplying Fractions
  • When teaching children how to multiply fractions,
    it is important to make the process meaningful.
    This may be done best by using a five-step
    process that helps children to visualize fraction
    multiplication, understand fraction
    multiplication, be able to do fraction
    multiplication, and be confident when multiplying
    fractions.

3
  • In the first step of this instructional process,
    children use a model to find answers to some
    fraction-multiplication examples.
  • In the second step (which really happens
    concurrently with step one) the children keep a
    record of the results from step one.

4
  • After enough examples have been completed the
    children move to the third step. They look for a
    pattern that suggests how to do the
    multiplication without the model.
  • In the fourth step the children hypothesize how
    to do the multiplication without the model.

5
  • This hypothesis really a first description of the
    fraction-multiplication algorithm (procedure).
  • The fifth step is to complete examples using the
    hypothesized procedure and then redo those
    examples with the model to check the correctness
    of the procedure.

6
  • Of course, this 5-step instructional process can
    only work if you have an effective (and
    believable) way to model the multiplication of
    fractions.
  • We will look at three procedures for modeling
    fraction multiplication that are found in the
    literature.

7
There are 3 approaches for modeling fraction
multiplication
  • A Fraction of a Fraction
  • Length X Length Area
  • Cross Shading

We will now examine each of these 3 approaches.
8
We will think of multiplying fractions as finding
a fraction of another fraction.
We use a fraction square to represent the
fraction .
9
Then, we shade of . We can see that it is
the same as .
But, of is the same as .
X
So,
10
To find the answer to , we will use the
model to find of .
We use a fraction square to represent the
fraction .
11
Then, we shade of . We can see that it is
the same as .
12
In this example, of has been shaded
What is the answer to ?
X
13
  • Modeling multiplication of fractions using the
    fraction of a fraction approach requires that the
    children understand the relationship of
    multiplication to the word of.
  • We can establish this understanding showing
    whole-number examples like 6 threes is the same
    as 6 X 3.

14
In the second method, we will think of
multiplying fractions as multiplying a length
times a length to get an area.
15
In the second method, we will think of
multiplying fractions as multiplying a length
times a length to get an area.
16
We think of the rectangle having those sides.
Its area is the product of those sides.
17
We can find another name for that area by seeing
what part of the square is shaded.
18
We have two names for the same area. They must
be equal.
19
Length X Length Area
This area is X
20
What is the answer to X ?
21
  • Modeling multiplication of fractions using the
    length times length equals area approach requires
    that the children understand how to find the area
    of a rectangle.
  • A great advantage to this approach is that the
    area model is consistently used for
    multiplication of whole numbers and decimals.
    Its use for fractions, then is merely an
    extension of previous experience.

22
In the third method, we will represent both
fractions on the same square.
is
23
The product of the two fractions is the part of
the square that is shaded both directions.
X
is
is
24
We will look at another example using cross
shading. We shade one direction.
25
Then we shade the other direction.
The answer to X is the part that is
shaded both directions.
26
  • Modeling multiplication of fractions using the
    cross shading approach does produce correct
    answers. However, to children, it is a nonsense
    method.
  • The rationale for the answer, because it is
    shaded both directions does not make sense. It
    would make as much sense to say that the answer
    is all the parts that are shaded only one
    direction or the part that is not shaded.

27
  • If the rationale for the answer does not make
    sense to the children--if it is not
    meaningful--it is simply another rote rule. For
    this reason, THE CROSS SHADING METHOD IS NOT
    RECOMMENDED. Teachers should choose to use
    either the fraction of a fraction method or the
    length times length equals area method when
    modeling multiplication of fractions.

28
  • With your partner, practice using the fraction of
    a fraction method to model multiplication of
    fractions until you are both comfortable enough
    to make a presentation using the method.
  • Also, practice using the length times length
    equals area method to model multiplication of
    fractions until you are both comfortable enough
    to make a presentation using this method.
  • When you are ready, make an appointment with your
    instructor to demonstrate each method.

29
The End
  • Dr. Benny Tucker
  • Ex. 5396
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