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Number and Operations Session 4, Part B: Area Models for Multiplication and Division
 
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Session 4, Part B:
Area Models for Multiplication and Division

In This Part: Multiplication with Manipulatives | Multiplication Model
Division with Manipulatives | Division Model

Let's look at the area model that you used to compute 18711.

Like the previous multiplication problems, we know that the rectangle's area is 187 and its height is 11. This problem asks us to determine its length.

The first step in this process is to create a model that represents an area of 187, using one flat, eight longs, and seven units:

Now use the flat and one of the longs to start the rectangle:

The rectangle we've built is composed of 11 tens, which, since multiplication is commutative, is equivalent to 10 elevens. Ten elevens is 110:

We have 77 left over. This 77 (composed of 7 tens and 7 ones) can be grouped in elevens as well. There are 7 of them:

The algorithm matches the model. You first subtracted 10 elevens and then subtracted 7 more elevens, for a total of 17 elevens. Thus, there are 17 elevens in 187. The length of the rectangle is 17, and the answer to the problem 18711 is 17.


 
 

The previous division problem is easy to visualize with an area model, but it becomes more complicated when you can't arrange the manipulatives neatly into a rectangle. For example, what happens when you try to divide 182 by 13?

In this case, if you use one flat, eight longs, and two units, there is no way to form these manipulatives directly into a rectangle with one side whose length is 13.

You can start the rectangle with the flat and three of the longs:

The rectangle you've built is 13 tens, which, since this type of multiplication is commutative, is equivalent to 10 thirteens. You have 52 left over.

In this case, you'll need to regroup the 52 (5 tens and 2 ones) to make thirteens. You can take one of the tens and trade it for 10 ones. Now you have 4 tens and 12 ones, and you can make groups of 13 to place to the right of the 10 you already have. Note 8


 

Problem B6

Solution  

Does the algorithm match the area model? Explain.



video thumbnail
 

Video Segment
In this video segment, Susan explains how her group resolved the division problem by using the quotative approach. Next, Professor Findell relates the group's method to an algorithm that is very similar to the long division algorithm.

Watch this segment after you've completed Problems B5 and B6.

If you are using a VCR, you can find the first segment on the session video approximately 8 minutes and 46 seconds after the Annenberg Media logo. The second part begins approximately 9 minutes and 25 seconds after the Annenberg Media logo.

 

Next > Part C: Colored-Chip Models

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