Classifying Multiplication Problems

On the Addition tab there is a link to an explanation of my system for classifying word problems. In brief, I first classify problems as examples of either addition, subtraction, multiplication, or division. Having done that, I further classify word problems by how I would teach children to solve them - that is, on the basis of what model I would help them to make.

In the case of multiplication, I only distinguish 3 models. Other authors make further distinctions. Consider for example these two problems.

From my perspective, the only distinction is that in the first example each fish tank contains 2 fish while in the second example each fish tank is associated with 2 dollars. I might model both problems using 3 cups to represent the fish tanks and 6 pennies to represent either the fish or the dollars. Whether the pennies are in the cups or not does not seem significant.

Or consider this problem.

From my perspective, this problem is yet another example of Easy Multiplication. As the diagram below makes clear, my rug is made up of 3 columns of squares, with each square having an area of 1 square foot. There are 2 of these squares in each column.

Of course, the concept of area is one that needs to be taught. But once children understand the concept they are able to model area problems as above. They then should see that the model for area problems is identical to the model for Easy Multiplication problems.