Multiplication as equal groups
The most common situation, and the only one we usually start with, is multiplication as repeated addition:
Another sub-type of the equal group structure is one that we usually associate with measuring.
In all these situations, we are replicating a set of a fixed size, many times.
We can represent this structure using groups, arrays or on a number line as well as in a number sentence.
When we know the size and number of groups, the problem is multiplication. We know the factors or parts, what we are looking for is the product or the whole amount.
When we know the product but we don’t know one of the factors or parts, the problem becomes a missing factor multiplication or division situation.
Division as equal groups
Even though we think of equal group multiplication as repeated addition, we tend not to think of equal group division as repeated subtraction. That’s because in this structure, we are far more likely to be engaged in sharing and grouping situations.
When we are sharing a known number of items between a known number of groups, we are performing partition division. We are finding how many items in each group.
When we are grouping a known number of items, and we know how many are to be in each group, we are performing quotation division. We are finding how many groups we can make.