Modeling Multiplication and Division
We discussed the concepts behind the basic operations multiplication and division in the beginning of the course. If we use those same ideas, we can explore multiplying and dividing using the base five pieces. Some problems lend themselves to certain models, so let's take a look at several.
Consider the problem 3five x 142five = ?
It could be "3five groups of 142five "...
We can also make use of an area model...
In either case, after building the model, we simply regroup the area into its minimal collection:
Let's look at division with the following example: 314five ÷ 11five = ?
Here's a rectangle model...
In the first example, the answer is represented by the number of groups of 11five we could form, and in the second example by the 'other' dimension of the rectangle. In both cases, the answer to the problem is
Just as in Modeling Addition and Subtraction, we have created models which will work in any base. In all four of the operations regrouping emerges as an important fundamental underlying concept. These models can be useful as frameworks upon which to base mental computation and estimation strategies. Here are some more multiplication and division problems to explore... make some paper 'pieces' or use simple sketches to guide your thinking.