Multiplying Fractions

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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.

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.

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.

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.

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.

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.

3 4 3 4 We will think of multiplying fractions as finding a fraction of another fraction. We use a fraction square to represent the fraction .

2 3 3 4 6 12 2 3 2 3 of 3 4 4 3 3 2 4 3 6 3 2 = X 12 4 3 Then, we shade of . We can see that it is the same as . But, of is the same as . X So,

3 1 X 5 2 3 1 5 2 3 5 3 5 To find the answer to , we will use the model to find of . We use a fraction square to represent the fraction .

1 3 2 5 3 10 1 3 of 5 2 So, 3 3 1 = X 10 5 2 Then, we shade of . We can see that it is the same as .

12 34 1 3 of 4 2 12 34 In this example, of has been shaded What is the answer to ? X

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.

3 This length is 4 In the second method, we will think of multiplying fractions as multiplying a length times a length to get an area.

3 4 2 This length is 3 In the second method, we will think of multiplying fractions as multiplying a length times a length to get an area.

3 4 2 3 2 3 This area is X 3 4 We think of the rectangle having those sides. Its area is the product of those sides.

3 4 2 3 2 3 This area is X 3 4 6 It is also 12 We can find another name for that area by seeing what part of the square is shaded.

3 4 2 6 2 3 = 3 X 12 3 4 2 3 This area is X 3 4 6 It is also 12 We have two names for the same area. They must be equal.

1 2 1 3 2 4 3 4 3 It is also 8 3 1 3 = X 8 2 4 Length X Length = Area This area is X

45 14 1 4 4 5 What is the answer to X ?

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.

1 2 3 is 4 In the third method, we will represent both fractions on the same square. is

3 1 3 8 2 4 1 2 3 4 The product of the two fractions is the part of the square that is shaded both directions. X = is is

4 5 4 5 We will look at another example using cross shading. We shade one direction.

2 3 4 5 2 3 2 4 3 5 8 2 4 X = 3 5 15 Then we shade the other direction. The answer to X is the part that is shaded both directions.

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.

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.

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.

The End Dr. Benny Tucker Ex. 5396

Multiplying Fractions