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Session 9, Part A: Models for the Multiplication and Division of Fractions
 
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Session 9, Part A:
Models for the Multiplication and Division of Fractions

In This Part: Area Model for Multiplication | Try It Yourself | Area Model for Division
The Common Denominator Model for Division | Translating the Process to Decimals

The area model for the division of fractions does not help to illustrate why the algorithm we're most familiar with (invert the divisor and then multiply) works. Unfortunately, no model can show that. Note 4

But here is a different division algorithm, one that we can explain with a model: Find the common denominator, find the equivalent fractions, and divide the numerators.

In order to understand the model for this algorithm, let's first go back to review some of the concepts of division. It is usually easier to compute if you think about division in a quotative way. Thus, you can say that 6 3 asks, "How many 3s are there in 6?"

Next, we need to understand the role of units in division.


 

Problem A5

Solution  

Which, if any, of these questions yields a different answer?

 

How many 3s are there in 6?

 

How many groups of 3 tens are there in 6 tens?

 

How many groups of 3 fives are there in 6 fives?

 

How many groups of 3 tenths are there in 6 tenths?

 

How many groups of 3 @s are there in 6 @s?

 

How many groups of 3 anythings are there in 6 anythings (as long as both anythings refer to the same unit)?


 
 

The point here is that the units of the problem do not matter -- if the units are the same entity, they disappear when you divide.

This brings us back to the new algorithm for division with fractions: To divide two fractions, find a common denominator and then divide the numerators.

Let's try a visual version of the problem we did before: 1/4 2/3. First, find a common denominator:

1/4 2/3 = 3/12 8/12

Next, divide the numerators:

38 = 3/8

Here is the model for this problem, called the common denominator model:

In this problem, in effect, the original question was "How many 2/3s are there in 1/4?" By finding a common denominator, we changed the question to "How many 8/12s are there in 3/12?" -- which is the same as asking "How many 8s are there in 3?" The answer to both questions is the same: "There is 3/8 of an 8 in 3."


 

Problem A6

Solution  

Use the common denominator model to divide 3/5 by 3/4.


 

Problem A7

Solution  

Why does 0.6 0.2 have the same answer as 6 2?


Next > Part A (Continued): Translating the Process to Decimals

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